Subgroups — C₂²×#384 #34010

Order 256.

2751 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂×#384	C₂	1	24	yes	maximal, direct product factor
4	C₂⁴×C₄	C₂²	1	4	yes	—
5	C₂⁴×C₄	C₂²	1	3	yes	—
6	C₂²×C₄⋊C₄	—	2	48	no	—
7	C₂²×C₄⋊C₄	C₂²	1	72	yes	—
8	C₂²×C₄²	C₄	1	16	yes	complemented
9	C₂²×C₄²	C₂²	1	12	yes	—
10	#384	C₂²	1	64	yes	direct product factor
11	C₂⁵	C₂³	1	1	yes	characteristic
12	C₂³×C₄	—	2	48	no	—
13	C₂³×C₄	C₂×C₄	1	24	yes	—
14	C₂³×C₄	D₄	1	24	yes	—
15	C₂³×C₄	Q₈	1	24	yes	—
16	C₂³×C₄	C₂³	1	24	yes	—
17	C₂³×C₄	C₂³	1	18	yes	—
18	C₂×C₄⋊C₄	—	2	288	no	—
19	C₂×C₄⋊C₄	C₂³	1	96	yes	—
20	C₂×C₄²	C₂×C₄	1	96	yes	complemented
21	C₂×C₄²	C₂³	1	16	yes	—
22	C₂⁴	C₂²×C₄	1	4	yes	—
23	C₂⁴	C₂×D₄	1	12	yes	—
24	C₂⁴	C₂×Q₈	1	12	yes	—
25	C₂⁴	C₂⁴	1	3	yes	—
26	C₂²×C₄	—	2	144	no	—
27	C₂²×C₄	—	4	64	no	—
28	C₂²×C₄	C₄⋊C₄	1	192	yes	complemented
29	C₂²×C₄	C₂²×C₄	1	72	yes	—
30	C₂²×C₄	C₂×D₄	1	72	yes	—
31	C₂²×C₄	C₂×Q₈	1	72	yes	—
32	C₂²×C₄	C₂⁴	1	16	yes	—
33	C₂²×C₄	C₂⁴	1	12	yes	—
34	C₄⋊C₄	—	2	192	no	—
35	C₄²	C₂²×C₄	1	64	yes	complemented
36	C₂³	C₂×C₄⋊C₄	1	48	yes	—
37	C₂³	#127	1	16	yes	—
38	C₂³	C₄⋊Q₈	1	48	yes	—
39	C₂³	C₂³×C₄	1	6	yes	—
40	C₂³	C₂²×D₄	1	18	yes	—
41	C₂³	C₂²×Q₈	1	18	yes	—
42	C₂³	C₂⁵	1	1	yes	characteristic
43	C₂×C₄	—	2	48	no	—
44	C₂×C₄	—	4	96	no	—
45	C₂×C₄	C₂×C₄⋊C₄	1	288	yes	complemented
46	C₂×C₄	C₂³×C₄	1	24	yes	—
47	C₂×C₄	C₂²×D₄	1	24	yes	—
48	C₂×C₄	C₂²×Q₈	1	24	yes	—
49	C₂²	#384	1	64	yes	direct product factor
50	C₂²	C₂²×C₄⋊C₄	1	36	yes	—
51	C₂²	C₂×#127	1	12	yes	—
52	C₂²	C₂×C₄⋊Q₈	1	36	yes	—
53	C₂²	C₂⁴×C₄	1	1	yes	characteristic
54	C₂²	C₂³×D₄	1	3	yes	—
55	C₂²	C₂³×Q₈	1	3	yes	—
56	C₄	—	4	16	no	—
57	C₄	C₂²×C₄⋊C₄	1	48	yes	complemented
58	C₂	C₂×#384	1	24	yes	direct product factor
59	C₂	C₂³×C₄⋊C₄	1	3	yes	—
60	C₂	C₂²×#127	1	1	yes	characteristic
61	C₂	C₂²×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂³
commutator	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ 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×#384	C₂	direct (N × H)
8. C₂²×C₄²	C₄	semidirect
10. #384	C₂²	direct (N × H)
20. C₂×C₄²	C₂×C₄	semidirect
28. C₂²×C₄	C₄⋊C₄	semidirect
35. C₄²	C₂²×C₄	semidirect
45. C₂×C₄	C₂×C₄⋊C₄	semidirect
49. C₂²	#384	direct (N × H)
57. C₄	C₂²×C₄⋊C₄	semidirect
58. C₂	C₂×#384	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄	2
C₂³×C₄²	2
C₂×#384	2

Subgroups — C22×#384 #34010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#384 #34010