Subgroups — #12039

Order 256.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2169	C₂	1	2	yes	maximal, complemented
2	C₈×D₈	C₂	1	1	yes	maximal, characteristic
3	#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₃₂	—	2	2	no	—
7	C₄×D₈	C₄	1	1	yes	characteristic
8	C₈×D₄	C₂²	1	2	yes	—
9	C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₂²⋊C₁₆	—	2	2	no	—
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#335	C₄	1	1	yes	characteristic
14	D₄⋊C₈	C₄	1	2	yes	—
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×D₈	C₈	1	1	yes	characteristic
17	C₂²×C₈	—	2	2	no	—
18	C₄×D₄	C₂×C₄	1	2	yes	—
19	M₃₂	—	4	4	no	—
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₆	D₄	1	1	yes	characteristic
24	#107	C₈	1	1	yes	characteristic
25	C₄⋊C₈	C₂×C₄	1	2	yes	—
26	#102	C₈	1	2	yes	—
27	C₂²⋊C₈	—	2	2	no	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	C₂³	1	1	yes	characteristic
31	C₂×D₄	C₂×C₈	1	2	yes	—
32	C₂²×C₄	—	2	2	no	—
33	D₈	—	2	2	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈	#41	1	1	yes	characteristic
42	C₄⋊C₄	C₂×C₈	1	2	yes	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₄²	C₂²×C₄	1	1	yes	characteristic
45	C₁₆	—	2	2	no	—
46	C₁₆	—	2	1	no	—
47	C₁₆	—	4	2	no	—
48	C₁₆	D₈	1	2	yes	complemented
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	C₄×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄	#131	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	2	1	no	—
60	C₈	—	2	1	no	—
61	C₈	—	4	2	no	—
62	C₈	M₃₂	1	2	yes	complemented
63	C₈	C₂×D₈	1	1	yes	characteristic
64	C₈	#135	1	1	yes	characteristic
65	C₂²	—	4	2	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	4	2	no	—
68	C₂²	C₈×D₄	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	C₄×D₈	1	1	yes	characteristic
72	C₄	#443	1	1	yes	characteristic
73	C₄	C₂×M₃₂	1	1	yes	characteristic
74	C₄	#504	1	1	yes	characteristic
75	C₂	—	8	2	no	—
76	C₂	C₈×D₈	1	1	yes	characteristic
77	C₂	#2169	1	1	yes	characteristic
78	C₂	#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₈×D₄
prime radical	C₂×D₈	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2169	C₂	semidirect
3. #1370	C₂	semidirect
4. D₄⋊C₁₆	C₂	semidirect
5. C₈×C₁₆	C₂	semidirect
48. C₁₆	D₈	semidirect
62. C₈	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2169	2
C₈×D₈	2
#1370	2
D₄⋊C₁₆	2
C₈×C₁₆	2