Subgroups — #3358

Order 128.

213 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#550	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#486	C₂	1	3	yes	maximal, complemented
4	#474	C₂	1	3	yes	maximal, complemented
5	#459	C₂	1	3	yes	maximal, complemented
6	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₈	C₂	1	3	yes	maximal, complemented
8	C₂×#41	—	2	1	no	—
9	C₂²×Q₈	—	2	1	no	—
10	C₂×D₈	C₂²	1	3	yes	—
11	C₄⋊Q₈	C₂²	1	3	yes	—
12	#127	C₂²	1	3	yes	—
13	C₂²⋊Q₈	—	2	3	no	—
14	#121	—	2	3	no	—
15	C₄×Q₈	C₂²	1	1	yes	characteristic
16	C₄×D₄	C₂²	1	3	yes	—
17	C₄×D₄	C₂²	1	3	yes	—
18	#107	C₂²	1	3	yes	—
19	C₄⋊C₈	C₂²	1	3	yes	—
20	#102	C₂²	1	1	yes	characteristic
21	#102	C₂²	1	3	yes	—
22	#102	C₂²	1	6	yes	—
23	C₄×C₈	C₂²	1	3	yes	—
24	#41	—	4	4	no	—
25	C₂×Q₈	—	2	3	no	—
26	C₂×Q₈	—	4	4	no	—
27	C₂×Q₈	D₄	1	1	yes	characteristic
28	C₂×D₄	—	2	3	no	—
29	C₂×D₄	—	2	3	no	—
30	C₂×D₄	C₂³	1	3	yes	—
31	C₂×D₄	C₂³	1	1	yes	characteristic
32	C₂²×C₄	—	2	3	no	—
33	C₂²×C₄	—	2	3	no	—
34	D₈	—	2	6	no	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈	C₂³	1	3	yes	—
37	C₄⋊C₄	—	2	3	no	—
38	C₄⋊C₄	D₄	1	3	yes	—
39	C₄⋊C₄	C₂³	1	3	yes	—
40	C₄⋊C₄	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄	—	2	3	no	—
42	C₂²⋊C₄	—	2	3	no	—
43	C₄²	C₂³	1	3	yes	—
44	C₂³	—	2	1	no	—
45	C₂³	—	2	3	no	—
46	Q₈	—	4	6	no	—
47	Q₈	D₈	1	4	yes	complemented
48	D₄	—	2	3	no	—
49	D₄	—	2	1	no	—
50	D₄	—	2	3	no	—
51	D₄	—	4	6	no	—
52	D₄	—	8	3	no	—
53	D₄	#41	1	2	yes	complemented
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	3	no	—
56	C₂×C₄	—	4	6	no	—
57	C₂×C₄	—	4	6	no	—
58	C₂×C₄	C₂×D₄	1	3	yes	—
59	C₂×C₄	C₂×D₄	1	3	yes	—
60	C₂×C₄	C₂⁴	1	1	yes	characteristic
61	C₈	—	2	3	no	—
62	C₈	—	2	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	3	no	—
67	C₂²	—	4	3	no	—
68	C₂²	—	4	3	no	—
69	C₂²	C₂²×D₄	1	1	yes	characteristic
70	C₄	—	2	3	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	3	no	—
73	C₄	C₂×D₈	1	6	yes	—
74	C₄	C₂×#41	1	1	yes	characteristic
75	C₄	2₋¹⁺⁴	1	1	yes	characteristic
76	C₂	—	4	2	no	—
77	C₂	—	8	3	no	—
78	C₂	#548	1	1	yes	characteristic
79	C₂	C₂²×D₈	1	1	yes	characteristic
80	C₂	#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
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. #550	C₂	semidirect
2. D₄×Q₈	C₂	semidirect
3. #486	C₂	semidirect
4. #474	C₂	semidirect
5. #459	C₂	semidirect
6. C₈×Q₈	C₂	semidirect
7. C₄×D₈	C₂	semidirect
47. Q₈	D₈	semidirect
53. D₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#550	2
D₄×Q₈	2
#486	2
#474	2
#459	2
C₈×Q₈	2
C₄×D₈	2