Subgroups — C₂²×#556 #62706

Order 256.

3923 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#126	C₂	1	4	yes	maximal, complemented
4	C₂²×#124	C₂	1	2	yes	maximal
5	C₂²×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Q₈	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×#117	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×Q₈	C₂²	1	2	yes	—
12	C₂³×D₄	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄	C₂²	1	2	yes	—
14	#556	C₂²	1	256	yes	direct product factor
15	C₂×C₄⋊Q₈	C₂²	1	24	yes	complemented
16	C₂×#126	C₂²	1	96	yes	complemented
17	C₂×#124	C₂²	1	48	yes	—
18	C₂×C₂²⋊Q₈	C₂²	1	48	yes	complemented
19	C₂×C₄×Q₈	C₂²	1	48	yes	complemented
20	C₂×C₄×Q₈	C₂²	1	24	yes	complemented
21	C₂×C₄×D₄	C₂²	1	24	yes	—
22	C₂×#117	C₂²	1	48	yes	complemented
23	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
30	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄	C₂²	1	4	yes	—
32	C₂²×C₄²	C₂²	1	4	yes	—
33	C₂²×C₄²	C₂²	1	1	yes	characteristic
34	C₂²×C₄²	C₂²	1	2	yes	—
35	C₂⁵	C₂³	1	2	yes	—
36	C₂²×Q₈	—	2	8	no	—
37	C₂²×Q₈	—	2	16	no	—
38	C₂²×Q₈	C₂³	1	12	yes	—
39	C₂²×Q₈	C₂³	1	24	yes	—
40	C₂²×D₄	—	2	8	no	—
41	C₂²×D₄	C₂³	1	12	yes	—
42	C₂³×C₄	—	2	16	no	—
43	C₂³×C₄	C₂³	1	24	yes	—
44	C₂³×C₄	C₂³	1	1	yes	characteristic
45	C₂³×C₄	C₂³	1	4	yes	—
46	C₂³×C₄	C₂³	1	2	yes	—
47	C₂³×C₄	C₂³	1	2	yes	—
48	C₂³×C₄	C₂³	1	2	yes	—
49	C₂³×C₄	C₂³	1	1	yes	characteristic
50	C₂³×C₄	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	C₂³	1	64	yes	complemented
52	#126	C₂³	1	256	yes	complemented
53	#124	C₂³	1	128	yes	—
54	C₂²⋊Q₈	C₂³	1	128	yes	complemented
55	C₄×Q₈	C₂³	1	128	yes	complemented
56	C₄×Q₈	C₂³	1	64	yes	complemented
57	C₄×D₄	C₂³	1	64	yes	—
58	#117	C₂³	1	128	yes	complemented
59	C₂×C₄⋊C₄	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄	C₂³	1	24	yes	—
61	C₂×C₄⋊C₄	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄	C₂³	1	48	yes	—
63	C₂×C₄⋊C₄	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄	C₂³	1	24	yes	—
65	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
66	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
67	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
68	C₂×C₄²	C₂³	1	48	yes	—
69	C₂×C₄²	C₂³	1	12	yes	—
70	C₂×C₄²	C₂³	1	24	yes	—
71	C₂⁴	—	2	8	no	—
72	C₂⁴	—	2	8	no	—
73	C₂⁴	—	2	8	no	—
74	C₂⁴	C₂⁴	1	12	yes	—
75	C₂⁴	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈	—	2	48	no	—
77	C₂×Q₈	—	2	96	no	—
78	C₂×Q₈	C₂⁴	1	16	yes	—
79	C₂×Q₈	C₂⁴	1	32	yes	—
80	C₂×D₄	—	2	48	no	—
81	C₂×D₄	C₂⁴	1	16	yes	—
82	C₂²×C₄	—	2	8	no	—
83	C₂²×C₄	—	2	8	no	—
84	C₂²×C₄	—	2	16	no	—
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	8	no	—
87	C₂²×C₄	—	2	96	no	—
88	C₂²×C₄	#41	1	8	yes	—
89	C₂²×C₄	#41	1	8	yes	—
90	C₂²×C₄	C₂⁴	1	12	yes	—
91	C₂²×C₄	C₂⁴	1	12	yes	—
92	C₂²×C₄	C₂⁴	1	6	yes	—
93	C₂²×C₄	C₂⁴	1	24	yes	—
94	C₂²×C₄	C₂⁴	1	12	yes	—
95	C₂²×C₄	C₂⁴	1	32	yes	—
96	C₂²×C₄	C₂⁴	1	6	yes	—
97	C₂²×C₄	C₂⁴	1	6	yes	—
98	C₄⋊C₄	C₂⁴	1	16	yes	—
99	C₄⋊C₄	C₂⁴	1	32	yes	—
100	C₄⋊C₄	C₂⁴	1	16	yes	—
101	C₄⋊C₄	C₂⁴	1	64	yes	—
102	C₄⋊C₄	C₂⁴	1	32	yes	—
103	C₄⋊C₄	C₂⁴	1	32	yes	—
104	C₂²⋊C₄	C₂⁴	1	64	yes	—
105	C₂²⋊C₄	C₂⁴	1	32	yes	—
106	C₂²⋊C₄	C₂⁴	1	64	yes	—
107	C₄²	C₂⁴	1	64	yes	—
108	C₄²	C₂⁴	1	16	yes	—
109	C₄²	C₂⁴	1	32	yes	—
110	C₂³	—	2	24	no	—
111	C₂³	—	2	24	no	—
112	C₂³	—	2	24	no	—
113	C₂³	—	4	32	no	—
114	C₂³	C₂×#41	1	4	yes	—
115	C₂³	C₂×#41	1	4	yes	—
116	C₂³	2₋¹⁺⁴	1	4	yes	—
117	C₂³	C₂⁵	1	8	yes	—
118	C₂³	C₂⁵	1	3	yes	—
119	Q₈	—	2	32	no	—
120	Q₈	—	2	64	no	—
121	D₄	—	2	32	no	—
122	C₂×C₄	—	2	24	no	—
123	C₂×C₄	—	2	24	no	—
124	C₂×C₄	—	2	48	no	—
125	C₂×C₄	—	2	24	no	—
126	C₂×C₄	—	2	12	no	—
127	C₂×C₄	—	2	64	no	—
128	C₂×C₄	C₂×#41	1	24	yes	—
129	C₂×C₄	C₂×#41	1	24	yes	—
130	C₂×C₄	C₂⁵	1	8	yes	—
131	C₂×C₄	C₂⁵	1	8	yes	—
132	C₂×C₄	C₂⁵	1	4	yes	—
133	C₂×C₄	C₂⁵	1	16	yes	—
134	C₂×C₄	C₂⁵	1	8	yes	—
135	C₂×C₄	C₂⁵	1	4	yes	—
136	C₂×C₄	C₂⁵	1	4	yes	—
137	C₂²	—	2	8	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	2	8	no	—
140	C₂²	—	4	48	no	—
141	C₂²	#556	1	16	yes	direct product factor
142	C₂²	C₂²×#41	1	6	yes	—
143	C₂²	C₂²×#41	1	6	yes	—
144	C₂²	C₂×2₋¹⁺⁴	1	6	yes	—
145	C₂²	C₂⁶	1	1	yes	characteristic
146	C₄	—	2	8	no	—
147	C₄	—	2	8	no	—
148	C₄	—	2	16	no	—
149	C₄	—	2	8	no	—
150	C₄	—	2	4	no	—
151	C₄	C₂²×#41	1	8	yes	—
152	C₄	C₂²×#41	1	8	yes	—
153	C₂	—	4	8	no	—
154	C₂	C₂×#556	1	12	yes	direct product factor
155	C₂	C₂³×#41	1	1	yes	characteristic
156	C₂	C₂³×#41	1	1	yes	characteristic
157	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#556	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈	C₂	semidirect
3. C₂²×#126	C₂	semidirect
5. C₂²×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₄×Q₈	C₂	semidirect
7. C₂²×C₄×Q₈	C₂	semidirect
9. C₂²×#117	C₂	semidirect
14. #556	C₂²	direct (N × H)
15. C₂×C₄⋊Q₈	C₂²	semidirect
16. C₂×#126	C₂²	semidirect
18. C₂×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₄×Q₈	C₂²	semidirect
20. C₂×C₄×Q₈	C₂²	semidirect
22. C₂×#117	C₂²	semidirect
51. C₄⋊Q₈	C₂³	semidirect
52. #126	C₂³	semidirect
54. C₂²⋊Q₈	C₂³	semidirect
55. C₄×Q₈	C₂³	semidirect
56. C₄×Q₈	C₂³	semidirect
58. #117	C₂³	semidirect
141. C₂²	#556	direct (N × H)
154. C₂	C₂×#556	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×#556	2
C₂²×C₄⋊Q₈	2
C₂²×#126	2
C₂²×#124	2
C₂²×C₂²⋊Q₈	2
C₂²×C₄×Q₈	2
C₂²×C₄×D₄	2
C₂²×#117	2

Subgroups — C22×#556 #62706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#556 #62706