Subgroups — C₃²×C₄×SD₃₂ #11769222

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#358	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#357	C₂	1	1	yes	maximal, characteristic
7	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×SD₃₂	C₃	1	4	yes	maximal, direct product factor
9	C₂×C₃²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	—	2	1	no	—
12	C₃²×C₄×D₄	—	2	1	no	—
13	C₃²×SD₃₂	C₄	1	4	yes	direct product factor
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
16	C₃²×#107	C₂²	1	1	yes	characteristic
17	C₃²×#107	C₂²	1	1	yes	characteristic
18	C₃²×#103	—	2	1	no	—
19	C₃²×#102	—	2	1	no	—
20	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×SD₃₂	C₂×C₃	1	4	yes	—
22	C₃×C₄×Q₁₆	C₂×C₃	1	4	yes	complemented
23	C₃×C₄×D₈	C₂×C₃	1	4	yes	—
24	C₃×#367	C₂×C₃	1	4	yes	complemented
25	C₃×#358	C₂×C₃	1	4	yes	complemented
26	C₃×#357	C₂×C₃	1	4	yes	—
27	C₃×C₄×C₁₆	C₂×C₃	1	4	yes	complemented
28	C₂×C₃²×Q₈	—	2	1	no	—
29	C₂×C₃²×D₄	—	2	1	no	—
30	C₂²×C₃²×C₄	—	4	1	no	—
31	C₃²×Q₁₆	—	2	1	no	—
32	C₃²×Q₁₆	C₂×C₄	1	2	yes	complemented
33	C₃²×D₈	—	2	1	no	—
34	C₃²×D₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
37	C₃²×C₄⋊C₄	—	2	1	no	—
38	C₃²×C₄⋊C₄	—	2	1	no	—
39	C₃²×C₄⋊C₄	—	4	1	no	—
40	C₃²×C₂²⋊C₄	—	4	1	no	—
41	C₃²×C₄²	—	4	1	no	—
42	C₃²×C₄²	D₄	1	1	yes	characteristic
43	C₃²×C₁₆	—	2	1	no	—
44	C₃²×C₁₆	C₂×C₄	1	2	yes	complemented
45	C₄×SD₃₂	C₃²	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
47	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
48	C₃×C₄×Q₈	—	2	4	no	—
49	C₃×C₄×D₄	—	2	4	no	—
50	C₃×SD₃₂	C₃×C₄	1	16	yes	direct product factor
51	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
52	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
53	C₃×#107	C₂²×C₃	1	4	yes	—
54	C₃×#107	C₂²×C₃	1	4	yes	—
55	C₃×#103	—	2	4	no	—
56	C₃×#102	—	2	4	no	—
57	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
58	C₂³×C₃²	—	4	1	no	—
59	C₃²×Q₈	—	2	2	no	—
60	C₃²×Q₈	—	4	1	no	—
61	C₃²×D₄	—	2	2	no	—
62	C₃²×D₄	—	4	1	no	—
63	C₂×C₃²×C₄	—	4	1	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	—	4	1	no	—
66	C₂×C₃²×C₄	—	8	2	no	—
67	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₃²×C₈	—	2	1	no	—
71	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₃²×C₈	#41	1	1	yes	characteristic
73	C₂×SD₃₂	C₂×C₃²	1	1	yes	characteristic
74	C₄×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
75	C₄×D₈	C₂×C₃²	1	1	yes	characteristic
76	#367	C₂×C₃²	1	1	yes	characteristic, complemented
77	#358	C₂×C₃²	1	1	yes	characteristic, complemented
78	#357	C₂×C₃²	1	1	yes	characteristic
79	C₄×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₂×C₃×Q₈	—	2	4	no	—
81	C₂×C₃×D₄	—	2	4	no	—
82	C₂²×C₃×C₄	—	4	4	no	—
83	C₃×Q₁₆	—	2	4	no	—
84	C₃×Q₁₆	C₂×C₃×C₄	1	8	yes	complemented
85	C₃×D₈	—	2	4	no	—
86	C₃×D₈	C₂×C₃×C₄	1	8	yes	—
87	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
88	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₄⋊C₄	—	2	4	no	—
91	C₃×C₄⋊C₄	—	4	4	no	—
92	C₃×C₂²⋊C₄	—	4	4	no	—
93	C₃×C₄²	—	4	4	no	—
94	C₃×C₄²	C₃×D₄	1	4	yes	—
95	C₃×C₁₆	—	2	4	no	—
96	C₃×C₁₆	C₂×C₃×C₄	1	8	yes	complemented
97	C₂²×C₃²	—	4	2	no	—
98	C₂²×C₃²	—	8	1	no	—
99	C₂²×C₃²	—	8	1	no	—
100	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
101	C₃²×C₄	—	2	1	no	—
102	C₃²×C₄	—	4	2	no	—
103	C₃²×C₄	—	8	1	no	—
104	C₃²×C₄	—	8	1	no	—
105	C₃²×C₄	SD₃₂	1	2	yes	direct product factor
106	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
107	C₃²×C₄	#135	1	1	yes	characteristic
108	C₂×Q₁₆	C₂²×C₃²	1	1	yes	characteristic
109	C₂×D₈	C₂²×C₃²	1	1	yes	characteristic
110	C₄×Q₈	—	2	1	no	—
111	C₄×D₄	—	2	1	no	—
112	SD₃₂	C₃²×C₄	1	4	yes	direct product factor
113	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
114	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
115	#107	C₂²×C₃²	1	1	yes	characteristic
116	#107	C₂²×C₃²	1	1	yes	characteristic
117	#103	—	2	1	no	—
118	#102	—	2	1	no	—
119	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
120	C₂³×C₃	—	4	4	no	—
121	C₃×Q₈	—	2	8	no	—
122	C₃×Q₈	—	4	4	no	—
123	C₃×D₄	—	2	8	no	—
124	C₃×D₄	—	4	4	no	—
125	C₂×C₃×C₄	—	4	4	no	—
126	C₂×C₃×C₄	—	4	4	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	—	8	8	no	—
129	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
130	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
131	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
132	C₃×C₈	—	2	4	no	—
133	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
134	C₃×C₈	C₃×#41	1	4	yes	—
135	C₂×C₃²	—	8	2	no	—
136	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
137	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
138	C₂×C₃²	#508	1	1	yes	characteristic
139	C₂×Q₈	—	2	1	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	Q₁₆	—	2	1	no	—
143	Q₁₆	C₂×C₃²×C₄	1	2	yes	complemented
144	D₈	—	2	1	no	—
145	D₈	C₂×C₃²×C₄	1	2	yes	—
146	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
147	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	4	1	no	—
153	C₄²	C₃²×D₄	1	1	yes	characteristic
154	C₁₆	—	2	1	no	—
155	C₁₆	C₂×C₃²×C₄	1	2	yes	complemented
156	C₂²×C₃	—	4	8	no	—
157	C₂²×C₃	—	8	4	no	—
158	C₂²×C₃	—	8	4	no	—
159	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	—	4	8	no	—
162	C₃×C₄	—	8	4	no	—
163	C₃×C₄	—	8	4	no	—
164	C₃×C₄	C₃×SD₃₂	1	8	yes	direct product factor
165	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
166	C₃×C₄	C₃×#135	1	4	yes	—
167	C₃²	C₄×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	—	4	1	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	4	1	no	—
171	D₄	—	2	2	no	—
172	D₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
178	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
179	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	C₂²×C₃²×C₄	1	1	yes	characteristic
182	C₈	C₃²×#41	1	1	yes	characteristic
183	C₂×C₃	—	8	8	no	—
184	C₂×C₃	C₃×C₄×D₈	1	4	yes	—
185	C₂×C₃	C₂×C₃×SD₃₂	1	4	yes	—
186	C₂×C₃	C₃×#508	1	4	yes	—
187	C₂²	—	4	2	no	—
188	C₂²	—	8	1	no	—
189	C₂²	—	8	1	no	—
190	C₂²	C₂×C₃²×D₈	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	8	1	no	—
194	C₄	—	8	1	no	—
195	C₄	C₃²×SD₃₂	1	2	yes	direct product factor
196	C₄	C₃²×C₄×D₄	1	1	yes	characteristic
197	C₄	C₃²×#135	1	1	yes	characteristic
198	C₃	C₃×C₄×SD₃₂	1	4	yes	direct product factor
199	C₂	—	8	2	no	—
200	C₂	C₃²×C₄×D₈	1	1	yes	characteristic
201	C₂	C₂×C₃²×SD₃₂	1	1	yes	characteristic
202	C₂	C₃²×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	D₈
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₂×D₈
prime radical	C₂×C₃²×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×Q₁₆	C₂	semidirect
4. C₃²×#367	C₂	semidirect
5. C₃²×#358	C₂	semidirect
7. C₃²×C₄×C₁₆	C₂	semidirect
8. C₃×C₄×SD₃₂	C₃	direct (N × H)
13. C₃²×SD₃₂	C₄	direct (N × H)
22. C₃×C₄×Q₁₆	C₂×C₃	semidirect
24. C₃×#367	C₂×C₃	semidirect
25. C₃×#358	C₂×C₃	semidirect
27. C₃×C₄×C₁₆	C₂×C₃	semidirect
32. C₃²×Q₁₆	C₂×C₄	semidirect
44. C₃²×C₁₆	C₂×C₄	semidirect
45. C₄×SD₃₂	C₃²	direct (N × H)
50. C₃×SD₃₂	C₃×C₄	direct (N × H)
74. C₄×Q₁₆	C₂×C₃²	semidirect
76. #367	C₂×C₃²	semidirect
77. #358	C₂×C₃²	semidirect
79. C₄×C₁₆	C₂×C₃²	semidirect
84. C₃×Q₁₆	C₂×C₃×C₄	semidirect
96. C₃×C₁₆	C₂×C₃×C₄	semidirect
105. C₃²×C₄	SD₃₂	direct (N × H)
112. SD₃₂	C₃²×C₄	direct (N × H)
143. Q₁₆	C₂×C₃²×C₄	semidirect
155. C₁₆	C₂×C₃²×C₄	semidirect
164. C₃×C₄	C₃×SD₃₂	direct (N × H)
167. C₃²	C₄×SD₃₂	direct (N × H)
195. C₄	C₃²×SD₃₂	direct (N × H)
198. C₃	C₃×C₄×SD₃₂	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×C₃²×SD₃₂	2
C₃²×C₄×Q₁₆	2
C₃²×C₄×D₈	2
C₃²×#367	2
C₃²×#358	2
C₃²×#357	2
C₃²×C₄×C₁₆	2
C₃×C₄×SD₃₂	3

Subgroups — C32×C4×SD32 #11769222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×SD₃₂ #11769222