Subgroups — C₂²×#532 #62666

Order 256.

6439 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	2	yes	maximal, complemented
2	C₂×#532	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#124	C₂	1	2	yes	maximal, complemented
6	C₂²×#121	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₄	C₂	1	4	yes	maximal, complemented
8	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41	—	2	32	no	—
10	C₂²×#41	C₂²	1	48	yes	complemented
11	C₂³×Q₈	C₂²	1	2	yes	—
12	C₂³×D₄	C₂²	1	4	yes	—
13	C₂³×D₄	C₂²	1	2	yes	—
14	C₂³×D₄	C₂²	1	4	yes	—
15	C₂⁴×C₄	C₂²	1	2	yes	—
16	C₂⁴×C₄	C₂²	1	4	yes	—
17	C₂⁴×C₄	C₂²	1	1	yes	characteristic
18	#532	C₂²	1	256	yes	direct product factor
19	C₂×C₄⋊Q₈	C₂²	1	24	yes	complemented
20	C₂×#127	C₂²	1	24	yes	complemented
21	C₂×#124	C₂²	1	48	yes	complemented
22	C₂×#121	C₂²	1	96	yes	complemented
23	C₂×C₄×D₄	C₂²	1	96	yes	complemented
24	C₂²×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄	C₂²	1	8	yes	—
26	C₂²×C₄²	C₂²	1	1	yes	characteristic
27	C₂²×C₄²	C₂²	1	1	yes	characteristic
28	C₂²×C₄²	C₂²	1	2	yes	—
29	C₂²×C₄²	C₂²	1	24	yes	complemented
30	C₂⁵	C₂³	1	4	yes	—
31	C₂⁵	C₂³	1	1	yes	characteristic
32	C₂×#41	—	2	384	no	—
33	C₂×#41	C₂³	1	128	yes	complemented
34	C₂²×Q₈	—	2	16	no	—
35	C₂²×Q₈	C₂³	1	24	yes	—
36	C₂²×D₄	—	2	32	no	—
37	C₂²×D₄	—	2	16	no	—
38	C₂²×D₄	—	2	32	no	—
39	C₂²×D₄	C₂³	1	48	yes	—
40	C₂²×D₄	C₂³	1	48	yes	—
41	C₂²×D₄	C₂³	1	24	yes	—
42	C₂³×C₄	—	2	32	no	—
43	C₂³×C₄	—	2	16	no	—
44	C₂³×C₄	D₄	1	16	yes	complemented
45	C₂³×C₄	C₂³	1	2	yes	—
46	C₂³×C₄	C₂³	1	4	yes	—
47	C₂³×C₄	C₂³	1	48	yes	—
48	C₂³×C₄	C₂³	1	24	yes	—
49	C₂³×C₄	C₂³	1	1	yes	characteristic
50	C₂³×C₄	C₂³	1	12	yes	—
51	C₂³×C₄	C₂³	1	2	yes	—
52	C₂³×C₄	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈	C₂³	1	64	yes	complemented
54	#127	C₂³	1	64	yes	complemented
55	#124	C₂³	1	128	yes	complemented
56	#121	C₂³	1	256	yes	complemented
57	C₄×D₄	C₂³	1	256	yes	complemented
58	C₂×C₄⋊C₄	C₂³	1	48	yes	—
59	C₂×C₂²⋊C₄	C₂³	1	96	yes	—
60	C₂×C₄²	C₂³	1	24	yes	—
61	C₂×C₄²	C₂³	1	12	yes	—
62	C₂×C₄²	C₂³	1	12	yes	—
63	C₂×C₄²	C₂³	1	64	yes	complemented
64	C₂⁴	—	2	8	no	—
65	C₂⁴	—	2	16	no	—
66	C₂⁴	—	2	32	no	—
67	C₂⁴	C₂×D₄	1	8	yes	—
68	C₂⁴	C₂⁴	1	24	yes	—
69	C₂⁴	C₂⁴	1	6	yes	—
70	C₂⁴	C₂⁴	1	1	yes	characteristic
71	#41	—	2	512	no	—
72	C₂×Q₈	—	2	96	no	—
73	C₂×Q₈	C₂⁴	1	32	yes	—
74	C₂×D₄	—	2	192	no	—
75	C₂×D₄	—	2	96	no	—
76	C₂×D₄	—	2	192	no	—
77	C₂×D₄	C₂⁴	1	64	yes	—
78	C₂×D₄	C₂⁴	1	64	yes	—
79	C₂×D₄	C₂⁴	1	32	yes	—
80	C₂²×C₄	—	2	16	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	—	2	192	no	—
83	C₂²×C₄	—	2	96	no	—
84	C₂²×C₄	C₂×D₄	1	8	yes	—
85	C₂²×C₄	C₂×D₄	1	8	yes	—
86	C₂²×C₄	C₂×D₄	1	96	yes	complemented
87	C₂²×C₄	#41	1	16	yes	complemented
88	C₂²×C₄	C₂⁴	1	12	yes	—
89	C₂²×C₄	C₂⁴	1	32	yes	—
90	C₂²×C₄	C₂⁴	1	24	yes	—
91	C₂²×C₄	C₂⁴	1	64	yes	—
92	C₂²×C₄	C₂⁴	1	6	yes	—
93	C₂²×C₄	C₂⁴	1	6	yes	—
94	C₂²×C₄	C₂⁴	1	16	yes	—
95	C₂²×C₄	C₂⁴	1	12	yes	—
96	C₄⋊C₄	C₂⁴	1	64	yes	—
97	C₂²⋊C₄	C₂⁴	1	128	yes	—
98	C₄²	C₂⁴	1	32	yes	—
99	C₄²	C₂⁴	1	16	yes	—
100	C₄²	C₂⁴	1	16	yes	—
101	C₂³	—	2	32	no	—
102	C₂³	—	2	24	no	—
103	C₂³	—	2	48	no	—
104	C₂³	—	2	96	no	—
105	C₂³	—	4	64	no	—
106	C₂³	C₂²×D₄	1	24	yes	—
107	C₂³	C₂²×D₄	1	4	yes	—
108	C₂³	C₂×#41	1	8	yes	—
109	C₂³	C₂⁵	1	16	yes	—
110	C₂³	C₂⁵	1	4	yes	—
111	C₂³	C₂⁵	1	3	yes	—
112	Q₈	—	2	64	no	—
113	D₄	—	2	128	no	—
114	D₄	—	2	64	no	—
115	D₄	—	2	128	no	—
116	C₂×C₄	—	2	48	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	2	128	no	—
119	C₂×C₄	—	2	64	no	—
120	C₂×C₄	C₂²×D₄	1	24	yes	—
121	C₂×C₄	C₂²×D₄	1	24	yes	—
122	C₂×C₄	C₂²×D₄	1	64	yes	complemented
123	C₂×C₄	C₂×#41	1	48	yes	complemented
124	C₂×C₄	C₂⁵	1	8	yes	—
125	C₂×C₄	C₂⁵	1	16	yes	—
126	C₂×C₄	C₂⁵	1	4	yes	—
127	C₂×C₄	C₂⁵	1	4	yes	—
128	C₂×C₄	C₂⁵	1	8	yes	—
129	C₂²	—	2	8	no	—
130	C₂²	—	2	48	no	—
131	C₂²	—	2	16	no	—
132	C₂²	—	2	32	no	—
133	C₂²	—	4	96	no	—
134	C₂²	#532	1	16	yes	direct product factor
135	C₂²	C₂³×D₄	1	8	yes	—
136	C₂²	C₂³×D₄	1	6	yes	—
137	C₂²	C₂²×#41	1	12	yes	—
138	C₂²	C₂⁶	1	1	yes	characteristic
139	C₄	—	2	16	no	—
140	C₄	—	2	8	no	—
141	C₄	C₂³×D₄	1	8	yes	—
142	C₄	C₂³×D₄	1	8	yes	—
143	C₄	C₂²×#41	1	16	yes	complemented
144	C₂	—	2	8	no	—
145	C₂	—	4	16	no	—
146	C₂	C₂×#532	1	12	yes	direct product factor
147	C₂	C₂⁴×D₄	1	1	yes	characteristic
148	C₂	C₂³×#41	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂⁴	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#41	C₂	semidirect
2. C₂×#532	C₂	direct (N × H)
3. C₂²×C₄⋊Q₈	C₂	semidirect
4. C₂²×#127	C₂	semidirect
5. C₂²×#124	C₂	semidirect
6. C₂²×#121	C₂	semidirect
7. C₂²×C₄×D₄	C₂	semidirect
8. C₂³×C₄²	C₂	semidirect
10. C₂²×#41	C₂²	semidirect
18. #532	C₂²	direct (N × H)
19. C₂×C₄⋊Q₈	C₂²	semidirect
20. C₂×#127	C₂²	semidirect
21. C₂×#124	C₂²	semidirect
22. C₂×#121	C₂²	semidirect
23. C₂×C₄×D₄	C₂²	semidirect
29. C₂²×C₄²	C₂²	semidirect
33. C₂×#41	C₂³	semidirect
44. C₂³×C₄	D₄	semidirect
53. C₄⋊Q₈	C₂³	semidirect
54. #127	C₂³	semidirect
55. #124	C₂³	semidirect
56. #121	C₂³	semidirect
57. C₄×D₄	C₂³	semidirect
63. C₂×C₄²	C₂³	semidirect
86. C₂²×C₄	C₂×D₄	semidirect
87. C₂²×C₄	#41	semidirect
122. C₂×C₄	C₂²×D₄	semidirect
123. C₂×C₄	C₂×#41	semidirect
134. C₂²	#532	direct (N × H)
143. C₄	C₂²×#41	semidirect
146. C₂	C₂×#532	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×#41	2
C₂×#532	2
C₂²×C₄⋊Q₈	2
C₂²×#127	2
C₂²×#124	2
C₂²×#121	2
C₂²×C₄×D₄	2
C₂³×C₄²	2

Subgroups — C22×#532 #62666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#532 #62666