Subgroups — ΓL(1,𝔽₈)×#2403 #72259827

Order 2688.

3128 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
2	C₂×ΓL(1,𝔽₈)×#120	C₂	1	4	yes	maximal, complemented
3	ΓL(1,𝔽₈)×#392	C₂	1	4	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#386	C₂	1	4	yes	maximal, complemented
5	ΓL(1,𝔽₈)×#379	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2403	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂⁵×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₂⁵×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
9	C₂²×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
10	C₂²×D₄×ΓL(1,𝔽₈)	—	2	16	no	—
11	C₂²×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
12	C₂²×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
13	C₂²×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
14	C₂³×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
15	ΓL(1,𝔽₈)×#120	—	2	16	no	—
16	C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
18	C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
19	ΓL(1,𝔽₈)×#95	C₂²	1	4	yes	complemented
20	C₂³×C₇×D₄	C₂×C₃	1	2	yes	complemented
21	C₂×C₇×#120	C₂×C₃	1	4	yes	complemented
22	C₇×#392	C₂×C₃	1	4	yes	complemented
23	C₇×#386	C₂×C₃	1	4	yes	complemented
24	C₇×#379	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#2403	—	7	1	no	maximal, Hall
26	C₂⁴×ΓL(1,𝔽₈)	—	2	8	no	—
27	C₂⁴×ΓL(1,𝔽₈)	—	2	4	no	—
28	C₂⁴×ΓL(1,𝔽₈)	—	2	8	no	—
29	C₂⁴×ΓL(1,𝔽₈)	—	2	8	no	—
30	C₂⁴×ΓL(1,𝔽₈)	—	4	2	no	—
31	C₂⁴×ΓL(1,𝔽₈)	—	4	4	no	—
32	C₂⁴×ΓL(1,𝔽₈)	D₄	1	4	yes	complemented
33	C₂⁴×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
34	C₂⁴×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
35	C₂⁴×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
36	C₂×D₄×ΓL(1,𝔽₈)	—	2	64	no	—
37	C₂×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
38	C₂×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
39	C₂×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
40	C₂×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
41	C₂×D₄×ΓL(1,𝔽₈)	—	2	16	no	—
42	C₂×D₄×ΓL(1,𝔽₈)	—	2	8	no	—
43	C₂×D₄×ΓL(1,𝔽₈)	—	4	4	no	—
44	C₂×D₄×ΓL(1,𝔽₈)	D₄	1	16	yes	complemented
45	C₂²×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
46	C₂²×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
47	C₂²×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
48	C₂²×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
49	C₂²⋊C₄×ΓL(1,𝔽₈)	—	2	8	no	—
50	C₂²⋊C₄×ΓL(1,𝔽₈)	—	2	16	no	—
51	C₂²⋊C₄×ΓL(1,𝔽₈)	—	4	2	no	—
52	C₂⁵×C₇	C₂²×C₃	1	1	yes	characteristic
53	C₂⁵×C₇	C₂²×C₃	1	2	yes	complemented
54	C₂²×C₇×D₄	—	2	8	no	—
55	C₂²×C₇×D₄	—	2	16	no	—
56	C₂²×C₇×D₄	C₂²×C₃	1	4	yes	—
57	C₂²×C₇×D₄	C₂²×C₃	1	4	yes	complemented
58	C₂²×C₇×D₄	C₂²×C₃	1	4	yes	complemented
59	C₂³×C₄×C₇	C₂²×C₃	1	2	yes	complemented
60	C₇×#120	—	2	16	no	—
61	C₂×C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
62	C₂×C₇×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
63	C₂×C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
64	C₇×#95	C₂²×C₃	1	4	yes	complemented
65	C₂³×C₃×D₄	—	7	2	no	—
66	C₂×C₃×#120	—	7	4	no	—
67	C₃×#392	—	7	4	no	—
68	C₃×#386	—	7	4	no	—
69	C₃×#379	—	7	1	no	—
70	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
71	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
72	C₂³×ΓL(1,𝔽₈)	—	2	16	no	—
73	C₂³×ΓL(1,𝔽₈)	—	2	16	no	—
74	C₂³×ΓL(1,𝔽₈)	—	2	8	no	—
75	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
76	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
77	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
78	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
79	C₂³×ΓL(1,𝔽₈)	—	2	8	no	—
80	C₂³×ΓL(1,𝔽₈)	—	4	4	no	—
81	C₂³×ΓL(1,𝔽₈)	—	4	16	no	—
82	C₂³×ΓL(1,𝔽₈)	—	4	32	no	—
83	C₂³×ΓL(1,𝔽₈)	—	4	8	no	—
84	C₂³×ΓL(1,𝔽₈)	—	4	8	no	—
85	C₂³×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
86	C₂³×ΓL(1,𝔽₈)	C₂×D₄	1	8	yes	—
87	C₂³×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	—
88	C₂³×ΓL(1,𝔽₈)	C₂×D₄	1	8	yes	complemented
89	C₂³×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
90	D₄×ΓL(1,𝔽₈)	—	2	32	no	—
91	D₄×ΓL(1,𝔽₈)	—	2	32	no	—
92	D₄×ΓL(1,𝔽₈)	—	4	16	no	—
93	C₂×C₄×ΓL(1,𝔽₈)	—	2	16	no	—
94	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
95	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
96	C₂×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
97	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
98	C₂×C₄×ΓL(1,𝔽₈)	C₂×D₄	1	8	yes	complemented
99	#2403	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
100	C₂⁴×C₇	—	2	8	no	—
101	C₂⁴×C₇	—	2	4	no	—
102	C₂⁴×C₇	—	2	8	no	—
103	C₂⁴×C₇	—	2	8	no	—
104	C₂⁴×C₇	—	4	2	no	—
105	C₂⁴×C₇	—	4	4	no	—
106	C₂⁴×C₇	C₃×D₄	1	4	yes	complemented
107	C₂⁴×C₇	C₂³×C₃	1	1	yes	characteristic
108	C₂⁴×C₇	C₂³×C₃	1	4	yes	—
109	C₂⁴×C₇	C₂³×C₃	1	2	yes	—
110	C₂×C₇×D₄	—	2	64	no	—
111	C₂×C₇×D₄	—	2	8	no	—
112	C₂×C₇×D₄	—	2	8	no	—
113	C₂×C₇×D₄	—	2	8	no	—
114	C₂×C₇×D₄	—	2	8	no	—
115	C₂×C₇×D₄	—	2	16	no	—
116	C₂×C₇×D₄	—	2	8	no	—
117	C₂×C₇×D₄	—	4	4	no	—
118	C₂×C₇×D₄	C₃×D₄	1	16	yes	complemented
119	C₂²×C₄×C₇	—	2	4	no	—
120	C₂²×C₄×C₇	—	2	8	no	—
121	C₂²×C₄×C₇	C₂³×C₃	1	4	yes	—
122	C₂²×C₄×C₇	C₂³×C₃	1	4	yes	—
123	C₇×C₂²⋊C₄	—	2	8	no	—
124	C₇×C₂²⋊C₄	—	2	16	no	—
125	C₇×C₂²⋊C₄	—	4	2	no	—
126	C₂⁵×C₃	—	7	1	no	—
127	C₂⁵×C₃	—	7	2	no	—
128	C₂²×C₃×D₄	—	7	4	no	—
129	C₂²×C₃×D₄	—	7	4	no	—
130	C₂²×C₃×D₄	—	7	4	no	—
131	C₂²×C₃×D₄	—	14	16	no	—
132	C₂²×C₃×D₄	—	14	8	no	—
133	C₂³×C₃×C₄	—	7	2	no	—
134	C₃×#120	—	14	16	no	—
135	C₂×C₃×C₂²⋊C₄	—	7	4	no	—
136	C₂×C₃×C₂²⋊C₄	—	7	8	no	—
137	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
138	C₃×#95	—	7	4	no	—
139	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
140	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
141	C₂²×ΓL(1,𝔽₈)	—	2	16	no	—
142	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
143	C₂²×ΓL(1,𝔽₈)	—	2	8	no	—
144	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
145	C₂²×ΓL(1,𝔽₈)	—	2	16	no	—
146	C₂²×ΓL(1,𝔽₈)	—	4	16	no	—
147	C₂²×ΓL(1,𝔽₈)	—	4	32	no	—
148	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
149	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
150	C₂²×ΓL(1,𝔽₈)	—	4	8	no	—
151	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
152	C₂²×ΓL(1,𝔽₈)	#120	1	8	yes	complemented
153	C₂²×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
154	C₂²×ΓL(1,𝔽₈)	C₂²×D₄	1	4	yes	—
155	C₂²×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	2	yes	—
156	C₄×ΓL(1,𝔽₈)	—	2	8	no	—
157	C₄×ΓL(1,𝔽₈)	—	4	4	no	—
158	C₂³×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
159	C₂×#120	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
160	#392	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
161	#386	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
162	#379	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
163	C₂³×C₇	—	2	4	no	—
164	C₂³×C₇	—	2	4	no	—
165	C₂³×C₇	—	2	16	no	—
166	C₂³×C₇	—	2	16	no	—
167	C₂³×C₇	—	2	8	no	—
168	C₂³×C₇	—	2	4	no	—
169	C₂³×C₇	—	2	4	no	—
170	C₂³×C₇	—	2	4	no	—
171	C₂³×C₇	—	2	2	no	—
172	C₂³×C₇	—	2	8	no	—
173	C₂³×C₇	—	4	4	no	—
174	C₂³×C₇	—	4	16	no	—
175	C₂³×C₇	—	4	32	no	—
176	C₂³×C₇	—	4	8	no	—
177	C₂³×C₇	—	4	8	no	—
178	C₂³×C₇	C₂×C₃×D₄	1	2	yes	—
179	C₂³×C₇	C₂×C₃×D₄	1	8	yes	—
180	C₂³×C₇	C₂×C₃×D₄	1	4	yes	—
181	C₂³×C₇	C₂×C₃×D₄	1	8	yes	complemented
182	C₂³×C₇	C₂⁴×C₃	1	1	yes	characteristic
183	C₇×D₄	—	2	32	no	—
184	C₇×D₄	—	2	32	no	—
185	C₇×D₄	—	4	16	no	—
186	C₂×C₄×C₇	—	2	16	no	—
187	C₂×C₄×C₇	—	2	4	no	—
188	C₂×C₄×C₇	—	2	4	no	—
189	C₂×C₄×C₇	—	2	8	no	—
190	C₂×C₄×C₇	—	2	4	no	—
191	C₂×C₄×C₇	C₂×C₃×D₄	1	8	yes	complemented
192	C₂⁴×C₃	—	7	4	no	—
193	C₂⁴×C₃	—	7	2	no	—
194	C₂⁴×C₃	—	7	1	no	—
195	C₂⁴×C₃	—	7	4	no	—
196	C₂⁴×C₃	—	14	8	no	—
197	C₂⁴×C₃	—	14	4	no	—
198	C₂⁴×C₃	—	14	8	no	—
199	C₂⁴×C₃	—	14	8	no	—
200	C₂⁴×C₃	—	28	2	no	—
201	C₂⁴×C₃	—	28	4	no	—
202	C₂×C₃×D₄	—	7	16	no	—
203	C₂×C₃×D₄	—	14	8	no	—
204	C₂×C₃×D₄	—	14	8	no	—
205	C₂×C₃×D₄	—	14	8	no	—
206	C₂×C₃×D₄	—	14	8	no	—
207	C₂×C₃×D₄	—	14	8	no	—
208	C₂×C₃×D₄	—	14	64	no	—
209	C₂×C₃×D₄	—	14	16	no	—
210	C₂×C₃×D₄	—	28	4	no	—
211	C₂²×C₃×C₄	—	7	4	no	—
212	C₂²×C₃×C₄	—	7	4	no	—
213	C₂²×C₃×C₄	—	14	8	no	—
214	C₂²×C₃×C₄	—	14	4	no	—
215	C₃×C₂²⋊C₄	—	14	16	no	—
216	C₃×C₂²⋊C₄	—	14	8	no	—
217	C₃×C₂²⋊C₄	—	28	2	no	—
218	C₂×ΓL(1,𝔽₈)	—	2	8	no	—
219	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
220	C₂×ΓL(1,𝔽₈)	—	4	8	no	—
221	C₂×ΓL(1,𝔽₈)	C₂×#120	1	2	yes	—
222	C₂×ΓL(1,𝔽₈)	#534	1	1	yes	characteristic
223	C₂×ΓL(1,𝔽₈)	D₄²	1	4	yes	—
224	C₂⁵	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
225	C₂⁵	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
226	C₂²×D₄	—	2	8	no	—
227	C₂²×D₄	—	2	16	no	—
228	C₂²×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
229	C₂²×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
230	C₂²×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
231	C₂³×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
232	#120	—	2	16	no	—
233	C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
234	C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
235	C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
236	#95	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
237	C₂²×C₇	—	2	4	no	—
238	C₂²×C₇	—	2	4	no	—
239	C₂²×C₇	—	2	16	no	—
240	C₂²×C₇	—	2	4	no	—
241	C₂²×C₇	—	2	8	no	—
242	C₂²×C₇	—	2	2	no	—
243	C₂²×C₇	—	2	16	no	—
244	C₂²×C₇	—	4	16	no	—
245	C₂²×C₇	—	4	32	no	—
246	C₂²×C₇	—	4	4	no	—
247	C₂²×C₇	—	4	4	no	—
248	C₂²×C₇	—	4	8	no	—
249	C₂²×C₇	—	4	4	no	—
250	C₂²×C₇	C₃×#120	1	8	yes	complemented
251	C₂²×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
252	C₂²×C₇	C₂²×C₃×D₄	1	4	yes	—
253	C₂²×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
254	C₄×C₇	—	2	8	no	—
255	C₄×C₇	—	4	4	no	—
256	C₂³×C₃	—	7	8	no	—
257	C₂³×C₃	—	7	4	no	—
258	C₂³×C₃	—	7	2	no	—
259	C₂³×C₃	—	7	8	no	—
260	C₂³×C₃	—	7	1	no	—
261	C₂³×C₃	—	14	4	no	—
262	C₂³×C₃	—	14	4	no	—
263	C₂³×C₃	—	14	8	no	—
264	C₂³×C₃	—	14	2	no	—
265	C₂³×C₃	—	14	4	no	—
266	C₂³×C₃	—	14	4	no	—
267	C₂³×C₃	—	14	4	no	—
268	C₂³×C₃	—	14	16	no	—
269	C₂³×C₃	—	14	8	no	—
270	C₂³×C₃	—	14	16	no	—
271	C₂³×C₃	—	28	8	no	—
272	C₂³×C₃	—	28	4	no	—
273	C₂³×C₃	—	28	16	no	—
274	C₂³×C₃	—	28	32	no	—
275	C₂³×C₃	—	28	8	no	—
276	C₃×D₄	—	14	32	no	—
277	C₃×D₄	—	14	32	no	—
278	C₃×D₄	—	28	16	no	—
279	C₂×C₃×C₄	—	7	8	no	—
280	C₂×C₃×C₄	—	14	4	no	—
281	C₂×C₃×C₄	—	14	4	no	—
282	C₂×C₃×C₄	—	14	16	no	—
283	C₂×C₃×C₄	—	14	8	no	—
284	C₂×C₃×C₄	—	14	4	no	—
285	ΓL(1,𝔽₈)	#2403	1	1	yes	characteristic, direct product factor, Hall
286	C₂⁴	—	2	8	no	—
287	C₂⁴	—	2	4	no	—
288	C₂⁴	—	2	8	no	—
289	C₂⁴	—	2	8	no	—
290	C₂⁴	—	4	2	no	—
291	C₂⁴	—	4	4	no	—
292	C₂⁴	D₄×ΓL(1,𝔽₈)	1	4	yes	complemented
293	C₂⁴	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
294	C₂⁴	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
295	C₂⁴	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
296	C₂×D₄	—	2	64	no	—
297	C₂×D₄	—	2	8	no	—
298	C₂×D₄	—	2	8	no	—
299	C₂×D₄	—	2	8	no	—
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	16	no	—
302	C₂×D₄	—	2	8	no	—
303	C₂×D₄	—	4	4	no	—
304	C₂×D₄	D₄×ΓL(1,𝔽₈)	1	16	yes	complemented
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
308	C₂²×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
309	C₂²⋊C₄	—	2	8	no	—
310	C₂²⋊C₄	—	2	16	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂×C₇	—	2	8	no	—
313	C₂×C₇	—	4	2	no	—
314	C₂×C₇	—	4	8	no	—
315	C₂×C₇	C₂×C₃×#120	1	2	yes	—
316	C₂×C₇	C₃×#534	1	1	yes	characteristic
317	C₂×C₇	C₃×D₄²	1	4	yes	—
318	C₂²×C₃	—	7	4	no	—
319	C₂²×C₃	—	7	1	no	—
320	C₂²×C₃	—	7	2	no	—
321	C₂²×C₃	—	7	8	no	—
322	C₂²×C₃	—	14	4	no	—
323	C₂²×C₃	—	14	4	no	—
324	C₂²×C₃	—	14	16	no	—
325	C₂²×C₃	—	14	4	no	—
326	C₂²×C₃	—	14	8	no	—
327	C₂²×C₃	—	14	16	no	—
328	C₂²×C₃	—	14	2	no	—
329	C₂²×C₃	—	28	4	no	—
330	C₂²×C₃	—	28	4	no	—
331	C₂²×C₃	—	28	8	no	—
332	C₂²×C₃	—	28	4	no	—
333	C₂²×C₃	—	28	16	no	—
334	C₂²×C₃	—	28	32	no	—
335	C₃×C₄	—	14	8	no	—
336	C₃×C₄	—	28	4	no	—
337	C₂³	—	2	4	no	—
338	C₂³	—	2	4	no	—
339	C₂³	—	2	16	no	—
340	C₂³	—	2	8	no	—
341	C₂³	—	2	4	no	—
342	C₂³	—	2	4	no	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	8	no	—
346	C₂³	—	2	16	no	—
347	C₂³	—	4	4	no	—
348	C₂³	—	4	16	no	—
349	C₂³	—	4	32	no	—
350	C₂³	—	4	8	no	—
351	C₂³	—	4	8	no	—
352	C₂³	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
353	C₂³	C₂×D₄×ΓL(1,𝔽₈)	1	8	yes	—
354	C₂³	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	—
355	C₂³	C₂×D₄×ΓL(1,𝔽₈)	1	8	yes	complemented
356	C₂³	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
357	D₄	—	2	32	no	—
358	D₄	—	2	32	no	—
359	D₄	—	4	16	no	—
360	C₂×C₄	—	2	16	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	C₂×D₄×ΓL(1,𝔽₈)	1	8	yes	complemented
366	C₇	C₃×#2403	1	1	yes	characteristic, complemented, Sylow
367	C₂×C₃	—	7	2	no	—
368	C₂×C₃	—	7	1	no	—
369	C₂×C₃	—	7	4	no	—
370	C₂×C₃	—	14	8	no	—
371	C₂×C₃	—	28	2	no	—
372	C₂×C₃	—	28	8	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	16	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	16	no	—
380	C₂²	—	4	16	no	—
381	C₂²	—	4	32	no	—
382	C₂²	—	4	4	no	—
383	C₂²	—	4	4	no	—
384	C₂²	—	4	8	no	—
385	C₂²	—	4	4	no	—
386	C₂²	ΓL(1,𝔽₈)×#120	1	8	yes	complemented
387	C₂²	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
388	C₂²	C₂²×D₄×ΓL(1,𝔽₈)	1	4	yes	—
389	C₂²	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	2	yes	—
390	C₄	—	2	8	no	—
391	C₄	—	4	4	no	—
392	C₃	—	7	1	no	Sylow
393	C₂	—	2	8	no	—
394	C₂	—	4	2	no	—
395	C₂	—	4	8	no	—
396	C₂	C₂×ΓL(1,𝔽₈)×#120	1	2	yes	—
397	C₂	ΓL(1,𝔽₈)×#534	1	1	yes	characteristic
398	C₂	D₄²×ΓL(1,𝔽₈)	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂³×C₇	C₂⁴×C₃
Fitting	C₇×#2403	C₃
hypercenter	#2403	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#2403
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #2403

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂×ΓL(1,𝔽₈)×#120	C₂	semidirect
3. ΓL(1,𝔽₈)×#392	C₂	semidirect
4. ΓL(1,𝔽₈)×#386	C₂	semidirect
5. ΓL(1,𝔽₈)×#379	C₂	semidirect
6. C₇×#2403	C₃	semidirect
8. C₂⁵×ΓL(1,𝔽₈)	C₂²	semidirect
12. C₂²×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
13. C₂²×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
14. C₂³×C₄×ΓL(1,𝔽₈)	C₂²	semidirect
16. C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
18. C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
19. ΓL(1,𝔽₈)×#95	C₂²	semidirect
20. C₂³×C₇×D₄	C₂×C₃	semidirect
21. C₂×C₇×#120	C₂×C₃	semidirect
22. C₇×#392	C₂×C₃	semidirect
23. C₇×#386	C₂×C₃	semidirect
24. C₇×#379	C₂×C₃	semidirect
32. C₂⁴×ΓL(1,𝔽₈)	D₄	semidirect
44. C₂×D₄×ΓL(1,𝔽₈)	D₄	semidirect
53. C₂⁵×C₇	C₂²×C₃	semidirect
57. C₂²×C₇×D₄	C₂²×C₃	semidirect
58. C₂²×C₇×D₄	C₂²×C₃	semidirect
59. C₂³×C₄×C₇	C₂²×C₃	semidirect
61. C₂×C₇×C₂²⋊C₄	C₂²×C₃	semidirect
62. C₂×C₇×C₂²⋊C₄	C₂²×C₃	semidirect
63. C₂×C₇×C₂²⋊C₄	C₂²×C₃	semidirect
64. C₇×#95	C₂²×C₃	semidirect
88. C₂³×ΓL(1,𝔽₈)	C₂×D₄	semidirect
98. C₂×C₄×ΓL(1,𝔽₈)	C₂×D₄	semidirect
99. #2403	ΓL(1,𝔽₈)	direct (N × H)
106. C₂⁴×C₇	C₃×D₄	semidirect
118. C₂×C₇×D₄	C₃×D₄	semidirect
152. C₂²×ΓL(1,𝔽₈)	#120	semidirect
158. C₂³×D₄	C₂×ΓL(1,𝔽₈)	semidirect
159. C₂×#120	C₂×ΓL(1,𝔽₈)	semidirect
160. #392	C₂×ΓL(1,𝔽₈)	semidirect
161. #386	C₂×ΓL(1,𝔽₈)	semidirect
162. #379	C₂×ΓL(1,𝔽₈)	semidirect
181. C₂³×C₇	C₂×C₃×D₄	semidirect
191. C₂×C₄×C₇	C₂×C₃×D₄	semidirect
225. C₂⁵	C₂²×ΓL(1,𝔽₈)	semidirect
229. C₂²×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
230. C₂²×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
231. C₂³×C₄	C₂²×ΓL(1,𝔽₈)	semidirect
233. C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
234. C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
235. C₂×C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
236. #95	C₂²×ΓL(1,𝔽₈)	semidirect
250. C₂²×C₇	C₃×#120	semidirect
285. ΓL(1,𝔽₈)	#2403	direct (N × H)
292. C₂⁴	D₄×ΓL(1,𝔽₈)	semidirect
304. C₂×D₄	D₄×ΓL(1,𝔽₈)	semidirect
355. C₂³	C₂×D₄×ΓL(1,𝔽₈)	semidirect
365. C₂×C₄	C₂×D₄×ΓL(1,𝔽₈)	semidirect
366. C₇	C₃×#2403	semidirect
386. C₂²	ΓL(1,𝔽₈)×#120	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)×#120	2
ΓL(1,𝔽₈)×#392	2
ΓL(1,𝔽₈)×#386	2
ΓL(1,𝔽₈)×#379	2
C₇×#2403	3
C₃×#2403	7

Subgroups — ΓL(1,𝔽8)×#2403 #72259827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#2403 #72259827