Subgroups — C₂²×C₈×D₄ #60130

Order 256.

1847 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₄	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄	C₄	1	1	yes	characteristic
8	C₂⁴×C₄	C₂²	1	2	yes	—
9	C₂³×C₈	—	2	16	no	—
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	2	yes	—
12	C₂³×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₈	C₂²	1	24	yes	complemented
14	C₂×C₄×D₄	C₄	1	12	yes	—
15	C₂×C₄×D₄	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄	C₄	1	2	yes	—
18	C₂²×C₄²	C₂²	1	1	yes	characteristic
19	C₈×D₄	C₂²	1	64	yes	direct product factor
20	C₂×C₄⋊C₈	C₂²	1	12	yes	complemented
21	C₂×C₂²⋊C₈	C₂²	1	24	yes	complemented
22	C₂×C₄×C₈	C₂²	1	12	yes	complemented
23	C₂⁵	C₂×C₄	1	2	yes	—
24	C₂²×D₄	C₈	1	16	yes	direct product factor
25	C₂²×D₄	C₂×C₄	1	12	yes	—
26	C₂³×C₄	—	2	8	no	—
27	C₂³×C₄	—	2	8	no	—
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄	C₂×C₄	1	2	yes	—
30	C₂³×C₄	C₂×C₄	1	12	yes	—
31	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄	C₂³	1	12	yes	—
33	C₂³×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₈	—	2	8	no	—
35	C₂²×C₈	—	2	4	no	—
36	C₂²×C₈	—	2	96	no	—
37	C₂²×C₈	D₄	1	8	yes	direct product factor
38	C₂²×C₈	C₂³	1	12	yes	—
39	C₂²×C₈	C₂³	1	6	yes	—
40	C₂²×C₈	C₂³	1	6	yes	—
41	C₂²×C₈	C₂³	1	32	yes	complemented
42	C₄×D₄	C₂×C₄	1	48	yes	—
43	C₄×D₄	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄	C₂×C₄	1	12	yes	—
45	C₂×C₂²⋊C₄	C₂×C₄	1	24	yes	—
46	C₂×C₄²	C₂×C₄	1	6	yes	—
47	C₂×C₄²	C₂³	1	6	yes	—
48	C₄⋊C₈	C₂³	1	16	yes	complemented
49	C₂²⋊C₈	C₂³	1	32	yes	complemented
50	C₄×C₈	C₂³	1	16	yes	complemented
51	C₂⁴	—	2	8	no	—
52	C₂⁴	—	2	8	no	—
53	C₂⁴	C₂×C₈	1	16	yes	complemented
54	C₂⁴	C₂²×C₄	1	12	yes	—
55	C₂⁴	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄	C₂×C₈	1	96	yes	direct product factor
57	C₂×D₄	C₂²×C₄	1	16	yes	—
58	C₂²×C₄	—	2	8	no	—
59	C₂²×C₄	—	2	4	no	—
60	C₂²×C₄	—	2	48	no	—
61	C₂²×C₄	—	2	24	no	—
62	C₂²×C₄	—	2	24	no	—
63	C₂²×C₄	C₂×C₈	1	8	yes	complemented
64	C₂²×C₄	C₂²×C₄	1	6	yes	—
65	C₂²×C₄	C₂²×C₄	1	12	yes	—
66	C₂²×C₄	C₂²×C₄	1	24	yes	—
67	C₂²×C₄	C₂²×C₄	1	3	yes	—
68	C₂²×C₄	C₂²×C₄	1	6	yes	—
69	C₂²×C₄	C₂×D₄	1	4	yes	—
70	C₂²×C₄	#41	1	4	yes	—
71	C₂²×C₄	C₂⁴	1	8	yes	—
72	C₂²×C₄	C₂⁴	1	3	yes	—
73	C₂×C₈	—	2	24	no	—
74	C₂×C₈	—	2	12	no	—
75	C₂×C₈	—	2	64	no	—
76	C₂×C₈	C₂×D₄	1	24	yes	direct product factor
77	C₂×C₈	C₂⁴	1	4	yes	—
78	C₂×C₈	C₂⁴	1	8	yes	—
79	C₂×C₈	C₂⁴	1	4	yes	—
80	C₄⋊C₄	C₂²×C₄	1	16	yes	—
81	C₂²⋊C₄	C₂²×C₄	1	32	yes	—
82	C₄²	C₂²×C₄	1	12	yes	—
83	C₄²	C₂⁴	1	4	yes	—
84	C₂³	—	2	24	no	—
85	C₂³	—	2	24	no	—
86	C₂³	—	2	64	no	—
87	C₂³	C₄×D₄	1	4	yes	—
88	C₂³	C₂²×C₈	1	4	yes	—
89	C₂³	C₂²×C₈	1	48	yes	complemented
90	C₂³	#131	1	4	yes	—
91	C₂³	C₂³×C₄	1	8	yes	—
92	C₂³	C₂³×C₄	1	3	yes	—
93	D₄	C₂²×C₈	1	64	yes	direct product factor
94	C₂×C₄	—	2	24	no	—
95	C₂×C₄	—	2	12	no	—
96	C₂×C₄	—	2	48	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	C₄×D₄	1	12	yes	—
100	C₂×C₄	C₂²×C₈	1	24	yes	complemented
101	C₂×C₄	C₂³×C₄	1	4	yes	—
102	C₂×C₄	C₂³×C₄	1	8	yes	—
103	C₂×C₄	C₂³×C₄	1	3	yes	—
104	C₂×C₄	C₂³×C₄	1	4	yes	—
105	C₂×C₄	C₂²×D₄	1	6	yes	—
106	C₂×C₄	C₂×#41	1	6	yes	—
107	C₂×C₄	C₂⁵	1	1	yes	characteristic
108	C₈	—	2	8	no	—
109	C₈	—	2	4	no	—
110	C₈	C₂²×D₄	1	8	yes	direct product factor
111	C₂²	—	2	8	no	—
112	C₂²	—	2	96	no	—
113	C₂²	—	2	8	no	—
114	C₂²	C₈×D₄	1	16	yes	direct product factor
115	C₂²	C₂×C₄×D₄	1	6	yes	—
116	C₂²	C₂³×C₈	1	6	yes	—
117	C₂²	C₂³×C₈	1	16	yes	complemented
118	C₂²	C₂×#131	1	6	yes	—
119	C₂²	C₂⁴×C₄	1	1	yes	characteristic
120	C₄	—	2	8	no	—
121	C₄	—	2	4	no	—
122	C₄	C₂×C₄×D₄	1	6	yes	—
123	C₄	C₂³×C₈	1	8	yes	complemented
124	C₄	C₂³×D₄	1	1	yes	characteristic
125	C₄	C₂²×#41	1	1	yes	characteristic
126	C₂	—	2	16	no	—
127	C₂	C₂×C₈×D₄	1	12	yes	direct product factor
128	C₂	C₂²×C₄×D₄	1	1	yes	characteristic
129	C₂	C₂⁴×C₈	1	1	yes	characteristic
130	C₂	C₂²×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂⁴	C₂²×C₄
prime radical	C₂³×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₈	C₂	semidirect
3. C₂×C₈×D₄	C₂	direct (N × H)
4. C₂²×C₄⋊C₈	C₂	semidirect
5. C₂²×C₂²⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈	C₂	semidirect
13. C₂³×C₈	C₂²	semidirect
19. C₈×D₄	C₂²	direct (N × H)
20. C₂×C₄⋊C₈	C₂²	semidirect
21. C₂×C₂²⋊C₈	C₂²	semidirect
22. C₂×C₄×C₈	C₂²	semidirect
24. C₂²×D₄	C₈	direct (N × H)
37. C₂²×C₈	D₄	direct (N × H)
41. C₂²×C₈	C₂³	semidirect
48. C₄⋊C₈	C₂³	semidirect
49. C₂²⋊C₈	C₂³	semidirect
50. C₄×C₈	C₂³	semidirect
53. C₂⁴	C₂×C₈	semidirect
56. C₂×D₄	C₂×C₈	direct (N × H)
63. C₂²×C₄	C₂×C₈	semidirect
76. C₂×C₈	C₂×D₄	direct (N × H)
89. C₂³	C₂²×C₈	semidirect
93. D₄	C₂²×C₈	direct (N × H)
100. C₂×C₄	C₂²×C₈	semidirect
110. C₈	C₂²×D₄	direct (N × H)
114. C₂²	C₈×D₄	direct (N × H)
117. C₂²	C₂³×C₈	semidirect
123. C₄	C₂³×C₈	semidirect
127. C₂	C₂×C₈×D₄	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×C₈	2
C₂²×C₄×D₄	2
C₂×C₈×D₄	2
C₂²×C₄⋊C₈	2
C₂²×C₂²⋊C₈	2
C₂²×C₄×C₈	2

Subgroups — C22×C8×D4 #60130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×D₄ #60130