Subgroups — #8806

Order 256.

231 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈	—	2	4	no	—
5	C₂×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₄³	C₂²	1	1	yes	characteristic
11	#335	—	2	8	no	—
12	C₂²×C₈	—	2	4	no	—
13	C₂²×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²	D₄	1	2	yes	—
18	C₂×C₄²	Q₈	1	2	yes	—
19	C₂×C₄²	C₂³	1	1	yes	characteristic
20	C₄⋊C₈	—	2	8	no	—
21	C₄⋊C₈	—	4	8	no	—
22	C₄×C₈	—	2	4	no	—
23	C₄×C₈	—	2	2	no	—
24	C₄×C₈	C₈	1	4	yes	complemented
25	C₄×C₈	D₄	1	4	yes	—
26	C₄×C₈	Q₈	1	4	yes	—
27	C₂²×C₄	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	—	4	4	no	—
35	C₂×C₈	—	4	8	no	—
36	C₂×C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₈	C₂×C₈	1	4	yes	—
39	C₂×C₈	M₁₆	1	4	yes	—
40	C₄²	—	2	4	no	—
41	C₄²	—	2	2	no	—
42	C₄²	C₂×C₈	1	2	yes	—
43	C₄²	M₁₆	1	2	yes	—
44	C₄²	D₈	1	4	yes	—
45	C₄²	Q₁₆	1	4	yes	—
46	C₄²	C₂×D₄	1	2	yes	—
47	C₄²	C₂×Q₈	1	2	yes	—
48	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	C₄⋊C₈	1	4	yes	—
53	C₂×C₄	#107	1	4	yes	—
54	C₂×C₄	#107	1	4	yes	—
55	C₂×C₄	#108	1	4	yes	—
56	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄	#127	1	1	yes	characteristic
58	C₂×C₄	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄	C₂×M₁₆	1	2	yes	—
62	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄	C₂×D₈	1	2	yes	—
64	C₂×C₄	C₂×Q₁₆	1	2	yes	—
65	C₈	—	2	4	no	—
66	C₈	—	8	4	no	—
67	C₈	C₄⋊C₈	1	8	yes	complemented
68	C₂²	#384	1	1	yes	characteristic
69	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²	#423	1	1	yes	characteristic
71	C₂²	C₂×#107	1	2	yes	—
72	C₂²	C₂×#108	1	2	yes	—
73	C₄	—	2	4	no	—
74	C₄	—	2	2	no	—
75	C₄	#335	1	8	yes	complemented
76	C₄	C₂×C₄⋊C₈	1	2	yes	—
77	C₄	#423	1	2	yes	—
78	C₄	#493	1	1	yes	characteristic
79	C₄	#494	1	1	yes	characteristic
80	C₄	#500	1	2	yes	—
81	C₂	C₂×#335	1	2	yes	—
82	C₂	#1567	1	2	yes	—
83	C₂	#1842	1	1	yes	characteristic
84	C₂	#1845	1	1	yes	characteristic
85	C₂	#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
24. C₄×C₈	C₈	semidirect
67. C₈	C₄⋊C₈	semidirect
75. C₄	#335	semidirect

Maximal subgroups

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

subgroup	index
#1842	2
C₄²×C₈	2
C₂×#335	2