Subgroups — #3499

Order 128.

738 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	D₄×Q₈	C₂	1	4	yes	maximal, complemented
3	#547	C₂	1	12	yes	maximal, complemented
4	#536	C₂	1	6	yes	maximal, complemented
5	#532	C₂	1	3	yes	maximal, complemented
6	C₂×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
7	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴	—	2	16	no	—
9	C₂×#41	C₂²	1	6	yes	—
10	C₂×#41	C₂²	1	12	yes	—
11	C₂×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈	C₂²	1	6	yes	—
13	C₂²×Q₈	C₂²	1	4	yes	—
14	C₄⋊Q₈	C₂²	1	3	yes	—
15	C₄⋊Q₈	C₂²	1	6	yes	—
16	C₄⋊Q₈	C₂²	1	12	yes	—
17	#127	C₂²	1	1	yes	characteristic
18	#124	C₂²	1	6	yes	—
19	#123	C₂²	1	24	yes	—
20	C₂²⋊Q₈	C₂²	1	24	yes	—
21	#121	C₂²	1	12	yes	—
22	C₄×Q₈	C₂²	1	2	yes	—
23	C₄×D₄	C₂²	1	6	yes	—
24	C₄×D₄	C₂²	1	12	yes	—
25	#117	C₂²	1	3	yes	—
26	C₂×C₄⋊C₄	C₂²	1	12	yes	—
27	C₂×C₄²	C₂²	1	3	yes	—
28	#41	—	2	48	no	—
29	#41	—	2	24	no	—
30	#41	D₄	1	8	yes	complemented
31	C₂×Q₈	—	2	16	no	—
32	C₂×Q₈	—	2	24	no	—
33	C₂×Q₈	C₂³	1	12	yes	—
34	C₂×Q₈	C₂³	1	6	yes	—
35	C₂×Q₈	C₂³	1	1	yes	characteristic
36	C₂×D₄	C₂³	1	4	yes	—
37	C₂×D₄	C₂³	1	12	yes	—
38	C₂×D₄	C₂³	1	2	yes	—
39	C₂×D₄	C₂³	1	3	yes	—
40	C₂²×C₄	C₂³	1	6	yes	—
41	C₂²×C₄	C₂³	1	12	yes	—
42	C₂²×C₄	C₂³	1	12	yes	—
43	C₂²×C₄	C₂³	1	3	yes	—
44	C₄⋊C₄	C₂³	1	12	yes	—
45	C₄⋊C₄	C₂³	1	24	yes	—
46	C₄⋊C₄	C₂³	1	6	yes	—
47	C₂²⋊C₄	C₂³	1	6	yes	—
48	C₂²⋊C₄	C₂³	1	24	yes	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₄²	C₂³	1	3	yes	—
51	C₄²	C₂³	1	6	yes	—
52	C₂³	C₂⁴	1	4	yes	—
53	C₂³	C₂⁴	1	3	yes	—
54	Q₈	—	2	24	no	—
55	Q₈	—	2	12	no	—
56	Q₈	C₂×D₄	1	4	yes	—
57	D₄	—	2	24	no	—
58	D₄	—	2	4	no	—
59	D₄	—	2	8	no	—
60	D₄	C₂×D₄	1	12	yes	—
61	C₂×C₄	—	2	24	no	—
62	C₂×C₄	—	2	24	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	C₂×D₄	1	12	yes	—
65	C₂×C₄	C₂⁴	1	6	yes	—
66	C₂×C₄	C₂⁴	1	12	yes	—
67	C₂×C₄	C₂⁴	1	1	yes	characteristic
68	C₂×C₄	C₂⁴	1	3	yes	—
69	C₂×C₄	C₂⁴	1	2	yes	—
70	C₂²	—	2	6	no	—
71	C₂²	—	2	4	no	—
72	C₂²	—	2	8	no	—
73	C₂²	C₂²×D₄	1	6	yes	—
74	C₂²	C₂⁵	1	1	yes	characteristic
75	C₄	—	2	12	no	—
76	C₄	—	2	6	no	—
77	C₄	C₂²×D₄	1	6	yes	—
78	C₄	C₂²×D₄	1	2	yes	—
79	C₄	2₋¹⁺⁴	1	4	yes	complemented
80	C₂	—	2	6	no	—
81	C₂	—	4	4	no	—
82	C₂	C₂³×D₄	1	1	yes	characteristic
83	C₂	C₂×2₋¹⁺⁴	1	2	yes	—

	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. D₄×Q₈	C₂	semidirect
3. #547	C₂	semidirect
4. #536	C₂	semidirect
5. #532	C₂	semidirect
6. C₂×C₄⋊Q₈	C₂	semidirect
7. C₄×#41	C₂	semidirect
30. #41	D₄	semidirect
79. 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
D₄×Q₈	2
#547	2
#536	2
#532	2
C₂×C₄⋊Q₈	2
C₄×#41	2