Subgroups — ΓL(1,𝔽₈)×#553 #13060714

Order 1344.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#126	C₂	1	2	yes	maximal, complemented
2	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#123	C₂	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#121	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#553	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
17	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
26	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
28	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
29	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
31	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
32	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
34	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
35	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
36	C₇×#126	C₂×C₃	1	2	yes	complemented
37	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
38	C₇×#123	C₂×C₃	1	2	yes	complemented
39	C₇×#121	C₂×C₃	1	2	yes	complemented
40	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
41	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
42	C₄×C₇×D₄	C₂×C₃	1	2	yes	complemented
43	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
44	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
45	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#553	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
50	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
52	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
53	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
54	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
55	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
56	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
57	C₂×C₄×ΓL(1,𝔽₈)	—	4	2	no	—
58	C₂×C₄×ΓL(1,𝔽₈)	—	4	2	no	—
59	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
60	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
61	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
62	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
63	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
64	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
65	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
66	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
67	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
68	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
75	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
86	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₄²×C₇	C₂²×C₃	1	2	yes	—
89	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
90	C₃×#126	—	7	2	no	—
91	C₃×#125	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×#121	—	7	2	no	—
94	C₃×#121	—	7	1	no	—
95	C₃×#121	—	7	1	no	—
96	C₃×C₄×D₄	—	7	2	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₃×C₄×D₄	—	7	1	no	—
99	C₃×#117	—	7	1	no	—
100	C₂×C₃×C₄⋊C₄	—	7	1	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
103	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
104	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
105	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
106	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
107	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
108	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
110	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
111	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
112	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
113	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
114	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
115	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
116	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
117	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
118	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
119	#553	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂³×C₇	C₂³×C₃	1	2	yes	—
122	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₇×D₄	—	2	4	no	—
124	C₇×D₄	—	2	2	no	—
125	C₇×D₄	—	2	2	no	—
126	C₇×D₄	—	4	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
133	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
135	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
138	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×D₄	—	7	1	no	—
140	C₂×C₃×D₄	—	7	2	no	—
141	C₂×C₃×D₄	—	7	1	no	—
142	C₂×C₃×D₄	—	7	2	no	—
143	C₂²×C₃×C₄	—	7	1	no	—
144	C₂²×C₃×C₄	—	7	1	no	—
145	C₂²×C₃×C₄	—	7	2	no	—
146	C₂²×C₃×C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	—	7	1	no	—
148	C₃×C₄⋊C₄	—	7	1	no	—
149	C₃×C₄⋊C₄	—	7	2	no	—
150	C₃×C₄⋊C₄	—	7	1	no	—
151	C₃×C₄⋊C₄	—	7	2	no	—
152	C₃×C₄⋊C₄	—	7	1	no	—
153	C₃×C₄⋊C₄	—	7	2	no	—
154	C₃×C₂²⋊C₄	—	7	2	no	—
155	C₃×C₂²⋊C₄	—	7	1	no	—
156	C₃×C₂²⋊C₄	—	7	2	no	—
157	C₃×C₂²⋊C₄	—	7	2	no	—
158	C₃×C₂²⋊C₄	—	7	1	no	—
159	C₃×C₂²⋊C₄	—	7	2	no	—
160	C₃×C₄²	—	7	2	no	—
161	C₃×C₄²	—	7	1	no	—
162	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
163	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
164	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
165	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
166	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
167	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
168	#126	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
169	#125	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
170	#123	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
171	#121	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
172	#121	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
173	#121	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
174	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
175	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
176	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
177	#117	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
178	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	—	2	1	no	—
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	—	2	1	no	—
183	C₂²×C₇	—	2	2	no	—
184	C₂²×C₇	—	2	2	no	—
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	C₃×#41	1	2	yes	—
188	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
189	C₄×C₇	—	2	1	no	—
190	C₄×C₇	—	2	1	no	—
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	—	2	1	no	—
193	C₄×C₇	—	2	1	no	—
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	2	2	no	—
196	C₄×C₇	C₃×#41	1	2	yes	—
197	C₂³×C₃	—	7	1	no	—
198	C₂³×C₃	—	7	1	no	—
199	C₂³×C₃	—	7	2	no	—
200	C₃×D₄	—	14	2	no	—
201	C₃×D₄	—	14	4	no	—
202	C₃×D₄	—	14	2	no	—
203	C₃×D₄	—	28	2	no	—
204	C₂×C₃×C₄	—	7	1	no	—
205	C₂×C₃×C₄	—	7	1	no	—
206	C₂×C₃×C₄	—	7	2	no	—
207	C₂×C₃×C₄	—	7	1	no	—
208	C₂×C₃×C₄	—	7	2	no	—
209	C₂×C₃×C₄	—	7	1	no	—
210	C₂×C₃×C₄	—	7	1	no	—
211	C₂×C₃×C₄	—	7	2	no	—
212	C₂×C₃×C₄	—	14	2	no	—
213	C₂×C₃×C₄	—	14	2	no	—
214	C₂×C₃×C₄	—	28	2	no	—
215	C₂×C₃×C₄	—	28	2	no	—
216	ΓL(1,𝔽₈)	#553	1	1	yes	characteristic, direct product factor, Hall
217	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
218	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
219	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
220	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
221	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
222	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
223	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
224	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
225	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
228	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
229	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
230	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
231	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
232	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
233	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
234	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
235	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
236	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
237	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
238	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
239	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
240	C₂×C₇	—	2	2	no	—
241	C₂×C₇	—	4	1	no	—
242	C₂×C₇	—	4	2	no	—
243	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
244	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
245	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂²×C₃	—	7	2	no	—
247	C₂²×C₃	—	7	1	no	—
248	C₂²×C₃	—	14	1	no	—
249	C₂²×C₃	—	14	1	no	—
250	C₂²×C₃	—	14	1	no	—
251	C₂²×C₃	—	14	2	no	—
252	C₂²×C₃	—	14	2	no	—
253	C₂²×C₃	—	14	1	no	—
254	C₂²×C₃	—	14	2	no	—
255	C₂²×C₃	—	14	1	no	—
256	C₃×C₄	—	7	2	no	—
257	C₃×C₄	—	14	1	no	—
258	C₃×C₄	—	14	1	no	—
259	C₃×C₄	—	14	2	no	—
260	C₃×C₄	—	14	1	no	—
261	C₃×C₄	—	14	2	no	—
262	C₃×C₄	—	14	2	no	—
263	C₃×C₄	—	14	1	no	—
264	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
265	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
266	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	—	2	2	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
276	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
277	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
278	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
279	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
280	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
281	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
282	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
283	C₇	C₃×#553	1	1	yes	characteristic, complemented, Sylow
284	C₂×C₃	—	7	1	no	—
285	C₂×C₃	—	7	1	no	—
286	C₂×C₃	—	7	1	no	—
287	C₂×C₃	—	14	2	no	—
288	C₂×C₃	—	28	1	no	—
289	C₂×C₃	—	28	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	#41×ΓL(1,𝔽₈)	1	2	yes	—
299	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	—
308	C₃	—	7	1	no	Sylow
309	C₂	—	2	2	no	—
310	C₂	—	4	1	no	—
311	C₂	—	4	2	no	—
312	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
313	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
314	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#553	C₃
hypercenter	#553	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#553
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #553

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#126	C₂	semidirect
2. ΓL(1,𝔽₈)×#125	C₂	semidirect
3. ΓL(1,𝔽₈)×#123	C₂	semidirect
4. ΓL(1,𝔽₈)×#121	C₂	semidirect
5. ΓL(1,𝔽₈)×#121	C₂	semidirect
6. ΓL(1,𝔽₈)×#121	C₂	semidirect
7. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
8. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
9. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
10. ΓL(1,𝔽₈)×#117	C₂	semidirect
11. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
12. C₇×#553	C₃	semidirect
36. C₇×#126	C₂×C₃	semidirect
37. C₇×#125	C₂×C₃	semidirect
38. C₇×#123	C₂×C₃	semidirect
39. C₇×#121	C₂×C₃	semidirect
40. C₇×#121	C₂×C₃	semidirect
41. C₇×#121	C₂×C₃	semidirect
42. C₄×C₇×D₄	C₂×C₃	semidirect
43. C₄×C₇×D₄	C₂×C₃	semidirect
44. C₄×C₇×D₄	C₂×C₃	semidirect
45. C₇×#117	C₂×C₃	semidirect
46. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
119. #553	ΓL(1,𝔽₈)	direct (N × H)
168. #126	C₂×ΓL(1,𝔽₈)	semidirect
169. #125	C₂×ΓL(1,𝔽₈)	semidirect
170. #123	C₂×ΓL(1,𝔽₈)	semidirect
171. #121	C₂×ΓL(1,𝔽₈)	semidirect
172. #121	C₂×ΓL(1,𝔽₈)	semidirect
173. #121	C₂×ΓL(1,𝔽₈)	semidirect
174. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
175. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
176. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
177. #117	C₂×ΓL(1,𝔽₈)	semidirect
178. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
216. ΓL(1,𝔽₈)	#553	direct (N × H)
283. C₇	C₃×#553	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×#126	2
ΓL(1,𝔽₈)×#125	2
ΓL(1,𝔽₈)×#123	2
ΓL(1,𝔽₈)×#121	2
C₄×D₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#117	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₇×#553	3
C₃×#553	7

Subgroups — ΓL(1,𝔽8)×#553 #13060714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#553 #13060714