Subgroups — C₃×#82033 #11899757

Order 1152.

742 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5771	C₂	1	2	yes	maximal, complemented
2	C₃²×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5195	C₂	1	2	yes	maximal, complemented
4	C₃×D₃×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₃×#4915	C₂	1	4	yes	maximal, complemented
6	C₃×#4912	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3417	—	3	1	no	maximal
10	#82033	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₃×#1010	—	2	2	no	—
12	C₃²×#135	C₂²	1	2	yes	—
13	C₃×#951	—	2	2	no	—
14	C₃×D₃×Q₁₆	C₂²	1	4	yes	—
15	C₃×#915	—	2	4	no	—
16	C₃×#912	C₂²	1	4	yes	—
17	C₂×C₃×Dic₁₂	C₂²	1	2	yes	—
18	C₃×#901	C₂²	1	1	yes	characteristic
19	C₃²×Q₃₂	C₂²	1	1	yes	characteristic
20	C₃²×SD₃₂	C₂²	1	2	yes	—
21	C₃²×D₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
23	C₃×#829	C₂²	1	4	yes	—
24	C₃×#827	C₂²	1	4	yes	—
25	C₃×#812	C₂²	1	1	yes	characteristic
26	C₃×Dic₂₄	C₂²	1	2	yes	—
27	C₃×Dic₂₄	C₂²	1	2	yes	—
28	C₃×#798	C₂²	1	2	yes	—
29	C₃×D₃×C₁₆	C₂²	1	2	yes	—
30	C₃×#578	—	3	2	no	—
31	#5771	C₂×C₃	1	2	yes	complemented
32	C₃×#510	—	3	2	no	—
33	C₃×#510	—	3	4	no	—
34	C₃×#508	—	2	1	no	—
35	C₃×#508	—	3	2	no	—
36	C₃×#508	D₃	1	1	yes	characteristic, complemented
37	C₃×#508	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×Q₃₂	—	3	2	no	—
39	C₂×C₃×Q₃₂	—	3	1	no	—
40	C₃×#504	—	3	1	no	—
41	#5195	C₂×C₃	1	2	yes	complemented
42	D₃×Q₃₂	C₂×C₃	1	2	yes	complemented
43	#4915	C₂×C₃	1	4	yes	complemented
44	#4912	C₂×C₃	1	2	yes	complemented
45	C₂×Dic₂₄	C₂×C₃	1	1	yes	characteristic, complemented
46	#4901	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃²×#41	—	2	2	no	—
48	C₃×D₃×Q₈	—	2	4	no	—
49	C₃×#237	—	2	4	no	—
50	C₃×#237	—	4	4	no	—
51	C₃×#235	—	4	2	no	—
52	C₃×#235	D₄	1	1	yes	characteristic
53	C₂×C₃×Dic₆	—	2	2	no	—
54	C₂×C₃×Dic₆	—	4	2	no	—
55	C₃²×Q₁₆	C₂³	1	2	yes	—
56	C₃²×SD₁₆	—	2	2	no	—
57	C₃²×D₈	C₂³	1	2	yes	—
58	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
59	C₃×#216	—	2	4	no	—
60	C₃×#214	—	2	4	no	—
61	C₃×#208	D₄	1	1	yes	characteristic
62	C₃×Dic₁₂	—	2	2	no	—
63	C₃×Dic₁₂	C₂³	1	4	yes	—
64	C₃×ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
65	C₃×D₃×C₈	C₂³	1	2	yes	—
66	C₃²×C₁₆	C₂³	1	1	yes	characteristic
67	C₃²×C₁₆	C₂³	1	1	yes	characteristic
68	C₃×C₃⋊C₁₆	C₂³	1	2	yes	—
69	#3417	—	3	1	no	Sylow
70	C₃×2₋¹⁺⁴	—	6	2	no	—
71	#1010	—	2	2	no	—
72	C₃×#137	—	6	4	no	—
73	C₃×#137	—	6	2	no	—
74	C₃×#135	—	2	2	no	—
75	C₃×#135	—	3	4	no	—
76	C₃×#135	C₂×D₃	1	2	yes	—
77	C₃×#135	C₂²×C₃	1	2	yes	—
78	C₂×C₃×Q₁₆	—	3	4	no	—
79	C₂×C₃×Q₁₆	—	3	2	no	—
80	C₃×#131	—	3	1	no	—
81	#951	—	2	2	no	—
82	D₃×Q₁₆	C₂²×C₃	1	4	yes	—
83	#915	—	2	4	no	—
84	#912	C₂²×C₃	1	4	yes	—
85	C₂×Dic₁₂	C₂²×C₃	1	2	yes	—
86	#901	C₂²×C₃	1	1	yes	characteristic
87	C₃×Q₃₂	—	2	1	no	—
88	C₃×Q₃₂	—	3	4	no	—
89	C₃×Q₃₂	—	3	2	no	—
90	C₃×Q₃₂	—	3	2	no	—
91	C₃×Q₃₂	C₂×D₃	1	1	yes	characteristic
92	C₃×Q₃₂	C₂²×C₃	1	1	yes	characteristic
93	C₃×SD₃₂	—	2	2	no	—
94	C₃×SD₃₂	—	3	4	no	—
95	C₃×SD₃₂	C₂×D₃	1	2	yes	—
96	C₃×SD₃₂	C₂²×C₃	1	2	yes	—
97	C₃×D₁₆	—	2	1	no	—
98	C₃×D₁₆	C₂×D₃	1	1	yes	characteristic
99	C₃×D₁₆	C₂²×C₃	1	1	yes	characteristic
100	C₃×M₃₂	—	3	1	no	—
101	C₃×M₃₂	—	3	2	no	—
102	C₂×C₃×C₁₆	—	2	1	no	—
103	C₂×C₃×C₁₆	—	3	2	no	—
104	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
105	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
106	#829	C₂²×C₃	1	4	yes	—
107	#827	C₂²×C₃	1	4	yes	—
108	#812	C₂²×C₃	1	1	yes	characteristic
109	Dic₂₄	C₂²×C₃	1	2	yes	—
110	Dic₂₄	C₂²×C₃	1	2	yes	—
111	#798	C₂²×C₃	1	2	yes	—
112	D₃×C₁₆	C₂²×C₃	1	2	yes	—
113	C₃²×Q₈	—	2	2	no	—
114	C₃²×D₄	—	2	2	no	—
115	C₃²×D₄	—	4	2	no	—
116	C₂×C₃²×C₄	—	4	2	no	—
117	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
118	C₃×#67	—	4	4	no	—
119	C₃×#67	D₈	1	2	yes	—
120	C₂×C₃×Dic₃	—	4	2	no	—
121	C₂×C₃×Dic₃	—	4	4	no	—
122	C₃×D₁₂	D₈	1	1	yes	characteristic
123	C₃×C₄×D₃	—	4	4	no	—
124	C₃×C₄×D₃	C₂×D₄	1	2	yes	—
125	C₃×Dic₆	—	2	4	no	—
126	C₃×Dic₆	—	4	2	no	—
127	C₃×Dic₆	—	4	4	no	—
128	C₃×Dic₆	D₈	1	1	yes	characteristic
129	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
130	C₃²×C₈	C₂⁴	1	1	yes	characteristic
131	C₃×C₃⋊C₈	C₂×D₄	1	2	yes	—
132	#578	—	3	2	no	—
133	#510	—	3	2	no	—
134	#510	—	3	4	no	—
135	#508	—	3	2	no	—
136	#508	C₃×D₃	1	1	yes	characteristic, complemented
137	C₂×Q₃₂	—	3	2	no	—
138	C₂×Q₃₂	—	3	1	no	—
139	#504	—	3	1	no	—
140	C₃×#41	—	2	2	no	—
141	C₃×#41	—	2	2	no	—
142	C₃×#41	—	3	1	no	—
143	C₃×#41	—	4	2	no	—
144	C₃×#41	—	6	4	no	—
145	C₃×#41	—	12	2	no	—
146	C₃×#41	—	12	4	no	—
147	C₂×C₃×Q₈	—	6	4	no	—
148	C₂×C₃×Q₈	—	6	2	no	—
149	C₂×C₃×Q₈	—	12	2	no	—
150	D₃×Q₈	—	2	4	no	—
151	#237	—	2	4	no	—
152	#237	—	4	4	no	—
153	#235	—	4	2	no	—
154	#235	C₃×D₄	1	1	yes	characteristic
155	C₂×Dic₆	—	2	2	no	—
156	C₂×Dic₆	—	4	2	no	—
157	C₃×Q₁₆	—	2	2	no	—
158	C₃×Q₁₆	—	3	4	no	—
159	C₃×Q₁₆	—	6	4	no	—
160	C₃×Q₁₆	—	6	2	no	—
161	C₃×Q₁₆	C₂²×D₃	1	2	yes	—
162	C₃×Q₁₆	C₂³×C₃	1	2	yes	—
163	C₃×SD₁₆	—	2	2	no	—
164	C₃×SD₁₆	—	2	2	no	—
165	C₃×SD₁₆	—	4	2	no	—
166	C₃×SD₁₆	—	6	4	no	—
167	C₃×D₈	—	2	2	no	—
168	C₃×D₈	C₂²×D₃	1	2	yes	—
169	C₃×D₈	C₂³×C₃	1	2	yes	—
170	C₃×M₁₆	—	3	1	no	—
171	C₃×M₁₆	—	3	2	no	—
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	—	3	2	no	—
174	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
176	#216	—	2	4	no	—
177	#214	—	2	4	no	—
178	#208	C₃×D₄	1	1	yes	characteristic
179	Dic₁₂	—	2	2	no	—
180	Dic₁₂	C₂³×C₃	1	4	yes	—
181	ΓL(1,𝔽₂₅)	C₃×D₄	1	2	yes	—
182	D₃×C₈	C₂³×C₃	1	2	yes	—
183	C₃×C₁₆	—	2	1	no	—
184	C₃×C₁₆	—	2	1	no	—
185	C₃×C₁₆	—	3	2	no	—
186	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
187	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
188	C₃×C₁₆	C₂³×C₃	1	1	yes	characteristic
189	C₃×C₁₆	C₂³×C₃	1	1	yes	characteristic
190	C₃⋊C₁₆	C₂³×C₃	1	2	yes	—
191	C₂²×C₃²	—	4	2	no	—
192	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
193	C₂×C₃×D₃	C₂×D₈	1	2	yes	—
194	C₃²×C₄	—	4	2	no	—
195	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
196	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
197	C₃×Dic₃	—	4	4	no	—
198	C₃×Dic₃	C₂×D₈	1	2	yes	—
199	2₋¹⁺⁴	—	6	2	no	—
200	#137	—	6	4	no	—
201	#137	—	6	2	no	—
202	#135	—	3	4	no	—
203	#135	C₂×C₃×D₃	1	2	yes	—
204	C₂×Q₁₆	—	3	4	no	—
205	C₂×Q₁₆	—	3	2	no	—
206	#131	—	3	1	no	—
207	Q₃₂	—	3	4	no	—
208	Q₃₂	—	3	2	no	—
209	Q₃₂	—	3	2	no	—
210	Q₃₂	C₂×C₃×D₃	1	1	yes	characteristic
211	SD₃₂	—	3	4	no	—
212	SD₃₂	C₂×C₃×D₃	1	2	yes	—
213	D₁₆	C₂×C₃×D₃	1	1	yes	characteristic
214	M₃₂	—	3	1	no	—
215	M₃₂	—	3	2	no	—
216	C₂×C₁₆	—	3	2	no	—
217	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
218	C₃×Q₈	—	2	2	no	—
219	C₃×Q₈	—	2	2	no	—
220	C₃×Q₈	—	3	1	no	—
221	C₃×Q₈	—	4	2	no	—
222	C₃×Q₈	—	6	4	no	—
223	C₃×Q₈	—	12	4	no	—
224	C₃×Q₈	—	12	2	no	—
225	C₃×D₄	—	2	2	no	—
226	C₃×D₄	—	2	2	no	—
227	C₃×D₄	—	3	2	no	—
228	C₃×D₄	—	3	1	no	—
229	C₃×D₄	—	4	2	no	—
230	C₃×D₄	—	4	2	no	—
231	C₃×D₄	—	4	2	no	—
232	C₃×D₄	—	8	2	no	—
233	C₃×D₄	—	12	4	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	3	2	no	—
236	C₂×C₃×C₄	—	4	2	no	—
237	C₂×C₃×C₄	—	4	2	no	—
238	C₂×C₃×C₄	—	8	2	no	—
239	C₂×C₃×C₄	—	12	4	no	—
240	C₂×C₃×C₄	—	12	4	no	—
241	C₂×C₃×C₄	—	12	2	no	—
242	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	#67	—	4	4	no	—
245	#67	C₃×D₈	1	2	yes	—
246	C₂×Dic₃	—	4	2	no	—
247	C₂×Dic₃	—	4	4	no	—
248	D₁₂	C₃×D₈	1	1	yes	characteristic
249	C₄×D₃	—	4	4	no	—
250	C₄×D₃	C₂×C₃×D₄	1	2	yes	—
251	Dic₆	—	2	4	no	—
252	Dic₆	—	4	2	no	—
253	Dic₆	—	4	4	no	—
254	Dic₆	C₃×D₈	1	1	yes	characteristic
255	C₃×C₈	—	2	1	no	—
256	C₃×C₈	—	2	1	no	—
257	C₃×C₈	—	3	2	no	—
258	C₃×C₈	D₃×D₄	1	1	yes	characteristic
259	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
260	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
261	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
262	C₃⋊C₈	C₂×C₃×D₄	1	2	yes	—
263	C₂×C₃²	—	2	1	no	—
264	C₂×C₃²	—	8	2	no	—
265	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
266	C₃×D₃	—	2	2	no	—
267	#41	—	2	2	no	—
268	#41	—	3	1	no	—
269	#41	—	6	4	no	—
270	#41	—	12	2	no	—
271	#41	—	12	4	no	—
272	C₂×Q₈	—	6	4	no	—
273	C₂×Q₈	—	6	2	no	—
274	C₂×Q₈	—	12	2	no	—
275	Q₁₆	—	3	4	no	—
276	Q₁₆	—	6	4	no	—
277	Q₁₆	—	6	2	no	—
278	Q₁₆	C₂²×C₃×D₃	1	2	yes	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	6	4	no	—
281	D₈	C₂²×C₃×D₃	1	2	yes	—
282	M₁₆	—	3	1	no	—
283	M₁₆	—	3	2	no	—
284	C₂×C₈	—	3	2	no	—
285	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
286	C₁₆	—	3	2	no	—
287	C₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
288	C₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	3	2	no	—
291	C₂²×C₃	—	4	2	no	—
292	C₂²×C₃	—	4	2	no	—
293	C₂²×C₃	—	8	2	no	—
294	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
295	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
296	C₂×D₃	C₂×C₃×D₈	1	2	yes	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	3	2	no	—
300	C₃×C₄	—	4	2	no	—
301	C₃×C₄	—	4	2	no	—
302	C₃×C₄	—	8	2	no	—
303	C₃×C₄	—	12	4	no	—
304	C₃×C₄	D₃×D₈	1	1	yes	characteristic
305	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
306	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
307	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
308	Dic₃	—	4	4	no	—
309	Dic₃	C₂×C₃×D₈	1	2	yes	—
310	C₃²	#3417	1	1	yes	characteristic, complemented, Sylow
311	Q₈	—	2	2	no	—
312	Q₈	—	3	1	no	—
313	Q₈	—	6	4	no	—
314	Q₈	—	12	4	no	—
315	Q₈	—	12	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	3	2	no	—
318	D₄	—	3	1	no	—
319	D₄	—	4	2	no	—
320	D₄	—	12	4	no	—
321	C₂×C₄	—	3	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	12	4	no	—
324	C₂×C₄	—	12	4	no	—
325	C₂×C₄	—	12	2	no	—
326	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
327	C₈	—	3	2	no	—
328	C₈	C₃×D₃×D₄	1	1	yes	characteristic
329	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	4	1	no	—
334	C₂×C₃	—	6	2	no	—
335	C₂×C₃	—	8	2	no	—
336	C₂×C₃	—	8	2	no	—
337	C₂×C₃	—	16	2	no	—
338	C₂×C₃	C₂×D₃×D₈	1	1	yes	characteristic
339	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
340	D₃	—	2	2	no	—
341	C₂²	—	3	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	C₃×D₃×D₈	1	1	yes	characteristic
344	C₄	—	3	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	12	4	no	—
347	C₄	C₃×D₃×D₈	1	1	yes	characteristic
348	C₄	C₂×C₃×D₃×D₄	1	1	yes	characteristic
349	C₃	—	2	1	no	—
350	C₃	#82033	1	1	yes	characteristic, direct product factor
351	C₃	C₃×#3417	1	1	yes	characteristic, complemented
352	C₂	—	2	1	no	—
353	C₂	—	6	2	no	—
354	C₂	—	8	2	no	—
355	C₂	C₂×C₃×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₃²×#508	C₂
socle	C₂×C₃²	C₂²×D₈
hypercenter	C₃×#508	D₃
hypocenter	C₃	C₃×#3417
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#508

Complements

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

N	H ≅ G/N	type
1. C₃×#5771	C₂	semidirect
2. C₃²×#508	C₂	semidirect
3. C₃×#5195	C₂	semidirect
4. C₃×D₃×Q₃₂	C₂	semidirect
5. C₃×#4915	C₂	semidirect
6. C₃×#4912	C₂	semidirect
7. C₂×C₃×Dic₂₄	C₂	semidirect
8. C₃×#4901	C₂	semidirect
10. #82033	C₃	direct (N × H)
31. #5771	C₂×C₃	semidirect
36. C₃×#508	D₃	semidirect
37. C₃×#508	C₂×C₃	semidirect
41. #5195	C₂×C₃	semidirect
42. D₃×Q₃₂	C₂×C₃	semidirect
43. #4915	C₂×C₃	semidirect
44. #4912	C₂×C₃	semidirect
45. C₂×Dic₂₄	C₂×C₃	semidirect
46. #4901	C₂×C₃	semidirect
136. #508	C₃×D₃	semidirect
310. C₃²	#3417	semidirect
350. C₃	#82033	direct (N × H)
351. C₃	C₃×#3417	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5771	2
C₃²×#508	2
C₃×#5195	2
C₃×D₃×Q₃₂	2
C₃×#4915	2
C₃×#4912	2
C₂×C₃×Dic₂₄	2
C₃×#4901	2
C₃×#3417	3
#82033	3

Subgroups — C3×#82033 #11899757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#82033 #11899757