Subgroups — C₂²×#395 #34534

Order 256.

3023 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₂×#395	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95	C₂	1	3	yes	maximal
5	C₂³×Q₈	—	2	24	no	—
6	C₂³×Q₈	C₂²	1	12	yes	—
7	C₂⁴×C₄	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄	C₂²	1	3	yes	—
9	C₂⁴×C₄	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄	—	2	24	no	—
11	C₂²×C₄⋊C₄	C₂²	1	36	yes	—
12	#395	C₂²	1	64	yes	direct product factor
13	C₂×#95	C₂²	1	36	yes	—
14	C₂⁵	C₂³	1	1	yes	characteristic
15	C₂²×Q₈	—	2	128	no	—
16	C₂²×Q₈	—	2	144	no	—
17	C₂²×Q₈	C₂³	1	16	yes	—
18	C₂³×C₄	—	2	12	no	—
19	C₂³×C₄	—	2	24	no	—
20	C₂³×C₄	—	2	12	no	—
21	C₂³×C₄	—	2	12	no	—
22	C₂³×C₄	D₄	1	24	yes	—
23	C₂³×C₄	Q₈	1	24	yes	complemented
24	C₂³×C₄	C₂³	1	6	yes	—
25	C₂³×C₄	C₂³	1	18	yes	—
26	C₂³×C₄	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄	—	2	144	no	—
28	C₂×C₄⋊C₄	C₂³	1	48	yes	—
29	#95	C₂³	1	48	yes	—
30	C₂⁴	C₂×D₄	1	12	yes	—
31	C₂⁴	C₂×Q₈	1	12	yes	—
32	C₂⁴	#41	1	4	yes	—
33	C₂⁴	C₂⁴	1	3	yes	—
34	C₂×Q₈	—	2	384	no	—
35	C₂×Q₈	—	2	96	no	—
36	C₂²×C₄	—	2	96	no	—
37	C₂²×C₄	—	2	96	no	—
38	C₂²×C₄	—	2	36	no	—
39	C₂²×C₄	—	2	72	no	—
40	C₂²×C₄	—	2	36	no	—
41	C₂²×C₄	—	2	36	no	—
42	C₂²×C₄	—	4	16	no	—
43	C₂²×C₄	C₂×D₄	1	72	yes	—
44	C₂²×C₄	C₂×Q₈	1	72	yes	complemented
45	C₂²×C₄	C₂⁴	1	4	yes	—
46	C₂²×C₄	C₂⁴	1	12	yes	—
47	C₂²×C₄	C₂⁴	1	12	yes	—
48	C₄⋊C₄	—	2	96	no	—
49	C₂³	#120	1	16	yes	—
50	C₂³	C₂²⋊Q₈	1	48	yes	—
51	C₂³	C₄⋊Q₈	1	48	yes	—
52	C₂³	C₂²×D₄	1	18	yes	—
53	C₂³	C₂²×Q₈	1	18	yes	—
54	C₂³	C₂×#41	1	6	yes	—
55	C₂³	C₂⁵	1	1	yes	characteristic
56	Q₈	—	2	128	no	—
57	C₂×C₄	—	2	144	no	—
58	C₂×C₄	—	2	144	no	—
59	C₂×C₄	—	2	12	no	—
60	C₂×C₄	—	2	24	no	—
61	C₂×C₄	—	2	12	no	—
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	4	24	no	—
64	C₂×C₄	C₂²×D₄	1	24	yes	—
65	C₂×C₄	C₂²×Q₈	1	24	yes	complemented
66	C₂²	#395	1	64	yes	direct product factor
67	C₂²	C₂×#120	1	12	yes	—
68	C₂²	C₂×C₂²⋊Q₈	1	36	yes	—
69	C₂²	C₂×C₄⋊Q₈	1	36	yes	—
70	C₂²	C₂³×D₄	1	3	yes	—
71	C₂²	C₂³×Q₈	1	3	yes	—
72	C₂²	C₂²×#41	1	1	yes	characteristic
73	C₄	—	2	24	no	—
74	C₄	—	2	24	no	—
75	C₄	—	4	4	no	—
76	C₂	C₂×#395	1	24	yes	direct product factor
77	C₂	C₂²×#120	1	1	yes	characteristic
78	C₂	C₂²×C₂²⋊Q₈	1	3	yes	—
79	C₂	C₂²×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵
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
3. C₂×#395	C₂	direct (N × H)
12. #395	C₂²	direct (N × H)
23. C₂³×C₄	Q₈	semidirect
44. C₂²×C₄	C₂×Q₈	semidirect
65. C₂×C₄	C₂²×Q₈	semidirect
66. C₂²	#395	direct (N × H)
76. C₂	C₂×#395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈	2
C₂³×C₄⋊C₄	2
C₂×#395	2
C₂²×#95	2

Subgroups — C22×#395 #34534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#395 #34534