Subgroups — C₂×D₄×C₃⋊D₃×ΓL(1,𝔽₈) #72245578

Order 6048.

3264 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₈)×#638	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
3	C₂×ΓL(1,𝔽₈)×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×D₄×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃×D₄×ΓL(1,𝔽₈)	—	3	4	no	maximal
9	C₂³×C₃²×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
10	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	—	2	8	no	—
11	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
13	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
14	C₃²×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
15	C₂×C₃²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)×#638	C₂²	1	8	yes	complemented
17	C₂×ΓL(1,𝔽₈)×C₃⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	complemented
19	C₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
20	C₂³×C₇×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
21	C₂³×D₃×ΓL(1,𝔽₈)	—	3	8	no	—
22	C₂×C₃²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₇×#638	C₂×C₃	1	2	yes	complemented
24	C₇×D₄×C₃⋊D₃	C₂×C₃	1	8	yes	complemented
25	C₂×C₇×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₄×C₇×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×C₃×D₄×ΓL(1,𝔽₈)	—	2	4	no	—
28	C₂×C₃×D₄×ΓL(1,𝔽₈)	D₃	1	4	yes	complemented
29	C₂×ΓL(1,𝔽₈)×#67	—	3	8	no	—
30	D₃×D₄×ΓL(1,𝔽₈)	—	3	32	no	—
31	C₂×ΓL(1,𝔽₈)×D₁₂	—	3	4	no	—
32	C₂×C₄×D₃×ΓL(1,𝔽₈)	—	3	4	no	—
33	C₂×C₃×D₄×C₃⋊D₃	—	7	1	no	maximal, Hall
34	C₂²×C₃²×ΓL(1,𝔽₈)	—	2	4	no	—
35	C₂²×C₃²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
36	C₂²×C₃²×ΓL(1,𝔽₈)	C₂³	1	4	yes	complemented
37	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	—	2	4	no	—
38	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	—	2	16	no	—
39	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	D₄	1	4	yes	direct product factor
40	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	C₂³	1	4	yes	complemented
42	C₃²×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	complemented
43	ΓL(1,𝔽₈)×C₃⋊Dic₃	C₂³	1	2	yes	complemented
44	C₂×D₃×C₇×D₄	—	3	4	no	—
45	C₂²×D₄×ΓL(1,𝔽₈)	—	9	1	no	—
46	C₂³×C₃²×C₇	C₂²×C₃	1	2	yes	complemented
47	C₂²×C₇×C₃⋊D₃	—	2	8	no	—
48	C₂²×C₇×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₇×C₃⋊D₃	C₂²×C₃	1	8	yes	complemented
50	C₂²×C₇×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
51	C₂³×C₃×ΓL(1,𝔽₈)	—	2	8	no	—
52	C₂³×C₃×ΓL(1,𝔽₈)	C₂×D₃	1	8	yes	complemented
53	C₂²×D₃×ΓL(1,𝔽₈)	—	3	4	no	—
54	C₂²×D₃×ΓL(1,𝔽₈)	—	3	8	no	—
55	C₂²×D₃×ΓL(1,𝔽₈)	—	3	32	no	—
56	C₂²×D₃×ΓL(1,𝔽₈)	—	6	32	no	—
57	C₃²×C₇×D₄	C₂²×C₃	1	4	yes	complemented
58	C₂×C₃²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
59	C₇×#638	C₂²×C₃	1	8	yes	complemented
60	C₂×C₇×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₇×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	4	yes	complemented
62	C₄×C₇×C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
63	C₃×D₄×ΓL(1,𝔽₈)	—	2	16	no	—
64	C₃×D₄×ΓL(1,𝔽₈)	C₂×D₃	1	16	yes	complemented
65	C₂×C₃×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
66	C₂×C₃×C₄×ΓL(1,𝔽₈)	C₂×D₃	1	4	yes	complemented
67	ΓL(1,𝔽₈)×#67	—	3	32	no	—
68	C₂×Dic₃×ΓL(1,𝔽₈)	—	3	4	no	—
69	ΓL(1,𝔽₈)×D₁₂	—	3	16	no	—
70	C₄×D₃×ΓL(1,𝔽₈)	—	3	16	no	—
71	C₂³×C₃×C₃⋊D₃	—	7	2	no	—
72	C₂×C₃³×D₄	—	7	1	no	—
73	C₂×C₃×#638	—	7	2	no	—
74	C₃×D₄×C₃⋊D₃	—	7	8	no	—
75	C₂×C₃×C₄⋊(C₃⋊D₃)	—	7	1	no	—
76	C₂×C₃×C₄×C₃⋊D₃	—	7	1	no	—
77	C₂×C₃²×ΓL(1,𝔽₈)	—	2	4	no	—
78	C₂×C₃²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	complemented
79	C₂×C₃²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
80	C₃⋊D₃×ΓL(1,𝔽₈)	—	2	4	no	—
81	C₃⋊D₃×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	direct product factor
82	C₂³×D₃×C₇	—	3	8	no	—
83	C₂⁴×ΓL(1,𝔽₈)	—	9	2	no	—
84	C₂×C₃×C₇×D₄	C₃×D₃	1	4	yes	complemented
85	C₂×C₇×#67	—	3	8	no	—
86	D₃×C₇×D₄	—	3	32	no	—
87	C₂×C₇×D₁₂	—	3	4	no	—
88	C₂×C₄×D₃×C₇	—	3	4	no	—
89	C₂×D₄×ΓL(1,𝔽₈)	—	2	4	no	—
90	C₂×D₄×ΓL(1,𝔽₈)	—	9	2	no	—
91	C₂×D₄×ΓL(1,𝔽₈)	—	9	8	no	—
92	C₂×D₄×ΓL(1,𝔽₈)	—	9	1	no	—
93	C₂×D₄×ΓL(1,𝔽₈)	C₃⋊D₃	1	1	yes	characteristic, direct product factor
94	C₂²×C₄×ΓL(1,𝔽₈)	—	9	1	no	—
95	C₂×D₄×C₃⋊D₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
96	C₂×C₃×D₃×D₄	—	21	4	no	—
97	C₂²×C₃²×C₇	—	2	4	no	—
98	C₂²×C₃²×C₇	C₂³×C₃	1	1	yes	characteristic
99	C₂²×C₃²×C₇	C₂³×C₃	1	4	yes	complemented
100	C₂×C₇×C₃⋊D₃	—	2	4	no	—
101	C₂×C₇×C₃⋊D₃	—	2	16	no	—
102	C₂×C₇×C₃⋊D₃	C₃×D₄	1	4	yes	complemented
103	C₂×C₇×C₃⋊D₃	C₂³×C₃	1	2	yes	—
104	C₂×C₇×C₃⋊D₃	C₂³×C₃	1	4	yes	complemented
105	C₂²×C₃×ΓL(1,𝔽₈)	—	2	4	no	—
106	C₂²×C₃×ΓL(1,𝔽₈)	—	2	16	no	—
107	C₂²×C₃×ΓL(1,𝔽₈)	—	2	16	no	—
108	C₂²×C₃×ΓL(1,𝔽₈)	—	4	16	no	—
109	C₂²×C₃×ΓL(1,𝔽₈)	C₂²×D₃	1	4	yes	—
110	C₂²×C₃×ΓL(1,𝔽₈)	C₂²×D₃	1	16	yes	complemented
111	C₂×D₃×ΓL(1,𝔽₈)	—	3	16	no	—
112	C₂×D₃×ΓL(1,𝔽₈)	—	3	8	no	—
113	C₂×D₃×ΓL(1,𝔽₈)	—	3	16	no	—
114	C₂×D₃×ΓL(1,𝔽₈)	—	6	64	no	—
115	C₂×D₃×ΓL(1,𝔽₈)	—	6	16	no	—
116	C₃²×C₄×C₇	C₂³×C₃	1	2	yes	complemented
117	C₇×C₃⋊Dic₃	C₂³×C₃	1	2	yes	complemented
118	C₃×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
119	C₃×C₄×ΓL(1,𝔽₈)	C₂²×D₃	1	8	yes	complemented
120	Dic₃×ΓL(1,𝔽₈)	—	3	8	no	—
121	C₂²×C₇×D₄	—	9	1	no	Hall
122	C₂³×C₃³	—	7	2	no	—
123	C₂²×C₃×C₃⋊D₃	—	7	1	no	—
124	C₂²×C₃×C₃⋊D₃	—	7	2	no	—
125	C₂²×C₃×C₃⋊D₃	—	7	8	no	—
126	C₂²×C₃×C₃⋊D₃	—	14	8	no	—
127	C₃³×D₄	—	7	4	no	—
128	C₂×C₃³×C₄	—	7	1	no	—
129	C₃×#638	—	7	8	no	—
130	C₂×C₃×C₃⋊Dic₃	—	7	1	no	—
131	C₃×C₄⋊(C₃⋊D₃)	—	7	4	no	—
132	C₃×C₄×C₃⋊D₃	—	7	4	no	—
133	C₃²×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
134	C₂³×C₃×C₇	C₂×C₃×D₃	1	8	yes	complemented
135	C₂²×D₃×C₇	—	3	4	no	—
136	C₂²×D₃×C₇	—	3	8	no	—
137	C₂²×D₃×C₇	—	3	32	no	—
138	C₂²×D₃×C₇	—	6	32	no	—
139	C₂³×ΓL(1,𝔽₈)	—	2	8	no	—
140	C₂³×ΓL(1,𝔽₈)	—	9	1	no	—
141	C₂³×ΓL(1,𝔽₈)	—	9	2	no	—
142	C₂³×ΓL(1,𝔽₈)	—	9	8	no	—
143	C₂³×ΓL(1,𝔽₈)	—	18	8	no	—
144	C₂³×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	1	2	yes	complemented
145	C₃×C₇×D₄	C₂×C₃×D₃	1	16	yes	complemented
146	C₂×C₃×C₄×C₇	C₂×C₃×D₃	1	4	yes	complemented
147	C₇×#67	—	3	32	no	—
148	C₂×C₇×Dic₃	—	3	4	no	—
149	C₇×D₁₂	—	3	16	no	—
150	C₄×D₃×C₇	—	3	16	no	—
151	D₄×ΓL(1,𝔽₈)	—	2	16	no	—
152	D₄×ΓL(1,𝔽₈)	—	9	8	no	—
153	D₄×ΓL(1,𝔽₈)	—	9	4	no	—
154	D₄×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	1	4	yes	direct product factor
155	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
156	C₂×C₄×ΓL(1,𝔽₈)	—	9	1	no	—
157	C₂×C₄×ΓL(1,𝔽₈)	—	9	4	no	—
158	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
159	C₂³×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
160	C₂³×C₃×D₃	—	21	8	no	—
161	C₂×C₃²×D₄	—	7	4	no	—
162	C₂×C₃²×D₄	—	14	4	no	—
163	C₂×C₃²×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
164	C₂×#638	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
165	D₄×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	1	8	yes	direct product factor
166	C₂×C₄⋊(C₃⋊D₃)	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
167	C₂×C₄×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
168	C₂×C₃×#67	—	21	8	no	—
169	C₃×D₃×D₄	—	21	32	no	—
170	C₂×C₃×D₁₂	—	21	4	no	—
171	C₂×C₃×C₄×D₃	—	21	4	no	—
172	C₂×C₃²×C₇	—	2	4	no	—
173	C₂×C₃²×C₇	C₂×C₃×D₄	1	2	yes	complemented
174	C₂×C₃²×C₇	C₂⁴×C₃	1	1	yes	characteristic
175	C₇×C₃⋊D₃	—	2	4	no	—
176	C₇×C₃⋊D₃	C₂×C₃×D₄	1	4	yes	complemented
177	C₂×C₃×ΓL(1,𝔽₈)	—	2	4	no	—
178	C₂×C₃×ΓL(1,𝔽₈)	—	2	8	no	—
179	C₂×C₃×ΓL(1,𝔽₈)	—	2	16	no	—
180	C₂×C₃×ΓL(1,𝔽₈)	—	4	16	no	—
181	C₂×C₃×ΓL(1,𝔽₈)	D₃×D₄	1	8	yes	complemented
182	C₂×C₃×ΓL(1,𝔽₈)	C₂³×D₃	1	4	yes	—
183	D₃×ΓL(1,𝔽₈)	—	3	16	no	—
184	D₃×ΓL(1,𝔽₈)	—	6	16	no	—
185	C₂⁴×C₇	—	9	2	no	—
186	C₂×C₇×D₄	—	9	2	no	—
187	C₂×C₇×D₄	—	9	8	no	—
188	C₂×C₇×D₄	—	9	1	no	—
189	C₂×C₇×D₄	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
190	C₂²×C₄×C₇	—	9	1	no	—
191	C₂²×C₃³	—	7	1	no	—
192	C₂²×C₃³	—	7	4	no	—
193	C₂²×C₃³	—	14	4	no	—
194	C₂×C₃×C₃⋊D₃	—	7	4	no	—
195	C₂×C₃×C₃⋊D₃	—	7	2	no	—
196	C₂×C₃×C₃⋊D₃	—	7	4	no	—
197	C₂×C₃×C₃⋊D₃	—	14	16	no	—
198	C₂×C₃×C₃⋊D₃	—	14	4	no	—
199	C₃³×C₄	—	7	2	no	—
200	C₃×C₃⋊Dic₃	—	7	2	no	—
201	C₂²×C₃×D₄	—	63	1	no	—
202	C₂×D₃×D₄	—	3	4	no	—
203	C₂²×C₃×C₇	—	2	16	no	—
204	C₂²×C₃×C₇	C₂²×C₃×D₃	1	4	yes	—
205	C₂²×C₃×C₇	C₂²×C₃×D₃	1	16	yes	complemented
206	C₂×D₃×C₇	—	3	8	no	—
207	C₂×D₃×C₇	—	3	16	no	—
208	C₂×D₃×C₇	—	3	16	no	—
209	C₂×D₃×C₇	—	6	64	no	—
210	C₂×D₃×C₇	—	6	16	no	—
211	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
212	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
213	C₂²×ΓL(1,𝔽₈)	—	2	16	no	—
214	C₂²×ΓL(1,𝔽₈)	—	4	16	no	—
215	C₂²×ΓL(1,𝔽₈)	—	9	2	no	—
216	C₂²×ΓL(1,𝔽₈)	—	9	4	no	—
217	C₂²×ΓL(1,𝔽₈)	—	9	4	no	—
218	C₂²×ΓL(1,𝔽₈)	—	18	16	no	—
219	C₂²×ΓL(1,𝔽₈)	—	18	4	no	—
220	C₂²×ΓL(1,𝔽₈)	C₂²×C₃⋊D₃	1	1	yes	characteristic
221	C₂²×ΓL(1,𝔽₈)	C₂²×C₃⋊D₃	1	4	yes	complemented
222	C₃×C₄×C₇	C₂²×C₃×D₃	1	8	yes	complemented
223	C₇×Dic₃	—	3	8	no	—
224	C₄×ΓL(1,𝔽₈)	—	2	8	no	—
225	C₄×ΓL(1,𝔽₈)	—	9	2	no	—
226	C₄×ΓL(1,𝔽₈)	C₂²×C₃⋊D₃	1	2	yes	complemented
227	C₂³×C₃²	—	7	8	no	—
228	C₂³×C₃²	—	14	8	no	—
229	C₂³×C₃²	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
230	C₂²×C₃⋊D₃	—	2	8	no	—
231	C₂²×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
232	C₂²×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
233	C₂²×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
234	C₂²×C₃×D₃	—	21	4	no	—
235	C₂²×C₃×D₃	—	21	8	no	—
236	C₂²×C₃×D₃	—	21	32	no	—
237	C₂²×C₃×D₃	—	42	32	no	—
238	C₃²×D₄	—	7	16	no	—
239	C₃²×D₄	—	14	16	no	—
240	C₃²×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
241	C₂×C₃²×C₄	—	7	4	no	—
242	C₂×C₃²×C₄	—	14	4	no	—
243	C₂×C₃²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
244	#638	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
245	C₂×C₃⋊Dic₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
246	C₄⋊(C₃⋊D₃)	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
247	C₄×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
248	C₃×#67	—	21	32	no	—
249	C₂×C₃×Dic₃	—	21	4	no	—
250	C₃×D₁₂	—	21	16	no	—
251	C₃×C₄×D₃	—	21	16	no	—
252	C₃²×C₇	C₂²×C₃×D₄	1	1	yes	characteristic, complemented
253	C₃×ΓL(1,𝔽₈)	—	2	4	no	—
254	C₃×ΓL(1,𝔽₈)	C₂×D₃×D₄	1	4	yes	complemented
255	C₂³×C₇	—	9	1	no	—
256	C₂³×C₇	—	9	2	no	—
257	C₂³×C₇	—	9	8	no	—
258	C₂³×C₇	—	18	8	no	—
259	C₂³×C₇	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
260	C₇×D₄	—	9	8	no	—
261	C₇×D₄	—	9	4	no	—
262	C₇×D₄	C₂×C₃×C₃⋊D₃	1	4	yes	complemented
263	C₂×C₄×C₇	—	9	1	no	—
264	C₂×C₄×C₇	—	9	4	no	—
265	C₂×C₄×C₇	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
266	C₂×C₃³	—	7	2	no	—
267	C₂×C₃³	—	7	1	no	—
268	C₂×C₃³	—	14	4	no	—
269	C₃×C₃⋊D₃	—	7	4	no	—
270	C₃×C₃⋊D₃	—	14	4	no	—
271	C₂⁴×C₃	—	63	2	no	—
272	C₂³×D₃	—	3	8	no	—
273	C₂×C₃×D₄	—	7	1	no	—
274	C₂×C₃×D₄	—	14	4	no	—
275	C₂×C₃×D₄	—	63	1	no	—
276	C₂×C₃×D₄	—	63	8	no	—
277	C₂×C₃×D₄	—	63	2	no	—
278	C₂×C₃×D₄	D₃×ΓL(1,𝔽₈)	1	4	yes	complemented
279	C₂²×C₃×C₄	—	63	1	no	—
280	C₂×#67	—	3	8	no	—
281	D₃×D₄	—	3	32	no	—
282	C₂×D₁₂	—	3	4	no	—
283	C₂×C₄×D₃	—	3	4	no	—
284	C₂×C₃×C₇	—	2	16	no	—
285	C₂×C₃×C₇	C₃×D₃×D₄	1	8	yes	complemented
286	C₂×C₃×C₇	C₂³×C₃×D₃	1	4	yes	—
287	D₃×C₇	—	3	16	no	—
288	D₃×C₇	—	6	16	no	—
289	C₂×ΓL(1,𝔽₈)	—	2	4	no	—
290	C₂×ΓL(1,𝔽₈)	—	2	4	no	—
291	C₂×ΓL(1,𝔽₈)	—	2	8	no	—
292	C₂×ΓL(1,𝔽₈)	—	4	16	no	—
293	C₂×ΓL(1,𝔽₈)	—	9	4	no	—
294	C₂×ΓL(1,𝔽₈)	—	18	4	no	—
295	C₂×ΓL(1,𝔽₈)	D₄×C₃⋊D₃	1	2	yes	direct product factor
296	C₂×ΓL(1,𝔽₈)	C₂³×C₃⋊D₃	1	1	yes	characteristic
297	C₂²×C₃²	—	2	4	no	—
298	C₂²×C₃²	—	7	4	no	—
299	C₂²×C₃²	—	7	16	no	—
300	C₂²×C₃²	—	14	16	no	—
301	C₂²×C₃²	—	14	4	no	—
302	C₂²×C₃²	—	14	16	no	—
303	C₂²×C₃²	—	28	16	no	—
304	C₂²×C₃²	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
305	C₂²×C₃²	C₂³×ΓL(1,𝔽₈)	1	4	yes	complemented
306	C₂×C₃⋊D₃	—	2	4	no	—
307	C₂×C₃⋊D₃	—	2	16	no	—
308	C₂×C₃⋊D₃	D₄×ΓL(1,𝔽₈)	1	4	yes	direct product factor
309	C₂×C₃⋊D₃	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
310	C₂×C₃⋊D₃	C₂³×ΓL(1,𝔽₈)	1	4	yes	complemented
311	C₂×C₃×D₃	—	21	8	no	—
312	C₂×C₃×D₃	—	21	16	no	—
313	C₂×C₃×D₃	—	21	16	no	—
314	C₂×C₃×D₃	—	42	64	no	—
315	C₂×C₃×D₃	—	42	16	no	—
316	C₃²×C₄	—	7	8	no	—
317	C₃²×C₄	—	14	8	no	—
318	C₃²×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	complemented
319	C₃⋊Dic₃	C₂³×ΓL(1,𝔽₈)	1	2	yes	complemented
320	C₃×Dic₃	—	21	8	no	—
321	C₂²×D₄	—	9	1	no	Sylow
322	C₂²×C₇	—	2	4	no	—
323	C₂²×C₇	—	9	2	no	—
324	C₂²×C₇	—	9	4	no	—
325	C₂²×C₇	—	9	4	no	—
326	C₂²×C₇	—	18	16	no	—
327	C₂²×C₇	—	18	4	no	—
328	C₂²×C₇	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
329	C₂²×C₇	C₂²×C₃×C₃⋊D₃	1	4	yes	complemented
330	C₄×C₇	—	9	2	no	—
331	C₄×C₇	C₂²×C₃×C₃⋊D₃	1	2	yes	complemented
332	C₃³	—	7	1	no	Sylow
333	C₂³×C₃	—	7	2	no	—
334	C₂³×C₃	—	14	8	no	—
335	C₂³×C₃	—	63	1	no	—
336	C₂³×C₃	—	63	2	no	—
337	C₂³×C₃	—	63	8	no	—
338	C₂³×C₃	—	126	8	no	—
339	C₂³×C₃	C₂×D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
340	C₂²×D₃	—	3	4	no	—
341	C₂²×D₃	—	3	8	no	—
342	C₂²×D₃	—	3	32	no	—
343	C₂²×D₃	—	6	32	no	—
344	C₃×D₄	—	7	4	no	—
345	C₃×D₄	—	14	16	no	—
346	C₃×D₄	—	63	4	no	—
347	C₃×D₄	—	63	8	no	—
348	C₃×D₄	C₂×D₃×ΓL(1,𝔽₈)	1	16	yes	complemented
349	C₂×C₃×C₄	—	7	1	no	—
350	C₂×C₃×C₄	—	14	4	no	—
351	C₂×C₃×C₄	—	63	4	no	—
352	C₂×C₃×C₄	—	63	1	no	—
353	C₂×C₃×C₄	C₂×D₃×ΓL(1,𝔽₈)	1	4	yes	complemented
354	#67	—	3	32	no	—
355	C₂×Dic₃	—	3	4	no	—
356	D₁₂	—	3	16	no	—
357	C₄×D₃	—	3	16	no	—
358	C₃×C₇	C₂×C₃×D₃×D₄	1	4	yes	complemented
359	ΓL(1,𝔽₈)	—	2	4	no	—
360	ΓL(1,𝔽₈)	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic, direct product factor
361	C₂×C₃²	—	2	4	no	—
362	C₂×C₃²	—	7	8	no	—
363	C₂×C₃²	—	7	4	no	—
364	C₂×C₃²	—	14	16	no	—
365	C₂×C₃²	—	14	8	no	—
366	C₂×C₃²	—	14	4	no	—
367	C₂×C₃²	—	28	16	no	—
368	C₂×C₃²	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	complemented
369	C₂×C₃²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
370	C₃⋊D₃	—	2	4	no	—
371	C₃⋊D₃	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	direct product factor
372	C₃×D₃	—	21	16	no	—
373	C₃×D₃	—	42	16	no	—
374	C₂⁴	—	9	2	no	—
375	C₂×D₄	—	9	2	no	—
376	C₂×D₄	—	9	8	no	—
377	C₂×D₄	—	9	1	no	—
378	C₂×D₄	C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
379	C₂²×C₄	—	9	1	no	—
380	C₂×C₇	—	2	4	no	—
381	C₂×C₇	—	9	4	no	—
382	C₂×C₇	—	18	4	no	—
383	C₂×C₇	C₃×D₄×C₃⋊D₃	1	2	yes	complemented
384	C₂×C₇	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
385	C₂²×C₃	—	2	16	no	—
386	C₂²×C₃	—	7	1	no	—
387	C₂²×C₃	—	7	4	no	—
388	C₂²×C₃	—	14	4	no	—
389	C₂²×C₃	—	14	4	no	—
390	C₂²×C₃	—	14	16	no	—
391	C₂²×C₃	—	28	16	no	—
392	C₂²×C₃	—	63	2	no	—
393	C₂²×C₃	—	63	4	no	—
394	C₂²×C₃	—	63	4	no	—
395	C₂²×C₃	—	126	4	no	—
396	C₂²×C₃	—	126	16	no	—
397	C₂²×C₃	C₂²×D₃×ΓL(1,𝔽₈)	1	4	yes	—
398	C₂²×C₃	C₂²×D₃×ΓL(1,𝔽₈)	1	16	yes	complemented
399	C₂×D₃	—	3	16	no	—
400	C₂×D₃	—	3	8	no	—
401	C₂×D₃	—	3	16	no	—
402	C₂×D₃	—	6	64	no	—
403	C₂×D₃	—	6	16	no	—
404	C₃×C₄	—	7	2	no	—
405	C₃×C₄	—	14	8	no	—
406	C₃×C₄	—	63	2	no	—
407	C₃×C₄	C₂²×D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
408	Dic₃	—	3	8	no	—
409	C₃²	—	7	4	no	—
410	C₃²	—	14	4	no	—
411	C₃²	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
412	C₂³	—	9	1	no	—
413	C₂³	—	9	2	no	—
414	C₂³	—	9	8	no	—
415	C₂³	—	18	8	no	—
416	C₂³	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	1	2	yes	complemented
417	D₄	—	9	8	no	—
418	D₄	—	9	4	no	—
419	D₄	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	1	4	yes	direct product factor
420	C₂×C₄	—	9	1	no	—
421	C₂×C₄	—	9	4	no	—
422	C₂×C₄	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
423	C₇	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
424	C₂×C₃	—	2	16	no	—
425	C₂×C₃	—	7	2	no	—
426	C₂×C₃	—	7	1	no	—
427	C₂×C₃	—	14	4	no	—
428	C₂×C₃	—	14	8	no	—
429	C₂×C₃	—	14	4	no	—
430	C₂×C₃	—	28	16	no	—
431	C₂×C₃	—	63	4	no	—
432	C₂×C₃	—	126	4	no	—
433	C₂×C₃	D₃×D₄×ΓL(1,𝔽₈)	1	8	yes	complemented
434	C₂×C₃	C₂³×D₃×ΓL(1,𝔽₈)	1	4	yes	—
435	D₃	—	3	16	no	—
436	D₃	—	6	16	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	9	4	no	—
439	C₂²	—	9	2	no	—
440	C₂²	—	9	4	no	—
441	C₂²	—	18	16	no	—
442	C₂²	—	18	4	no	—
443	C₂²	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
444	C₂²	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	1	4	yes	complemented
445	C₄	—	9	2	no	—
446	C₄	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	1	2	yes	complemented
447	C₃	—	7	1	no	—
448	C₃	—	14	4	no	—
449	C₃	C₂×D₃×D₄×ΓL(1,𝔽₈)	1	4	yes	complemented
450	C₂	—	2	4	no	—
451	C₂	—	9	4	no	—
452	C₂	—	18	4	no	—
453	C₂	D₄×C₃⋊D₃×ΓL(1,𝔽₈)	1	2	yes	direct product factor
454	C₂	C₂³×C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₃²×C₇	C₂⁴×C₃
Frattini	C₂	C₂³×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₂×C₃²×C₇×D₄	C₂×C₃
socle	C₂²×C₃²×C₇	C₂³×C₃
hypercenter	C₂×D₄	C₃⋊D₃×ΓL(1,𝔽₈)
hypocenter	C₃²×C₇	C₂²×C₃×D₄
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×ΓL(1,𝔽₈)×#638	C₂	semidirect
2. C₂³×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×ΓL(1,𝔽₈)×C₄⋊(C₃⋊D₃)	C₂	semidirect
4. C₂×C₃²×D₄×ΓL(1,𝔽₈)	C₂	semidirect
5. D₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	direct (N × H)
6. C₂×C₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂	semidirect
7. C₂×C₇×D₄×C₃⋊D₃	C₃	semidirect
9. C₂³×C₃²×ΓL(1,𝔽₈)	C₂²	semidirect
12. C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	semidirect
13. C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	semidirect
14. C₃²×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
15. C₂×C₃²×C₄×ΓL(1,𝔽₈)	C₂²	semidirect
16. ΓL(1,𝔽₈)×#638	C₂²	semidirect
17. C₂×ΓL(1,𝔽₈)×C₃⋊Dic₃	C₂²	semidirect
18. ΓL(1,𝔽₈)×C₄⋊(C₃⋊D₃)	C₂²	semidirect
19. C₄×C₃⋊D₃×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₂³×C₇×C₃⋊D₃	C₂×C₃	semidirect
22. C₂×C₃²×C₇×D₄	C₂×C₃	semidirect
23. C₂×C₇×#638	C₂×C₃	semidirect
24. C₇×D₄×C₃⋊D₃	C₂×C₃	semidirect
25. C₂×C₇×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
26. C₂×C₄×C₇×C₃⋊D₃	C₂×C₃	semidirect
28. C₂×C₃×D₄×ΓL(1,𝔽₈)	D₃	semidirect
36. C₂²×C₃²×ΓL(1,𝔽₈)	C₂³	semidirect
39. C₂×C₃⋊D₃×ΓL(1,𝔽₈)	D₄	direct (N × H)
41. C₂×C₃⋊D₃×ΓL(1,𝔽₈)	C₂³	semidirect
42. C₃²×C₄×ΓL(1,𝔽₈)	C₂³	semidirect
43. ΓL(1,𝔽₈)×C₃⋊Dic₃	C₂³	semidirect
46. C₂³×C₃²×C₇	C₂²×C₃	semidirect
49. C₂²×C₇×C₃⋊D₃	C₂²×C₃	semidirect
50. C₂²×C₇×C₃⋊D₃	C₂²×C₃	semidirect
52. C₂³×C₃×ΓL(1,𝔽₈)	C₂×D₃	semidirect
57. C₃²×C₇×D₄	C₂²×C₃	semidirect
58. C₂×C₃²×C₄×C₇	C₂²×C₃	semidirect
59. C₇×#638	C₂²×C₃	semidirect
60. C₂×C₇×C₃⋊Dic₃	C₂²×C₃	semidirect
61. C₇×C₄⋊(C₃⋊D₃)	C₂²×C₃	semidirect
62. C₄×C₇×C₃⋊D₃	C₂²×C₃	semidirect
64. C₃×D₄×ΓL(1,𝔽₈)	C₂×D₃	semidirect
66. C₂×C₃×C₄×ΓL(1,𝔽₈)	C₂×D₃	semidirect
78. C₂×C₃²×ΓL(1,𝔽₈)	C₂×D₄	semidirect
81. C₃⋊D₃×ΓL(1,𝔽₈)	C₂×D₄	direct (N × H)
84. C₂×C₃×C₇×D₄	C₃×D₃	semidirect
93. C₂×D₄×ΓL(1,𝔽₈)	C₃⋊D₃	direct (N × H)
95. C₂×D₄×C₃⋊D₃	ΓL(1,𝔽₈)	direct (N × H)
99. C₂²×C₃²×C₇	C₂³×C₃	semidirect
102. C₂×C₇×C₃⋊D₃	C₃×D₄	semidirect
104. C₂×C₇×C₃⋊D₃	C₂³×C₃	semidirect
110. C₂²×C₃×ΓL(1,𝔽₈)	C₂²×D₃	semidirect
116. C₃²×C₄×C₇	C₂³×C₃	semidirect
117. C₇×C₃⋊Dic₃	C₂³×C₃	semidirect
119. C₃×C₄×ΓL(1,𝔽₈)	C₂²×D₃	semidirect
133. C₃²×ΓL(1,𝔽₈)	C₂²×D₄	semidirect
134. C₂³×C₃×C₇	C₂×C₃×D₃	semidirect
144. C₂³×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	semidirect
145. C₃×C₇×D₄	C₂×C₃×D₃	semidirect
146. C₂×C₃×C₄×C₇	C₂×C₃×D₃	semidirect
154. D₄×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	direct (N × H)
158. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₃⋊D₃	semidirect
159. C₂³×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	semidirect
163. C₂×C₃²×D₄	C₂×ΓL(1,𝔽₈)	semidirect
164. C₂×#638	C₂×ΓL(1,𝔽₈)	semidirect
165. D₄×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	direct (N × H)
166. C₂×C₄⋊(C₃⋊D₃)	C₂×ΓL(1,𝔽₈)	semidirect
167. C₂×C₄×C₃⋊D₃	C₂×ΓL(1,𝔽₈)	semidirect
173. C₂×C₃²×C₇	C₂×C₃×D₄	semidirect
176. C₇×C₃⋊D₃	C₂×C₃×D₄	semidirect
181. C₂×C₃×ΓL(1,𝔽₈)	D₃×D₄	semidirect
189. C₂×C₇×D₄	C₃×C₃⋊D₃	semidirect
205. C₂²×C₃×C₇	C₂²×C₃×D₃	semidirect
221. C₂²×ΓL(1,𝔽₈)	C₂²×C₃⋊D₃	semidirect
222. C₃×C₄×C₇	C₂²×C₃×D₃	semidirect
226. C₄×ΓL(1,𝔽₈)	C₂²×C₃⋊D₃	semidirect
229. C₂³×C₃²	C₂²×ΓL(1,𝔽₈)	semidirect
232. C₂²×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	semidirect
233. C₂²×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	semidirect
240. C₃²×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
243. C₂×C₃²×C₄	C₂²×ΓL(1,𝔽₈)	semidirect
244. #638	C₂²×ΓL(1,𝔽₈)	semidirect
245. C₂×C₃⋊Dic₃	C₂²×ΓL(1,𝔽₈)	semidirect
246. C₄⋊(C₃⋊D₃)	C₂²×ΓL(1,𝔽₈)	semidirect
247. C₄×C₃⋊D₃	C₂²×ΓL(1,𝔽₈)	semidirect
252. C₃²×C₇	C₂²×C₃×D₄	semidirect
254. C₃×ΓL(1,𝔽₈)	C₂×D₃×D₄	semidirect
259. C₂³×C₇	C₂×C₃×C₃⋊D₃	semidirect
262. C₇×D₄	C₂×C₃×C₃⋊D₃	semidirect
265. C₂×C₄×C₇	C₂×C₃×C₃⋊D₃	semidirect
278. C₂×C₃×D₄	D₃×ΓL(1,𝔽₈)	semidirect
285. C₂×C₃×C₇	C₃×D₃×D₄	semidirect
295. C₂×ΓL(1,𝔽₈)	D₄×C₃⋊D₃	direct (N × H)
305. C₂²×C₃²	C₂³×ΓL(1,𝔽₈)	semidirect
308. C₂×C₃⋊D₃	D₄×ΓL(1,𝔽₈)	direct (N × H)
310. C₂×C₃⋊D₃	C₂³×ΓL(1,𝔽₈)	semidirect
318. C₃²×C₄	C₂³×ΓL(1,𝔽₈)	semidirect
319. C₃⋊Dic₃	C₂³×ΓL(1,𝔽₈)	semidirect
329. C₂²×C₇	C₂²×C₃×C₃⋊D₃	semidirect
331. C₄×C₇	C₂²×C₃×C₃⋊D₃	semidirect
339. C₂³×C₃	C₂×D₃×ΓL(1,𝔽₈)	semidirect
348. C₃×D₄	C₂×D₃×ΓL(1,𝔽₈)	semidirect
353. C₂×C₃×C₄	C₂×D₃×ΓL(1,𝔽₈)	semidirect
358. C₃×C₇	C₂×C₃×D₃×D₄	semidirect
360. ΓL(1,𝔽₈)	C₂×D₄×C₃⋊D₃	direct (N × H)
368. C₂×C₃²	C₂×D₄×ΓL(1,𝔽₈)	semidirect
371. C₃⋊D₃	C₂×D₄×ΓL(1,𝔽₈)	direct (N × H)
378. C₂×D₄	C₃⋊D₃×ΓL(1,𝔽₈)	direct (N × H)
383. C₂×C₇	C₃×D₄×C₃⋊D₃	semidirect
398. C₂²×C₃	C₂²×D₃×ΓL(1,𝔽₈)	semidirect
407. C₃×C₄	C₂²×D₃×ΓL(1,𝔽₈)	semidirect
411. C₃²	C₂²×D₄×ΓL(1,𝔽₈)	semidirect
416. C₂³	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	semidirect
419. D₄	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	direct (N × H)
422. C₂×C₄	C₂×C₃⋊D₃×ΓL(1,𝔽₈)	semidirect
423. C₇	C₂×C₃×D₄×C₃⋊D₃	semidirect
433. C₂×C₃	D₃×D₄×ΓL(1,𝔽₈)	semidirect
444. C₂²	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	semidirect
446. C₄	C₂²×C₃⋊D₃×ΓL(1,𝔽₈)	semidirect
449. C₃	C₂×D₃×D₄×ΓL(1,𝔽₈)	semidirect
453. C₂	D₄×C₃⋊D₃×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×ΓL(1,𝔽₈)×#638	2
C₂³×C₃⋊D₃×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)×C₄⋊(C₃⋊D₃)	2
C₂×C₃²×D₄×ΓL(1,𝔽₈)	2
D₄×C₃⋊D₃×ΓL(1,𝔽₈)	2
C₂×C₄×C₃⋊D₃×ΓL(1,𝔽₈)	2
C₂×C₇×D₄×C₃⋊D₃	3
C₂×D₃×D₄×ΓL(1,𝔽₈)	3
C₂×C₃×D₄×C₃⋊D₃	7

Subgroups — C2×D4×C3⋊D3×ΓL(1,𝔽8) #72245578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₃⋊D₃×ΓL(1,𝔽₈) #72245578