Subgroups — C₂²×#399 #34540

Order 256.

2919 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂³×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₂×#399	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄	C₂²	1	3	yes	—
7	C₂⁴×C₄	C₂²	1	3	yes	—
8	C₂²×C₄⋊C₄	—	2	24	no	—
9	C₂²×C₄⋊C₄	C₂²	1	36	yes	complemented
10	C₂²×C₂²⋊C₄	—	2	24	no	—
11	C₂²×C₂²⋊C₄	C₂²	1	36	yes	—
12	#399	C₂²	1	64	yes	direct product factor
13	C₂×#95	C₂²	1	12	yes	complemented
14	C₂⁵	—	2	12	no	—
15	C₂⁵	—	2	12	no	—
16	C₂⁵	Q₈	1	8	yes	—
17	C₂⁵	C₂³	1	6	yes	—
18	C₂⁵	C₂³	1	1	yes	characteristic
19	C₂³×C₄	—	2	24	no	—
20	C₂³×C₄	—	2	12	no	—
21	C₂³×C₄	—	2	24	no	—
22	C₂³×C₄	D₄	1	24	yes	—
23	C₂³×C₄	C₂³	1	18	yes	—
24	C₂³×C₄	C₂³	1	18	yes	—
25	C₂×C₄⋊C₄	—	2	144	no	—
26	C₂×C₄⋊C₄	C₂³	1	48	yes	complemented
27	C₂×C₂²⋊C₄	—	2	144	no	—
28	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
29	#95	C₂³	1	16	yes	complemented
30	C₂⁴	—	2	96	no	—
31	C₂⁴	—	2	36	no	—
32	C₂⁴	—	2	36	no	—
33	C₂⁴	—	4	48	no	—
34	C₂⁴	—	4	16	no	—
35	C₂⁴	C₂×D₄	1	12	yes	—
36	C₂⁴	C₂×Q₈	1	24	yes	—
37	C₂⁴	C₂×Q₈	1	4	yes	—
38	C₂⁴	#41	1	12	yes	—
39	C₂⁴	C₂⁴	1	4	yes	—
40	C₂⁴	C₂⁴	1	3	yes	—
41	C₂²×C₄	—	2	96	no	—
42	C₂²×C₄	—	2	72	no	—
43	C₂²×C₄	—	2	36	no	—
44	C₂²×C₄	—	2	72	no	—
45	C₂²×C₄	—	4	48	no	—
46	C₂²×C₄	C₂×D₄	1	72	yes	—
47	C₂²×C₄	C₂⁴	1	12	yes	—
48	C₂²×C₄	C₂⁴	1	12	yes	—
49	C₄⋊C₄	—	2	96	no	—
50	C₂²⋊C₄	—	2	96	no	—
51	C₂³	—	2	144	no	—
52	C₂³	—	2	12	no	—
53	C₂³	—	2	12	no	—
54	C₂³	—	4	72	no	—
55	C₂³	—	4	24	no	—
56	C₂³	—	4	128	no	—
57	C₂³	C₂²⋊Q₈	1	48	yes	—
58	C₂³	#123	1	48	yes	—
59	C₂³	#127	1	16	yes	—
60	C₂³	C₂²×D₄	1	18	yes	—
61	C₂³	C₂²×Q₈	1	8	yes	—
62	C₂³	C₂²×Q₈	1	6	yes	—
63	C₂³	C₂×#41	1	18	yes	—
64	C₂³	C₂⁵	1	1	yes	characteristic
65	C₂×C₄	—	2	144	no	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	—	2	24	no	—
69	C₂×C₄	—	4	72	no	—
70	C₂×C₄	C₂²×D₄	1	24	yes	—
71	C₂²	—	2	24	no	—
72	C₂²	—	4	96	no	—
73	C₂²	—	4	12	no	—
74	C₂²	—	4	4	no	—
75	C₂²	#399	1	64	yes	direct product factor
76	C₂²	C₂×C₂²⋊Q₈	1	36	yes	—
77	C₂²	C₂×#123	1	36	yes	—
78	C₂²	C₂×#127	1	12	yes	—
79	C₂²	C₂³×D₄	1	3	yes	—
80	C₂²	C₂³×Q₈	1	1	yes	characteristic
81	C₂²	C₂²×#41	1	3	yes	—
82	C₄	—	2	24	no	—
83	C₄	—	4	12	no	—
84	C₂	—	4	8	no	—
85	C₂	C₂×#399	1	24	yes	direct product factor
86	C₂	C₂²×C₂²⋊Q₈	1	3	yes	—
87	C₂	C₂²×#123	1	3	yes	—
88	C₂	C₂²×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁶	C₂²
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. C₂³×C₄⋊C₄	C₂	semidirect
3. C₂×#399	C₂	direct (N × H)
4. C₂²×#95	C₂	semidirect
9. C₂²×C₄⋊C₄	C₂²	semidirect
12. #399	C₂²	direct (N × H)
13. C₂×#95	C₂²	semidirect
26. C₂×C₄⋊C₄	C₂³	semidirect
29. #95	C₂³	semidirect
75. C₂²	#399	direct (N × H)
85. C₂	C₂×#399	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×C₂²⋊C₄	2
C₂×#399	2
C₂²×#95	2

Subgroups — C22×#399 #34540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#399 #34540