Subgroups — #3549

Order 128.

629 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₄)	C₂	1	4	yes	maximal, complemented
2	#555	C₂	1	6	yes	maximal, complemented
3	D₄²	C₂	1	6	yes	maximal, complemented
4	#535	C₂	1	12	yes	maximal, complemented
5	#532	C₂	1	3	yes	maximal, complemented
6	C₂×#41	C₂²	1	6	yes	complemented
7	C₂²×D₄	C₂²	1	12	yes	complemented
8	C₄⋊Q₈	C₂²	1	3	yes	complemented
9	#127	C₂²	1	3	yes	—
10	#127	C₂²	1	8	yes	complemented
11	#124	C₂²	1	6	yes	complemented
12	#124	C₂²	1	24	yes	complemented
13	#121	C₂²	1	12	yes	complemented
14	#121	C₂²	1	24	yes	complemented
15	#120	C₂²	1	8	yes	complemented
16	#120	C₂²	1	24	yes	complemented
17	C₄×D₄	C₂²	1	12	yes	complemented
18	#117	C₂²	1	12	yes	complemented
19	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
20	C₂⁴	C₂³	1	8	yes	—
21	#41	—	4	12	no	—
22	C₂×Q₈	C₂³	1	6	yes	—
23	C₂×D₄	—	4	24	no	—
24	C₂×D₄	C₂³	1	12	yes	—
25	C₂×D₄	C₂³	1	6	yes	—
26	C₂×D₄	C₂³	1	24	yes	—
27	C₂×D₄	C₂³	1	12	yes	—
28	C₂²×C₄	C₂³	1	3	yes	—
29	C₂²×C₄	C₂³	1	12	yes	—
30	C₄⋊C₄	C₂³	1	12	yes	—
31	C₂²⋊C₄	C₂³	1	24	yes	—
32	C₂²⋊C₄	C₂³	1	24	yes	—
33	C₄²	C₂³	1	3	yes	—
34	C₄²	C₂³	1	1	yes	characteristic
35	C₄²	C₂³	1	8	yes	—
36	C₂³	—	4	24	no	—
37	C₂³	C₂⁴	1	12	yes	—
38	C₂³	C₂⁴	1	1	yes	characteristic
39	Q₈	—	4	6	no	—
40	D₄	—	2	24	no	—
41	D₄	—	4	24	no	—
42	D₄	—	4	12	no	—
43	D₄	—	4	6	no	—
44	C₂×C₄	—	2	24	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	C₂⁴	1	3	yes	—
47	C₂×C₄	C₂⁴	1	12	yes	—
48	C₂×C₄	C₂⁴	1	3	yes	—
49	C₂²	—	2	3	no	—
50	C₂²	—	2	12	no	—
51	C₂²	—	2	24	no	—
52	C₂²	—	8	16	no	—
53	C₂²	C₂⁵	1	1	yes	characteristic
54	C₄	—	2	12	no	—
55	C₄	—	2	3	no	—
56	C₄	H(5,𝔽₂)	1	6	yes	—
57	C₂	—	4	1	no	—
58	C₂	—	4	12	no	—
59	C₂	C₂×H(5,𝔽₂)	1	3	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. H(3,𝔽₄)	C₂	semidirect
2. #555	C₂	semidirect
3. D₄²	C₂	semidirect
4. #535	C₂	semidirect
5. #532	C₂	semidirect
6. C₂×#41	C₂²	semidirect
7. C₂²×D₄	C₂²	semidirect
8. C₄⋊Q₈	C₂²	semidirect
10. #127	C₂²	semidirect
11. #124	C₂²	semidirect
12. #124	C₂²	semidirect
13. #121	C₂²	semidirect
14. #121	C₂²	semidirect
15. #120	C₂²	semidirect
16. #120	C₂²	semidirect
17. C₄×D₄	C₂²	semidirect
18. #117	C₂²	semidirect
19. C₂×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₄)	2
#555	2
D₄²	2
#535	2
#532	2