Subgroups — #7127

Order 256.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1944	C₂	1	1	yes	maximal, characteristic
2	#1377	C₂	1	1	yes	maximal, characteristic
3	#1275	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	#431	—	2	1	no	—
6	C₂×#107	C₄	1	1	yes	characteristic
7	C₂×#106	C₄	1	1	yes	characteristic
8	C₂×#97	C₂²	1	1	yes	characteristic
9	#389	—	2	1	no	—
10	#340	—	2	1	no	—
11	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈	D₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄	—	2	1	no	—
14	C₂×C₄⋊C₄	—	2	1	no	—
15	C₂×C₄⋊C₄	—	4	1	no	—
16	C₂×C₄²	D₄	1	1	yes	characteristic
17	M₃₂	—	2	2	no	—
18	M₃₂	—	4	1	no	—
19	C₂×C₁₆	—	4	1	no	—
20	#107	—	4	1	no	—
21	#106	—	2	1	no	—
22	#106	C₈	1	2	yes	complemented
23	C₄⋊C₈	—	4	1	no	—
24	C₂²⋊C₈	—	4	1	no	—
25	#97	—	4	1	no	—
26	C₄×C₈	—	2	2	no	—
27	#95	—	4	1	no	—
28	C₂²×C₄	—	4	1	no	—
29	C₂²×C₄	—	4	1	no	—
30	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄	D₈	1	1	yes	characteristic
32	C₂²×C₄	SD₁₆	1	1	yes	characteristic
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₈	SD₁₆	1	1	yes	characteristic
41	C₂×C₈	Q₁₆	1	1	yes	characteristic
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	8	2	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	2	1	no	—
49	C₁₆	—	4	2	no	—
50	C₂³	#102	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	—	8	1	no	—
58	C₂×C₄	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄	#99	1	1	yes	characteristic
63	C₂×C₄	#101	1	1	yes	characteristic
64	C₂×C₄	#103	1	1	yes	characteristic
65	C₂×C₄	#104	1	1	yes	characteristic
66	C₂×C₄	#104	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	—	8	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	D₄⋊C₈	1	1	yes	characteristic
75	C₂²	#327	1	1	yes	characteristic
76	C₂²	#332	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	4	1	no	—
79	C₄	—	8	2	no	—
80	C₄	—	16	1	no	—
81	C₄	#324	1	1	yes	characteristic
82	C₄	Q₈⋊C₈	1	1	yes	characteristic
83	C₄	#328	1	1	yes	characteristic
84	C₄	#330	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	—	2	1	no	—
87	C₂	#1271	1	1	yes	characteristic
88	C₂	#1339	1	1	yes	characteristic
89	C₂	#1342	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#327
commutator	C₂×C₈	C₂×C₈
Frattini, cosocle	C₂×#97	C₂²
prime radical	C₂³	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
22. #106	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1944	2
#1377	2
#1275	2