Subgroups — #3502

Order 128.

482 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	4	yes	maximal, complemented
2	#557	C₂	1	8	yes	maximal, complemented
3	#557	C₂	1	4	yes	maximal, complemented
4	#544	C₂	1	6	yes	maximal
5	#533	C₂	1	6	yes	maximal
6	#519	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈	C₂²	1	6	yes	—
10	C₄⋊Q₈	C₂²	1	6	yes	—
11	C₄⋊Q₈	C₂²	1	24	yes	—
12	#125	C₂²	1	12	yes	—
13	#125	C₂²	1	24	yes	—
14	C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₂²⋊Q₈	C₂²	1	12	yes	—
16	C₄×Q₈	C₂²	1	4	yes	—
17	C₄×Q₈	C₂²	1	4	yes	—
18	C₄×Q₈	C₂²	1	8	yes	—
19	C₄×Q₈	C₂²	1	8	yes	—
20	C₄×Q₈	C₂²	1	12	yes	—
21	C₄×Q₈	C₂²	1	12	yes	—
22	#117	C₂²	1	12	yes	—
23	C₂×C₄⋊C₄	C₂²	1	3	yes	—
24	C₂×C₄²	C₂²	1	3	yes	—
25	C₂×Q₈	Q₈	1	8	yes	complemented
26	C₂×Q₈	C₂³	1	4	yes	—
27	C₂×Q₈	C₂³	1	12	yes	—
28	C₂×Q₈	C₂³	1	4	yes	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₄	C₂³	1	3	yes	—
31	C₂²×C₄	C₂³	1	3	yes	—
32	C₄⋊C₄	C₂³	1	12	yes	—
33	C₄⋊C₄	C₂³	1	12	yes	—
34	C₄⋊C₄	C₂³	1	12	yes	—
35	C₄⋊C₄	C₂³	1	8	yes	—
36	C₄⋊C₄	C₂³	1	4	yes	—
37	C₄⋊C₄	C₂³	1	12	yes	—
38	C₄⋊C₄	C₂³	1	24	yes	—
39	C₂²⋊C₄	C₂³	1	8	yes	—
40	C₄²	C₂³	1	6	yes	—
41	C₄²	C₂³	1	6	yes	—
42	C₄²	C₂³	1	24	yes	—
43	C₂³	C₂⁴	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	Q₈	—	4	12	no	—
46	Q₈	C₂×Q₈	1	16	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	3	no	—
49	C₂×C₄	C₂×Q₈	1	12	yes	—
50	C₂×C₄	C₂⁴	1	1	yes	characteristic
51	C₂×C₄	C₂⁴	1	8	yes	—
52	C₂×C₄	C₂⁴	1	8	yes	—
53	C₂×C₄	C₂⁴	1	6	yes	—
54	C₂×C₄	C₂⁴	1	6	yes	—
55	C₂×C₄	C₂⁴	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	C₂²×Q₈	1	2	yes	—
59	C₂²	C₂⁵	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	2	8	no	—
62	C₄	—	2	6	no	—
63	C₄	—	2	8	no	—
64	C₄	C₂²×Q₈	1	12	yes	—
65	C₄	2₋¹⁺⁴	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	C₂³×Q₈	1	1	yes	characteristic
68	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
69	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂³	C₂⁴
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. Q₈²	C₂	semidirect
2. #557	C₂	semidirect
3. #557	C₂	semidirect
25. C₂×Q₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈²	2
#557	2
#544	2
#533	2
#519	2
C₂×C₄×Q₈	2