Subgroups — ΓL(1,𝔽₈)×#538 #13060699

Order 1344.

500 conjugacy classes of subgroups.

The subgroup lattice has 300 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)×#124	C₂	1	2	yes	maximal, complemented
3	ΓL(1,𝔽₈)×#123	C₂	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	ΓL(1,𝔽₈)×#121	C₂	1	2	yes	maximal, complemented
6	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	ΓL(1,𝔽₈)×#120	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#538	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂⁴×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
15	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
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	complemented
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
23	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
24	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
25	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
26	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
27	C₂²⋊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	2	yes	complemented
30	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
31	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
32	C₇×#126	C₂×C₃	1	2	yes	complemented
33	C₇×#124	C₂×C₃	1	2	yes	complemented
34	C₇×#123	C₂×C₃	1	2	yes	complemented
35	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₇×#121	C₂×C₃	1	2	yes	complemented
37	C₇×#121	C₂×C₃	1	1	yes	characteristic, complemented
38	C₇×#120	C₂×C₃	1	2	yes	complemented
39	C₄×C₇×D₄	C₂×C₃	1	2	yes	complemented
40	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#538	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
43	C₂³×ΓL(1,𝔽₈)	—	4	1	no	—
44	C₂³×ΓL(1,𝔽₈)	—	4	1	no	—
45	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
46	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
47	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
48	Q₈×ΓL(1,𝔽₈)	—	4	1	no	—
49	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
50	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
51	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
52	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
53	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
54	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
55	C₂×C₄×ΓL(1,𝔽₈)	—	4	2	no	—
56	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
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	1	yes	characteristic
60	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
61	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
62	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
64	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
65	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
66	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
67	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	complemented
71	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
72	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
73	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
75	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
81	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₄²×C₇	C₂²×C₃	1	2	yes	complemented
83	C₃×#126	—	7	2	no	—
84	C₃×#124	—	7	2	no	—
85	C₃×#123	—	7	2	no	—
86	C₃×C₂²⋊Q₈	—	7	1	no	—
87	C₃×#121	—	7	2	no	—
88	C₃×#121	—	7	1	no	—
89	C₃×#120	—	7	2	no	—
90	C₃×C₄×D₄	—	7	2	no	—
91	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
93	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
94	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
95	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
96	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
97	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
98	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
99	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
100	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
101	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
102	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
103	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
105	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
106	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
107	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
108	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
109	#538	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³×C₇	—	2	2	no	—
111	C₂³×C₇	—	4	1	no	—
112	C₂³×C₇	—	4	1	no	—
113	C₂³×C₇	C₂³×C₃	1	2	yes	—
114	C₂³×C₇	C₂³×C₃	1	2	yes	—
115	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
116	C₇×Q₈	—	4	1	no	—
117	C₇×D₄	—	2	4	no	—
118	C₇×D₄	—	4	2	no	—
119	C₇×D₄	—	4	2	no	—
120	C₇×D₄	—	4	1	no	—
121	C₂×C₄×C₇	—	4	1	no	—
122	C₂×C₄×C₇	—	4	1	no	—
123	C₂×C₄×C₇	—	4	2	no	—
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	C₂⁴×C₃	—	7	1	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂×C₃×D₄	—	7	2	no	—
133	C₂×C₃×D₄	—	7	2	no	—
134	C₂×C₃×D₄	—	7	2	no	—
135	C₂×C₃×D₄	—	7	1	no	—
136	C₂²×C₃×C₄	—	7	1	no	—
137	C₂²×C₃×C₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₄²	—	7	2	no	—
151	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
152	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
153	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
154	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
155	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
157	#126	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
158	#124	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
159	#123	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
160	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
161	#121	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
162	#121	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
163	#120	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
164	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
165	C₂×C₂²⋊C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	4	4	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	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	2	no	—
183	C₂³×C₃	—	7	1	no	—
184	C₂³×C₃	—	7	2	no	—
185	C₂³×C₃	—	7	2	no	—
186	C₂³×C₃	—	14	2	no	—
187	C₂³×C₃	—	28	1	no	—
188	C₂³×C₃	—	28	1	no	—
189	C₃×Q₈	—	28	1	no	—
190	C₃×D₄	—	14	4	no	—
191	C₃×D₄	—	28	2	no	—
192	C₃×D₄	—	28	2	no	—
193	C₃×D₄	—	28	1	no	—
194	C₂×C₃×C₄	—	7	2	no	—
195	C₂×C₃×C₄	—	7	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	2	no	—
200	C₂×C₃×C₄	—	28	1	no	—
201	C₂×C₃×C₄	—	28	1	no	—
202	C₂×C₃×C₄	—	28	2	no	—
203	ΓL(1,𝔽₈)	#538	1	1	yes	characteristic, direct product factor, Hall
204	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
205	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
206	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
207	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
208	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
209	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
210	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
211	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
212	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
213	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
214	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
215	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
216	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
217	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
218	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
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	2	yes	—
222	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
223	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
224	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	4	2	no	—
227	C₂×C₇	—	4	2	no	—
228	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₇	C₃×H(5,𝔽₂)	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	2	no	—
236	C₂²×C₃	—	14	2	no	—
237	C₂²×C₃	—	14	2	no	—
238	C₂²×C₃	—	14	2	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₂²×C₃	—	28	4	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	C₃×C₄	—	14	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
252	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
253	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
263	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
264	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
265	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
266	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
267	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
268	C₇	C₃×#538	1	1	yes	characteristic, complemented, Sylow
269	C₂×C₃	—	7	1	no	—
270	C₂×C₃	—	7	1	no	—
271	C₂×C₃	—	7	1	no	—
272	C₂×C₃	—	14	2	no	—
273	C₂×C₃	—	28	2	no	—
274	C₂×C₃	—	28	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	4	no	—
284	C₂²	#41×ΓL(1,𝔽₈)	1	2	yes	—
285	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₃	—	7	1	no	Sylow
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
297	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic
298	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×#538	C₃
hypercenter	#538	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#538
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #538

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)×#124	C₂	semidirect
3. ΓL(1,𝔽₈)×#123	C₂	semidirect
4. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
5. ΓL(1,𝔽₈)×#121	C₂	semidirect
6. ΓL(1,𝔽₈)×#121	C₂	semidirect
7. ΓL(1,𝔽₈)×#120	C₂	semidirect
8. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
9. C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
10. C₇×#538	C₃	semidirect
12. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
16. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
19. C₂²×C₄×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
21. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
22. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
23. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
29. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
30. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
31. C₄²×ΓL(1,𝔽₈)	C₂²	semidirect
32. C₇×#126	C₂×C₃	semidirect
33. C₇×#124	C₂×C₃	semidirect
34. C₇×#123	C₂×C₃	semidirect
35. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
36. C₇×#121	C₂×C₃	semidirect
37. C₇×#121	C₂×C₃	semidirect
38. C₇×#120	C₂×C₃	semidirect
39. C₄×C₇×D₄	C₂×C₃	semidirect
40. C₂×C₇×C₂²⋊C₄	C₂×C₃	semidirect
63. C₂×C₇×Q₈	C₂²×C₃	semidirect
67. C₂×C₇×D₄	C₂²×C₃	semidirect
70. C₂²×C₄×C₇	C₂²×C₃	semidirect
71. C₇×C₄⋊C₄	C₂²×C₃	semidirect
72. C₇×C₄⋊C₄	C₂²×C₃	semidirect
73. C₇×C₄⋊C₄	C₂²×C₃	semidirect
74. C₇×C₄⋊C₄	C₂²×C₃	semidirect
80. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
81. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
82. C₄²×C₇	C₂²×C₃	semidirect
109. #538	ΓL(1,𝔽₈)	direct (N × H)
157. #126	C₂×ΓL(1,𝔽₈)	semidirect
158. #124	C₂×ΓL(1,𝔽₈)	semidirect
159. #123	C₂×ΓL(1,𝔽₈)	semidirect
160. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
161. #121	C₂×ΓL(1,𝔽₈)	semidirect
162. #121	C₂×ΓL(1,𝔽₈)	semidirect
163. #120	C₂×ΓL(1,𝔽₈)	semidirect
164. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
165. C₂×C₂²⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
203. ΓL(1,𝔽₈)	#538	direct (N × H)
205. C₂×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
209. C₂×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
212. C₂²×C₄	C₂²×ΓL(1,𝔽₈)	semidirect
213. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
214. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
215. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
216. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
222. C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
223. C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
224. C₄²	C₂²×ΓL(1,𝔽₈)	semidirect
268. C₇	C₃×#538	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)×#124	2
ΓL(1,𝔽₈)×#123	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
ΓL(1,𝔽₈)×#121	2
ΓL(1,𝔽₈)×#120	2
C₄×D₄×ΓL(1,𝔽₈)	2
C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	2
C₇×#538	3
C₃×#538	7

Subgroups — ΓL(1,𝔽8)×#538 #13060699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#538 #13060699