Subgroups — #12900

Order 256.

191 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	4	yes	maximal
2	#2150	C₂	1	2	yes	maximal
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	#364	—	2	8	no	—
11	C₄⋊C₁₆	—	2	4	no	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²	—	2	2	no	—
16	C₂×C₄²	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	M₃₂	—	2	8	no	—
19	C₂×C₁₆	—	2	4	no	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	C₈	1	2	yes	—
22	C₄×C₈	C₈	1	2	yes	—
23	C₄×C₈	D₄	1	4	yes	—
24	C₄×C₈	D₄	1	2	yes	—
25	C₄×C₈	Q₈	1	4	yes	—
26	C₄×C₈	Q₈	1	2	yes	—
27	C₂²×C₄	—	2	2	no	—
28	C₂²×C₄	M₁₆	1	2	yes	—
29	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₈	C₄⋊C₄	1	4	yes	—
33	C₂×C₈	C₂×C₈	1	4	yes	—
34	C₂×C₈	M₁₆	1	4	yes	—
35	C₂×C₈	C₂×D₄	1	2	yes	—
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈	C₂×Q₈	1	2	yes	—
39	C₄²	—	2	8	no	—
40	C₄²	—	2	2	no	—
41	C₄²	C₄⋊C₄	1	4	yes	—
42	C₄²	C₄⋊C₄	1	2	yes	—
43	C₄²	C₂×C₈	1	1	yes	characteristic
44	C₄²	C₂×C₈	1	1	yes	characteristic
45	C₁₆	—	4	4	no	—
46	C₂³	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₄	—	2	8	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	C₄⋊C₈	1	2	yes	—
51	C₂×C₄	C₄⋊C₈	1	2	yes	—
52	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄	C₂×M₁₆	1	2	yes	—
56	C₈	—	2	2	no	—
57	C₈	C₄⋊C₈	1	8	yes	—
58	C₈	#127	1	1	yes	characteristic
59	C₈	C₄⋊Q₈	1	2	yes	—
60	C₈	C₄⋊Q₈	1	1	yes	characteristic
61	C₂²	—	2	2	no	—
62	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²	#423	1	2	yes	—
64	C₄	—	2	8	no	—
65	C₄	—	2	2	no	—
66	C₄	#364	1	4	yes	complemented
67	C₄	#384	1	1	yes	characteristic
68	C₄	C₂×C₄⋊C₈	1	2	yes	—
69	C₄	#423	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	#1842	1	1	yes	characteristic
72	C₂	C₂×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
66. C₄	#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
#2150	2
C₄²×C₈	2