Subgroups — #3444

Order 128.

606 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	3	yes	maximal, complemented
2	#532	C₂	1	3	yes	maximal, complemented
3	#529	C₂	1	12	yes	maximal, complemented
4	#525	C₂	1	6	yes	maximal, complemented
5	C₄×#41	C₂	1	6	yes	maximal, complemented
6	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	C₂²	1	6	yes	—
8	C₂³×C₄	C₂²	1	3	yes	—
9	C₄⋊Q₈	C₂²	1	3	yes	complemented
10	#127	C₂²	1	1	yes	characteristic, complemented
11	#126	C₂²	1	8	yes	complemented
12	#125	C₂²	1	6	yes	complemented
13	#124	C₂²	1	6	yes	complemented
14	#123	C₂²	1	12	yes	—
15	C₂²⋊Q₈	C₂²	1	12	yes	—
16	#121	C₂²	1	12	yes	—
17	#120	C₂²	1	4	yes	—
18	C₄×Q₈	C₂²	1	12	yes	complemented
19	C₄×D₄	C₂²	1	24	yes	—
20	C₄×D₄	C₂²	1	12	yes	complemented
21	#117	C₂²	1	6	yes	—
22	#117	C₂²	1	12	yes	complemented
23	C₂×C₄²	C₂²	1	6	yes	—
24	C₂×C₄²	C₂²	1	3	yes	—
25	C₂×C₄²	C₂²	1	3	yes	—
26	C₂×C₄²	C₂²	1	4	yes	complemented
27	C₂⁴	C₂³	1	1	yes	characteristic
28	#41	—	2	24	no	—
29	C₂×Q₈	C₂³	1	6	yes	—
30	C₂×D₄	C₂³	1	12	yes	—
31	C₂×D₄	C₂³	1	6	yes	—
32	C₂²×C₄	—	2	12	no	—
33	C₂²×C₄	C₂³	1	6	yes	—
34	C₂²×C₄	C₂³	1	6	yes	—
35	C₂²×C₄	C₂³	1	3	yes	—
36	C₂²×C₄	C₂³	1	12	yes	—
37	C₂²×C₄	C₂³	1	3	yes	—
38	C₄⋊C₄	C₂³	1	12	yes	—
39	C₄⋊C₄	C₂³	1	24	yes	—
40	C₂²⋊C₄	C₂³	1	12	yes	—
41	C₂²⋊C₄	C₂³	1	24	yes	—
42	C₄²	C₂³	1	3	yes	—
43	C₄²	C₂³	1	6	yes	—
44	C₄²	C₂³	1	3	yes	—
45	C₄²	C₂³	1	2	yes	—
46	C₄²	C₂³	1	1	yes	characteristic
47	C₄²	C₂³	1	12	yes	—
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	—	2	6	no	—
50	C₂³	C₂⁴	1	4	yes	—
51	C₂³	C₂⁴	1	3	yes	—
52	Q₈	—	2	12	no	—
53	D₄	—	2	24	no	—
54	D₄	—	2	12	no	—
55	C₂×C₄	—	2	6	no	—
56	C₂×C₄	—	2	12	no	—
57	C₂×C₄	—	2	12	no	—
58	C₂×C₄	—	2	24	no	—
59	C₂×C₄	#41	1	12	yes	complemented
60	C₂×C₄	C₂⁴	1	6	yes	—
61	C₂×C₄	C₂⁴	1	12	yes	—
62	C₂×C₄	C₂⁴	1	3	yes	—
63	C₂×C₄	C₂⁴	1	3	yes	—
64	C₂²	—	2	8	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	2	12	no	—
67	C₂²	C₂×#41	1	6	yes	—
68	C₂²	C₂⁵	1	1	yes	characteristic
69	C₄	—	2	6	no	—
70	C₄	—	2	12	no	—
71	C₄	C₂×#41	1	6	yes	—
72	C₄	C₂×#41	1	6	yes	—
73	C₂	—	2	6	no	—
74	C₂	—	4	4	no	—
75	C₂	C₂²×#41	1	3	yes	—

	subgroup	G/H
center	C₄²	C₂³
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #533	C₂	semidirect
2. #532	C₂	semidirect
3. #529	C₂	semidirect
4. #525	C₂	semidirect
5. C₄×#41	C₂	semidirect
6. C₂²×C₄²	C₂	semidirect
9. C₄⋊Q₈	C₂²	semidirect
10. #127	C₂²	semidirect
11. #126	C₂²	semidirect
12. #125	C₂²	semidirect
13. #124	C₂²	semidirect
18. C₄×Q₈	C₂²	semidirect
20. C₄×D₄	C₂²	semidirect
22. #117	C₂²	semidirect
26. C₂×C₄²	C₂²	semidirect
59. C₂×C₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#532	2
#529	2
#525	2
C₄×#41	2
C₂²×C₄²	2