Subgroups — C₃×#70718 #11785506

Order 1152.

557 conjugacy classes of subgroups.

The subgroup lattice has 363 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×#5132	C₂	1	2	yes	maximal
4	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃⋊#338	C₂	1	2	yes	maximal, complemented
6	C₃×#1813	—	3	1	no	maximal
7	#70718	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×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃×#901	—	2	2	no	—
14	C₃×#901	—	2	2	no	—
15	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
16	C₂×C₃×D₃×C₈	C₄	1	2	yes	—
17	C₃²×#108	—	2	2	no	—
18	C₃×#824	—	2	2	no	—
19	C₃×#823	—	2	2	no	—
20	C₃×#819	—	2	2	no	—
21	C₂×C₃×#131	—	3	1	no	—
22	C₂×#901	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#108	—	2	1	no	—
24	C₂×C₃×#108	—	3	1	no	—
25	C₂×C₃×#108	D₃	1	1	yes	characteristic, complemented
26	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#413	—	3	2	no	—
28	#5132	C₂×C₃	1	2	yes	—
29	C₂×#819	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×#338	—	3	2	no	—
31	C₃⋊#338	C₂×C₃	1	2	yes	complemented
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×#67	Q₈	1	2	yes	—
34	C₃×#235	—	2	1	no	—
35	C₃×#235	—	2	1	no	—
36	C₃×#235	—	2	1	no	—
37	C₃×#235	—	2	1	no	—
38	C₂×C₃×D₁₂	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
40	C₂×C₃×Dic₆	D₄	1	1	yes	characteristic
41	C₃²×M₁₆	—	2	2	no	—
42	C₃²×M₁₆	—	2	2	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	2	yes	—
47	C₂×C₃²×C₈	D₄	1	2	yes	—
48	C₃×#208	—	2	1	no	—
49	C₃×#208	—	2	1	no	—
50	C₃×#208	—	2	2	no	—
51	C₃×#208	—	2	2	no	—
52	C₂×C₃×C₃⋊C₈	—	2	2	no	—
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
54	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
55	C₃×D₃×C₈	—	2	4	no	—
56	#1813	—	3	1	no	Sylow
57	C₂×C₃×#41	—	3	1	no	—
58	C₂×#235	C₂²×C₃	1	1	yes	characteristic
59	C₃×#131	—	6	2	no	—
60	C₃×#131	—	6	2	no	—
61	C₂×C₃×M₁₆	—	2	2	no	—
62	C₂×C₃×M₁₆	—	3	1	no	—
63	C₂×C₃×M₁₆	—	3	2	no	—
64	C₂×C₃×M₁₆	—	3	2	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	—	3	2	no	—
69	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂×#208	C₂²×C₃	1	2	yes	—
72	C₂×#208	C₂²×C₃	1	1	yes	characteristic
73	#901	—	2	2	no	—
74	#901	—	2	2	no	—
75	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
76	C₂×D₃×C₈	C₃×C₄	1	2	yes	—
77	C₃×#108	—	2	4	no	—
78	C₃×#108	—	2	2	no	—
79	C₃×#108	—	2	2	no	—
80	C₃×#108	—	6	2	no	—
81	C₃×#101	—	6	2	no	—
82	C₃×#100	—	6	2	no	—
83	#824	—	2	2	no	—
84	#823	—	2	2	no	—
85	#819	—	2	2	no	—
86	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₃×D₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	#41	1	1	yes	characteristic
93	C₂×C₃²×C₄	#41	1	1	yes	characteristic
94	C₃×#67	—	2	2	no	—
95	C₃×#67	—	2	2	no	—
96	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
97	C₃×D₁₂	—	2	1	no	—
98	C₃×D₁₂	—	2	1	no	—
99	C₃×C₄×D₃	—	2	2	no	—
100	C₃×C₄×D₃	—	2	2	no	—
101	C₃×Dic₆	—	2	1	no	—
102	C₃×Dic₆	—	2	1	no	—
103	C₃²×C₈	—	4	2	no	—
104	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
105	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
106	C₃×C₃⋊C₈	—	2	2	no	—
107	C₃×C₃⋊C₈	—	4	2	no	—
108	C₂×#131	—	3	1	no	—
109	C₂×#108	—	3	1	no	—
110	C₂×#108	C₃×D₃	1	1	yes	characteristic, complemented
111	#413	—	3	2	no	—
112	#338	—	3	2	no	—
113	C₃×#41	—	6	1	no	—
114	C₃×#41	—	6	1	no	—
115	C₃×#41	—	6	1	no	—
116	C₃×#41	—	6	1	no	—
117	C₂×C₃×Q₈	—	3	1	no	—
118	C₂×C₃×D₄	—	3	2	no	—
119	C₂×C₃×D₄	—	3	1	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	3	2	no	—
122	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂×#67	C₃×Q₈	1	2	yes	—
125	#235	—	2	1	no	—
126	#235	—	2	1	no	—
127	#235	—	2	1	no	—
128	#235	—	2	1	no	—
129	C₂×D₁₂	C₃×D₄	1	1	yes	characteristic
130	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
131	C₂×Dic₆	C₃×D₄	1	1	yes	characteristic
132	C₃×M₁₆	—	2	4	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	2	no	—
135	C₃×M₁₆	—	2	2	no	—
136	C₃×M₁₆	—	2	2	no	—
137	C₃×M₁₆	—	4	2	no	—
138	C₃×M₁₆	—	6	1	no	—
139	C₃×M₁₆	—	6	1	no	—
140	C₃×M₁₆	—	6	4	no	—
141	C₃×M₁₆	—	6	2	no	—
142	C₃×M₁₆	—	6	2	no	—
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	—	3	2	no	—
150	C₂×C₃×C₈	—	4	2	no	—
151	C₂×C₃×C₈	—	6	4	no	—
152	C₂×C₃×C₈	—	6	2	no	—
153	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
154	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
155	C₂×C₃×C₈	D₁₂	1	2	yes	—
156	C₂×C₃×C₈	#67	1	2	yes	—
157	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
159	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
160	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
161	#208	—	2	1	no	—
162	#208	—	2	1	no	—
163	#208	—	2	2	no	—
164	#208	—	2	2	no	—
165	C₂×C₃⋊C₈	—	2	2	no	—
166	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
167	ΓL(1,𝔽₂₅)	—	2	4	no	—
168	D₃×C₈	—	2	4	no	—
169	C₂²×C₃²	—	4	1	no	—
170	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₃×D₃	—	2	2	no	—
174	C₂×C₃×D₃	—	4	2	no	—
175	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₃²×C₄	#117	1	1	yes	characteristic
177	C₃²×C₄	#121	1	1	yes	characteristic
178	C₃²×C₄	#121	1	1	yes	characteristic
179	C₃×Dic₃	—	2	2	no	—
180	C₂×#41	—	3	1	no	—
181	#131	—	6	2	no	—
182	#131	—	6	2	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	3	2	no	—
185	C₂×M₁₆	—	3	2	no	—
186	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
187	C₂²×C₈	—	3	2	no	—
188	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
189	#108	—	2	2	no	—
190	#108	—	6	2	no	—
191	#101	—	6	2	no	—
192	#100	—	6	2	no	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	3	2	no	—
195	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
196	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
197	C₂²×D₃	C₃×C₄⋊C₄	1	2	yes	—
198	C₃×Q₈	—	6	1	no	—
199	C₃×Q₈	—	6	1	no	—
200	C₃×D₄	—	6	2	no	—
201	C₃×D₄	—	6	2	no	—
202	C₃×D₄	—	6	1	no	—
203	C₃×D₄	—	6	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	3	2	no	—
211	C₂×C₃×C₄	—	6	2	no	—
212	C₂×C₃×C₄	—	6	2	no	—
213	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
215	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	#237	1	1	yes	characteristic
217	C₂×C₃×C₄	#239	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
224	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
225	#67	—	2	2	no	—
226	#67	—	2	2	no	—
227	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
228	D₁₂	—	2	1	no	—
229	D₁₂	—	2	1	no	—
230	C₄×D₃	—	2	2	no	—
231	C₄×D₃	—	2	2	no	—
232	Dic₆	—	2	1	no	—
233	Dic₆	—	2	1	no	—
234	C₃×C₈	—	2	2	no	—
235	C₃×C₈	—	2	2	no	—
236	C₃×C₈	—	4	4	no	—
237	C₃×C₈	—	4	2	no	—
238	C₃×C₈	—	4	2	no	—
239	C₃×C₈	—	6	2	no	—
240	C₃×C₈	—	12	2	no	—
241	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
242	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
243	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
244	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
245	C₃⋊C₈	—	2	2	no	—
246	C₃⋊C₈	—	4	2	no	—
247	C₂×C₃²	—	2	1	no	—
248	C₂×C₃²	—	2	1	no	—
249	C₂×C₃²	—	2	1	no	—
250	C₂×C₃²	#380	1	1	yes	characteristic
251	C₃×D₃	—	4	2	no	—
252	#41	—	6	1	no	—
253	#41	—	6	1	no	—
254	#41	—	6	1	no	—
255	#41	—	6	1	no	—
256	C₂×Q₈	—	3	1	no	—
257	C₂×D₄	—	3	2	no	—
258	C₂×D₄	—	3	1	no	—
259	C₂²×C₄	—	3	2	no	—
260	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	6	1	no	—
264	M₁₆	—	6	1	no	—
265	M₁₆	—	6	4	no	—
266	M₁₆	—	6	2	no	—
267	M₁₆	—	6	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	3	2	no	—
270	C₂×C₈	—	6	4	no	—
271	C₂×C₈	—	6	2	no	—
272	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
273	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
274	C₂×C₈	C₃×D₁₂	1	2	yes	—
275	C₂×C₈	C₃×#67	1	2	yes	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	—	4	1	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	6	2	no	—
283	C₂²×C₃	—	12	2	no	—
284	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
285	C₂²×C₃	#897	1	1	yes	characteristic
286	C₂²×C₃	#946	1	1	yes	characteristic
287	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
288	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
290	C₂×D₃	—	2	2	no	—
291	C₂×D₃	—	4	2	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	6	2	no	—
297	C₃×C₄	#892	1	1	yes	characteristic
298	C₃×C₄	#895	1	1	yes	characteristic
299	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
300	C₃×C₄	#938	1	1	yes	characteristic
301	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
302	C₃×C₄	C₃×#117	1	1	yes	characteristic
303	C₃×C₄	C₃×#121	1	1	yes	characteristic
304	C₃×C₄	C₃×#121	1	1	yes	characteristic
305	Dic₃	—	2	2	no	—
306	C₃²	#1813	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	3	2	no	—
308	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
309	Q₈	—	6	1	no	—
310	Q₈	—	6	1	no	—
311	D₄	—	6	2	no	—
312	D₄	—	6	2	no	—
313	D₄	—	6	1	no	—
314	D₄	—	6	1	no	—
315	C₂×C₄	—	3	2	no	—
316	C₂×C₄	—	6	2	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
320	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
321	C₂×C₄	C₃×#237	1	1	yes	characteristic
322	C₂×C₄	C₃×#239	1	1	yes	characteristic
323	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
324	C₈	—	4	2	no	—
325	C₈	—	6	2	no	—
326	C₈	—	12	2	no	—
327	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
328	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
329	C₂×C₃	—	2	2	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	4	1	no	—
339	C₂×C₃	—	12	2	no	—
340	C₂×C₃	#4987	1	1	yes	characteristic
341	C₂×C₃	C₃×#380	1	1	yes	characteristic
342	D₃	—	4	2	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	6	2	no	—
345	C₂²	—	12	2	no	—
346	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
347	C₂²	C₃×#897	1	1	yes	characteristic
348	C₂²	C₃×#946	1	1	yes	characteristic
349	C₄	—	6	2	no	—
350	C₄	C₃×#892	1	1	yes	characteristic
351	C₄	C₃×#895	1	1	yes	characteristic
352	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
353	C₄	C₃×#938	1	1	yes	characteristic
354	C₃	—	2	1	no	—
355	C₃	#70718	1	1	yes	characteristic, direct product factor
356	C₃	C₃×#1813	1	1	yes	characteristic, complemented
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	12	2	no	—
361	C₂	C₃×#4987	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#892
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1813
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ 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×#819	C₂	semidirect
5. C₃×C₃⋊#338	C₂	semidirect
7. #70718	C₃	direct (N × H)
25. C₂×C₃×#108	D₃	semidirect
26. C₂×C₃×#108	C₂×C₃	semidirect
29. C₂×#819	C₂×C₃	semidirect
31. C₃⋊#338	C₂×C₃	semidirect
110. C₂×#108	C₃×D₃	semidirect
306. C₃²	#1813	semidirect
355. C₃	#70718	direct (N × H)
356. C₃	C₃×#1813	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×#5132	2
C₂×C₃×#819	2
C₃×C₃⋊#338	2
C₃×#1813	3
#70718	3

Subgroups — C3×#70718 #11785506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70718 #11785506