Subgroups — #3487

Order 128.

768 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	#550	C₂	1	8	yes	maximal, complemented
4	#548	C₂	1	8	yes	maximal, complemented
5	#535	C₂	1	6	yes	maximal, complemented
6	#525	C₂	1	6	yes	maximal, complemented
7	#519	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴	—	2	8	no	—
9	C₂×#41	—	2	8	no	—
10	C₂×#41	C₂²	1	6	yes	—
11	C₂×#41	C₂²	1	8	yes	—
12	C₂×#41	C₂²	1	4	yes	complemented
13	C₂²×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×Q₈	C₂²	1	4	yes	complemented
15	C₂²×D₄	C₂²	1	3	yes	—
16	C₂³×C₄	C₂²	1	3	yes	—
17	#127	C₂²	1	12	yes	complemented
18	#124	C₂²	1	12	yes	complemented
19	#123	C₂²	1	4	yes	—
20	C₂²⋊Q₈	C₂²	1	12	yes	complemented
21	#121	C₂²	1	12	yes	complemented
22	#121	C₂²	1	24	yes	complemented
23	#120	C₂²	1	12	yes	—
24	C₄×Q₈	C₂²	1	8	yes	complemented
25	C₄×D₄	C₂²	1	24	yes	complemented
26	#117	C₂²	1	6	yes	—
27	C₂⁴	C₂³	1	3	yes	—
28	#41	—	2	32	no	—
29	#41	—	2	16	no	—
30	#41	—	2	24	no	—
31	C₂×Q₈	—	2	16	no	—
32	C₂×Q₈	D₄	1	8	yes	complemented
33	C₂×Q₈	C₂³	1	6	yes	—
34	C₂×Q₈	C₂³	1	4	yes	—
35	C₂×D₄	—	2	12	no	—
36	C₂×D₄	C₂³	1	4	yes	—
37	C₂×D₄	C₂³	1	12	yes	—
38	C₂×D₄	C₂³	1	6	yes	—
39	C₂×D₄	C₂³	1	24	yes	—
40	C₂²×C₄	—	2	12	no	—
41	C₂²×C₄	C₂³	1	12	yes	—
42	C₂²×C₄	C₂³	1	12	yes	—
43	C₂²×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₄	C₂³	1	3	yes	—
45	C₄⋊C₄	C₂³	1	12	yes	—
46	C₄⋊C₄	C₂³	1	8	yes	—
47	C₂²⋊C₄	C₂³	1	4	yes	—
48	C₂²⋊C₄	C₂³	1	8	yes	—
49	C₂²⋊C₄	C₂³	1	24	yes	—
50	C₄²	C₂³	1	12	yes	—
51	C₂³	—	2	6	no	—
52	C₂³	—	4	6	no	—
53	C₂³	C₂⁴	1	6	yes	—
54	C₂³	C₂⁴	1	4	yes	—
55	C₂³	C₂⁴	1	1	yes	characteristic
56	Q₈	—	2	12	no	—
57	Q₈	C₂×D₄	1	16	yes	—
58	D₄	—	2	8	no	—
59	D₄	—	2	24	no	—
60	D₄	—	2	12	no	—
61	D₄	—	4	24	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	C₂×D₄	1	12	yes	—
66	C₂×C₄	C₂⁴	1	8	yes	—
67	C₂×C₄	C₂⁴	1	4	yes	—
68	C₂×C₄	C₂⁴	1	2	yes	—
69	C₂×C₄	C₂⁴	1	6	yes	—
70	C₂²	—	2	2	no	—
71	C₂²	—	2	4	no	—
72	C₂²	—	2	6	no	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	12	no	—
75	C₂²	—	4	12	no	—
76	C₂²	C₂²×D₄	1	2	yes	—
77	C₂²	C₂⁵	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	2	8	no	—
81	C₄	C₂²×D₄	1	12	yes	—
82	C₂	—	2	2	no	—
83	C₂	—	4	4	no	—
84	C₂	—	4	6	no	—
85	C₂	C₂³×D₄	1	1	yes	characteristic
86	C₂	#585	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. C₂²×#41	C₂	semidirect
3. #550	C₂	semidirect
4. #548	C₂	semidirect
5. #535	C₂	semidirect
6. #525	C₂	semidirect
7. #519	C₂	semidirect
12. C₂×#41	C₂²	semidirect
14. C₂²×Q₈	C₂²	semidirect
17. #127	C₂²	semidirect
18. #124	C₂²	semidirect
20. C₂²⋊Q₈	C₂²	semidirect
21. #121	C₂²	semidirect
22. #121	C₂²	semidirect
24. C₄×Q₈	C₂²	semidirect
25. C₄×D₄	C₂²	semidirect
32. C₂×Q₈	D₄	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
C₂²×#41	2
#550	2
#548	2
#535	2
#525	2
#519	2