Subgroups — C₄²×#41 #33993

Order 256.

2043 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41	C₂	1	3	yes	maximal
2	C₄²×Q₈	C₂	1	4	yes	maximal, complemented
3	C₄²×D₄	C₂	1	12	yes	maximal, complemented
4	C₄×#117	C₂	1	9	yes	maximal, complemented
5	C₂×C₄³	C₂	1	3	yes	maximal, complemented
6	C₂²×#41	C₂²	1	1	yes	characteristic
7	C₄×#41	C₄	1	48	yes	direct product factor
8	C₂×C₄×Q₈	C₂²	1	6	yes	—
9	C₂×C₄×D₄	C₂²	1	18	yes	—
10	C₂×#117	C₂²	1	9	yes	—
11	C₂²×C₄²	C₂²	1	12	yes	—
12	C₂²×C₄²	C₂²	1	9	yes	—
13	C₄×C₄⋊C₄	C₂²	1	36	yes	—
14	C₄×C₂²⋊C₄	C₂²	1	36	yes	—
15	#376	C₂²	1	6	yes	—
16	C₄³	C₂²	1	9	yes	—
17	C₄³	C₂²	1	12	yes	—
18	C₄³	C₂²	1	1	yes	characteristic
19	C₂×#41	C₂×C₄	1	24	yes	—
20	C₂²×Q₈	C₂³	1	1	yes	characteristic
21	C₂²×D₄	C₂³	1	3	yes	—
22	C₂³×C₄	C₂³	1	18	yes	—
23	C₂³×C₄	C₂³	1	3	yes	—
24	C₄×Q₈	C₂×C₄	1	48	yes	complemented
25	C₄×D₄	C₂×C₄	1	144	yes	complemented
26	#117	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄	C₂³	1	18	yes	—
29	C₂×C₄²	—	2	48	no	—
30	C₂×C₄²	C₂×C₄	1	72	yes	complemented
31	C₂×C₄²	C₂³	1	36	yes	—
32	C₂×C₄²	C₂³	1	9	yes	—
33	C₂×C₄²	C₂³	1	18	yes	—
34	C₂×C₄²	C₂³	1	3	yes	—
35	C₂×C₄²	C₂³	1	4	yes	—
36	#95	C₂³	1	24	yes	—
37	C₂⁴	C₂⁴	1	3	yes	—
38	#41	C₄²	1	64	yes	direct product factor
39	C₂×Q₈	C₂²×C₄	1	12	yes	—
40	C₂×D₄	C₂²×C₄	1	36	yes	—
41	C₂²×C₄	—	2	72	no	—
42	C₂²×C₄	C₂²×C₄	1	72	yes	—
43	C₂²×C₄	C₂²×C₄	1	36	yes	—
44	C₂²×C₄	C₂⁴	1	18	yes	—
45	C₂²×C₄	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄	C₂⁴	1	3	yes	—
47	C₂²×C₄	C₂⁴	1	6	yes	—
48	C₄⋊C₄	C₂²×C₄	1	72	yes	—
49	C₂²⋊C₄	C₂²×C₄	1	72	yes	—
50	C₄²	—	2	72	no	—
51	C₄²	C₂²×C₄	1	12	yes	—
52	C₄²	C₂²×C₄	1	36	yes	—
53	C₄²	C₂²×C₄	1	72	yes	—
54	C₄²	#41	1	16	yes	direct product factor
55	C₂³	—	2	12	no	—
56	C₂³	C₂³×C₄	1	18	yes	—
57	C₂³	C₂⁵	1	1	yes	characteristic
58	Q₈	C₂×C₄²	1	16	yes	complemented
59	D₄	C₂×C₄²	1	48	yes	complemented
60	C₂×C₄	—	2	36	no	—
61	C₂×C₄	—	2	144	no	—
62	C₂×C₄	C₂×C₄²	1	48	yes	complemented
63	C₂×C₄	C₂³×C₄	1	36	yes	—
64	C₂×C₄	C₂³×C₄	1	12	yes	—
65	C₂×C₄	C₂³×C₄	1	6	yes	—
66	C₂×C₄	C₂³×C₄	1	18	yes	—
67	C₂×C₄	C₂×#41	1	24	yes	—
68	C₂²	—	2	36	no	—
69	C₂²	C₂²×C₄²	1	12	yes	—
70	C₂²	C₂⁴×C₄	1	3	yes	—
71	C₂²	C₂²×#41	1	4	yes	—
72	C₄	—	2	36	no	—
73	C₄	C₂²×C₄²	1	4	yes	—
74	C₄	C₂²×C₄²	1	12	yes	—
75	C₄	C₄×#41	1	24	yes	direct product factor
76	C₂	—	2	12	no	—
77	C₂	C₂³×C₄²	1	1	yes	characteristic
78	C₂	C₂×C₄×#41	1	6	yes	—

	subgroup	G/H
center	C₄³	C₂²
commutator	C₂	C₂³×C₄²
Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈	C₂	semidirect
3. C₄²×D₄	C₂	semidirect
4. C₄×#117	C₂	semidirect
5. C₂×C₄³	C₂	semidirect
7. C₄×#41	C₄	direct (N × H)
24. C₄×Q₈	C₂×C₄	semidirect
25. C₄×D₄	C₂×C₄	semidirect
26. #117	C₂×C₄	semidirect
30. C₂×C₄²	C₂×C₄	semidirect
38. #41	C₄²	direct (N × H)
54. C₄²	#41	direct (N × H)
58. Q₈	C₂×C₄²	semidirect
59. D₄	C₂×C₄²	semidirect
62. C₂×C₄	C₂×C₄²	semidirect
75. C₄	C₄×#41	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×C₄×#41	2
C₄²×Q₈	2
C₄²×D₄	2
C₄×#117	2
C₂×C₄³	2

Subgroups — C42×#41 #33993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#41 #33993