Subgroups — #62980

Order 256.

3057 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3568	C₂	1	9	yes	maximal, complemented
2	#3564	C₂	1	27	yes	maximal, complemented
3	#3555	C₂	1	27	yes	maximal, complemented
4	#563	C₂²	1	27	yes	complemented
5	#563	C₂²	1	54	yes	complemented
6	H(3,𝔽₄)	C₂²	1	3	yes	complemented
7	Q₈²	C₂²	1	9	yes	complemented
8	#557	C₂²	1	27	yes	complemented
9	#556	C₂²	1	54	yes	complemented
10	#556	C₂²	1	27	yes	complemented
11	#554	C₂²	1	27	yes	complemented
12	#552	C₂²	1	54	yes	complemented
13	#551	C₂²	1	54	yes	complemented
14	#544	C₂²	1	27	yes	complemented
15	#543	C₂²	1	9	yes	complemented
16	#542	C₂²	1	54	yes	—
17	#541	C₂²	1	54	yes	complemented
18	#541	C₂²	1	54	yes	complemented
19	#539	C₂²	1	27	yes	—
20	#538	C₂²	1	27	yes	—
21	#533	C₂²	1	27	yes	complemented
22	#529	C₂²	1	27	yes	—
23	C₂×#126	C₂²	1	9	yes	complemented
24	C₄⋊Q₈	C₂³	1	27	yes	—
25	C₄⋊Q₈	C₂³	1	27	yes	—
26	C₄⋊Q₈	C₂³	1	54	yes	—
27	#126	C₂³	1	54	yes	—
28	#126	C₂³	1	18	yes	—
29	#126	C₂³	1	54	yes	—
30	#126	C₂³	1	54	yes	—
31	#126	C₂³	1	54	yes	—
32	#125	C₂³	1	27	yes	—
33	#125	C₂³	1	54	yes	—
34	#125	C₂³	1	54	yes	—
35	#125	C₂³	1	54	yes	—
36	#124	C₂³	1	27	yes	—
37	#124	C₂³	1	54	yes	—
38	#123	C₂³	1	54	yes	—
39	#123	C₂³	1	54	yes	—
40	C₂²⋊Q₈	C₂³	1	27	yes	—
41	C₂²⋊Q₈	C₂³	1	27	yes	—
42	C₂²⋊Q₈	C₂³	1	27	yes	—
43	C₂²⋊Q₈	C₂³	1	54	yes	—
44	C₂²⋊Q₈	C₂³	1	54	yes	—
45	C₂²⋊Q₈	C₂³	1	54	yes	—
46	#121	C₂³	1	27	yes	—
47	#120	C₂³	1	18	yes	—
48	C₄×Q₈	C₂³	1	27	yes	—
49	C₄×Q₈	C₂³	1	27	yes	—
50	C₄×Q₈	C₂³	1	27	yes	—
51	C₄×Q₈	C₂³	1	54	yes	—
52	C₄×D₄	C₂³	1	54	yes	—
53	C₄×D₄	C₂³	1	27	yes	—
54	#117	C₂³	1	54	yes	—
55	#117	C₂³	1	54	yes	—
56	C₂×C₄⋊C₄	C₂³	1	27	yes	—
57	C₂×C₂²⋊C₄	C₂³	1	27	yes	—
58	C₂×C₄²	C₂³	1	9	yes	—
59	C₂⁴	C₂⁴	1	3	yes	—
60	C₂×Q₈	C₂⁴	1	9	yes	—
61	C₂×Q₈	C₂⁴	1	27	yes	—
62	C₂×Q₈	C₂⁴	1	9	yes	—
63	C₂×D₄	C₂⁴	1	27	yes	—
64	C₂²×C₄	C₂⁴	1	27	yes	—
65	C₂²×C₄	C₂⁴	1	27	yes	—
66	C₄⋊C₄	C₂⁴	1	27	yes	—
67	C₄⋊C₄	C₂⁴	1	54	yes	—
68	C₄⋊C₄	C₂⁴	1	54	yes	—
69	C₄⋊C₄	C₂⁴	1	27	yes	—
70	C₄⋊C₄	C₂⁴	1	27	yes	—
71	C₄⋊C₄	C₂⁴	1	54	yes	—
72	C₄⋊C₄	C₂⁴	1	27	yes	—
73	C₂²⋊C₄	C₂⁴	1	54	yes	—
74	C₂²⋊C₄	C₂⁴	1	54	yes	—
75	C₂²⋊C₄	C₂⁴	1	54	yes	—
76	C₄²	C₂⁴	1	27	yes	—
77	C₄²	C₂⁴	1	9	yes	—
78	C₄²	C₂⁴	1	54	yes	—
79	C₂³	—	4	9	no	—
80	C₂³	C₂⁵	1	9	yes	—
81	Q₈	—	4	9	no	—
82	Q₈	—	4	27	no	—
83	Q₈	—	4	9	no	—
84	D₄	—	4	27	no	—
85	C₂×C₄	—	4	27	no	—
86	C₂×C₄	—	4	27	no	—
87	C₂×C₄	C₂⁵	1	27	yes	—
88	C₂×C₄	C₂⁵	1	27	yes	—
89	C₂²	—	2	27	no	—
90	C₂²	—	16	3	no	—
91	C₂²	C₂⁶	1	1	yes	characteristic
92	C₄	—	2	27	no	—
93	C₄	—	2	27	no	—
94	C₂	—	4	9	no	—
95	C₂	2₋¹⁺⁶	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
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. #3568	C₂	semidirect
2. #3564	C₂	semidirect
3. #3555	C₂	semidirect
4. #563	C₂²	semidirect
5. #563	C₂²	semidirect
6. H(3,𝔽₄)	C₂²	semidirect
7. Q₈²	C₂²	semidirect
8. #557	C₂²	semidirect
9. #556	C₂²	semidirect
10. #556	C₂²	semidirect
11. #554	C₂²	semidirect
12. #552	C₂²	semidirect
13. #551	C₂²	semidirect
14. #544	C₂²	semidirect
15. #543	C₂²	semidirect
17. #541	C₂²	semidirect
18. #541	C₂²	semidirect
21. #533	C₂²	semidirect
23. C₂×#126	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3568	2
#3564	2
#3555	2