Subgroups — C₃²×D₃×M₃₂ #13162342

Order 1728.

644 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃³×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#798	C₂	1	2	yes	maximal, complemented
5	C₃²×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₃²×M₃₂	—	3	1	no	maximal
7	C₃×D₃×M₃₂	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₃⋊C₈	C₄	1	1	yes	characteristic
11	C₃²×D₃×C₈	C₄	1	1	yes	characteristic
12	C₃²×D₃×C₈	C₄	1	1	yes	characteristic
13	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic
14	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic
15	C₃³×C₁₆	C₂²	1	2	yes	complemented
16	C₃²×C₃⋊C₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₃²×C₈	—	3	1	no	—
18	C₂×C₃×D₃×C₈	C₂×C₃	1	4	yes	—
19	C₃²×M₃₂	—	2	4	no	—
20	C₃²×M₃₂	—	3	1	no	—
21	C₃²×M₃₂	—	3	2	no	—
22	C₃²×M₃₂	D₃	1	1	yes	characteristic, direct product factor
23	C₃²×M₃₂	C₂×C₃	1	4	yes	complemented
24	C₂×C₃²×C₁₆	—	3	2	no	—
25	C₃×#812	C₂×C₃	1	4	yes	complemented
26	C₃×#798	C₂×C₃	1	8	yes	complemented
27	C₃×D₃×C₁₆	C₂×C₃	1	8	yes	complemented
28	C₂²×C₃²×D₃	C₈	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	C₈	1	1	yes	characteristic
31	C₃²×C₄×D₃	C₈	1	1	yes	characteristic
32	C₃²×C₄×D₃	C₈	1	1	yes	characteristic
33	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
35	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃³×C₈	C₂³	1	1	yes	characteristic
37	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₃²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×M₃₂	—	3	4	no	—
40	D₃×M₃₂	C₃²	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	—	3	1	no	—
42	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
43	C₂×C₃²×C₈	—	2	4	no	—
44	C₂×C₃²×C₈	—	3	1	no	—
45	C₂×C₃²×C₈	—	3	1	no	—
46	C₂×C₃²×C₈	—	3	1	no	—
47	C₂×C₃²×C₈	—	3	1	no	—
48	C₂×C₃²×C₈	—	3	1	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
51	C₂×C₃×C₃⋊C₈	C₃×C₄	1	4	yes	—
52	C₃×D₃×C₈	C₃×C₄	1	4	yes	—
53	C₃×D₃×C₈	C₃×C₄	1	4	yes	—
54	C₃×D₃×C₈	C₂²×C₃	1	4	yes	—
55	C₃×D₃×C₈	C₂²×C₃	1	4	yes	—
56	C₃²×C₁₆	—	2	8	no	—
57	C₃²×C₁₆	—	3	2	no	—
58	C₃²×C₁₆	C₂×D₃	1	2	yes	complemented
59	C₃²×C₁₆	C₂²×C₃	1	8	yes	complemented
60	C₃×C₃⋊C₁₆	C₂²×C₃	1	8	yes	complemented
61	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃²×D₃	—	2	2	no	—
63	C₂×C₃²×D₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃²×D₃	C₂×C₈	1	1	yes	characteristic
65	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
66	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃²×Dic₃	C₂×C₈	1	1	yes	characteristic
68	C₃²×Dic₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₃×C₈	—	3	4	no	—
70	C₂×D₃×C₈	C₂×C₃²	1	1	yes	characteristic
71	C₃×M₃₂	—	2	4	no	—
72	C₃×M₃₂	—	3	4	no	—
73	C₃×M₃₂	—	3	8	no	—
74	C₃×M₃₂	C₃×D₃	1	4	yes	direct product factor
75	C₃×M₃₂	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₂×C₃×C₁₆	—	3	8	no	—
77	#812	C₂×C₃²	1	1	yes	characteristic, complemented
78	#798	C₂×C₃²	1	2	yes	complemented
79	D₃×C₁₆	C₂×C₃²	1	2	yes	complemented
80	C₂³×C₃²	—	3	1	no	—
81	C₂²×C₃×D₃	C₃×C₈	1	4	yes	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	3	1	no	—
84	C₂×C₃²×C₄	—	3	1	no	—
85	C₂×C₃²×C₄	—	3	1	no	—
86	C₂×C₃²×C₄	—	3	1	no	—
87	C₂×C₃²×C₄	—	3	1	no	—
88	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₃×Dic₃	C₃×C₈	1	4	yes	—
91	C₃×C₄×D₃	C₃×C₈	1	4	yes	—
92	C₃×C₄×D₃	C₃×C₈	1	4	yes	—
93	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
94	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
95	C₃²×C₈	—	2	4	no	—
96	C₃²×C₈	—	2	4	no	—
97	C₃²×C₈	—	3	1	no	—
98	C₃²×C₈	—	3	1	no	—
99	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
100	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
101	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₃²×C₈	C₂³×C₃	1	4	yes	—
103	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
104	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
105	C₂×M₃₂	—	3	1	no	Sylow
106	C₂×C₃³	—	2	1	no	—
107	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
108	C₃²×D₃	—	2	1	no	—
109	C₃²×D₃	M₃₂	1	2	yes	direct product factor
110	C₂²×C₃×C₄	—	3	4	no	—
111	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	3	4	no	—
114	C₂×C₃×C₈	—	3	4	no	—
115	C₂×C₃×C₈	—	3	4	no	—
116	C₂×C₃×C₈	—	3	4	no	—
117	C₂×C₃×C₈	—	3	4	no	—
118	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
119	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₃²×C₄	1	1	yes	characteristic
121	D₃×C₈	C₃²×C₄	1	1	yes	characteristic
122	D₃×C₈	C₃²×C₄	1	1	yes	characteristic
123	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic
124	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic
125	C₃×C₁₆	—	2	8	no	—
126	C₃×C₁₆	—	3	8	no	—
127	C₃×C₁₆	C₂×C₃×D₃	1	8	yes	complemented
128	C₃×C₁₆	C₂²×C₃²	1	2	yes	complemented
129	C₃⋊C₁₆	C₂²×C₃²	1	2	yes	complemented
130	C₂²×C₃²	—	2	4	no	—
131	C₂²×C₃²	—	3	1	no	—
132	C₂²×C₃²	—	3	1	no	—
133	C₂²×C₃²	—	6	2	no	—
134	C₂²×C₃²	D₃×C₈	1	1	yes	characteristic
135	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
136	C₂×C₃×D₃	—	2	8	no	—
137	C₂×C₃×D₃	C₂×C₃×C₈	1	4	yes	—
138	C₂×C₃×D₃	C₂×C₃×C₈	1	4	yes	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	—	3	1	no	—
142	C₃²×C₄	—	3	1	no	—
143	C₃²×C₄	D₃×C₈	1	1	yes	characteristic
144	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
145	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
146	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
147	C₃×Dic₃	C₂×C₃×C₈	1	4	yes	—
148	C₃×Dic₃	C₂×C₃×C₈	1	4	yes	—
149	C₂²×C₈	—	3	1	no	—
150	M₃₂	—	3	1	no	—
151	M₃₂	—	3	2	no	—
152	M₃₂	C₃²×D₃	1	1	yes	characteristic, direct product factor
153	C₂×C₁₆	—	3	2	no	—
154	C₃³	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₃	—	3	4	no	—
156	C₂²×D₃	C₃²×C₈	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	3	4	no	—
161	C₂×C₃×C₄	—	3	4	no	—
162	C₂×C₃×C₄	—	3	4	no	—
163	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
164	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
165	C₂×Dic₃	C₃²×C₈	1	1	yes	characteristic
166	C₄×D₃	C₃²×C₈	1	1	yes	characteristic
167	C₄×D₃	C₃²×C₈	1	1	yes	characteristic
168	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
169	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	2	4	no	—
172	C₃×C₈	—	3	4	no	—
173	C₃×C₈	—	3	4	no	—
174	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
175	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
176	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
177	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
178	C₃⋊C₈	C₂×C₃²×C₄	1	1	yes	characteristic
179	C₃⋊C₈	C₂×C₃²×C₄	1	1	yes	characteristic
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	3	2	no	—
184	C₂×C₃²	—	4	4	no	—
185	C₂×C₃²	—	6	1	no	—
186	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
187	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
188	C₃×D₃	—	2	4	no	—
189	C₃×D₃	C₃×M₃₂	1	8	yes	direct product factor
190	C₂²×C₄	—	3	1	no	—
191	C₂×C₈	—	3	1	no	—
192	C₂×C₈	—	3	1	no	—
193	C₂×C₈	—	3	1	no	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	3	1	no	—
196	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
197	C₁₆	—	3	2	no	—
198	C₁₆	C₂×C₃²×D₃	1	2	yes	complemented
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	3	4	no	—
201	C₂²×C₃	—	3	4	no	—
202	C₂²×C₃	—	6	8	no	—
203	C₂²×C₃	C₃×D₃×C₈	1	4	yes	—
204	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₂×C₃²×C₈	1	1	yes	characteristic
207	C₂×D₃	C₂×C₃²×C₈	1	1	yes	characteristic
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	3	4	no	—
211	C₃×C₄	—	3	4	no	—
212	C₃×C₄	C₃×D₃×C₈	1	4	yes	—
213	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
214	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
215	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
216	Dic₃	C₂×C₃²×C₈	1	1	yes	characteristic
217	Dic₃	C₂×C₃²×C₈	1	1	yes	characteristic
218	C₃²	—	2	4	no	—
219	C₃²	D₃×M₃₂	1	1	yes	characteristic, direct product factor
220	C₃²	C₂×C₃×M₃₂	1	4	yes	complemented
221	C₂³	—	3	1	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	3	1	no	—
225	C₂×C₄	—	3	1	no	—
226	C₂×C₄	—	3	1	no	—
227	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
228	C₈	—	3	1	no	—
229	C₈	—	3	1	no	—
230	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
231	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	3	8	no	—
236	C₂×C₃	—	4	4	no	—
237	C₂×C₃	—	6	4	no	—
238	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
239	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
240	D₃	—	2	1	no	—
241	D₃	C₃²×M₃₂	1	2	yes	direct product factor
242	C₂²	—	3	1	no	—
243	C₂²	—	3	1	no	—
244	C₂²	—	6	2	no	—
245	C₂²	C₃²×D₃×C₈	1	1	yes	characteristic
246	C₄	—	3	1	no	—
247	C₄	—	3	1	no	—
248	C₄	C₃²×D₃×C₈	1	1	yes	characteristic
249	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
250	C₃	—	2	4	no	—
251	C₃	C₃×D₃×M₃₂	1	4	yes	direct product factor
252	C₃	C₂×C₃²×M₃₂	1	1	yes	characteristic, complemented
253	C₂	—	2	1	no	—
254	C₂	—	3	2	no	—
255	C₂	—	6	1	no	—
256	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₈	C₂²×C₃²×D₃
Fitting	C₃³×M₃₂	C₂
socle	C₂×C₃³	C₂²×C₈
prime radical	C₂²×C₃²×D₃	C₈
cosocle	C₃×C₈	C₂³×C₃²
hypercenter	C₃²×M₃₂	D₃
hypocenter	C₃	C₂×C₃²×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₈ ◁ C₃²×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₃³×M₃₂	C₂	semidirect
3. C₃²×#812	C₂	semidirect
4. C₃²×#798	C₂	semidirect
5. C₃²×D₃×C₁₆	C₂	semidirect
7. C₃×D₃×M₃₂	C₃	direct (N × H)
15. C₃³×C₁₆	C₂²	semidirect
16. C₃²×C₃⋊C₁₆	C₂²	semidirect
22. C₃²×M₃₂	D₃	direct (N × H)
23. C₃²×M₃₂	C₂×C₃	semidirect
25. C₃×#812	C₂×C₃	semidirect
26. C₃×#798	C₂×C₃	semidirect
27. C₃×D₃×C₁₆	C₂×C₃	semidirect
40. D₃×M₃₂	C₃²	direct (N × H)
58. C₃²×C₁₆	C₂×D₃	semidirect
59. C₃²×C₁₆	C₂²×C₃	semidirect
60. C₃×C₃⋊C₁₆	C₂²×C₃	semidirect
74. C₃×M₃₂	C₃×D₃	direct (N × H)
75. C₃×M₃₂	C₂×C₃²	semidirect
77. #812	C₂×C₃²	semidirect
78. #798	C₂×C₃²	semidirect
79. D₃×C₁₆	C₂×C₃²	semidirect
109. C₃²×D₃	M₃₂	direct (N × H)
127. C₃×C₁₆	C₂×C₃×D₃	semidirect
128. C₃×C₁₆	C₂²×C₃²	semidirect
129. C₃⋊C₁₆	C₂²×C₃²	semidirect
152. M₃₂	C₃²×D₃	direct (N × H)
154. C₃³	C₂×M₃₂	semidirect
189. C₃×D₃	C₃×M₃₂	direct (N × H)
198. C₁₆	C₂×C₃²×D₃	semidirect
219. C₃²	D₃×M₃₂	direct (N × H)
220. C₃²	C₂×C₃×M₃₂	semidirect
241. D₃	C₃²×M₃₂	direct (N × H)
251. C₃	C₃×D₃×M₃₂	direct (N × H)
252. C₃	C₂×C₃²×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃×C₈	2
C₃³×M₃₂	2
C₃²×#812	2
C₃²×#798	2
C₃²×D₃×C₁₆	2
C₂×C₃²×M₃₂	3
C₃×D₃×M₃₂	3

Subgroups — C32×D3×M32 #13162342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×M₃₂ #13162342