Subgroups — C₂×C₈×2₋¹⁺⁴ #10574538

Order 512.

8274 conjugacy classes of subgroups.

The subgroup lattice has 175 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₈×2₋¹⁺⁴	C₂	1	32	yes	maximal, direct product factor
3	C₂×#2988	C₂	1	10	yes	maximal, complemented
4	C₂×#2982	C₂	1	5	yes	maximal, complemented
5	C₂×C₈×#41	C₂	1	10	yes	maximal, complemented
6	C₂²×C₈×Q₈	C₂	1	5	yes	maximal, complemented
7	C₂²×2₋¹⁺⁴	C₄	1	1	yes	characteristic
8	C₄×2₋¹⁺⁴	C₄	1	16	yes	—
9	C₄×2₋¹⁺⁴	C₂²	1	16	yes	—
10	C₂×#520	C₄	1	10	yes	—
11	C₂×#519	C₄	1	5	yes	—
12	C₂×C₄×#41	C₂²	1	10	yes	—
13	C₂²×C₄×Q₈	C₂²	1	5	yes	—
14	#2988	C₂²	1	160	yes	complemented
15	#2982	C₂²	1	80	yes	complemented
16	C₈×#41	C₂²	1	160	yes	complemented
17	C₂×C₈×Q₈	C₂²	1	10	yes	—
18	C₂×C₈×Q₈	C₂²	1	20	yes	—
19	C₂×C₈×Q₈	C₂²	1	10	yes	—
20	C₂×C₈×Q₈	C₂²	1	80	yes	complemented
21	C₂×C₈×D₄	C₂²	1	10	yes	—
22	C₂×C₈×D₄	C₂²	1	30	yes	—
23	C₂×#431	C₂²	1	30	yes	—
24	C₂²×C₄⋊C₈	C₂²	1	15	yes	—
25	C₂²×C₄×C₈	C₂²	1	15	yes	—
26	C₂×2₋¹⁺⁴	C₈	1	32	yes	direct product factor
27	C₂×2₋¹⁺⁴	C₂×C₄	1	16	yes	—
28	C₂²×#41	C₂×C₄	1	10	yes	—
29	C₂³×Q₈	C₂×C₄	1	5	yes	—
30	C₂³×C₈	C₂³	1	5	yes	—
31	C₂³×C₈	C₂³	1	15	yes	—
32	#520	C₂×C₄	1	160	yes	—
33	#519	C₂×C₄	1	80	yes	—
34	C₄×#41	C₂×C₄	1	80	yes	—
35	C₄×#41	C₂³	1	80	yes	—
36	C₂×C₄×Q₈	C₂×C₄	1	20	yes	—
37	C₂×C₄×Q₈	C₂×C₄	1	40	yes	—
38	C₂×C₄×Q₈	C₂×C₄	1	10	yes	—
39	C₂×C₄×Q₈	C₂³	1	10	yes	—
40	C₂×C₄×Q₈	C₂³	1	40	yes	—
41	C₂×C₄×D₄	C₂×C₄	1	30	yes	—
42	C₂×C₄×D₄	C₂³	1	10	yes	—
43	C₂×#117	C₂×C₄	1	30	yes	—
44	C₂²×C₄⋊C₄	C₂×C₄	1	15	yes	—
45	C₂²×C₄²	C₂³	1	15	yes	—
46	C₈×Q₈	C₂³	1	80	yes	—
47	C₈×Q₈	C₂³	1	160	yes	—
48	C₈×Q₈	C₂³	1	80	yes	—
49	C₈×D₄	C₂³	1	80	yes	—
50	C₈×D₄	C₂³	1	240	yes	—
51	#431	C₂³	1	240	yes	—
52	C₂×C₄⋊C₈	C₂³	1	30	yes	—
53	C₂×C₄⋊C₈	C₂³	1	120	yes	—
54	C₂×C₄⋊C₈	C₂³	1	10	yes	—
55	C₂×C₂²⋊C₈	C₂³	1	20	yes	—
56	C₂×C₄×C₈	C₂³	1	30	yes	—
57	C₂×C₄×C₈	C₂³	1	120	yes	—
58	C₂×C₄×C₈	C₂³	1	10	yes	—
59	2₋¹⁺⁴	C₂×C₈	1	256	yes	direct product factor
60	C₂×#41	C₂×C₈	1	160	yes	complemented
61	C₂×#41	C₂²×C₄	1	80	yes	—
62	C₂²×Q₈	C₂×C₈	1	80	yes	complemented
63	C₂²×Q₈	C₂²×C₄	1	10	yes	—
64	C₂²×Q₈	C₂²×C₄	1	40	yes	—
65	C₂²×D₄	C₂²×C₄	1	10	yes	—
66	C₂³×C₄	C₂²×C₄	1	15	yes	—
67	C₂³×C₄	C₂²×C₄	1	15	yes	—
68	C₂³×C₄	C₂⁴	1	5	yes	—
69	C₂²×C₈	—	2	20	no	—
70	C₂²×C₈	—	4	30	no	—
71	C₂²×C₈	C₂⁴	1	10	yes	—
72	C₂²×C₈	C₂⁴	1	5	yes	—
73	C₂²×C₈	C₂⁴	1	20	yes	—
74	C₂²×C₈	C₂⁴	1	60	yes	—
75	C₂²×C₈	C₂⁴	1	1	yes	characteristic
76	C₄×Q₈	C₂²×C₄	1	40	yes	—
77	C₄×Q₈	C₂²×C₄	1	160	yes	—
78	C₄×Q₈	C₂²×C₄	1	80	yes	—
79	C₄×Q₈	C₂⁴	1	40	yes	—
80	C₄×D₄	C₂²×C₄	1	240	yes	—
81	C₄×D₄	C₂²×C₄	1	40	yes	—
82	C₄×D₄	C₂⁴	1	40	yes	—
83	#117	C₂²×C₄	1	240	yes	—
84	C₂×C₄⋊C₄	C₂²×C₄	1	30	yes	—
85	C₂×C₄⋊C₄	C₂²×C₄	1	120	yes	—
86	C₂×C₄⋊C₄	C₂²×C₄	1	10	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₄	1	20	yes	—
88	C₂×C₄²	C₂²×C₄	1	30	yes	—
89	C₂×C₄²	C₂²×C₄	1	60	yes	—
90	C₂×C₄²	C₂⁴	1	10	yes	—
91	C₂×C₄²	C₂⁴	1	60	yes	—
92	C₄⋊C₈	C₂⁴	1	120	yes	—
93	C₄⋊C₈	C₂⁴	1	40	yes	—
94	C₂²⋊C₈	C₂⁴	1	80	yes	—
95	C₄×C₈	C₂⁴	1	40	yes	—
96	C₄×C₈	C₂⁴	1	120	yes	—
97	C₂⁴	C₂³×C₄	1	5	yes	—
98	#41	C₂²×C₈	1	640	yes	complemented
99	C₂×Q₈	C₂²×C₈	1	80	yes	—
100	C₂×Q₈	C₂²×C₈	1	320	yes	complemented
101	C₂×Q₈	C₂³×C₄	1	40	yes	—
102	C₂×D₄	C₂²×C₈	1	80	yes	—
103	C₂×D₄	C₂³×C₄	1	40	yes	—
104	C₂²×C₄	—	2	10	no	—
105	C₂²×C₄	—	2	10	no	—
106	C₂²×C₄	—	4	30	no	—
107	C₂²×C₄	C₂²×C₈	1	120	yes	—
108	C₂²×C₄	C₂³×C₄	1	5	yes	—
109	C₂²×C₄	C₂³×C₄	1	10	yes	—
110	C₂²×C₄	C₂³×C₄	1	10	yes	—
111	C₂²×C₄	C₂³×C₄	1	60	yes	—
112	C₂²×C₄	C₂³×C₄	1	10	yes	—
113	C₂²×C₄	C₂³×C₄	1	60	yes	—
114	C₂²×C₄	C₂⁵	1	10	yes	—
115	C₂²×C₄	C₂⁵	1	1	yes	characteristic
116	C₂×C₈	—	2	10	no	—
117	C₂×C₈	—	2	20	no	—
118	C₂×C₈	—	2	40	no	—
119	C₂×C₈	—	4	60	no	—
120	C₂×C₈	2₋¹⁺⁴	1	4	yes	direct product factor
121	C₂×C₈	C₂⁵	1	10	yes	—
122	C₂×C₈	C₂⁵	1	20	yes	—
123	C₂×C₈	C₂⁵	1	2	yes	—
124	C₄⋊C₄	C₂³×C₄	1	120	yes	—
125	C₄⋊C₄	C₂³×C₄	1	40	yes	—
126	C₂²⋊C₄	C₂³×C₄	1	80	yes	—
127	C₄²	C₂³×C₄	1	20	yes	—
128	C₄²	C₂³×C₄	1	120	yes	—
129	C₄²	C₂⁵	1	20	yes	—
130	C₂³	—	2	10	no	—
131	C₂³	—	2	10	no	—
132	C₂³	C₂³×C₈	1	20	yes	—
133	C₂³	C₂⁴×C₄	1	10	yes	—
134	C₂³	C₂⁴×C₄	1	1	yes	characteristic
135	Q₈	C₂³×C₈	1	160	yes	—
136	D₄	C₂³×C₈	1	160	yes	—
137	C₂×C₄	—	2	10	no	—
138	C₂×C₄	—	2	20	no	—
139	C₂×C₄	—	2	10	no	—
140	C₂×C₄	—	2	10	no	—
141	C₂×C₄	—	2	20	no	—
142	C₂×C₄	—	4	60	no	—
143	C₂×C₄	C₂³×C₈	1	40	yes	—
144	C₂×C₄	C₂³×C₈	1	240	yes	—
145	C₂×C₄	C₂⁴×C₄	1	10	yes	—
146	C₂×C₄	C₂⁴×C₄	1	20	yes	—
147	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
148	C₂×C₄	C₂⁴×C₄	1	20	yes	—
149	C₂×C₄	C₂×2₋¹⁺⁴	1	2	yes	—
150	C₂×C₄	#585	1	2	yes	—
151	C₂×C₄	C₂⁶	1	1	yes	characteristic
152	C₈	—	2	10	no	—
153	C₈	—	2	20	no	—
154	C₈	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
155	C₂²	—	2	10	no	—
156	C₂²	—	2	40	no	—
157	C₂²	—	2	10	no	—
158	C₂²	C₄×2₋¹⁺⁴	1	2	yes	—
159	C₂²	C₂⁴×C₈	1	20	yes	—
160	C₂²	C₂⁴×C₈	1	2	yes	—
161	C₂²	#3573	1	2	yes	—
162	C₂²	C₂⁵×C₄	1	1	yes	characteristic
163	C₄	—	2	20	no	—
164	C₄	—	2	10	no	—
165	C₄	C₄×2₋¹⁺⁴	1	2	yes	—
166	C₄	C₂⁴×C₈	1	40	yes	—
167	C₄	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
168	C₄	C₂×#585	1	1	yes	characteristic
169	C₂	—	2	20	no	—
170	C₂	C₈×2₋¹⁺⁴	1	4	yes	direct product factor
171	C₂	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic
172	C₂	C₂⁵×C₈	1	1	yes	characteristic
173	C₂	C₂×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂	C₂⁵×C₈
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₈×2₋¹⁺⁴	C₂	direct (N × H)
3. C₂×#2988	C₂	semidirect
4. C₂×#2982	C₂	semidirect
5. C₂×C₈×#41	C₂	semidirect
6. C₂²×C₈×Q₈	C₂	semidirect
14. #2988	C₂²	semidirect
15. #2982	C₂²	semidirect
16. C₈×#41	C₂²	semidirect
20. C₂×C₈×Q₈	C₂²	semidirect
26. C₂×2₋¹⁺⁴	C₈	direct (N × H)
59. 2₋¹⁺⁴	C₂×C₈	direct (N × H)
60. C₂×#41	C₂×C₈	semidirect
62. C₂²×Q₈	C₂×C₈	semidirect
98. #41	C₂²×C₈	semidirect
100. C₂×Q₈	C₂²×C₈	semidirect
120. C₂×C₈	2₋¹⁺⁴	direct (N × H)
154. C₈	C₂×2₋¹⁺⁴	direct (N × H)
170. C₂	C₈×2₋¹⁺⁴	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×C₄×2₋¹⁺⁴	2
C₈×2₋¹⁺⁴	2
C₂×#2988	2
C₂×#2982	2
C₂×C₈×#41	2
C₂²×C₈×Q₈	2

Subgroups — C2×C8×2−1+4 #10574538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×2₋¹⁺⁴ #10574538