Subgroups — #7125

Order 256.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1946	C₂	1	1	yes	maximal, characteristic
2	#1375	C₂	1	1	yes	maximal, characteristic
3	#1281	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	#431	—	2	1	no	—
6	C₂×#108	C₄	1	1	yes	characteristic
7	#427	C₄	1	1	yes	characteristic
8	#424	—	2	1	no	—
9	C₂×#97	C₂²	1	1	yes	characteristic
10	#339	—	2	1	no	—
11	C₂×M₁₆	—	2	1	no	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	#117	—	2	1	no	—
15	C₂×C₄²	D₄	1	1	yes	characteristic
16	M₃₂	—	4	1	no	—
17	M₃₂	—	4	1	no	—
18	C₂×C₁₆	—	2	2	no	—
19	#108	—	4	1	no	—
20	#107	—	2	1	no	—
21	#106	C₈	1	2	yes	complemented
22	C₄⋊C₈	—	4	1	no	—
23	C₄⋊C₈	—	4	1	no	—
24	C₄⋊C₈	—	4	1	no	—
25	C₂²⋊C₈	—	2	2	no	—
26	#97	—	4	1	no	—
27	C₄×C₈	—	4	1	no	—
28	C₂²×C₄	—	2	1	no	—
29	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	M₁₆	—	4	1	no	—
31	M₁₆	—	4	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₈	D₈	1	1	yes	characteristic
41	C₂×C₈	SD₁₆	1	1	yes	characteristic
42	C₄⋊C₄	—	2	2	no	—
43	C₂²⋊C₄	—	4	1	no	—
44	C₄²	—	2	1	no	—
45	C₄²	—	4	1	no	—
46	C₄²	SD₁₆	1	1	yes	characteristic
47	C₄²	Q₁₆	1	1	yes	characteristic
48	C₁₆	—	4	2	no	—
49	C₂³	#104	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	2	no	—
53	C₂×C₄	—	8	1	no	—
54	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄	#99	1	1	yes	characteristic
56	C₂×C₄	#101	1	1	yes	characteristic
57	C₂×C₄	#102	1	1	yes	characteristic
58	C₂×C₄	#103	1	1	yes	characteristic
59	C₂×C₄	#104	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	—	2	1	no	—
64	C₈	—	8	2	no	—
65	C₈	—	8	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	D₄⋊C₈	1	1	yes	characteristic
68	C₂²	#328	1	1	yes	characteristic
69	C₂²	#330	1	1	yes	characteristic
70	C₄	—	4	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	8	2	no	—
73	C₄	#324	1	1	yes	characteristic
74	C₄	Q₈⋊C₈	1	1	yes	characteristic
75	C₄	#327	1	1	yes	characteristic
76	C₄	#332	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	#1271	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#327
commutator	C₂×C₈	C₂×C₈
Frattini, cosocle	C₂×#97	C₂²
socle	C₂	#1271
prime radical	C₂³	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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
21. #106	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1946	2
#1375	2
#1281	2