Subgroups — ΓL(1,𝔽₈)×#3291 #72229607

Order 2688.

920 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#577	C₂	1	4	yes	maximal, complemented
2	ΓL(1,𝔽₈)×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#471	C₂	1	4	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#456	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)×#453	C₂	1	2	yes	maximal, complemented
6	ΓL(1,𝔽₈)×#443	C₂	1	2	yes	maximal, complemented
7	C₇×#3291	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×2₋¹⁺⁴	C₂²	1	2	yes	—
9	ΓL(1,𝔽₈)×H(5,𝔽₂)	C₂²	1	2	yes	complemented
10	ΓL(1,𝔽₈)×#137	—	2	4	no	—
11	ΓL(1,𝔽₈)×#137	C₂²	1	4	yes	complemented
12	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	—	2	4	no	—
13	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	ΓL(1,𝔽₈)×#135	—	2	4	no	—
15	ΓL(1,𝔽₈)×#135	C₂²	1	2	yes	—
16	C₂×SD₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
17	C₂×SD₁₆×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
18	ΓL(1,𝔽₈)×#131	C₂²	1	4	yes	complemented
19	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₈)×#127	C₂²	1	1	yes	characteristic
21	ΓL(1,𝔽₈)×#108	C₂²	1	2	yes	complemented
22	ΓL(1,𝔽₈)×#104	C₂²	1	4	yes	complemented
23	ΓL(1,𝔽₈)×#101	C₂²	1	2	yes	complemented
24	ΓL(1,𝔽₈)×#100	C₂²	1	2	yes	complemented
25	C₄×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
26	C₇×#577	C₂×C₃	1	4	yes	complemented
27	C₇×#492	C₂×C₃	1	1	yes	characteristic, complemented
28	C₇×#471	C₂×C₃	1	4	yes	complemented
29	C₇×#456	C₂×C₃	1	2	yes	complemented
30	C₇×#453	C₂×C₃	1	2	yes	complemented
31	C₇×#443	C₂×C₃	1	2	yes	complemented
32	C₃×#3291	—	7	1	no	maximal, Hall
33	#41×ΓL(1,𝔽₈)	—	2	4	no	—
34	#41×ΓL(1,𝔽₈)	—	2	4	no	—
35	#41×ΓL(1,𝔽₈)	—	4	2	no	—
36	#41×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
37	C₂×Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
38	C₂×Q₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
39	C₂×D₄×ΓL(1,𝔽₈)	—	2	4	no	—
40	C₂×D₄×ΓL(1,𝔽₈)	—	4	1	no	—
41	C₂×D₄×ΓL(1,𝔽₈)	—	4	2	no	—
42	C₂×D₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
43	Q₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
44	Q₁₆×ΓL(1,𝔽₈)	D₄	1	2	yes	complemented
45	SD₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
46	SD₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
47	SD₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
48	SD₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
49	SD₁₆×ΓL(1,𝔽₈)	D₄	1	4	yes	complemented
50	D₈×ΓL(1,𝔽₈)	—	2	4	no	—
51	D₈×ΓL(1,𝔽₈)	D₄	1	2	yes	complemented
52	M₁₆×ΓL(1,𝔽₈)	—	2	4	no	—
53	M₁₆×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
54	C₂×C₈×ΓL(1,𝔽₈)	—	2	4	no	—
55	C₂×C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
56	C₄⋊C₄×ΓL(1,𝔽₈)	—	4	1	no	—
57	C₄²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
58	C₇×2₋¹⁺⁴	C₂²×C₃	1	2	yes	—
59	C₇×H(5,𝔽₂)	C₂²×C₃	1	2	yes	complemented
60	C₇×#137	—	2	4	no	—
61	C₇×#137	C₂²×C₃	1	4	yes	complemented
62	C₇×AGL(1,ℤ/8)	—	2	4	no	—
63	C₇×AGL(1,ℤ/8)	C₂²×C₃	1	4	yes	—
64	C₇×#135	—	2	4	no	—
65	C₇×#135	C₂²×C₃	1	2	yes	—
66	C₂×C₇×SD₁₆	—	2	4	no	—
67	C₂×C₇×SD₁₆	C₂²×C₃	1	4	yes	complemented
68	C₇×#131	C₂²×C₃	1	4	yes	complemented
69	C₇×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₇×#127	C₂²×C₃	1	1	yes	characteristic
71	C₇×#108	C₂²×C₃	1	2	yes	complemented
72	C₇×#104	C₂²×C₃	1	4	yes	complemented
73	C₇×#101	C₂²×C₃	1	2	yes	complemented
74	C₇×#100	C₂²×C₃	1	2	yes	complemented
75	C₄×C₇×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
76	C₃×#577	—	7	4	no	—
77	C₃×#492	—	7	1	no	—
78	C₃×#471	—	7	4	no	—
79	C₃×#456	—	7	2	no	—
80	C₃×#453	—	7	2	no	—
81	C₃×#443	—	7	2	no	—
82	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
83	C₂³×ΓL(1,𝔽₈)	—	4	2	no	—
84	Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
85	Q₈×ΓL(1,𝔽₈)	—	4	2	no	—
86	Q₈×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	—
87	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
88	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
89	D₄×ΓL(1,𝔽₈)	—	4	4	no	—
90	D₄×ΓL(1,𝔽₈)	—	4	2	no	—
91	D₄×ΓL(1,𝔽₈)	—	8	2	no	—
92	D₄×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	—
93	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
94	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
95	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
96	C₂×C₄×ΓL(1,𝔽₈)	—	4	2	no	—
97	C₂×C₄×ΓL(1,𝔽₈)	—	4	4	no	—
98	C₂×C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
99	C₈×ΓL(1,𝔽₈)	—	2	4	no	—
100	C₈×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
101	C₈×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
102	#3291	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
103	C₇×#41	—	2	4	no	—
104	C₇×#41	—	2	4	no	—
105	C₇×#41	—	4	2	no	—
106	C₇×#41	C₂³×C₃	1	4	yes	—
107	C₂×C₇×Q₈	—	2	4	no	—
108	C₂×C₇×Q₈	C₂³×C₃	1	2	yes	—
109	C₂×C₇×D₄	—	2	4	no	—
110	C₂×C₇×D₄	—	4	1	no	—
111	C₂×C₇×D₄	—	4	2	no	—
112	C₂×C₇×D₄	C₂³×C₃	1	2	yes	—
113	C₇×Q₁₆	—	2	4	no	—
114	C₇×Q₁₆	C₃×D₄	1	2	yes	complemented
115	C₇×SD₁₆	—	2	4	no	—
116	C₇×SD₁₆	—	2	4	no	—
117	C₇×SD₁₆	—	2	4	no	—
118	C₇×SD₁₆	—	2	4	no	—
119	C₇×SD₁₆	C₃×D₄	1	4	yes	complemented
120	C₇×D₈	—	2	4	no	—
121	C₇×D₈	C₃×D₄	1	2	yes	complemented
122	C₇×M₁₆	—	2	4	no	—
123	C₇×M₁₆	C₂³×C₃	1	4	yes	—
124	C₂×C₇×C₈	—	2	4	no	—
125	C₂×C₇×C₈	C₂³×C₃	1	2	yes	—
126	C₇×C₄⋊C₄	—	4	1	no	—
127	C₄²×C₇	C₂³×C₃	1	1	yes	characteristic
128	C₃×2₋¹⁺⁴	—	7	2	no	—
129	C₃×H(5,𝔽₂)	—	7	2	no	—
130	C₃×#137	—	7	4	no	—
131	C₃×#137	—	14	4	no	—
132	C₃×AGL(1,ℤ/8)	—	7	4	no	—
133	C₃×AGL(1,ℤ/8)	—	14	4	no	—
134	C₃×#135	—	7	2	no	—
135	C₃×#135	—	14	4	no	—
136	C₂×C₃×SD₁₆	—	7	4	no	—
137	C₂×C₃×SD₁₆	—	14	4	no	—
138	C₃×#131	—	7	4	no	—
139	C₃×C₄⋊Q₈	—	7	1	no	—
140	C₃×#127	—	7	1	no	—
141	C₃×#108	—	7	2	no	—
142	C₃×#104	—	7	4	no	—
143	C₃×#101	—	7	2	no	—
144	C₃×#100	—	7	2	no	—
145	C₃×C₄×C₈	—	7	1	no	—
146	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
147	C₂²×ΓL(1,𝔽₈)	—	4	2	no	—
148	C₂²×ΓL(1,𝔽₈)	—	8	2	no	—
149	C₂²×ΓL(1,𝔽₈)	—	16	2	no	—
150	C₂²×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
151	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
152	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
153	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
154	C₄×ΓL(1,𝔽₈)	C₂²×D₄	1	2	yes	—
155	#577	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
156	#492	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
157	#471	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
158	#456	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
159	#453	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
160	#443	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
161	C₂³×C₇	—	2	2	no	—
162	C₂³×C₇	—	4	2	no	—
163	C₇×Q₈	—	2	4	no	—
164	C₇×Q₈	—	4	2	no	—
165	C₇×Q₈	C₂×C₃×D₄	1	4	yes	—
166	C₇×D₄	—	2	4	no	—
167	C₇×D₄	—	2	4	no	—
168	C₇×D₄	—	4	4	no	—
169	C₇×D₄	—	4	2	no	—
170	C₇×D₄	—	8	2	no	—
171	C₇×D₄	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	4	no	—
174	C₂×C₄×C₇	—	2	2	no	—
175	C₂×C₄×C₇	—	4	2	no	—
176	C₂×C₄×C₇	—	4	4	no	—
177	C₂×C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
178	C₇×C₈	—	2	4	no	—
179	C₇×C₈	C₂×C₃×D₄	1	2	yes	—
180	C₇×C₈	C₂×C₃×D₄	1	2	yes	—
181	C₃×#41	—	7	4	no	—
182	C₃×#41	—	14	4	no	—
183	C₃×#41	—	14	4	no	—
184	C₃×#41	—	28	2	no	—
185	C₂×C₃×Q₈	—	7	2	no	—
186	C₂×C₃×Q₈	—	14	4	no	—
187	C₂×C₃×D₄	—	7	2	no	—
188	C₂×C₃×D₄	—	14	4	no	—
189	C₂×C₃×D₄	—	28	1	no	—
190	C₂×C₃×D₄	—	28	2	no	—
191	C₃×Q₁₆	—	7	2	no	—
192	C₃×Q₁₆	—	14	4	no	—
193	C₃×SD₁₆	—	7	4	no	—
194	C₃×SD₁₆	—	14	4	no	—
195	C₃×SD₁₆	—	14	4	no	—
196	C₃×SD₁₆	—	14	4	no	—
197	C₃×SD₁₆	—	14	4	no	—
198	C₃×D₈	—	7	2	no	—
199	C₃×D₈	—	14	4	no	—
200	C₃×M₁₆	—	7	4	no	—
201	C₃×M₁₆	—	14	4	no	—
202	C₂×C₃×C₈	—	7	2	no	—
203	C₂×C₃×C₈	—	14	4	no	—
204	C₃×C₄⋊C₄	—	28	1	no	—
205	C₃×C₄²	—	7	1	no	—
206	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
207	C₂×ΓL(1,𝔽₈)	—	4	4	no	—
208	C₂×ΓL(1,𝔽₈)	—	8	2	no	—
209	C₂×ΓL(1,𝔽₈)	D₄²	1	1	yes	characteristic
210	2₋¹⁺⁴	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
211	H(5,𝔽₂)	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
212	#137	—	2	4	no	—
213	#137	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
214	AGL(1,ℤ/8)	—	2	4	no	—
215	AGL(1,ℤ/8)	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
216	#135	—	2	4	no	—
217	#135	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×SD₁₆	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
220	#131	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
221	C₄⋊Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
222	#127	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
223	#108	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
224	#104	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
225	#101	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
226	#100	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
227	C₄×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
228	C₂²×C₇	—	2	4	no	—
229	C₂²×C₇	—	4	2	no	—
230	C₂²×C₇	—	8	2	no	—
231	C₂²×C₇	—	16	2	no	—
232	C₂²×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₄×C₇	—	2	2	no	—
234	C₄×C₇	—	2	4	no	—
235	C₄×C₇	—	4	2	no	—
236	C₄×C₇	C₂²×C₃×D₄	1	2	yes	—
237	C₂³×C₃	—	14	2	no	—
238	C₂³×C₃	—	28	2	no	—
239	C₃×Q₈	—	7	4	no	—
240	C₃×Q₈	—	14	4	no	—
241	C₃×Q₈	—	28	2	no	—
242	C₃×D₄	—	7	4	no	—
243	C₃×D₄	—	14	4	no	—
244	C₃×D₄	—	14	4	no	—
245	C₃×D₄	—	28	4	no	—
246	C₃×D₄	—	28	2	no	—
247	C₃×D₄	—	56	2	no	—
248	C₂×C₃×C₄	—	7	1	no	—
249	C₂×C₃×C₄	—	14	4	no	—
250	C₂×C₃×C₄	—	14	1	no	—
251	C₂×C₃×C₄	—	14	2	no	—
252	C₂×C₃×C₄	—	28	2	no	—
253	C₂×C₃×C₄	—	28	4	no	—
254	C₃×C₈	—	7	2	no	—
255	C₃×C₈	—	7	2	no	—
256	C₃×C₈	—	14	4	no	—
257	ΓL(1,𝔽₈)	#3291	1	1	yes	characteristic, direct product factor, Hall
258	#41	—	2	4	no	—
259	#41	—	2	4	no	—
260	#41	—	4	2	no	—
261	#41	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
262	C₂×Q₈	—	2	4	no	—
263	C₂×Q₈	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
268	Q₁₆	—	2	4	no	—
269	Q₁₆	D₄×ΓL(1,𝔽₈)	1	2	yes	complemented
270	SD₁₆	—	2	4	no	—
271	SD₁₆	—	2	4	no	—
272	SD₁₆	—	2	4	no	—
273	SD₁₆	—	2	4	no	—
274	SD₁₆	D₄×ΓL(1,𝔽₈)	1	4	yes	complemented
275	D₈	—	2	4	no	—
276	D₈	D₄×ΓL(1,𝔽₈)	1	2	yes	complemented
277	M₁₆	—	2	4	no	—
278	M₁₆	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄²	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
283	C₂×C₇	—	2	1	no	—
284	C₂×C₇	—	4	4	no	—
285	C₂×C₇	—	8	2	no	—
286	C₂×C₇	C₃×D₄²	1	1	yes	characteristic
287	C₂²×C₃	—	7	1	no	—
288	C₂²×C₃	—	14	4	no	—
289	C₂²×C₃	—	28	2	no	—
290	C₂²×C₃	—	56	2	no	—
291	C₂²×C₃	—	112	2	no	—
292	C₃×C₄	—	7	2	no	—
293	C₃×C₄	—	14	4	no	—
294	C₃×C₄	—	14	2	no	—
295	C₃×C₄	—	28	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	4	2	no	—
298	Q₈	—	2	4	no	—
299	Q₈	—	4	2	no	—
300	Q₈	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	—
301	D₄	—	2	4	no	—
302	D₄	—	2	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₈	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
315	C₈	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
316	C₇	C₃×#3291	1	1	yes	characteristic, complemented, Sylow
317	C₂×C₃	—	7	1	no	—
318	C₂×C₃	—	14	1	no	—
319	C₂×C₃	—	28	4	no	—
320	C₂×C₃	—	56	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	16	2	no	—
325	C₂²	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	4	no	—
328	C₄	—	4	2	no	—
329	C₄	C₂²×D₄×ΓL(1,𝔽₈)	1	2	yes	—
330	C₃	—	7	1	no	Sylow
331	C₂	—	2	1	no	—
332	C₂	—	4	4	no	—
333	C₂	—	8	2	no	—
334	C₂	D₄²×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄²×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#3291	C₃
socle	C₂×C₇	C₃×D₄²
hypercenter	#3291	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#3291
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3291

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#577	C₂	semidirect
2. ΓL(1,𝔽₈)×#492	C₂	semidirect
3. ΓL(1,𝔽₈)×#471	C₂	semidirect
4. ΓL(1,𝔽₈)×#456	C₂	semidirect
5. ΓL(1,𝔽₈)×#453	C₂	semidirect
6. ΓL(1,𝔽₈)×#443	C₂	semidirect
7. C₇×#3291	C₃	semidirect
9. ΓL(1,𝔽₈)×H(5,𝔽₂)	C₂²	semidirect
11. ΓL(1,𝔽₈)×#137	C₂²	semidirect
17. C₂×SD₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
18. ΓL(1,𝔽₈)×#131	C₂²	semidirect
19. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂²	semidirect
21. ΓL(1,𝔽₈)×#108	C₂²	semidirect
22. ΓL(1,𝔽₈)×#104	C₂²	semidirect
23. ΓL(1,𝔽₈)×#101	C₂²	semidirect
24. ΓL(1,𝔽₈)×#100	C₂²	semidirect
25. C₄×C₈×ΓL(1,𝔽₈)	C₂²	semidirect
26. C₇×#577	C₂×C₃	semidirect
27. C₇×#492	C₂×C₃	semidirect
28. C₇×#471	C₂×C₃	semidirect
29. C₇×#456	C₂×C₃	semidirect
30. C₇×#453	C₂×C₃	semidirect
31. C₇×#443	C₂×C₃	semidirect
44. Q₁₆×ΓL(1,𝔽₈)	D₄	semidirect
49. SD₁₆×ΓL(1,𝔽₈)	D₄	semidirect
51. D₈×ΓL(1,𝔽₈)	D₄	semidirect
59. C₇×H(5,𝔽₂)	C₂²×C₃	semidirect
61. C₇×#137	C₂²×C₃	semidirect
67. C₂×C₇×SD₁₆	C₂²×C₃	semidirect
68. C₇×#131	C₂²×C₃	semidirect
69. C₇×C₄⋊Q₈	C₂²×C₃	semidirect
71. C₇×#108	C₂²×C₃	semidirect
72. C₇×#104	C₂²×C₃	semidirect
73. C₇×#101	C₂²×C₃	semidirect
74. C₇×#100	C₂²×C₃	semidirect
75. C₄×C₇×C₈	C₂²×C₃	semidirect
102. #3291	ΓL(1,𝔽₈)	direct (N × H)
114. C₇×Q₁₆	C₃×D₄	semidirect
119. C₇×SD₁₆	C₃×D₄	semidirect
121. C₇×D₈	C₃×D₄	semidirect
155. #577	C₂×ΓL(1,𝔽₈)	semidirect
156. #492	C₂×ΓL(1,𝔽₈)	semidirect
157. #471	C₂×ΓL(1,𝔽₈)	semidirect
158. #456	C₂×ΓL(1,𝔽₈)	semidirect
159. #453	C₂×ΓL(1,𝔽₈)	semidirect
160. #443	C₂×ΓL(1,𝔽₈)	semidirect
211. H(5,𝔽₂)	C₂²×ΓL(1,𝔽₈)	semidirect
213. #137	C₂²×ΓL(1,𝔽₈)	semidirect
219. C₂×SD₁₆	C₂²×ΓL(1,𝔽₈)	semidirect
220. #131	C₂²×ΓL(1,𝔽₈)	semidirect
221. C₄⋊Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
223. #108	C₂²×ΓL(1,𝔽₈)	semidirect
224. #104	C₂²×ΓL(1,𝔽₈)	semidirect
225. #101	C₂²×ΓL(1,𝔽₈)	semidirect
226. #100	C₂²×ΓL(1,𝔽₈)	semidirect
227. C₄×C₈	C₂²×ΓL(1,𝔽₈)	semidirect
257. ΓL(1,𝔽₈)	#3291	direct (N × H)
269. Q₁₆	D₄×ΓL(1,𝔽₈)	semidirect
274. SD₁₆	D₄×ΓL(1,𝔽₈)	semidirect
276. D₈	D₄×ΓL(1,𝔽₈)	semidirect
316. C₇	C₃×#3291	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)×#577	2
ΓL(1,𝔽₈)×#492	2
ΓL(1,𝔽₈)×#471	2
ΓL(1,𝔽₈)×#456	2
ΓL(1,𝔽₈)×#453	2
ΓL(1,𝔽₈)×#443	2
C₇×#3291	3
C₃×#3291	7

Subgroups — ΓL(1,𝔽8)×#3291 #72229607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#3291 #72229607