Subgroups — #3497

Order 128.

746 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	#550	C₂	1	4	yes	maximal, complemented
3	D₄×Q₈	C₂	1	4	yes	maximal, complemented
4	#548	C₂	1	8	yes	maximal, complemented
5	#537	C₂	1	6	yes	maximal, complemented
6	#532	C₂	1	6	yes	maximal, complemented
7	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴	—	2	16	no	—
9	C₂×#41	C₂²	1	12	yes	—
10	C₂×#41	C₂²	1	8	yes	—
11	C₂²×Q₈	C₂²	1	8	yes	—
12	C₂²×Q₈	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈	C₂²	1	6	yes	—
14	#127	C₂²	1	6	yes	—
15	#124	C₂²	1	12	yes	—
16	C₂²⋊Q₈	C₂²	1	24	yes	—
17	#121	C₂²	1	4	yes	—
18	#121	C₂²	1	12	yes	—
19	#121	C₂²	1	24	yes	—
20	C₄×Q₈	C₂²	1	4	yes	—
21	C₄×Q₈	C₂²	1	4	yes	—
22	C₄×D₄	C₂²	1	12	yes	—
23	C₄×D₄	C₂²	1	12	yes	—
24	C₂×C₄⋊C₄	C₂²	1	3	yes	—
25	C₂×C₄²	C₂²	1	3	yes	—
26	#41	—	2	32	no	—
27	#41	—	2	48	no	—
28	C₂×Q₈	—	2	32	no	—
29	C₂×Q₈	D₄	1	8	yes	complemented
30	C₂×Q₈	C₂³	1	12	yes	—
31	C₂×Q₈	C₂³	1	4	yes	—
32	C₂×D₄	C₂³	1	4	yes	—
33	C₂×D₄	C₂³	1	8	yes	—
34	C₂×D₄	C₂³	1	12	yes	—
35	C₂×D₄	C₂³	1	12	yes	—
36	C₂²×C₄	C₂³	1	1	yes	characteristic
37	C₂²×C₄	C₂³	1	24	yes	—
38	C₂²×C₄	C₂³	1	3	yes	—
39	C₂²×C₄	C₂³	1	3	yes	—
40	C₄⋊C₄	C₂³	1	12	yes	—
41	C₄⋊C₄	C₂³	1	4	yes	—
42	C₄⋊C₄	C₂³	1	12	yes	—
43	C₂²⋊C₄	C₂³	1	8	yes	—
44	C₂²⋊C₄	C₂³	1	24	yes	—
45	C₄²	C₂³	1	6	yes	—
46	C₄²	C₂³	1	6	yes	—
47	C₂³	C₂⁴	1	8	yes	—
48	C₂³	C₂⁴	1	1	yes	characteristic
49	Q₈	—	2	24	no	—
50	Q₈	C₂×D₄	1	16	yes	—
51	D₄	—	2	16	no	—
52	D₄	—	2	24	no	—
53	D₄	—	2	8	no	—
54	D₄	—	4	12	no	—
55	C₂×C₄	—	2	48	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	4	3	no	—
58	C₂×C₄	C₂×D₄	1	12	yes	—
59	C₂×C₄	C₂⁴	1	1	yes	characteristic
60	C₂×C₄	C₂⁴	1	6	yes	—
61	C₂×C₄	C₂⁴	1	8	yes	—
62	C₂×C₄	C₂⁴	1	6	yes	—
63	C₂×C₄	C₂⁴	1	1	yes	characteristic
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	8	no	—
67	C₂²	—	2	8	no	—
68	C₂²	—	2	8	no	—
69	C₂²	C₂²×D₄	1	2	yes	—
70	C₂²	C₂⁵	1	1	yes	characteristic
71	C₄	—	2	8	no	—
72	C₄	—	2	1	no	—
73	C₄	—	2	6	no	—
74	C₄	C₂²×D₄	1	12	yes	—
75	C₄	2₋¹⁺⁴	1	2	yes	complemented
76	C₂	—	2	2	no	—
77	C₂	—	4	8	no	—
78	C₂	C₂³×D₄	1	1	yes	characteristic
79	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
80	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴	C₂	semidirect
2. #550	C₂	semidirect
3. D₄×Q₈	C₂	semidirect
4. #548	C₂	semidirect
5. #537	C₂	semidirect
6. #532	C₂	semidirect
7. C₂×C₄×Q₈	C₂	semidirect
29. C₂×Q₈	D₄	semidirect
75. C₄	2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴	2
#550	2
D₄×Q₈	2
#548	2
#537	2
#532	2
C₂×C₄×Q₈	2