Subgroups — C₃×#83696 #11895455

Order 1152.

772 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#557	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#5090	C₂	1	3	yes	maximal
4	C₃×#5079	C₂	1	3	yes	maximal
5	C₃×C₄×#216	C₂	1	3	yes	maximal
6	C₃×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₃×#5021	C₂	1	3	yes	maximal
8	C₃×#3363	—	3	1	no	maximal
9	#83696	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃²×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₃²×#125	—	2	3	no	—
12	C₃²×C₄×Q₈	—	2	1	no	—
13	C₃²×C₄×Q₈	C₂²	1	3	yes	—
14	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₃×Q₈×Dic₃	—	2	1	no	—
16	C₂×C₃×#216	C₂²	1	3	yes	—
17	C₃×#888	—	2	3	no	—
18	C₃×#869	C₂²	1	3	yes	—
19	C₃×C₄×Dic₆	C₂²	1	3	yes	—
20	C₃×#835	C₂²	1	6	yes	—
21	C₃×#835	C₂²	1	3	yes	—
22	C₃×#810	C₂²	1	1	yes	characteristic
23	C₃×#807	C₂²	1	3	yes	—
24	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	3	yes	—
26	C₃×Q₈²	—	3	1	no	—
27	C₃×#557	—	2	1	no	—
28	C₃×#557	D₃	1	1	yes	characteristic
29	C₃×#557	C₂×C₃	1	1	yes	characteristic
30	Q₈×Dic₆	C₂×C₃	1	1	yes	characteristic
31	C₃×#487	—	3	3	no	—
32	C₃×#477	—	3	3	no	—
33	C₃×#462	—	3	3	no	—
34	C₃×C₈×Q₈	—	3	1	no	—
35	C₃×C₄×Q₁₆	—	3	3	no	—
36	#5090	C₂×C₃	1	3	yes	—
37	#5079	C₂×C₃	1	3	yes	—
38	C₄×#216	C₂×C₃	1	3	yes	—
39	Q₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
40	#5021	C₂×C₃	1	3	yes	—
41	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
42	C₂×C₃²×Q₈	C₂³	1	3	yes	—
43	C₂×C₃×Dic₆	—	2	3	no	—
44	C₂×C₃×Dic₆	C₂³	1	1	yes	characteristic
45	C₃²×C₄⋊C₄	—	2	3	no	—
46	C₃²×C₄⋊C₄	—	2	3	no	—
47	C₃²×C₄⋊C₄	—	2	3	no	—
48	C₃²×C₄⋊C₄	D₄	1	3	yes	—
49	C₃²×C₄⋊C₄	C₂³	1	1	yes	characteristic
50	C₃²×C₄²	—	2	3	no	—
51	C₃²×C₄²	C₂³	1	3	yes	—
52	C₃×#216	—	2	6	no	—
53	C₃×C₄⋊Dic₃	—	2	3	no	—
54	C₃×C₄⋊Dic₃	C₂³	1	3	yes	—
55	C₃×Dic₃⋊C₄	—	2	3	no	—
56	C₃×C₄×Dic₃	—	2	3	no	—
57	C₂×C₃×C₃⋊C₈	C₂³	1	3	yes	—
58	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
59	#3363	—	3	1	no	Sylow
60	C₂×C₃×Q₁₆	—	3	3	no	—
61	C₃×C₄⋊Q₈	—	2	3	no	—
62	C₃×C₄⋊Q₈	—	3	3	no	—
63	C₃×C₄⋊Q₈	—	6	3	no	—
64	C₃×C₄⋊Q₈	C₂×D₃	1	3	yes	—
65	C₃×C₄⋊Q₈	C₂²×C₃	1	3	yes	—
66	C₃×#125	—	2	6	no	—
67	C₃×#125	—	2	3	no	—
68	C₃×#125	—	2	3	no	—
69	C₃×C₄×Q₈	—	2	2	no	—
70	C₃×C₄×Q₈	—	2	1	no	—
71	C₃×C₄×Q₈	—	2	1	no	—
72	C₃×C₄×Q₈	—	2	3	no	—
73	C₃×C₄×Q₈	—	2	1	no	—
74	C₃×C₄×Q₈	—	3	3	no	—
75	C₃×C₄×Q₈	—	6	1	no	—
76	C₃×C₄×Q₈	C₂×D₃	1	3	yes	—
77	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₃×C₄×Q₈	C₂²×C₃	1	3	yes	—
79	C₃×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
80	Q₈×Dic₃	—	2	1	no	—
81	C₂×#216	C₂²×C₃	1	3	yes	—
82	#888	—	2	3	no	—
83	#869	C₂²×C₃	1	3	yes	—
84	C₄×Dic₆	C₂²×C₃	1	3	yes	—
85	C₃×#107	—	3	3	no	—
86	C₃×C₄⋊C₈	—	3	3	no	—
87	C₃×#103	—	3	3	no	—
88	C₃×#103	—	3	6	no	—
89	C₃×#103	—	3	1	no	—
90	C₃×C₄×C₈	—	3	3	no	—
91	#835	C₂²×C₃	1	6	yes	—
92	#835	C₂²×C₃	1	3	yes	—
93	#810	C₂²×C₃	1	1	yes	characteristic
94	#807	C₂²×C₃	1	3	yes	—
95	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	3	yes	—
96	C₄×C₃⋊C₈	C₂²×C₃	1	3	yes	—
97	C₃²×Q₈	—	2	3	no	—
98	C₃²×Q₈	—	2	3	no	—
99	C₃²×Q₈	Q₁₆	1	4	yes	complemented
100	C₂×C₃²×C₄	—	2	1	no	—
101	C₂×C₃²×C₄	—	2	3	no	—
102	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
103	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
104	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	2	1	no	—
106	C₂×C₃×Dic₃	—	2	3	no	—
107	C₃×Dic₆	—	2	1	no	—
108	C₃×Dic₆	—	8	3	no	—
109	C₃×Dic₆	#41	1	2	yes	—
110	C₃×C₃⋊C₈	—	2	3	no	—
111	C₃×C₃⋊C₈	—	2	1	no	—
112	Q₈²	—	3	1	no	—
113	#557	C₃×D₃	1	1	yes	characteristic
114	#487	—	3	3	no	—
115	#477	—	3	3	no	—
116	#462	—	3	3	no	—
117	C₈×Q₈	—	3	1	no	—
118	C₄×Q₁₆	—	3	3	no	—
119	C₂×C₃×Q₈	—	2	1	no	—
120	C₂×C₃×Q₈	—	2	3	no	—
121	C₂×C₃×Q₈	—	3	1	no	—
122	C₂×C₃×Q₈	—	6	3	no	—
123	C₂×C₃×Q₈	#67	1	1	yes	characteristic
124	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
125	C₂×C₃×Q₈	C₂²×D₃	1	3	yes	—
126	C₂×C₃×Q₈	C₂³×C₃	1	3	yes	—
127	C₂×Dic₆	—	2	3	no	—
128	C₂×Dic₆	C₂³×C₃	1	1	yes	characteristic
129	C₃×Q₁₆	—	6	6	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	3	3	no	—
132	C₃×C₄⋊C₄	—	2	6	no	—
133	C₃×C₄⋊C₄	—	2	6	no	—
134	C₃×C₄⋊C₄	—	2	6	no	—
135	C₃×C₄⋊C₄	—	2	3	no	—
136	C₃×C₄⋊C₄	—	2	3	no	—
137	C₃×C₄⋊C₄	—	2	3	no	—
138	C₃×C₄⋊C₄	—	2	3	no	—
139	C₃×C₄⋊C₄	—	2	3	no	—
140	C₃×C₄⋊C₄	—	2	3	no	—
141	C₃×C₄⋊C₄	—	2	3	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	—	3	3	no	—
144	C₃×C₄⋊C₄	—	6	3	no	—
145	C₃×C₄⋊C₄	—	6	3	no	—
146	C₃×C₄⋊C₄	#67	1	3	yes	—
147	C₃×C₄⋊C₄	C₃×D₄	1	3	yes	—
148	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
150	C₃×C₄²	—	2	6	no	—
151	C₃×C₄²	—	2	3	no	—
152	C₃×C₄²	—	2	3	no	—
153	C₃×C₄²	—	2	3	no	—
154	C₃×C₄²	—	6	3	no	—
155	C₃×C₄²	C₂²×D₃	1	3	yes	—
156	C₃×C₄²	C₂³×C₃	1	3	yes	—
157	#216	—	2	6	no	—
158	C₄⋊Dic₃	—	2	3	no	—
159	C₄⋊Dic₃	C₂³×C₃	1	3	yes	—
160	Dic₃⋊C₄	—	2	3	no	—
161	C₄×Dic₃	—	2	3	no	—
162	C₂×C₃⋊C₈	C₂³×C₃	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₂³×C₃	1	3	yes	—
164	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
165	C₃²×C₄	—	2	3	no	—
166	C₃²×C₄	—	4	3	no	—
167	C₃²×C₄	—	4	1	no	—
168	C₃²×C₄	C₂×Q₁₆	1	6	yes	—
169	C₃²×C₄	C₂×#41	1	1	yes	characteristic
170	C₃²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
171	C₃×Dic₃	—	2	2	no	—
172	C₃×Dic₃	—	4	3	no	—
173	C₂×Q₁₆	—	3	3	no	—
174	C₄⋊Q₈	—	3	3	no	—
175	C₄⋊Q₈	—	6	3	no	—
176	C₄⋊Q₈	C₂×C₃×D₃	1	3	yes	—
177	#125	—	2	3	no	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	—	3	3	no	—
180	C₄×Q₈	—	6	1	no	—
181	C₄×Q₈	C₂×C₃×D₃	1	3	yes	—
182	C₄×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
183	#107	—	3	3	no	—
184	C₄⋊C₈	—	3	3	no	—
185	#103	—	3	3	no	—
186	#103	—	3	6	no	—
187	#103	—	3	1	no	—
188	C₄×C₈	—	3	3	no	—
189	C₃×Q₈	—	2	3	no	—
190	C₃×Q₈	—	2	3	no	—
191	C₃×Q₈	—	2	3	no	—
192	C₃×Q₈	—	2	3	no	—
193	C₃×Q₈	—	2	4	no	—
194	C₃×Q₈	—	3	2	no	—
195	C₃×Q₈	—	4	3	no	—
196	C₃×Q₈	—	4	3	no	—
197	C₃×Q₈	—	6	1	no	—
198	C₃×Q₈	—	24	3	no	—
199	C₃×Q₈	#216	1	4	yes	complemented
200	C₃×Q₈	C₃×Q₁₆	1	4	yes	complemented
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	6	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	3	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	3	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	3	no	—
209	C₂×C₃×C₄	—	2	3	no	—
210	C₂×C₃×C₄	—	6	1	no	—
211	C₂×C₃×C₄	—	6	3	no	—
212	C₂×C₃×C₄	C₂×#67	1	3	yes	—
213	C₂×C₃×C₄	C₂×#67	1	3	yes	—
214	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
215	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
216	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
218	C₂×Dic₃	—	2	1	no	—
219	C₂×Dic₃	—	2	3	no	—
220	Dic₆	—	2	1	no	—
221	Dic₆	—	8	3	no	—
222	Dic₆	C₃×#41	1	2	yes	—
223	C₃×C₈	—	6	3	no	—
224	C₃×C₈	—	6	1	no	—
225	C₃⋊C₈	—	2	3	no	—
226	C₃⋊C₈	—	2	1	no	—
227	C₂×C₃²	#548	1	1	yes	characteristic
228	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
229	C₂×C₃²	#577	1	1	yes	characteristic
230	C₂×Q₈	—	3	1	no	—
231	C₂×Q₈	—	6	3	no	—
232	C₂×Q₈	C₃×#67	1	1	yes	characteristic
233	C₂×Q₈	C₂²×C₃×D₃	1	3	yes	—
234	Q₁₆	—	6	6	no	—
235	C₂×C₈	—	3	1	no	—
236	C₂×C₈	—	3	3	no	—
237	C₄⋊C₄	—	2	3	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	2	3	no	—
240	C₄⋊C₄	—	3	3	no	—
241	C₄⋊C₄	—	6	3	no	—
242	C₄⋊C₄	—	6	3	no	—
243	C₄⋊C₄	C₃×#67	1	3	yes	—
244	C₄⋊C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₄²	—	2	3	no	—
246	C₄²	—	6	3	no	—
247	C₄²	C₂²×C₃×D₃	1	3	yes	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
250	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
251	C₃×C₄	—	2	3	no	—
252	C₃×C₄	—	2	3	no	—
253	C₃×C₄	—	2	6	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	4	3	no	—
257	C₃×C₄	—	4	6	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	3	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	4	3	no	—
263	C₃×C₄	—	6	2	no	—
264	C₃×C₄	—	12	3	no	—
265	C₃×C₄	C₂×#216	1	6	yes	—
266	C₃×C₄	C₂×C₃×Q₁₆	1	6	yes	—
267	C₃×C₄	D₃×#41	1	1	yes	characteristic
268	C₃×C₄	#1010	1	1	yes	characteristic
269	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
270	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
271	Dic₃	—	2	2	no	—
272	Dic₃	—	4	3	no	—
273	C₃²	#3363	1	1	yes	characteristic, complemented, Sylow
274	Q₈	—	2	3	no	—
275	Q₈	—	2	3	no	—
276	Q₈	—	3	2	no	—
277	Q₈	—	6	1	no	—
278	Q₈	—	24	3	no	—
279	Q₈	C₃×#216	1	4	yes	complemented
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	3	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	3	no	—
284	C₂×C₄	C₂×C₃×#67	1	3	yes	—
285	C₂×C₄	C₂×C₃×#67	1	3	yes	—
286	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
287	C₈	—	6	3	no	—
288	C₈	—	6	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	C₂²×#216	1	1	yes	characteristic
293	C₂×C₃	#5831	1	1	yes	characteristic
294	C₂×C₃	#5837	1	1	yes	characteristic
295	C₂×C₃	C₃×#548	1	1	yes	characteristic
296	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
297	C₂×C₃	C₃×#577	1	1	yes	characteristic
298	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
299	C₄	—	2	3	no	—
300	C₄	—	4	3	no	—
301	C₄	—	4	1	no	—
302	C₄	—	6	2	no	—
303	C₄	—	12	3	no	—
304	C₄	C₂×C₃×#216	1	6	yes	—
305	C₄	C₃×D₃×#41	1	1	yes	characteristic
306	C₄	C₃×#1010	1	1	yes	characteristic
307	C₃	—	2	1	no	—
308	C₃	#83696	1	1	yes	characteristic, direct product factor
309	C₃	C₃×#3363	1	1	yes	characteristic, complemented
310	C₂	C₂²×C₃×#216	1	1	yes	characteristic
311	C₂	C₃×#5831	1	1	yes	characteristic
312	C₂	C₃×#5837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#557	C₂
socle, prime radical	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×#557	D₃
hypocenter	C₃	C₃×#3363
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ C₃×#557

Complements

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

N	H ≅ G/N	type
9. #83696	C₃	direct (N × H)
99. C₃²×Q₈	Q₁₆	semidirect
199. C₃×Q₈	#216	semidirect
200. C₃×Q₈	C₃×Q₁₆	semidirect
273. C₃²	#3363	semidirect
279. Q₈	C₃×#216	semidirect
308. C₃	#83696	direct (N × H)
309. C₃	C₃×#3363	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#557	2
C₃×Q₈×Dic₆	2
C₃×#5090	2
C₃×#5079	2
C₃×C₄×#216	2
C₃×Q₈×C₃⋊C₈	2
C₃×#5021	2
C₃×#3363	3
#83696	3

Subgroups — C3×#83696 #11895455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#83696 #11895455