Subgroups — #3380

Order 128.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554	C₂	1	2	yes	maximal, complemented
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#499	C₂	1	2	yes	maximal, complemented
4	#478	C₂	1	4	yes	maximal, complemented
5	#476	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₈	C₂	1	2	yes	maximal
8	C₄×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂×D₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	2	yes	—
11	#125	C₂²	1	4	yes	—
12	C₂²⋊Q₈	—	2	2	no	—
13	C₂²⋊Q₈	—	2	4	no	—
14	C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₄×D₄	C₂²	1	2	yes	—
16	C₄×D₄	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄	—	2	2	no	—
18	#107	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	2	yes	—
20	#107	C₂²	1	2	yes	—
21	#106	C₂²	1	4	yes	—
22	C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₄⋊C₈	C₂²	1	2	yes	—
24	#102	C₂²	1	4	yes	—
25	#102	C₂²	1	2	yes	—
26	C₄×C₈	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₈	—	2	2	no	—
29	C₂×Q₈	D₄	1	1	yes	characteristic
30	C₂×D₄	C₂³	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	—	2	4	no	—
33	D₈	Q₈	1	4	yes	complemented
34	C₂×C₈	C₂³	1	2	yes	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₄⋊C₄	D₄	1	2	yes	—
42	C₄⋊C₄	C₂³	1	2	yes	—
43	C₄⋊C₄	C₂³	1	4	yes	—
44	C₂²⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₄²	C₂³	1	2	yes	—
47	C₄²	C₂³	1	1	yes	characteristic
48	C₂³	—	2	2	no	—
49	Q₈	—	2	2	no	—
50	Q₈	—	8	2	no	—
51	D₄	—	2	2	no	—
52	D₄	C₂×Q₈	1	4	yes	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	4	4	no	—
56	C₂×C₄	—	8	4	no	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×D₄	1	2	yes	—
60	C₂×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄	C₂⁴	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	2	1	no	—
64	C₈	C₂×Q₈	1	2	yes	—
65	C₂²	—	2	4	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	4	2	no	—
68	C₂²	C₂²×D₄	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	4	4	no	—
74	C₄	#135	1	2	yes	—
75	C₄	C₂²×Q₈	1	1	yes	characteristic
76	C₄	2₋¹⁺⁴	1	1	yes	characteristic
77	C₂	—	4	4	no	—
78	C₂	D₄×Q₈	1	1	yes	characteristic
79	C₂	C₂×#135	1	1	yes	characteristic
80	C₂	#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #554	C₂	semidirect
2. #500	C₂	semidirect
3. #499	C₂	semidirect
4. #478	C₂	semidirect
5. #476	C₂	semidirect
6. C₈×Q₈	C₂	semidirect
33. D₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#554	2
#500	2
#499	2
#478	2
#476	2
C₈×Q₈	2
C₄×D₈	2