Subgroups — C₂×Q₈×M₃₂ #7622643

Order 512.

1484 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₃₂	C₂	1	16	yes	maximal, direct product factor
3	C₂×#2183	C₂	1	6	yes	maximal, complemented
4	C₂×Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×#2150	C₂	1	3	yes	maximal
6	C₂×C₄×M₃₂	C₂	1	3	yes	maximal
7	C₂²×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₂²×M₃₂	C₂²	1	3	yes	—
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₄	1	3	yes	—
11	C₂×C₈×Q₈	C₄	1	8	yes	—
12	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Q₈	C₂²	1	3	yes	—
15	C₂×C₈×Q₈	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₈	C₄	1	3	yes	—
17	C₂²×C₄×C₈	C₂²	1	3	yes	—
18	#2183	C₂²	1	48	yes	complemented
19	Q₈×C₁₆	C₂²	1	16	yes	complemented
20	#2150	C₂²	1	24	yes	—
21	C₂×C₄⋊C₁₆	C₂²	1	6	yes	—
22	C₂×C₄⋊C₁₆	C₂²	1	6	yes	—
23	C₄×M₃₂	C₂²	1	24	yes	—
24	C₂×#346	C₂²	1	6	yes	—
25	C₂×C₄×C₁₆	C₂²	1	6	yes	—
26	C₂³×Q₈	C₈	1	1	yes	characteristic
27	C₂³×C₈	C₂×C₄	1	3	yes	—
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₈	1	3	yes	—
30	C₂×C₄×Q₈	C₈	1	8	yes	—
31	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₂×C₄	1	3	yes	—
33	C₂×C₄×Q₈	C₂×C₄	1	8	yes	—
34	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄⋊C₄	C₈	1	3	yes	—
36	C₂²×C₄²	C₂×C₄	1	3	yes	—
37	C₂×M₃₂	—	2	12	no	—
38	C₂×M₃₂	Q₈	1	8	yes	direct product factor
39	C₂×M₃₂	C₂³	1	12	yes	—
40	C₂×M₃₂	C₂³	1	4	yes	—
41	C₂²×C₁₆	C₂³	1	6	yes	—
42	C₂²×C₁₆	C₂³	1	2	yes	—
43	C₈×Q₈	C₂×C₄	1	4	yes	—
44	C₈×Q₈	C₂×C₄	1	12	yes	—
45	C₈×Q₈	C₂×C₄	1	24	yes	—
46	C₈×Q₈	C₂×C₄	1	8	yes	—
47	C₈×Q₈	C₂³	1	4	yes	—
48	C₈×Q₈	C₂³	1	12	yes	—
49	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
50	C₂×C₄⋊C₈	C₂×C₄	1	6	yes	—
51	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
52	C₂×C₄⋊C₈	C₂×C₄	1	24	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
56	C₂×C₄×C₈	C₂³	1	12	yes	—
57	C₂×C₄×C₈	C₂³	1	3	yes	—
58	C₂×C₄×C₈	C₂³	1	3	yes	—
59	C₄⋊C₁₆	C₂³	1	24	yes	—
60	C₄⋊C₁₆	C₂³	1	24	yes	—
61	#346	C₂³	1	24	yes	—
62	C₄×C₁₆	C₂³	1	24	yes	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	C₂×C₈	1	8	yes	—
65	C₂²×Q₈	C₂×C₈	1	3	yes	—
66	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
67	C₂³×C₄	C₂×C₈	1	3	yes	—
68	C₂³×C₄	C₂×C₈	1	3	yes	—
69	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	—	2	12	no	—
71	C₂²×C₈	C₂²×C₄	1	12	yes	—
72	C₂²×C₈	C₂²×C₄	1	3	yes	—
73	C₂²×C₈	C₂²×C₄	1	3	yes	—
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	2	yes	—
76	C₂²×C₈	C₂×Q₈	1	4	yes	—
77	C₂²×C₈	#41	1	4	yes	—
78	C₂²×C₈	C₂⁴	1	2	yes	—
79	C₂²×C₈	C₂⁴	1	1	yes	characteristic
80	C₄×Q₈	C₂×C₈	1	12	yes	—
81	C₄×Q₈	C₂×C₈	1	24	yes	—
82	C₄×Q₈	C₂×C₈	1	4	yes	—
83	C₄×Q₈	C₂×C₈	1	8	yes	—
84	C₄×Q₈	C₂²×C₄	1	12	yes	—
85	C₄×Q₈	C₂²×C₄	1	4	yes	—
86	C₂×C₄⋊C₄	C₂×C₈	1	6	yes	—
87	C₂×C₄⋊C₄	C₂×C₈	1	3	yes	—
88	C₂×C₄⋊C₄	C₂×C₈	1	3	yes	—
89	C₂×C₄⋊C₄	C₂×C₈	1	24	yes	—
90	C₂×C₄²	C₂×C₈	1	3	yes	—
91	C₂×C₄²	C₂×C₈	1	3	yes	—
92	C₂×C₄²	C₂×C₈	1	12	yes	—
93	C₂×C₄²	C₂²×C₄	1	3	yes	—
94	C₂×C₄²	C₂²×C₄	1	3	yes	—
95	C₂×C₄²	C₂²×C₄	1	12	yes	—
96	M₃₂	—	2	24	no	—
97	M₃₂	C₂×Q₈	1	16	yes	direct product factor
98	C₂×C₁₆	—	2	12	no	—
99	C₂×C₁₆	C₂×Q₈	1	8	yes	complemented
100	C₂×C₁₆	C₂⁴	1	12	yes	—
101	C₂×C₁₆	C₂⁴	1	4	yes	—
102	C₄⋊C₈	C₂²×C₄	1	12	yes	—
103	C₄⋊C₈	C₂²×C₄	1	24	yes	—
104	C₄⋊C₈	C₂²×C₄	1	12	yes	—
105	C₄×C₈	C₂²×C₄	1	12	yes	—
106	C₄×C₈	C₂²×C₄	1	12	yes	—
107	C₄×C₈	C₂²×C₄	1	6	yes	—
108	C₄×C₈	C₂²×C₄	1	6	yes	—
109	C₄×C₈	C₂⁴	1	6	yes	—
110	C₄×C₈	C₂⁴	1	6	yes	—
111	C₂⁴	C₂²×C₈	1	1	yes	characteristic
112	C₂×Q₈	—	2	32	no	—
113	C₂×Q₈	—	2	12	no	—
114	C₂×Q₈	M₃₂	1	8	yes	direct product factor
115	C₂×Q₈	C₂²×C₈	1	12	yes	—
116	C₂×Q₈	C₂²×C₈	1	4	yes	—
117	C₂²×C₄	—	2	12	no	—
118	C₂²×C₄	—	4	6	no	—
119	C₂²×C₄	C₄×Q₈	1	4	yes	—
120	C₂²×C₄	C₂²×C₈	1	3	yes	—
121	C₂²×C₄	C₂²×C₈	1	3	yes	—
122	C₂²×C₄	C₂²×C₈	1	3	yes	—
123	C₂²×C₄	C₂²×C₈	1	12	yes	—
124	C₂²×C₄	C₂²×C₈	1	12	yes	—
125	C₂²×C₄	C₂²×C₈	1	2	yes	—
126	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	3	yes	—
128	C₂²×C₄	#131	1	4	yes	—
129	C₂²×C₄	C₂³×C₄	1	2	yes	—
130	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₈	—	2	24	no	—
132	C₂×C₈	—	2	6	no	—
133	C₂×C₈	—	2	6	no	—
134	C₂×C₈	C₄×Q₈	1	4	yes	—
135	C₂×C₈	C₄×Q₈	1	8	yes	—
136	C₂×C₈	C₂³×C₄	1	6	yes	—
137	C₂×C₈	C₂³×C₄	1	6	yes	—
138	C₂×C₈	C₂³×C₄	1	2	yes	—
139	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₈	C₂²×Q₈	1	2	yes	—
141	C₂×C₈	C₂²×Q₈	1	4	yes	—
142	C₂×C₈	C₂×#41	1	2	yes	—
143	C₂×C₈	C₂×#41	1	4	yes	—
144	C₂×C₈	C₂⁵	1	1	yes	characteristic
145	C₄⋊C₄	C₂²×C₈	1	24	yes	—
146	C₄⋊C₄	C₂²×C₈	1	12	yes	—
147	C₄⋊C₄	C₂²×C₈	1	12	yes	—
148	C₄²	C₂²×C₈	1	12	yes	—
149	C₄²	C₂²×C₈	1	6	yes	—
150	C₄²	C₂²×C₈	1	12	yes	—
151	C₄²	C₂²×C₈	1	6	yes	—
152	C₄²	C₂³×C₄	1	6	yes	—
153	C₄²	C₂³×C₄	1	6	yes	—
154	C₁₆	—	2	12	no	—
155	C₁₆	C₂²×Q₈	1	8	yes	complemented
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	C₈×Q₈	1	4	yes	—
159	C₂³	C₂³×C₈	1	2	yes	—
160	C₂³	C₂³×C₈	1	1	yes	characteristic
161	Q₈	—	2	24	no	—
162	Q₈	C₂×M₃₂	1	16	yes	direct product factor
163	C₂×C₄	—	2	6	no	—
164	C₂×C₄	—	2	6	no	—
165	C₂×C₄	—	2	6	no	—
166	C₂×C₄	—	2	24	no	—
167	C₂×C₄	—	4	12	no	—
168	C₂×C₄	C₈×Q₈	1	8	yes	—
169	C₂×C₄	C₈×Q₈	1	4	yes	—
170	C₂×C₄	C₂×M₃₂	1	12	yes	—
171	C₂×C₄	C₂×C₄×Q₈	1	4	yes	—
172	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
173	C₂×C₄	C₂³×C₈	1	6	yes	—
174	C₂×C₄	C₂³×C₈	1	6	yes	—
175	C₂×C₄	C₂³×C₈	1	6	yes	—
176	C₂×C₄	C₂³×C₈	1	6	yes	—
177	C₂×C₄	C₂³×C₈	1	2	yes	—
178	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
179	C₂×C₄	C₂×#131	1	4	yes	—
180	C₂×C₄	C₂×#131	1	2	yes	—
181	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
182	C₈	—	2	6	no	—
183	C₈	—	2	6	no	—
184	C₈	C₂×C₄×Q₈	1	4	yes	—
185	C₈	C₂×C₄×Q₈	1	2	yes	—
186	C₈	C₂³×Q₈	1	1	yes	characteristic
187	C₈	C₂²×#41	1	1	yes	characteristic
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	8	no	—
191	C₂²	C₂×C₈×Q₈	1	4	yes	—
192	C₂²	C₂×C₈×Q₈	1	2	yes	—
193	C₂²	C₂²×M₃₂	1	2	yes	—
194	C₂²	#3407	1	2	yes	—
195	C₂²	C₂⁴×C₈	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	2	6	no	—
198	C₄	—	2	6	no	—
199	C₄	C₂×C₈×Q₈	1	4	yes	—
200	C₄	C₂×C₈×Q₈	1	2	yes	—
201	C₄	C₂²×M₃₂	1	12	yes	—
202	C₄	C₂²×C₄×Q₈	1	1	yes	characteristic
203	C₄	C₂²×#131	1	1	yes	characteristic
204	C₂	—	2	4	no	—
205	C₂	Q₈×M₃₂	1	4	yes	direct product factor
206	C₂	C₂²×C₈×Q₈	1	1	yes	characteristic
207	C₂	C₂³×M₃₂	1	1	yes	characteristic
208	C₂	C₂×#3407	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. Q₈×M₃₂	C₂	direct (N × H)
3. C₂×#2183	C₂	semidirect
4. C₂×Q₈×C₁₆	C₂	semidirect
18. #2183	C₂²	semidirect
19. Q₈×C₁₆	C₂²	semidirect
38. C₂×M₃₂	Q₈	direct (N × H)
97. M₃₂	C₂×Q₈	direct (N × H)
99. C₂×C₁₆	C₂×Q₈	semidirect
114. C₂×Q₈	M₃₂	direct (N × H)
155. C₁₆	C₂²×Q₈	semidirect
162. Q₈	C₂×M₃₂	direct (N × H)
205. C₂	Q₈×M₃₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₈×Q₈	2
Q₈×M₃₂	2
C₂×#2183	2
C₂×Q₈×C₁₆	2
C₂×#2150	2
C₂×C₄×M₃₂	2

Subgroups — C2×Q8×M32 #7622643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×M₃₂ #7622643