Subgroups — #3504

Order 128.

494 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557	C₂	1	4	yes	maximal, complemented
2	#554	C₂	1	12	yes	maximal, complemented
3	#544	C₂	1	6	yes	maximal, complemented
4	#533	C₂	1	3	yes	maximal, complemented
5	C₂×#125	C₂	1	3	yes	maximal, complemented
6	#520	C₂	1	2	yes	maximal
7	C₄×#41	C₂	1	1	yes	maximal, characteristic
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈	C₂²	1	6	yes	—
10	C₄⋊Q₈	C₂²	1	12	yes	—
11	#125	C₂²	1	3	yes	—
12	#125	C₂²	1	6	yes	—
13	#125	C₂²	1	12	yes	—
14	#125	C₂²	1	1	yes	characteristic
15	#125	C₂²	1	12	yes	—
16	C₂²⋊Q₈	C₂²	1	12	yes	—
17	C₂²⋊Q₈	C₂²	1	24	yes	—
18	C₄×Q₈	C₂²	1	2	yes	—
19	C₄×Q₈	C₂²	1	4	yes	—
20	C₄×Q₈	C₂²	1	12	yes	—
21	C₄×D₄	C₂²	1	6	yes	—
22	C₄×D₄	C₂²	1	12	yes	—
23	#117	C₂²	1	3	yes	—
24	#117	C₂²	1	6	yes	—
25	C₂×C₄⋊C₄	C₂²	1	6	yes	—
26	C₂×C₄⋊C₄	C₂²	1	12	yes	—
27	C₂×C₄²	C₂²	1	3	yes	—
28	#41	Q₈	1	8	yes	complemented
29	C₂×Q₈	C₂³	1	12	yes	—
30	C₂×Q₈	C₂³	1	1	yes	characteristic
31	C₂×D₄	C₂³	1	3	yes	—
32	C₂²×C₄	C₂³	1	6	yes	—
33	C₂²×C₄	C₂³	1	12	yes	—
34	C₂²×C₄	C₂³	1	3	yes	—
35	C₄⋊C₄	C₂³	1	6	yes	—
36	C₄⋊C₄	C₂³	1	12	yes	—
37	C₄⋊C₄	C₂³	1	12	yes	—
38	C₄⋊C₄	C₂³	1	24	yes	—
39	C₄⋊C₄	C₂³	1	2	yes	—
40	C₄⋊C₄	C₂³	1	6	yes	—
41	C₄⋊C₄	C₂³	1	4	yes	—
42	C₄⋊C₄	C₂³	1	12	yes	—
43	C₂²⋊C₄	C₂³	1	12	yes	—
44	C₂²⋊C₄	C₂³	1	6	yes	—
45	C₄²	C₂³	1	1	yes	characteristic
46	C₄²	C₂³	1	3	yes	—
47	C₄²	C₂³	1	6	yes	—
48	C₄²	C₂³	1	12	yes	—
49	C₂³	C₂⁴	1	3	yes	—
50	Q₈	—	4	12	no	—
51	Q₈	C₂×Q₈	1	4	yes	—
52	D₄	C₂×Q₈	1	12	yes	—
53	C₂×C₄	—	4	6	no	—
54	C₂×C₄	—	4	12	no	—
55	C₂×C₄	C₂×Q₈	1	12	yes	—
56	C₂×C₄	C₂⁴	1	2	yes	—
57	C₂×C₄	C₂⁴	1	4	yes	—
58	C₂×C₄	C₂⁴	1	6	yes	—
59	C₂×C₄	C₂⁴	1	12	yes	—
60	C₂×C₄	C₂⁴	1	3	yes	—
61	C₂×C₄	C₂⁴	1	1	yes	characteristic
62	C₂²	—	2	6	no	—
63	C₂²	C₂²×Q₈	1	6	yes	—
64	C₂²	C₂⁵	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	2	12	no	—
67	C₄	—	2	6	no	—
68	C₄	—	2	4	no	—
69	C₄	C₂²×Q₈	1	2	yes	—
70	C₄	C₂²×Q₈	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	C₂³×Q₈	1	1	yes	characteristic
73	C₂	#585	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂×#41	C₂²
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. #557	C₂	semidirect
2. #554	C₂	semidirect
3. #544	C₂	semidirect
4. #533	C₂	semidirect
5. C₂×#125	C₂	semidirect
28. #41	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#557	2
#554	2
#544	2
#533	2
C₂×#125	2
#520	2
C₄×#41	2