Subgroups — C₃×#70720 #11785571

Order 1152.

581 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5133	C₂	1	2	yes	maximal
4	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4554	C₂	1	2	yes	maximal, complemented
6	C₃×#1815	—	3	1	no	maximal
7	#70720	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#235	C₂²	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃×#901	—	2	2	no	—
14	C₃×#901	C₄	1	4	yes	complemented
15	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
16	C₂×C₃×D₃×C₈	—	2	1	no	—
17	C₃²×#108	—	2	2	no	—
18	C₃×#821	—	2	2	no	—
19	C₃×#817	—	2	1	no	—
20	C₃×#817	—	2	1	no	—
21	C₃×#816	—	2	2	no	—
22	C₂×C₃×#131	—	3	1	no	—
23	C₂×#901	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#108	—	2	1	no	—
25	C₂×C₃×#108	D₃	1	1	yes	characteristic, complemented
26	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×C₃×#106	—	3	1	no	—
28	C₃×#417	—	3	2	no	—
29	#5133	C₂×C₃	1	2	yes	—
30	C₂×#817	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#343	—	3	2	no	—
32	#4554	C₂×C₃	1	2	yes	complemented
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×#67	—	2	1	no	—
35	C₂²×C₃×Dic₃	—	2	2	no	—
36	C₃×#235	—	2	1	no	—
37	C₃×#235	—	2	1	no	—
38	C₃×#235	C₂×C₄	1	2	yes	—
39	C₃×#235	Q₈	1	2	yes	—
40	C₂×C₃×D₁₂	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	2	1	no	—
42	C₂×C₃×Dic₆	D₄	1	1	yes	characteristic
43	C₃²×M₁₆	—	2	2	no	—
44	C₃²×M₁₆	—	2	2	no	—
45	C₂×C₃²×C₈	—	2	2	no	—
46	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
48	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
49	C₂×C₃²×C₈	D₄	1	2	yes	—
50	C₃×C₄⋊Dic₃	—	2	2	no	—
51	C₃×C₄⋊Dic₃	—	2	2	no	—
52	C₃×#208	—	2	1	no	—
53	C₃×#208	C₂×C₄	1	2	yes	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
56	C₃×D₃×C₈	—	2	4	no	—
57	#1815	—	3	1	no	Sylow
58	C₂×C₃×#41	—	3	1	no	—
59	C₂×#235	C₂²×C₃	1	1	yes	characteristic
60	C₃×#131	—	3	4	no	—
61	C₃×#131	—	6	2	no	—
62	C₂×C₃×M₁₆	—	2	2	no	—
63	C₂×C₃×M₁₆	—	3	1	no	—
64	C₂×C₃×M₁₆	—	6	1	no	—
65	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
66	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×C₈	—	2	1	no	—
68	C₂²×C₃×C₈	—	6	1	no	—
69	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	—	3	2	no	—
72	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
73	C₂×#208	C₂²×C₃	1	1	yes	characteristic
74	#901	—	2	2	no	—
75	#901	C₃×C₄	1	4	yes	complemented
76	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
77	C₂×D₃×C₈	—	2	1	no	—
78	C₃×#108	—	2	4	no	—
79	C₃×#108	—	2	2	no	—
80	C₃×#108	—	2	2	no	—
81	C₃×#106	—	6	1	no	—
82	C₃×#106	—	6	1	no	—
83	C₃×#103	—	6	2	no	—
84	C₃×#102	—	6	2	no	—
85	#821	—	2	2	no	—
86	#817	—	2	1	no	—
87	#817	—	2	1	no	—
88	#816	—	2	2	no	—
89	C₂³×C₃²	#41	1	1	yes	characteristic
90	C₂²×C₃×D₃	—	2	1	no	—
91	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₃²×C₄	#41	1	1	yes	characteristic
97	C₃×#67	—	2	2	no	—
98	C₃×#67	—	2	2	no	—
99	C₂×C₃×Dic₃	—	2	1	no	—
100	C₂×C₃×Dic₃	—	4	2	no	—
101	C₂×C₃×Dic₃	—	8	2	no	—
102	C₃×D₁₂	—	2	1	no	—
103	C₃×D₁₂	C₄⋊C₄	1	2	yes	—
104	C₃×C₄×D₃	—	2	2	no	—
105	C₃×C₄×D₃	—	2	2	no	—
106	C₃×Dic₆	—	2	1	no	—
107	C₃×Dic₆	C₄⋊C₄	1	2	yes	—
108	C₃²×C₈	—	4	2	no	—
109	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
110	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
111	C₃×C₃⋊C₈	—	2	2	no	—
112	C₂×#131	—	3	1	no	—
113	C₂×#108	C₃×D₃	1	1	yes	characteristic, complemented
114	C₂×#106	—	3	1	no	—
115	#417	—	3	2	no	—
116	#343	—	3	2	no	—
117	C₃×#41	—	3	2	no	—
118	C₃×#41	—	3	2	no	—
119	C₃×#41	—	6	1	no	—
120	C₃×#41	—	6	1	no	—
121	C₂×C₃×Q₈	—	3	1	no	—
122	C₂×C₃×D₄	—	3	1	no	—
123	C₂×C₃×D₄	—	6	1	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	—	6	1	no	—
126	C₂²×C₃×C₄	—	6	2	no	—
127	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂×#67	—	2	1	no	—
130	C₂²×Dic₃	—	2	2	no	—
131	#235	—	2	1	no	—
132	#235	—	2	1	no	—
133	#235	C₂×C₃×C₄	1	2	yes	—
134	#235	C₃×Q₈	1	2	yes	—
135	C₂×D₁₂	C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×D₃	—	2	1	no	—
137	C₂×Dic₆	C₃×D₄	1	1	yes	characteristic
138	C₃×M₁₆	—	2	4	no	—
139	C₃×M₁₆	—	2	2	no	—
140	C₃×M₁₆	—	2	2	no	—
141	C₃×M₁₆	—	2	2	no	—
142	C₃×M₁₆	—	2	2	no	—
143	C₃×M₁₆	—	3	2	no	—
144	C₃×M₁₆	—	4	2	no	—
145	C₃×M₁₆	—	6	1	no	—
146	C₃×M₁₆	—	6	4	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	2	no	—
153	C₂×C₃×C₈	—	4	2	no	—
154	C₂×C₃×C₈	—	6	1	no	—
155	C₂×C₃×C₈	—	6	4	no	—
156	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
157	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
158	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
159	C₂×C₃×C₈	#67	1	2	yes	—
160	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
161	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
162	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
163	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
164	C₃×C₄⋊C₄	—	6	2	no	—
165	C₃×C₄⋊C₄	—	6	2	no	—
166	C₄⋊Dic₃	—	2	2	no	—
167	C₄⋊Dic₃	—	2	2	no	—
168	#208	—	2	1	no	—
169	#208	C₂×C₃×C₄	1	2	yes	—
170	C₂×C₃⋊C₈	—	2	1	no	—
171	ΓL(1,𝔽₂₅)	—	2	4	no	—
172	D₃×C₈	—	2	4	no	—
173	C₂²×C₃²	—	2	2	no	—
174	C₂²×C₃²	#117	1	1	yes	characteristic
175	C₂²×C₃²	#121	1	1	yes	characteristic
176	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
177	C₂×C₃×D₃	—	2	2	no	—
178	C₂×C₃×D₃	—	4	2	no	—
179	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₃²×C₄	#121	1	1	yes	characteristic
182	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
183	C₃×Dic₃	—	2	2	no	—
184	C₃×Dic₃	—	8	2	no	—
185	C₂×#41	—	3	1	no	—
186	#131	—	3	4	no	—
187	#131	—	6	2	no	—
188	C₂×M₁₆	—	3	1	no	—
189	C₂×M₁₆	—	6	1	no	—
190	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
191	C₂²×C₈	—	6	1	no	—
192	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	3	2	no	—
194	#108	—	2	2	no	—
195	#106	—	6	1	no	—
196	#106	—	6	1	no	—
197	#103	—	6	2	no	—
198	#102	—	6	2	no	—
199	C₂³×C₃	—	2	1	no	—
200	C₂³×C₃	—	6	1	no	—
201	C₂³×C₃	#239	1	1	yes	characteristic
202	C₂³×C₃	C₃×#41	1	1	yes	characteristic
203	C₂²×D₃	—	2	1	no	—
204	C₃×Q₈	—	3	2	no	—
205	C₃×Q₈	—	6	1	no	—
206	C₃×D₄	—	3	2	no	—
207	C₃×D₄	—	6	2	no	—
208	C₃×D₄	—	6	2	no	—
209	C₃×D₄	—	6	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	6	2	no	—
218	C₂×C₃×C₄	—	6	2	no	—
219	C₂×C₃×C₄	—	12	2	no	—
220	C₂×C₃×C₄	—	24	2	no	—
221	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
223	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	#237	1	1	yes	characteristic
225	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
232	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
233	#67	—	2	2	no	—
234	#67	—	2	2	no	—
235	C₂×Dic₃	—	2	1	no	—
236	C₂×Dic₃	—	4	2	no	—
237	C₂×Dic₃	—	8	2	no	—
238	D₁₂	—	2	1	no	—
239	D₁₂	C₃×C₄⋊C₄	1	2	yes	—
240	C₄×D₃	—	2	2	no	—
241	C₄×D₃	—	2	2	no	—
242	Dic₆	—	2	1	no	—
243	Dic₆	C₃×C₄⋊C₄	1	2	yes	—
244	C₃×C₈	—	2	2	no	—
245	C₃×C₈	—	2	2	no	—
246	C₃×C₈	—	4	4	no	—
247	C₃×C₈	—	4	2	no	—
248	C₃×C₈	—	4	2	no	—
249	C₃×C₈	—	6	2	no	—
250	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
251	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
252	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
253	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
254	C₃⋊C₈	—	2	2	no	—
255	C₂×C₃²	—	2	1	no	—
256	C₂×C₃²	—	2	1	no	—
257	C₂×C₃²	#380	1	1	yes	characteristic
258	C₂×C₃²	#471	1	2	yes	—
259	C₃×D₃	—	4	2	no	—
260	#41	—	3	2	no	—
261	#41	—	3	2	no	—
262	#41	—	6	1	no	—
263	#41	—	6	1	no	—
264	C₂×Q₈	—	3	1	no	—
265	C₂×D₄	—	3	1	no	—
266	C₂×D₄	—	6	1	no	—
267	C₂²×C₄	—	6	1	no	—
268	C₂²×C₄	—	6	2	no	—
269	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	3	2	no	—
273	M₁₆	—	6	1	no	—
274	M₁₆	—	6	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	6	1	no	—
277	C₂×C₈	—	6	4	no	—
278	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
279	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
280	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
281	C₂×C₈	C₃×#67	1	2	yes	—
282	C₄⋊C₄	—	6	2	no	—
283	C₄⋊C₄	—	6	2	no	—
284	C₂²×C₃	—	2	2	no	—
285	C₂²×C₃	—	2	2	no	—
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	2	1	no	—
289	C₂²×C₃	—	4	2	no	—
290	C₂²×C₃	—	6	2	no	—
291	C₂²×C₃	—	12	2	no	—
292	C₂²×C₃	#892	1	1	yes	characteristic
293	C₂²×C₃	#895	1	1	yes	characteristic
294	C₂²×C₃	#946	1	1	yes	characteristic
295	C₂²×C₃	C₃×#117	1	1	yes	characteristic
296	C₂²×C₃	C₃×#121	1	1	yes	characteristic
297	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂×D₃	—	2	2	no	—
299	C₂×D₃	—	4	2	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	6	2	no	—
305	C₃×C₄	—	24	2	no	—
306	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
307	C₃×C₄	#897	1	1	yes	characteristic
308	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	C₃×C₄	#938	1	1	yes	characteristic
310	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
311	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
312	C₃×C₄	C₃×#121	1	1	yes	characteristic
313	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
314	Dic₃	—	2	2	no	—
315	Dic₃	—	8	2	no	—
316	C₃²	#1815	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	6	1	no	—
318	C₂³	C₃×#239	1	1	yes	characteristic
319	Q₈	—	3	2	no	—
320	Q₈	—	6	1	no	—
321	D₄	—	3	2	no	—
322	D₄	—	6	2	no	—
323	D₄	—	6	2	no	—
324	D₄	—	6	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	12	2	no	—
329	C₂×C₄	—	24	2	no	—
330	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
332	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
333	C₂×C₄	C₃×#237	1	1	yes	characteristic
334	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
335	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	6	2	no	—
338	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
339	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
340	C₂×C₃	—	2	1	no	—
341	C₂×C₃	—	2	1	no	—
342	C₂×C₃	—	2	1	no	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	2	no	—
346	C₂×C₃	—	4	1	no	—
347	C₂×C₃	—	4	1	no	—
348	C₂×C₃	—	12	2	no	—
349	C₂×C₃	#4898	1	2	yes	—
350	C₂×C₃	#4987	1	1	yes	characteristic
351	C₂×C₃	C₃×#380	1	1	yes	characteristic
352	C₂×C₃	C₃×#471	1	2	yes	—
353	D₃	—	4	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	6	2	no	—
356	C₂²	—	12	2	no	—
357	C₂²	C₃×#892	1	1	yes	characteristic
358	C₂²	C₃×#895	1	1	yes	characteristic
359	C₂²	C₃×#946	1	1	yes	characteristic
360	C₄	—	6	2	no	—
361	C₄	—	24	2	no	—
362	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
363	C₄	C₃×#897	1	1	yes	characteristic
364	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
365	C₄	C₃×#938	1	1	yes	characteristic
366	C₃	—	2	1	no	—
367	C₃	#70720	1	1	yes	characteristic, direct product factor
368	C₃	C₃×#1815	1	1	yes	characteristic, complemented
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	12	2	no	—
372	C₂	C₃×#4898	1	2	yes	—
373	C₂	C₃×#4987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#895
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	#121
prime radical	C₂×C₃×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1815
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
4. C₂×C₃×#817	C₂	semidirect
5. C₃×#4554	C₂	semidirect
7. #70720	C₃	direct (N × H)
14. C₃×#901	C₄	semidirect
25. C₂×C₃×#108	D₃	semidirect
26. C₂×C₃×#108	C₂×C₃	semidirect
30. C₂×#817	C₂×C₃	semidirect
32. #4554	C₂×C₃	semidirect
75. #901	C₃×C₄	semidirect
113. C₂×#108	C₃×D₃	semidirect
316. C₃²	#1815	semidirect
367. C₃	#70720	direct (N × H)
368. C₃	C₃×#1815	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#901	2
C₂×C₃²×#108	2
C₃×#5133	2
C₂×C₃×#817	2
C₃×#4554	2
C₃×#1815	3
#70720	3

Subgroups — C3×#70720 #11785571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70720 #11785571