Subgroups — C₂×C₃×D₃×Q₃₂ #11899472

Order 1152.

761 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₃²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#829	C₂	1	2	yes	maximal, complemented
4	C₃×D₃×Q₃₂	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₃×Q₃₂	—	3	1	no	maximal
8	C₂×D₃×Q₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃×D₃×Q₈	—	2	2	no	—
10	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₃×#216	—	2	2	no	—
12	C₃×D₃×Q₁₆	—	2	4	no	—
13	C₃×D₃×Q₁₆	C₂²	1	8	yes	—
14	C₂×C₃×Dic₁₂	C₂²	1	2	yes	—
15	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
16	C₃²×Q₃₂	C₂²	1	4	yes	complemented
17	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#829	C₂²	1	8	yes	complemented
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₃×Dic₂₄	C₂²	1	4	yes	complemented
21	C₃×D₃×C₁₆	C₂²	1	4	yes	—
22	C₂²×C₃×Q₁₆	—	3	2	no	—
23	C₂×D₃×Q₁₆	C₂×C₃	1	2	yes	—
24	C₂×C₃×Q₃₂	—	2	1	no	—
25	C₂×C₃×Q₃₂	—	3	2	no	—
26	C₂×C₃×Q₃₂	—	3	8	no	—
27	C₂×C₃×Q₃₂	—	3	1	no	—
28	C₂×C₃×Q₃₂	D₃	1	1	yes	characteristic, direct product factor
29	C₂×C₃×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃×C₁₆	—	3	1	no	—
31	C₂×#829	C₂×C₃	1	2	yes	complemented
32	D₃×Q₃₂	C₂×C₃	1	8	yes	direct product factor
33	C₂×Dic₂₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×D₃×C₁₆	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃²×Q₈	—	2	2	no	—
36	C₃×D₃×Q₈	—	2	8	no	—
37	C₃×D₃×Q₈	—	4	4	no	—
38	C₂×C₃×C₄×D₃	—	4	2	no	—
39	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
40	C₂×C₃×Dic₆	—	2	2	no	—
41	C₂×C₃×Dic₆	—	4	2	no	—
42	C₃²×Q₁₆	—	2	2	no	—
43	C₃²×Q₁₆	C₂³	1	4	yes	—
44	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
45	C₃×#216	—	2	8	no	—
46	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₃×Dic₁₂	—	2	2	no	—
48	C₃×Dic₁₂	C₂³	1	4	yes	—
49	C₃×D₃×C₈	D₄	1	2	yes	—
50	C₃×D₃×C₈	C₂³	1	2	yes	—
51	C₃²×C₁₆	C₂³	1	2	yes	—
52	C₃×C₃⋊C₁₆	C₂³	1	2	yes	—
53	C₂²×Q₃₂	—	3	1	no	Sylow
54	C₂²×C₃×Q₈	—	6	2	no	—
55	C₂×D₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₁₆	—	2	2	no	—
57	C₂×C₃×Q₁₆	—	3	8	no	—
58	C₂×C₃×Q₁₆	—	3	2	no	—
59	C₂×C₃×Q₁₆	—	6	4	no	—
60	C₂×C₃×Q₁₆	—	6	2	no	—
61	C₂×C₃×Q₁₆	C₂×D₃	1	2	yes	—
62	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
63	C₂²×C₃×C₈	—	3	1	no	—
64	C₂×#216	—	2	2	no	—
65	D₃×Q₁₆	—	2	4	no	—
66	D₃×Q₁₆	C₂²×C₃	1	8	yes	—
67	C₂×Dic₁₂	C₂²×C₃	1	2	yes	—
68	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
69	C₃×Q₃₂	—	2	4	no	—
70	C₃×Q₃₂	—	3	8	no	—
71	C₃×Q₃₂	—	3	4	no	—
72	C₃×Q₃₂	C₂×D₃	1	4	yes	direct product factor
73	C₃×Q₃₂	C₂²×C₃	1	4	yes	complemented
74	C₂×C₃×C₁₆	—	2	1	no	—
75	C₂×C₃×C₁₆	—	3	1	no	—
76	C₂×C₃×C₁₆	—	3	4	no	—
77	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
79	#829	C₂²×C₃	1	8	yes	complemented
80	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
81	Dic₂₄	C₂²×C₃	1	4	yes	complemented
82	D₃×C₁₆	C₂²×C₃	1	4	yes	—
83	C₂²×C₃×D₃	D₈	1	1	yes	characteristic
84	C₃²×Q₈	—	2	4	no	—
85	C₃²×Q₈	—	4	2	no	—
86	C₂×C₃²×C₄	—	4	2	no	—
87	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₂×C₃×Dic₃	D₈	1	1	yes	characteristic
90	C₃×C₄×D₃	—	4	8	no	—
91	C₃×C₄×D₃	D₈	1	2	yes	—
92	C₃×C₄×D₃	C₂×D₄	1	2	yes	—
93	C₃×Dic₆	—	2	4	no	—
94	C₃×Dic₆	—	4	2	no	—
95	C₃×Dic₆	—	4	8	no	—
96	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₃²×C₈	C₂⁴	1	1	yes	characteristic
98	C₃×C₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂²×Q₁₆	—	3	2	no	—
100	C₂×Q₃₂	—	3	2	no	—
101	C₂×Q₃₂	—	3	8	no	—
102	C₂×Q₃₂	—	3	1	no	—
103	C₂×Q₃₂	C₃×D₃	1	1	yes	characteristic, direct product factor
104	C₂²×C₁₆	—	3	1	no	—
105	C₂×C₃×Q₈	—	2	2	no	—
106	C₂×C₃×Q₈	—	2	2	no	—
107	C₂×C₃×Q₈	—	4	2	no	—
108	C₂×C₃×Q₈	—	6	8	no	—
109	C₂×C₃×Q₈	—	6	2	no	—
110	C₂×C₃×Q₈	—	12	2	no	—
111	C₂×C₃×Q₈	—	12	4	no	—
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂²×C₃×C₄	—	12	2	no	—
114	D₃×Q₈	—	2	8	no	—
115	D₃×Q₈	—	4	4	no	—
116	C₂×C₄×D₃	—	4	2	no	—
117	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
118	C₂×Dic₆	—	2	2	no	—
119	C₂×Dic₆	—	4	2	no	—
120	C₃×Q₁₆	—	2	2	no	—
121	C₃×Q₁₆	—	2	2	no	—
122	C₃×Q₁₆	—	2	4	no	—
123	C₃×Q₁₆	—	3	4	no	—
124	C₃×Q₁₆	—	4	2	no	—
125	C₃×Q₁₆	—	6	8	no	—
126	C₃×Q₁₆	—	6	2	no	—
127	C₃×Q₁₆	C₂²×D₃	1	4	yes	—
128	C₃×Q₁₆	C₂³×C₃	1	4	yes	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	3	2	no	—
132	C₂×C₃×C₈	—	3	2	no	—
133	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
135	#216	—	2	8	no	—
136	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
137	Dic₁₂	—	2	2	no	—
138	Dic₁₂	C₂³×C₃	1	4	yes	—
139	D₃×C₈	C₃×D₄	1	2	yes	—
140	D₃×C₈	C₂³×C₃	1	2	yes	—
141	C₃×C₁₆	—	2	2	no	—
142	C₃×C₁₆	—	3	2	no	—
143	C₃×C₁₆	C₂²×D₃	1	2	yes	—
144	C₃×C₁₆	C₂³×C₃	1	2	yes	—
145	C₃⋊C₁₆	C₂³×C₃	1	2	yes	—
146	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
147	C₂×C₃×D₃	Q₃₂	1	4	yes	direct product factor
148	C₂×C₃×D₃	C₂×D₈	1	2	yes	—
149	C₃²×C₄	—	4	4	no	—
150	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
151	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₃×Dic₃	—	4	4	no	—
153	C₃×Dic₃	C₂×D₈	1	2	yes	—
154	C₂²×Q₈	—	6	2	no	—
155	C₂×Q₁₆	—	3	8	no	—
156	C₂×Q₁₆	—	3	2	no	—
157	C₂×Q₁₆	—	6	4	no	—
158	C₂×Q₁₆	—	6	2	no	—
159	C₂×Q₁₆	C₂×C₃×D₃	1	2	yes	—
160	C₂²×C₈	—	3	1	no	—
161	Q₃₂	—	3	8	no	—
162	Q₃₂	—	3	4	no	—
163	Q₃₂	C₂×C₃×D₃	1	4	yes	direct product factor
164	C₂×C₁₆	—	3	1	no	—
165	C₂×C₁₆	—	3	4	no	—
166	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
167	C₂³×C₃	—	3	1	no	—
168	C₂²×D₃	C₃×D₈	1	1	yes	characteristic
169	C₃×Q₈	—	2	4	no	—
170	C₃×Q₈	—	2	4	no	—
171	C₃×Q₈	—	4	4	no	—
172	C₃×Q₈	—	4	2	no	—
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	6	4	no	—
175	C₃×Q₈	—	8	2	no	—
176	C₃×Q₈	—	12	8	no	—
177	C₃×Q₈	—	12	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	—	3	2	no	—
181	C₂×C₃×C₄	—	3	2	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	2	no	—
184	C₂×C₃×C₄	—	8	2	no	—
185	C₂×C₃×C₄	—	12	8	no	—
186	C₂×C₃×C₄	—	12	2	no	—
187	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×Dic₃	—	4	2	no	—
190	C₂×Dic₃	C₃×D₈	1	1	yes	characteristic
191	C₄×D₃	—	4	8	no	—
192	C₄×D₃	C₃×D₈	1	2	yes	—
193	C₄×D₃	C₂×C₃×D₄	1	2	yes	—
194	Dic₆	—	2	4	no	—
195	Dic₆	—	4	2	no	—
196	Dic₆	—	4	8	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	3	2	no	—
200	C₃×C₈	D₃×D₄	1	1	yes	characteristic
201	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
202	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
203	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
204	C₃⋊C₈	C₂×C₃×D₄	1	2	yes	—
205	C₂×C₃²	C₂×Q₃₂	1	2	yes	complemented
206	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
207	C₃×D₃	C₂×Q₃₂	1	4	yes	direct product factor
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	6	8	no	—
210	C₂×Q₈	—	6	2	no	—
211	C₂×Q₈	—	12	2	no	—
212	C₂×Q₈	—	12	4	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	12	2	no	—
215	Q₁₆	—	2	2	no	—
216	Q₁₆	—	3	4	no	—
217	Q₁₆	—	6	8	no	—
218	Q₁₆	—	6	2	no	—
219	Q₁₆	C₂²×C₃×D₃	1	4	yes	—
220	C₂×C₈	—	3	1	no	—
221	C₂×C₈	—	3	2	no	—
222	C₂×C₈	—	3	2	no	—
223	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
224	C₁₆	—	3	2	no	—
225	C₁₆	C₂²×C₃×D₃	1	2	yes	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	3	4	no	—
228	C₂²×C₃	—	3	2	no	—
229	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
231	C₂×D₃	C₃×Q₃₂	1	4	yes	direct product factor
232	C₂×D₃	C₂×C₃×D₈	1	2	yes	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	3	2	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	—	4	4	no	—
238	C₃×C₄	—	8	4	no	—
239	C₃×C₄	—	12	4	no	—
240	C₃×C₄	D₃×D₈	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
242	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
243	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
244	Dic₃	—	4	4	no	—
245	Dic₃	C₂×C₃×D₈	1	2	yes	—
246	C₃²	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	3	1	no	—
248	Q₈	—	2	4	no	—
249	Q₈	—	4	2	no	—
250	Q₈	—	6	4	no	—
251	Q₈	—	12	8	no	—
252	Q₈	—	12	2	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	12	8	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
260	C₈	—	3	2	no	—
261	C₈	C₃×D₃×D₄	1	1	yes	characteristic
262	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	3	4	no	—
266	C₂×C₃	D₃×Q₃₂	1	2	yes	direct product factor
267	C₂×C₃	C₂×C₃×Q₃₂	1	2	yes	complemented
268	C₂×C₃	C₂×D₃×D₈	1	1	yes	characteristic
269	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
270	D₃	C₂×C₃×Q₃₂	1	4	yes	direct product factor
271	C₂²	—	3	2	no	—
272	C₂²	—	3	4	no	—
273	C₂²	C₃×D₃×D₈	1	1	yes	characteristic
274	C₄	—	3	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	12	4	no	—
277	C₄	C₃×D₃×D₈	1	1	yes	characteristic
278	C₄	C₂×C₃×D₃×D₄	1	1	yes	characteristic
279	C₃	—	2	1	no	—
280	C₃	C₂×D₃×Q₃₂	1	1	yes	characteristic, direct product factor
281	C₃	C₂²×C₃×Q₃₂	1	1	yes	characteristic, complemented
282	C₂	—	3	4	no	—
283	C₂	C₃×D₃×Q₃₂	1	2	yes	direct product factor
284	C₂	C₂×C₃×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₃₂	C₂
socle	C₂²×C₃²	C₂×D₈
prime radical	C₂²×C₃×D₃	D₈
hypercenter	C₂×C₃×Q₃₂	D₃
hypocenter	C₃	C₂²×C₃×Q₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×Q₃₂	C₂	semidirect
3. C₂×C₃×#829	C₂	semidirect
4. C₃×D₃×Q₃₂	C₂	direct (N × H)
5. C₂×C₃×Dic₂₄	C₂	semidirect
8. C₂×D₃×Q₃₂	C₃	direct (N × H)
16. C₃²×Q₃₂	C₂²	semidirect
18. C₃×#829	C₂²	semidirect
20. C₃×Dic₂₄	C₂²	semidirect
28. C₂×C₃×Q₃₂	D₃	direct (N × H)
29. C₂×C₃×Q₃₂	C₂×C₃	semidirect
31. C₂×#829	C₂×C₃	semidirect
32. D₃×Q₃₂	C₂×C₃	direct (N × H)
33. C₂×Dic₂₄	C₂×C₃	semidirect
72. C₃×Q₃₂	C₂×D₃	direct (N × H)
73. C₃×Q₃₂	C₂²×C₃	semidirect
79. #829	C₂²×C₃	semidirect
81. Dic₂₄	C₂²×C₃	semidirect
103. C₂×Q₃₂	C₃×D₃	direct (N × H)
147. C₂×C₃×D₃	Q₃₂	direct (N × H)
163. Q₃₂	C₂×C₃×D₃	direct (N × H)
205. C₂×C₃²	C₂×Q₃₂	semidirect
207. C₃×D₃	C₂×Q₃₂	direct (N × H)
231. C₂×D₃	C₃×Q₃₂	direct (N × H)
246. C₃²	C₂²×Q₃₂	semidirect
266. C₂×C₃	D₃×Q₃₂	direct (N × H)
267. C₂×C₃	C₂×C₃×Q₃₂	semidirect
270. D₃	C₂×C₃×Q₃₂	direct (N × H)
280. C₃	C₂×D₃×Q₃₂	direct (N × H)
281. C₃	C₂²×C₃×Q₃₂	semidirect
283. C₂	C₃×D₃×Q₃₂	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×C₃×D₃×Q₁₆	2
C₂×C₃²×Q₃₂	2
C₂×C₃×#829	2
C₃×D₃×Q₃₂	2
C₂×C₃×Dic₂₄	2
C₂×C₃×D₃×C₁₆	2
C₂²×C₃×Q₃₂	3
C₂×D₃×Q₃₂	3

Subgroups — C2×C3×D3×Q32 #11899472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×Q₃₂ #11899472