Subgroups — ΓL(1,𝔽₈)×#552 #13060713

Order 1344.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 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	2	yes	maximal, complemented
3	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)×#123	C₂	1	2	yes	maximal
5	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
9	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	ΓL(1,𝔽₈)×#117	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#552	C₃	1	1	yes	maximal, characteristic, complemented, Hall
13	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
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	2	yes	—
27	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
28	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
30	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
31	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
33	C₄²×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	2	yes	complemented
38	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
39	C₇×#123	C₂×C₃	1	2	yes	—
40	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
42	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
43	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
44	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
45	C₇×#117	C₂×C₃	1	2	yes	complemented
46	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#552	—	7	1	no	maximal, Hall
48	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
51	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
52	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
53	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
54	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
56	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
57	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
58	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
59	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
60	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
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×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
67	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄²×C₇	C₂²×C₃	1	2	yes	—
85	C₄²×C₇	C₂²×C₃	1	2	yes	—
86	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
87	C₃×#126	—	7	2	no	—
88	C₃×#125	—	7	2	no	—
89	C₃×#125	—	7	1	no	—
90	C₃×#123	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	1	no	—
93	C₃×C₄×Q₈	—	7	1	no	—
94	C₃×C₄×D₄	—	7	1	no	—
95	C₃×#117	—	7	1	no	—
96	C₃×#117	—	7	2	no	—
97	C₂×C₃×C₄⋊C₄	—	7	1	no	—
98	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
100	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
102	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
103	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
104	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
105	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
106	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
107	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
109	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
110	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
111	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
112	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
113	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
114	#552	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
117	C₇×Q₈	—	2	2	no	—
118	C₇×D₄	—	2	2	no	—
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
126	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
127	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
130	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×D₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₂²×C₃×C₄	—	7	1	no	—
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₃×C₄⋊C₄	—	7	1	no	—
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	1	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₄²	—	7	2	no	—
152	C₃×C₄²	—	7	2	no	—
153	C₃×C₄²	—	7	1	no	—
154	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
155	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
156	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
157	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
158	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
159	#126	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
160	#125	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
161	#125	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
162	#123	C₂×ΓL(1,𝔽₈)	1	2	yes	—
163	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
164	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
165	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
166	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
167	#117	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
168	#117	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
169	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	C₃×#41	1	2	yes	—
176	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	C₃×#41	1	2	yes	—
186	C₂³×C₃	—	7	1	no	—
187	C₂³×C₃	—	7	1	no	—
188	C₃×Q₈	—	14	2	no	—
189	C₃×D₄	—	14	2	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	1	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	14	2	no	—
200	C₂×C₃×C₄	—	14	2	no	—
201	C₂×C₃×C₄	—	14	4	no	—
202	ΓL(1,𝔽₈)	#552	1	1	yes	characteristic, direct product factor, Hall
203	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
204	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
205	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
206	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
207	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
208	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
211	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
212	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
213	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
214	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
216	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
217	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
218	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
219	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
220	C₂²⋊C₄	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	2	yes	—
223	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
224	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
225	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
226	C₂×C₇	—	2	2	no	—
227	C₂×C₇	—	4	1	no	—
228	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
230	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂²×C₃	—	7	2	no	—
232	C₂²×C₃	—	7	1	no	—
233	C₂²×C₃	—	14	1	no	—
234	C₂²×C₃	—	14	1	no	—
235	C₂²×C₃	—	14	1	no	—
236	C₂²×C₃	—	14	1	no	—
237	C₂²×C₃	—	14	1	no	—
238	C₃×C₄	—	7	2	no	—
239	C₃×C₄	—	14	1	no	—
240	C₃×C₄	—	14	1	no	—
241	C₃×C₄	—	14	2	no	—
242	C₃×C₄	—	14	2	no	—
243	C₃×C₄	—	14	1	no	—
244	C₃×C₄	—	14	2	no	—
245	C₃×C₄	—	14	1	no	—
246	C₃×C₄	—	14	2	no	—
247	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
248	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	D₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
255	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
256	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
257	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
258	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
259	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
260	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
261	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
262	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
263	C₇	C₃×#552	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	7	1	no	—
265	C₂×C₃	—	7	1	no	—
266	C₂×C₃	—	7	1	no	—
267	C₂×C₃	—	14	2	no	—
268	C₂×C₃	—	28	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	#41×ΓL(1,𝔽₈)	1	2	yes	—
275	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	—
285	C₃	—	7	1	no	Sylow
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
289	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
290	C₂	ΓL(1,𝔽₈)×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#552	C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#552	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#552
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #552

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)×#125	C₂	semidirect
5. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
6. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
7. C₄×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
9. ΓL(1,𝔽₈)×#117	C₂	semidirect
10. ΓL(1,𝔽₈)×#117	C₂	semidirect
11. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
12. C₇×#552	C₃	semidirect
36. C₇×#126	C₂×C₃	semidirect
37. C₇×#125	C₂×C₃	semidirect
38. C₇×#125	C₂×C₃	semidirect
40. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
41. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
42. C₄×C₇×Q₈	C₂×C₃	semidirect
44. C₇×#117	C₂×C₃	semidirect
45. C₇×#117	C₂×C₃	semidirect
46. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
114. #552	ΓL(1,𝔽₈)	direct (N × H)
159. #126	C₂×ΓL(1,𝔽₈)	semidirect
160. #125	C₂×ΓL(1,𝔽₈)	semidirect
161. #125	C₂×ΓL(1,𝔽₈)	semidirect
163. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
164. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
165. C₄×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
167. #117	C₂×ΓL(1,𝔽₈)	semidirect
168. #117	C₂×ΓL(1,𝔽₈)	semidirect
169. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
202. ΓL(1,𝔽₈)	#552	direct (N × H)
263. C₇	C₃×#552	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)×C₂²⋊Q₈	2
C₄×Q₈×ΓL(1,𝔽₈)	2
C₄×D₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#117	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₇×#552	3
C₃×#552	7

Subgroups — ΓL(1,𝔽8)×#552 #13060713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#552 #13060713