Subgroups — #12907

Order 256.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	2	yes	maximal
2	C₂×#366	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1846	C₂	1	2	yes	maximal
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#107	C₂²	1	4	yes	—
8	C₂×#106	—	2	2	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#382	—	2	2	no	—
11	#367	—	2	4	no	—
12	#366	—	2	4	no	—
13	C₄×C₁₆	C₄	1	4	yes	complemented
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄	—	2	4	no	—
17	C₂×C₄²	D₄	1	1	yes	characteristic
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	D₄	1	2	yes	—
21	C₂×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₆	Q₈	1	2	yes	—
23	C₂×C₁₆	Q₈	1	2	yes	—
24	#107	—	2	4	no	—
25	#107	—	2	4	no	—
26	#106	—	4	4	no	—
27	C₄×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₈	Q₈	1	2	yes	—
29	#95	—	4	4	no	—
30	C₂²×C₄	—	4	4	no	—
31	C₂²×C₄	D₈	1	2	yes	—
32	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈	C₄⋊C₄	1	2	yes	—
34	C₂×C₈	D₈	1	2	yes	—
35	C₂×C₈	Q₁₆	1	2	yes	—
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₄⋊C₄	—	4	4	no	—
43	C₄⋊C₄	—	4	4	no	—
44	C₄²	C₄⋊C₄	1	2	yes	—
45	C₄²	Q₁₆	1	2	yes	—
46	C₁₆	C₄⋊C₄	1	4	yes	—
47	C₁₆	C₄⋊C₄	1	4	yes	—
48	C₂³	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	—	8	4	no	—
51	C₂×C₄	—	8	8	no	—
52	C₂×C₄	#107	1	4	yes	—
53	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	#127	1	1	yes	characteristic
55	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
59	C₈	#107	1	4	yes	—
60	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈	C₄⋊Q₈	1	1	yes	characteristic
63	C₈	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²	C₂×#107	1	1	yes	characteristic
65	C₂²	#493	1	1	yes	characteristic
66	C₂²	#500	1	1	yes	characteristic
67	C₂²	#508	1	4	yes	—
68	C₄	—	2	4	no	—
69	C₄	—	16	4	no	—
70	C₄	#384	1	1	yes	characteristic
71	C₄	C₂×#107	1	1	yes	characteristic
72	C₄	#494	1	1	yes	characteristic
73	C₄	#500	1	1	yes	characteristic
74	C₂	#1845	1	1	yes	characteristic
75	C₂	#2158	1	2	yes	—
76	C₂	#2249	1	2	yes	—
77	C₂	#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ 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
13. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367	2
C₂×#366	2
C₂×C₄×C₁₆	2
#1846	2