Subgroups — C₂²×#2811 #7633619

Order 512.

8514 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2811	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#395	C₂	1	6	yes	maximal
3	C₂²×#391	C₂	1	9	yes	maximal
4	C₂⁴×Q₈	C₂²	1	2	yes	—
5	C₂⁴×Q₈	C₂²	1	3	yes	—
6	C₂³×C₄⋊C₄	C₂²	1	9	yes	—
7	C₂³×C₄²	C₂²	1	3	yes	—
8	#2811	C₂²	1	256	yes	direct product factor
9	C₂×#395	C₂²	1	144	yes	—
10	C₂×#391	C₂²	1	216	yes	—
11	C₂²×#95	C₂²	1	18	yes	—
12	C₂³×Q₈	—	4	12	no	—
13	C₂³×Q₈	—	4	12	no	—
14	C₂³×Q₈	—	4	24	no	—
15	C₂³×Q₈	Q₈	1	48	yes	complemented
16	C₂³×Q₈	C₂³	1	24	yes	—
17	C₂³×Q₈	C₂³	1	36	yes	—
18	C₂⁴×C₄	C₂³	1	6	yes	—
19	C₂⁴×C₄	C₂³	1	9	yes	—
20	C₂²×C₄⋊C₄	—	4	36	no	—
21	C₂²×C₄⋊C₄	C₂³	1	108	yes	—
22	C₂²×C₄²	—	4	12	no	—
23	C₂²×C₄²	C₂³	1	36	yes	—
24	#395	C₂³	1	384	yes	—
25	#391	C₂³	1	576	yes	—
26	C₂×#95	C₂³	1	216	yes	—
27	C₂⁵	C₂⁴	1	1	yes	characteristic
28	C₂²×Q₈	—	4	192	no	—
29	C₂²×Q₈	—	4	72	no	—
30	C₂²×Q₈	—	4	72	no	—
31	C₂²×Q₈	—	4	144	no	—
32	C₂²×Q₈	—	4	128	no	—
33	C₂²×Q₈	C₂×Q₈	1	288	yes	complemented
34	C₂²×Q₈	C₂⁴	1	32	yes	—
35	C₂²×Q₈	C₂⁴	1	48	yes	—
36	C₂³×C₄	—	2	36	no	—
37	C₂³×C₄	—	2	36	no	—
38	C₂³×C₄	—	2	72	no	—
39	C₂³×C₄	C₂×Q₈	1	72	yes	—
40	C₂³×C₄	C₂⁴	1	36	yes	—
41	C₂³×C₄	C₂⁴	1	54	yes	—
42	C₂×C₄⋊C₄	—	4	216	no	—
43	C₂×C₄⋊C₄	C₂⁴	1	144	yes	—
44	C₂×C₄²	—	4	72	no	—
45	C₂×C₄²	C₂⁴	1	48	yes	—
46	#95	C₂⁴	1	288	yes	—
47	C₂⁴	C₂²×Q₈	1	12	yes	—
48	C₂⁴	H(5,𝔽₂)	1	4	yes	—
49	C₂⁴	H(5,𝔽₂)	1	12	yes	—
50	C₂⁴	C₂⁵	1	3	yes	—
51	C₂×Q₈	—	4	576	no	—
52	C₂×Q₈	—	4	48	no	—
53	C₂×Q₈	—	4	48	no	—
54	C₂×Q₈	—	4	96	no	—
55	C₂×Q₈	—	4	384	no	—
56	C₂×Q₈	C₂²×Q₈	1	192	yes	complemented
57	C₂²×C₄	—	2	288	no	—
58	C₂²×C₄	—	2	108	no	—
59	C₂²×C₄	—	2	108	no	—
60	C₂²×C₄	—	2	216	no	—
61	C₂²×C₄	—	4	96	no	—
62	C₂²×C₄	C₂²×Q₈	1	216	yes	—
63	C₂²×C₄	C₂⁵	1	24	yes	—
64	C₂²×C₄	C₂⁵	1	36	yes	—
65	C₄⋊C₄	—	4	144	no	—
66	C₄²	—	4	48	no	—
67	C₂³	#543	1	48	yes	—
68	C₂³	Q₈²	1	48	yes	—
69	C₂³	H(3,𝔽₄)	1	16	yes	—
70	C₂³	C₂³×Q₈	1	18	yes	—
71	C₂³	C₂×H(5,𝔽₂)	1	6	yes	—
72	C₂³	C₂×H(5,𝔽₂)	1	18	yes	—
73	C₂³	C₂⁶	1	1	yes	characteristic
74	Q₈	—	4	192	no	—
75	Q₈	—	4	128	no	—
76	C₂×C₄	—	2	36	no	—
77	C₂×C₄	—	2	36	no	—
78	C₂×C₄	—	2	432	no	—
79	C₂×C₄	—	2	72	no	—
80	C₂×C₄	—	4	144	no	—
81	C₂×C₄	C₂³×Q₈	1	72	yes	—
82	C₂²	#2811	1	64	yes	direct product factor
83	C₂²	C₂×#543	1	36	yes	—
84	C₂²	C₂×Q₈²	1	36	yes	—
85	C₂²	C₂×H(3,𝔽₄)	1	12	yes	—
86	C₂²	C₂⁴×Q₈	1	3	yes	—
87	C₂²	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²	C₂²×H(5,𝔽₂)	1	3	yes	—
89	C₄	—	2	72	no	—
90	C₄	—	4	24	no	—
91	C₂	C₂×#2811	1	24	yes	direct product factor
92	C₂	C₂²×#543	1	3	yes	—
93	C₂	C₂²×H(3,𝔽₄)	1	1	yes	characteristic
94	C₂	C₂²×Q₈²	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2811	C₂	direct (N × H)
8. #2811	C₂²	direct (N × H)
15. C₂³×Q₈	Q₈	semidirect
33. C₂²×Q₈	C₂×Q₈	semidirect
56. C₂×Q₈	C₂²×Q₈	semidirect
82. C₂²	#2811	direct (N × H)
91. C₂	C₂×#2811	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2811	2
C₂²×#395	2
C₂²×#391	2

Subgroups — C22×#2811 #7633619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2811 #7633619