Subgroups — C₂²×#3502 #10585962

Order 512.

33442 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3502	C₂	1	96	yes	maximal, direct product factor
2	C₂²×Q₈²	C₂	1	4	yes	maximal, complemented
3	C₂²×#557	C₂	1	8	yes	maximal, complemented
4	C₂²×#557	C₂	1	4	yes	maximal, complemented
5	C₂²×#544	C₂	1	6	yes	maximal
6	C₂²×#533	C₂	1	6	yes	maximal
7	C₂²×#519	C₂	1	2	yes	maximal
8	C₂³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×Q₈	C₂²	1	1	yes	characteristic
10	#3502	C₂²	1	1024	yes	direct product factor
11	C₂×Q₈²	C₂²	1	192	yes	complemented
12	C₂×#557	C₂²	1	384	yes	complemented
13	C₂×#557	C₂²	1	192	yes	complemented
14	C₂×#544	C₂²	1	288	yes	—
15	C₂×#533	C₂²	1	288	yes	—
16	C₂²×C₄⋊Q₈	C₂²	1	6	yes	—
17	C₂²×C₄⋊Q₈	C₂²	1	6	yes	—
18	C₂²×C₄⋊Q₈	C₂²	1	24	yes	—
19	C₂²×#125	C₂²	1	12	yes	—
20	C₂²×#125	C₂²	1	24	yes	—
21	C₂²×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₂²×C₂²⋊Q₈	C₂²	1	12	yes	—
23	C₂×#519	C₂²	1	96	yes	—
24	C₂²×C₄×Q₈	C₂²	1	48	yes	—
25	C₂²×C₄×Q₈	C₂²	1	4	yes	—
26	C₂²×C₄×Q₈	C₂²	1	4	yes	—
27	C₂²×C₄×Q₈	C₂²	1	8	yes	—
28	C₂²×C₄×Q₈	C₂²	1	8	yes	—
29	C₂²×C₄×Q₈	C₂²	1	12	yes	—
30	C₂²×C₄×Q₈	C₂²	1	12	yes	—
31	C₂²×#117	C₂²	1	12	yes	—
32	C₂³×C₄⋊C₄	C₂²	1	3	yes	—
33	C₂³×C₄²	C₂²	1	3	yes	—
34	C₂³×Q₈	Q₈	1	32	yes	complemented
35	C₂³×Q₈	C₂³	1	4	yes	—
36	C₂³×Q₈	C₂³	1	12	yes	—
37	C₂³×Q₈	C₂³	1	4	yes	—
38	C₂³×Q₈	C₂³	1	24	yes	—
39	C₂⁴×C₄	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄	C₂³	1	3	yes	—
41	C₂⁴×C₄	C₂³	1	3	yes	—
42	Q₈²	C₂³	1	1024	yes	complemented
43	#557	C₂³	1	2048	yes	complemented
44	#557	C₂³	1	1024	yes	complemented
45	#544	C₂³	1	1536	yes	—
46	#533	C₂³	1	1536	yes	—
47	C₂×C₄⋊Q₈	C₂³	1	144	yes	—
48	C₂×C₄⋊Q₈	C₂³	1	144	yes	—
49	C₂×C₄⋊Q₈	C₂³	1	576	yes	—
50	C₂×#125	C₂³	1	288	yes	—
51	C₂×#125	C₂³	1	576	yes	—
52	C₂×C₂²⋊Q₈	C₂³	1	96	yes	—
53	C₂×C₂²⋊Q₈	C₂³	1	288	yes	—
54	#519	C₂³	1	512	yes	—
55	C₂×C₄×Q₈	C₂³	1	96	yes	—
56	C₂×C₄×Q₈	C₂³	1	96	yes	—
57	C₂×C₄×Q₈	C₂³	1	256	yes	—
58	C₂×C₄×Q₈	C₂³	1	192	yes	—
59	C₂×C₄×Q₈	C₂³	1	192	yes	—
60	C₂×C₄×Q₈	C₂³	1	288	yes	—
61	C₂×C₄×Q₈	C₂³	1	288	yes	—
62	C₂×#117	C₂³	1	288	yes	—
63	C₂²×C₄⋊C₄	C₂³	1	12	yes	—
64	C₂²×C₄⋊C₄	C₂³	1	72	yes	—
65	C₂²×C₄⋊C₄	C₂³	1	12	yes	—
66	C₂²×C₄⋊C₄	C₂³	1	12	yes	—
67	C₂²×C₄⋊C₄	C₂³	1	8	yes	—
68	C₂²×C₄⋊C₄	C₂³	1	4	yes	—
69	C₂²×C₄⋊C₄	C₂³	1	12	yes	—
70	C₂²×C₄⋊C₄	C₂³	1	24	yes	—
71	C₂²×C₂²⋊C₄	C₂³	1	8	yes	—
72	C₂²×C₄²	C₂³	1	6	yes	—
73	C₂²×C₄²	C₂³	1	6	yes	—
74	C₂²×C₄²	C₂³	1	72	yes	—
75	C₂²×C₄²	C₂³	1	24	yes	—
76	C₂⁵	C₂⁴	1	1	yes	characteristic
77	C₂²×Q₈	—	2	32	no	—
78	C₂²×Q₈	—	4	48	no	—
79	C₂²×Q₈	C₂×Q₈	1	64	yes	—
80	C₂²×Q₈	C₂×Q₈	1	384	yes	complemented
81	C₂²×Q₈	C₂⁴	1	48	yes	—
82	C₂²×Q₈	C₂⁴	1	144	yes	—
83	C₂²×Q₈	C₂⁴	1	48	yes	—
84	C₂²×Q₈	C₂⁴	1	64	yes	—
85	C₂³×C₄	—	2	8	no	—
86	C₂³×C₄	—	4	12	no	—
87	C₂³×C₄	C₂×Q₈	1	48	yes	—
88	C₂³×C₄	C₂⁴	1	12	yes	—
89	C₂³×C₄	C₂⁴	1	8	yes	—
90	C₂³×C₄	C₂⁴	1	8	yes	—
91	C₂³×C₄	C₂⁴	1	6	yes	—
92	C₂³×C₄	C₂⁴	1	1	yes	characteristic
93	C₂³×C₄	C₂⁴	1	36	yes	—
94	C₂³×C₄	C₂⁴	1	36	yes	—
95	C₂³×C₄	C₂⁴	1	6	yes	—
96	C₂³×C₄	C₂⁴	1	1	yes	characteristic
97	C₄⋊Q₈	C₂⁴	1	384	yes	—
98	C₄⋊Q₈	C₂⁴	1	384	yes	—
99	C₄⋊Q₈	C₂⁴	1	1536	yes	—
100	#125	C₂⁴	1	768	yes	—
101	#125	C₂⁴	1	1536	yes	—
102	C₂²⋊Q₈	C₂⁴	1	256	yes	—
103	C₂²⋊Q₈	C₂⁴	1	768	yes	—
104	C₄×Q₈	C₂⁴	1	256	yes	—
105	C₄×Q₈	C₂⁴	1	256	yes	—
106	C₄×Q₈	C₂⁴	1	512	yes	—
107	C₄×Q₈	C₂⁴	1	512	yes	—
108	C₄×Q₈	C₂⁴	1	768	yes	—
109	C₄×Q₈	C₂⁴	1	768	yes	—
110	#117	C₂⁴	1	768	yes	—
111	C₂×C₄⋊C₄	C₂⁴	1	144	yes	—
112	C₂×C₄⋊C₄	C₂⁴	1	192	yes	—
113	C₂×C₄⋊C₄	C₂⁴	1	144	yes	—
114	C₂×C₄⋊C₄	C₂⁴	1	144	yes	—
115	C₂×C₄⋊C₄	C₂⁴	1	96	yes	—
116	C₂×C₄⋊C₄	C₂⁴	1	48	yes	—
117	C₂×C₄⋊C₄	C₂⁴	1	144	yes	—
118	C₂×C₄⋊C₄	C₂⁴	1	288	yes	—
119	C₂×C₂²⋊C₄	C₂⁴	1	96	yes	—
120	C₂×C₄²	C₂⁴	1	288	yes	—
121	C₂×C₄²	C₂⁴	1	72	yes	—
122	C₂×C₄²	C₂⁴	1	72	yes	—
123	C₂×C₄²	C₂⁴	1	192	yes	—
124	C₂⁴	—	2	4	no	—
125	C₂⁴	—	2	4	no	—
126	C₂⁴	C₂²×Q₈	1	8	yes	—
127	C₂⁴	C₂⁵	1	6	yes	—
128	C₂⁴	C₂⁵	1	1	yes	characteristic
129	C₂×Q₈	—	2	192	no	—
130	C₂×Q₈	—	4	288	no	—
131	C₂×Q₈	C₂²×Q₈	1	384	yes	—
132	C₂×Q₈	C₂²×Q₈	1	512	yes	complemented
133	C₂×Q₈	C₂⁵	1	64	yes	—
134	C₂×Q₈	C₂⁵	1	192	yes	—
135	C₂×Q₈	C₂⁵	1	64	yes	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	32	no	—
138	C₂²×C₄	—	2	24	no	—
139	C₂²×C₄	—	2	32	no	—
140	C₂²×C₄	—	2	48	no	—
141	C₂²×C₄	—	4	72	no	—
142	C₂²×C₄	C₂²×Q₈	1	48	yes	—
143	C₂²×C₄	C₂²×Q₈	1	288	yes	—
144	C₂²×C₄	2₋¹⁺⁴	1	8	yes	—
145	C₂²×C₄	C₂⁵	1	6	yes	—
146	C₂²×C₄	C₂⁵	1	16	yes	—
147	C₂²×C₄	C₂⁵	1	48	yes	—
148	C₂²×C₄	C₂⁵	1	48	yes	—
149	C₂²×C₄	C₂⁵	1	36	yes	—
150	C₂²×C₄	C₂⁵	1	48	yes	—
151	C₂²×C₄	C₂⁵	1	36	yes	—
152	C₂²×C₄	C₂⁵	1	48	yes	—
153	C₂²×C₄	C₂⁵	1	6	yes	—
154	C₄⋊C₄	C₂⁵	1	192	yes	—
155	C₄⋊C₄	C₂⁵	1	192	yes	—
156	C₄⋊C₄	C₂⁵	1	192	yes	—
157	C₄⋊C₄	C₂⁵	1	128	yes	—
158	C₄⋊C₄	C₂⁵	1	64	yes	—
159	C₄⋊C₄	C₂⁵	1	192	yes	—
160	C₄⋊C₄	C₂⁵	1	384	yes	—
161	C₂²⋊C₄	C₂⁵	1	128	yes	—
162	C₄²	C₂⁵	1	96	yes	—
163	C₄²	C₂⁵	1	96	yes	—
164	C₄²	C₂⁵	1	384	yes	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	12	no	—
167	C₂³	—	2	32	no	—
168	C₂³	C₂³×Q₈	1	24	yes	—
169	C₂³	C₂³×Q₈	1	4	yes	—
170	C₂³	C₂×2₋¹⁺⁴	1	4	yes	—
171	C₂³	#585	1	4	yes	—
172	C₂³	C₂⁶	1	4	yes	—
173	C₂³	C₂⁶	1	3	yes	—
174	Q₈	—	2	128	no	—
175	Q₈	—	4	192	no	—
176	Q₈	C₂³×Q₈	1	256	yes	—
177	C₂×C₄	—	2	12	no	—
178	C₂×C₄	—	2	96	no	—
179	C₂×C₄	—	2	72	no	—
180	C₂×C₄	—	2	96	no	—
181	C₂×C₄	—	2	32	no	—
182	C₂×C₄	—	4	48	no	—
183	C₂×C₄	C₂³×Q₈	1	144	yes	—
184	C₂×C₄	C₂³×Q₈	1	192	yes	—
185	C₂×C₄	C₂×2₋¹⁺⁴	1	24	yes	—
186	C₂×C₄	C₂⁶	1	4	yes	—
187	C₂×C₄	C₂⁶	1	32	yes	—
188	C₂×C₄	C₂⁶	1	32	yes	—
189	C₂×C₄	C₂⁶	1	24	yes	—
190	C₂×C₄	C₂⁶	1	24	yes	—
191	C₂×C₄	C₂⁶	1	4	yes	—
192	C₂²	—	2	48	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	#3502	1	16	yes	direct product factor
196	C₂²	C₂⁴×Q₈	1	8	yes	—
197	C₂²	C₂⁴×Q₈	1	6	yes	—
198	C₂²	C₂²×2₋¹⁺⁴	1	6	yes	—
199	C₂²	C₂×#585	1	6	yes	—
200	C₂²	C₂⁷	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	2	32	no	—
203	C₄	—	2	24	no	—
204	C₄	—	2	32	no	—
205	C₄	C₂⁴×Q₈	1	48	yes	—
206	C₄	C₂²×2₋¹⁺⁴	1	8	yes	—
207	C₂	—	2	8	no	—
208	C₂	C₂×#3502	1	12	yes	direct product factor
209	C₂	C₂⁵×Q₈	1	1	yes	characteristic
210	C₂	C₂³×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂	C₂²×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁵	C₂⁴
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×#3502	C₂	direct (N × H)
2. C₂²×Q₈²	C₂	semidirect
3. C₂²×#557	C₂	semidirect
4. C₂²×#557	C₂	semidirect
10. #3502	C₂²	direct (N × H)
11. C₂×Q₈²	C₂²	semidirect
12. C₂×#557	C₂²	semidirect
13. C₂×#557	C₂²	semidirect
34. C₂³×Q₈	Q₈	semidirect
42. Q₈²	C₂³	semidirect
43. #557	C₂³	semidirect
44. #557	C₂³	semidirect
80. C₂²×Q₈	C₂×Q₈	semidirect
132. C₂×Q₈	C₂²×Q₈	semidirect
195. C₂²	#3502	direct (N × H)
208. C₂	C₂×#3502	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×#3502	2
C₂²×Q₈²	2
C₂²×#557	2
C₂²×#544	2
C₂²×#533	2
C₂²×#519	2
C₂³×C₄×Q₈	2

Subgroups — C22×#3502 #10585962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3502 #10585962