Subgroups — C₂×ΓL(1,𝔽₈)×AGL(1,ℤ/8) #13060734

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	C₂	1	8	yes	maximal, direct product factor
4	C₂×SD₁₆×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×AGL(1,ℤ/8)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂⁴×ΓL(1,𝔽₈)	—	2	1	no	—
9	#41×ΓL(1,𝔽₈)	—	2	2	no	—
10	#41×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄×ΓL(1,𝔽₈)	—	2	1	no	—
13	C₂×D₄×ΓL(1,𝔽₈)	—	2	1	no	—
14	C₂×D₄×ΓL(1,𝔽₈)	—	2	2	no	—
15	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
17	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
19	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	SD₁₆×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
21	D₈×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
22	M₁₆×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
23	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
24	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×AGL(1,ℤ/8)	C₂×C₃	1	8	yes	complemented
27	C₂×C₇×SD₁₆	C₂×C₃	1	2	yes	complemented
28	C₂×C₇×D₈	C₂×C₃	1	2	yes	complemented
29	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×AGL(1,ℤ/8)	—	7	1	no	maximal, Hall
31	C₂³×ΓL(1,𝔽₈)	—	2	1	no	—
32	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
33	C₂³×ΓL(1,𝔽₈)	—	2	4	no	—
34	C₂³×ΓL(1,𝔽₈)	—	4	4	no	—
35	C₂³×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
36	Q₈×ΓL(1,𝔽₈)	—	2	1	no	—
37	Q₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	complemented
38	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
39	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
40	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
41	D₄×ΓL(1,𝔽₈)	—	2	1	no	—
42	D₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
43	D₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	complemented
44	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
45	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
46	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
47	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
48	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
50	C₈×ΓL(1,𝔽₈)	C₂³	1	4	yes	complemented
51	C₂⁴×C₇	—	2	1	no	—
52	C₇×#41	—	2	2	no	—
53	C₇×#41	C₂²×C₃	1	4	yes	complemented
54	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
55	C₂×C₇×D₄	—	2	1	no	—
56	C₂×C₇×D₄	—	2	1	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
59	C₂×C₇×D₄	C₂²×C₃	1	4	yes	complemented
60	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
63	C₇×SD₁₆	C₂²×C₃	1	8	yes	complemented
64	C₇×D₈	C₂²×C₃	1	8	yes	complemented
65	C₇×M₁₆	C₂²×C₃	1	4	yes	complemented
66	C₂×C₇×C₈	C₂²×C₃	1	2	yes	complemented
67	C₂×C₃×#41	—	7	1	no	—
68	C₂²×C₃×D₄	—	7	1	no	—
69	C₃×AGL(1,ℤ/8)	—	7	8	no	—
70	C₂×C₃×SD₁₆	—	7	2	no	—
71	C₂×C₃×D₈	—	7	2	no	—
72	C₂×C₃×M₁₆	—	7	1	no	—
73	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
74	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
75	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
76	C₂²×ΓL(1,𝔽₈)	—	4	8	no	—
77	C₂²×ΓL(1,𝔽₈)	—	4	2	no	—
78	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
79	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
80	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
81	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
82	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
83	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
84	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
85	C₂×AGL(1,ℤ/8)	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₇	—	2	1	no	—
87	C₂³×C₇	—	2	2	no	—
88	C₂³×C₇	—	2	4	no	—
89	C₂³×C₇	—	4	4	no	—
90	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
91	C₇×Q₈	—	2	1	no	—
92	C₇×Q₈	C₂³×C₃	1	2	yes	complemented
93	C₇×D₄	—	2	4	no	—
94	C₇×D₄	—	2	2	no	—
95	C₇×D₄	—	2	4	no	—
96	C₇×D₄	—	2	1	no	—
97	C₇×D₄	C₂³×C₃	1	4	yes	—
98	C₇×D₄	C₂³×C₃	1	2	yes	complemented
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	4	no	—
101	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
103	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
105	C₇×C₈	C₂³×C₃	1	4	yes	complemented
106	C₂⁴×C₃	—	14	1	no	—
107	C₃×#41	—	7	4	no	—
108	C₃×#41	—	14	2	no	—
109	C₂×C₃×Q₈	—	7	1	no	—
110	C₂×C₃×D₄	—	7	2	no	—
111	C₂×C₃×D₄	—	7	1	no	—
112	C₂×C₃×D₄	—	7	4	no	—
113	C₂×C₃×D₄	—	14	1	no	—
114	C₂×C₃×D₄	—	14	2	no	—
115	C₂×C₃×D₄	—	14	1	no	—
116	C₂²×C₃×C₄	—	7	1	no	—
117	C₂²×C₃×C₄	—	14	1	no	—
118	C₃×SD₁₆	—	7	8	no	—
119	C₃×D₈	—	7	8	no	—
120	C₃×M₁₆	—	7	4	no	—
121	C₂×C₃×C₈	—	7	2	no	—
122	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
123	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
124	C₂×ΓL(1,𝔽₈)	—	4	4	no	—
125	C₂×ΓL(1,𝔽₈)	AGL(1,ℤ/8)	1	2	yes	direct product factor
126	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
127	C₂×#41	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
128	C₂²×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
129	AGL(1,ℤ/8)	C₂×ΓL(1,𝔽₈)	1	8	yes	direct product factor
130	C₂×SD₁₆	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
131	C₂×D₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
132	C₂×M₁₆	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	2	4	no	—
136	C₂²×C₇	—	4	8	no	—
137	C₂²×C₇	—	4	4	no	—
138	C₂²×C₇	—	4	2	no	—
139	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
140	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
141	C₄×C₇	—	2	2	no	—
142	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
143	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
144	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
145	C₂³×C₃	—	7	1	no	—
146	C₂³×C₃	—	14	1	no	—
147	C₂³×C₃	—	14	4	no	—
148	C₂³×C₃	—	14	2	no	—
149	C₂³×C₃	—	28	4	no	—
150	C₃×Q₈	—	7	2	no	—
151	C₃×Q₈	—	14	1	no	—
152	C₃×D₄	—	7	4	no	—
153	C₃×D₄	—	7	2	no	—
154	C₃×D₄	—	14	2	no	—
155	C₃×D₄	—	14	4	no	—
156	C₃×D₄	—	14	1	no	—
157	C₃×D₄	—	14	4	no	—
158	C₂×C₃×C₄	—	7	1	no	—
159	C₂×C₃×C₄	—	7	2	no	—
160	C₂×C₃×C₄	—	7	1	no	—
161	C₂×C₃×C₄	—	7	2	no	—
162	C₂×C₃×C₄	—	14	4	no	—
163	C₂×C₃×C₄	—	14	1	no	—
164	C₃×C₈	—	7	4	no	—
165	ΓL(1,𝔽₈)	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Hall
166	C₂⁴	—	2	1	no	—
167	#41	—	2	2	no	—
168	#41	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
169	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
174	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
175	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
178	SD₁₆	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
179	D₈	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
180	M₁₆	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
181	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	—	4	4	no	—
184	C₂×C₇	—	4	2	no	—
185	C₂×C₇	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
186	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
187	C₂²×C₃	—	7	2	no	—
188	C₂²×C₃	—	7	1	no	—
189	C₂²×C₃	—	14	2	no	—
190	C₂²×C₃	—	14	2	no	—
191	C₂²×C₃	—	14	4	no	—
192	C₂²×C₃	—	28	4	no	—
193	C₂²×C₃	—	28	2	no	—
194	C₂²×C₃	—	28	8	no	—
195	C₃×C₄	—	7	2	no	—
196	C₃×C₄	—	7	1	no	—
197	C₃×C₄	—	7	1	no	—
198	C₃×C₄	—	14	2	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	4	4	no	—
203	C₂³	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
204	Q₈	—	2	1	no	—
205	Q₈	C₂³×ΓL(1,𝔽₈)	1	2	yes	complemented
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	1	no	—
210	D₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
211	D₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	complemented
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	2	yes	—
215	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
216	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
217	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
218	C₈	C₂³×ΓL(1,𝔽₈)	1	4	yes	complemented
219	C₇	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
220	C₂×C₃	—	7	1	no	—
221	C₂×C₃	—	7	2	no	—
222	C₂×C₃	—	14	2	no	—
223	C₂×C₃	—	28	4	no	—
224	C₂×C₃	—	28	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
232	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
235	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
236	C₄	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
237	C₃	—	7	1	no	Sylow
238	C₂	—	2	2	no	—
239	C₂	—	4	4	no	—
240	C₂	—	4	2	no	—
241	C₂	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	1	2	yes	direct product factor
242	C₂	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₂×C₇×AGL(1,ℤ/8)	C₃
socle	C₂²×C₇	C₂×C₃×D₄
hypercenter	C₂×AGL(1,ℤ/8)	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. C₂×#41×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂²×D₄×ΓL(1,𝔽₈)	C₂	semidirect
3. ΓL(1,𝔽₈)×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂×SD₁₆×ΓL(1,𝔽₈)	C₂	semidirect
5. C₂×D₈×ΓL(1,𝔽₈)	C₂	semidirect
6. C₂×M₁₆×ΓL(1,𝔽₈)	C₂	semidirect
7. C₂×C₇×AGL(1,ℤ/8)	C₃	semidirect
10. #41×ΓL(1,𝔽₈)	C₂²	semidirect
11. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
16. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₂×D₄×ΓL(1,𝔽₈)	C₂²	semidirect
20. SD₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
21. D₈×ΓL(1,𝔽₈)	C₂²	semidirect
22. M₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
23. C₂×C₈×ΓL(1,𝔽₈)	C₂²	semidirect
24. C₂×C₇×#41	C₂×C₃	semidirect
25. C₂²×C₇×D₄	C₂×C₃	semidirect
26. C₇×AGL(1,ℤ/8)	C₂×C₃	semidirect
27. C₂×C₇×SD₁₆	C₂×C₃	semidirect
28. C₂×C₇×D₈	C₂×C₃	semidirect
29. C₂×C₇×M₁₆	C₂×C₃	semidirect
37. Q₈×ΓL(1,𝔽₈)	C₂³	semidirect
43. D₄×ΓL(1,𝔽₈)	C₂³	semidirect
50. C₈×ΓL(1,𝔽₈)	C₂³	semidirect
53. C₇×#41	C₂²×C₃	semidirect
54. C₂×C₇×Q₈	C₂²×C₃	semidirect
59. C₂×C₇×D₄	C₂²×C₃	semidirect
60. C₂×C₇×D₄	C₂²×C₃	semidirect
63. C₇×SD₁₆	C₂²×C₃	semidirect
64. C₇×D₈	C₂²×C₃	semidirect
65. C₇×M₁₆	C₂²×C₃	semidirect
66. C₂×C₇×C₈	C₂²×C₃	semidirect
85. C₂×AGL(1,ℤ/8)	ΓL(1,𝔽₈)	direct (N × H)
92. C₇×Q₈	C₂³×C₃	semidirect
98. C₇×D₄	C₂³×C₃	semidirect
105. C₇×C₈	C₂³×C₃	semidirect
125. C₂×ΓL(1,𝔽₈)	AGL(1,ℤ/8)	direct (N × H)
127. C₂×#41	C₂×ΓL(1,𝔽₈)	semidirect
128. C₂²×D₄	C₂×ΓL(1,𝔽₈)	semidirect
129. AGL(1,ℤ/8)	C₂×ΓL(1,𝔽₈)	direct (N × H)
130. C₂×SD₁₆	C₂×ΓL(1,𝔽₈)	semidirect
131. C₂×D₈	C₂×ΓL(1,𝔽₈)	semidirect
132. C₂×M₁₆	C₂×ΓL(1,𝔽₈)	semidirect
165. ΓL(1,𝔽₈)	C₂×AGL(1,ℤ/8)	direct (N × H)
168. #41	C₂²×ΓL(1,𝔽₈)	semidirect
169. C₂×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
174. C₂×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
175. C₂×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
178. SD₁₆	C₂²×ΓL(1,𝔽₈)	semidirect
179. D₈	C₂²×ΓL(1,𝔽₈)	semidirect
180. M₁₆	C₂²×ΓL(1,𝔽₈)	semidirect
181. C₂×C₈	C₂²×ΓL(1,𝔽₈)	semidirect
185. C₂×C₇	C₃×AGL(1,ℤ/8)	semidirect
205. Q₈	C₂³×ΓL(1,𝔽₈)	semidirect
211. D₄	C₂³×ΓL(1,𝔽₈)	semidirect
218. C₈	C₂³×ΓL(1,𝔽₈)	semidirect
219. C₇	C₂×C₃×AGL(1,ℤ/8)	semidirect
241. C₂	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×ΓL(1,𝔽₈)	2
C₂²×D₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×AGL(1,ℤ/8)	2
C₂×SD₁₆×ΓL(1,𝔽₈)	2
C₂×D₈×ΓL(1,𝔽₈)	2
C₂×M₁₆×ΓL(1,𝔽₈)	2
C₂×C₇×AGL(1,ℤ/8)	3
C₂×C₃×AGL(1,ℤ/8)	7

Subgroups — C2×ΓL(1,𝔽8)×AGL(1,ℤ/8) #13060734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×AGL(1,ℤ/8) #13060734