Subgroups — C₃×#70847 #11780884

Order 1152.

475 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#5015	C₂	1	2	yes	maximal, complemented
3	C₃×#4695	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4485	C₂	1	2	yes	maximal, complemented
5	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1667	—	3	1	no	maximal
7	#70847	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	—	2	1	no	—
9	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×#213	—	2	2	no	—
11	C₂×C₃×D₂₄	—	2	1	no	—
12	C₂×C₃×D₂₄	—	2	1	no	—
13	C₃×#874	C₂²	1	2	yes	—
14	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₂²⋊C₈	—	2	1	no	—
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#809	—	2	2	no	—
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×#493	—	3	1	no	—
20	C₃×#459	—	3	2	no	—
21	C₃×C₈×D₄	—	2	1	no	—
22	C₃×C₈×D₄	D₃	1	1	yes	characteristic, complemented
23	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
24	#5015	C₂×C₃	1	2	yes	complemented
25	#4695	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#335	—	3	1	no	—
27	C₃×#331	—	3	2	no	—
28	#4485	C₂×C₃	1	2	yes	complemented
29	#4458	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	—	2	1	no	—
32	C₂×C₃×D₁₂	—	2	2	no	—
33	C₂×C₃×D₁₂	—	2	2	no	—
34	C₂×C₃×D₁₂	—	2	2	no	—
35	C₂×C₃²×C₈	—	2	1	no	—
36	C₂×C₃²×C₈	—	4	2	no	—
37	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
39	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₃²×C₂²⋊C₄	—	2	1	no	—
41	C₃²×C₄²	C₂³	1	1	yes	characteristic
42	C₃×#213	—	4	4	no	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₃×D₂₄	—	4	2	no	—
45	C₃×D₂₄	—	4	2	no	—
46	#1667	—	3	1	no	Sylow
47	C₂×C₃×D₈	—	6	2	no	—
48	C₂×C₃×D₈	—	6	1	no	—
49	C₂×C₃×D₈	—	6	1	no	—
50	C₂²×C₃×C₈	—	2	2	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₃×#127	—	3	2	no	—
54	C₃×C₄×D₄	—	2	1	no	—
55	C₃×C₄×D₄	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×#213	—	2	2	no	—
58	C₂×D₂₄	—	2	1	no	—
59	C₂×D₂₄	—	2	1	no	—
60	#874	C₂²×C₃	1	2	yes	—
61	C₃×C₄⋊C₈	—	2	1	no	—
62	C₃×C₄⋊C₈	—	3	2	no	—
63	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₃×#102	—	6	2	no	—
66	C₃×C₂²⋊C₈	—	2	2	no	—
67	C₃×C₂²⋊C₈	—	2	1	no	—
68	C₃×C₂²⋊C₈	—	2	1	no	—
69	C₃×C₄×C₈	—	2	1	no	—
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
72	#809	—	2	2	no	—
73	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
74	C₂³×C₃²	—	2	1	no	—
75	C₂²×C₃×D₃	—	4	2	no	—
76	C₃²×D₄	—	2	1	no	—
77	C₃²×D₄	D₈	1	2	yes	complemented
78	C₂×C₃²×C₄	—	2	1	no	—
79	C₂×C₃²×C₄	—	4	1	no	—
80	C₂×C₃²×C₄	—	4	1	no	—
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	1	yes	characteristic
84	C₃×D₁₂	—	4	2	no	—
85	C₃×D₁₂	—	4	2	no	—
86	C₃×D₁₂	—	4	2	no	—
87	C₃×D₁₂	—	4	2	no	—
88	C₃×D₁₂	—	8	2	no	—
89	C₃²×C₈	—	2	1	no	—
90	C₃²×C₈	—	4	1	no	—
91	C₃²×C₈	D₈	1	2	yes	complemented
92	C₃×C₃⋊C₈	—	4	2	no	—
93	#493	—	3	1	no	—
94	#459	—	3	2	no	—
95	C₈×D₄	C₃×D₃	1	1	yes	characteristic, complemented
96	#335	—	3	1	no	—
97	#331	—	3	2	no	—
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	—	6	2	no	—
100	C₂×C₃×D₄	—	6	2	no	—
101	C₂×C₃×D₄	—	6	2	no	—
102	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
103	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂×D₁₂	—	2	2	no	—
108	C₂×D₁₂	—	2	2	no	—
109	C₂×D₁₂	—	2	2	no	—
110	C₃×D₈	—	12	2	no	—
111	C₃×D₈	—	12	4	no	—
112	C₃×D₈	—	12	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	4	4	no	—
119	C₂×C₃×C₈	—	4	2	no	—
120	C₂×C₃×C₈	—	4	2	no	—
121	C₂×C₃×C₈	—	6	2	no	—
122	C₂×C₃×C₈	#67	1	1	yes	characteristic
123	C₂×C₃×C₈	#67	1	1	yes	characteristic
124	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
128	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
129	C₃×C₂²⋊C₄	—	2	2	no	—
130	C₃×C₂²⋊C₄	—	2	1	no	—
131	C₃×C₂²⋊C₄	—	2	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
134	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
135	#213	—	4	4	no	—
136	C₂×C₃⋊C₈	—	2	2	no	—
137	D₂₄	—	4	2	no	—
138	D₂₄	—	4	2	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	—	4	1	no	—
141	C₂²×C₃²	—	4	1	no	—
142	C₂²×C₃²	#121	1	1	yes	characteristic
143	C₂×C₃×D₃	—	8	2	no	—
144	C₂×C₃×D₃	—	8	2	no	—
145	C₂×C₃×D₃	—	8	2	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	4	1	no	—
148	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
149	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
150	C₃²×C₄	#135	1	1	yes	characteristic
151	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×D₈	—	6	2	no	—
153	C₂×D₈	—	6	1	no	—
154	C₂×D₈	—	6	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	#127	—	3	2	no	—
157	C₄×D₄	C₂×C₃×D₃	1	1	yes	characteristic
158	C₄⋊C₈	—	3	2	no	—
159	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
160	#102	—	6	2	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
163	C₂³×C₃	—	2	2	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	—	12	2	no	—
167	C₂²×D₃	—	4	2	no	—
168	C₃×D₄	—	2	2	no	—
169	C₃×D₄	—	2	1	no	—
170	C₃×D₄	—	2	1	no	—
171	C₃×D₄	—	2	2	no	—
172	C₃×D₄	—	12	2	no	—
173	C₃×D₄	—	12	2	no	—
174	C₃×D₄	—	12	2	no	—
175	C₃×D₄	—	12	2	no	—
176	C₃×D₄	—	24	2	no	—
177	C₃×D₄	D₂₄	1	2	yes	complemented
178	C₃×D₄	C₃×D₈	1	2	yes	complemented
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
192	C₂×C₃×C₄	#235	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
197	D₁₂	—	4	2	no	—
198	D₁₂	—	4	2	no	—
199	D₁₂	—	4	2	no	—
200	D₁₂	—	4	2	no	—
201	D₁₂	—	8	2	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	4	1	no	—
206	C₃×C₈	—	4	2	no	—
207	C₃×C₈	—	4	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	12	2	no	—
210	C₃×C₈	#213	1	2	yes	complemented
211	C₃×C₈	C₃×D₈	1	2	yes	complemented
212	C₃⋊C₈	—	4	2	no	—
213	C₂×C₃²	—	4	2	no	—
214	C₂×C₃²	#459	1	1	yes	characteristic
215	C₂×C₃²	#466	1	1	yes	characteristic
216	C₂×C₃²	#472	1	1	yes	characteristic
217	C₃×D₃	—	16	2	no	—
218	C₂×D₄	—	6	2	no	—
219	C₂×D₄	—	6	2	no	—
220	C₂×D₄	—	6	2	no	—
221	C₂×D₄	C₃×D₁₂	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	D₈	—	12	2	no	—
224	D₈	—	12	4	no	—
225	D₈	—	12	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	6	2	no	—
229	C₂×C₈	C₃×#67	1	1	yes	characteristic
230	C₂×C₈	C₃×#67	1	1	yes	characteristic
231	C₄⋊C₄	C₃×D₁₂	1	1	yes	characteristic
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	1	no	—
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	—	4	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	24	2	no	—
245	C₂²×C₃	—	24	2	no	—
246	C₂²×C₃	—	24	2	no	—
247	C₂²×C₃	#930	1	1	yes	characteristic
248	C₂²×C₃	C₃×#121	1	1	yes	characteristic
249	C₂×D₃	—	8	2	no	—
250	C₂×D₃	—	8	2	no	—
251	C₂×D₃	—	8	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
263	C₃×C₄	#904	1	1	yes	characteristic
264	C₃×C₄	C₂×#213	1	1	yes	characteristic
265	C₃×C₄	#949	1	1	yes	characteristic
266	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
267	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
268	C₃×C₄	C₃×#135	1	1	yes	characteristic
269	C₃×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₃²	#1667	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	2	1	no	—
272	C₂³	—	12	2	no	—
273	D₄	—	2	1	no	—
274	D₄	—	12	2	no	—
275	D₄	—	12	2	no	—
276	D₄	—	12	2	no	—
277	D₄	—	12	2	no	—
278	D₄	—	24	2	no	—
279	D₄	C₃×D₂₄	1	2	yes	complemented
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
284	C₂×C₄	C₃×#235	1	1	yes	characteristic
285	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	12	2	no	—
289	C₈	C₃×#213	1	2	yes	complemented
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	4	4	no	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	—	4	2	no	—
296	C₂×C₃	—	48	2	no	—
297	C₂×C₃	#5015	1	1	yes	characteristic
298	C₂×C₃	#5115	1	1	yes	characteristic
299	C₂×C₃	#5142	1	1	yes	characteristic
300	C₂×C₃	C₃×#459	1	1	yes	characteristic
301	C₂×C₃	C₃×#466	1	1	yes	characteristic
302	C₂×C₃	C₃×#472	1	1	yes	characteristic
303	D₃	—	16	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	24	2	no	—
308	C₂²	—	24	2	no	—
309	C₂²	—	24	2	no	—
310	C₂²	C₃×#930	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	C₂×C₃×D₂₄	1	1	yes	characteristic
314	C₄	C₃×#904	1	1	yes	characteristic
315	C₄	C₂×C₃×#213	1	1	yes	characteristic
316	C₄	C₃×#949	1	1	yes	characteristic
317	C₃	—	2	1	no	—
318	C₃	#70847	1	1	yes	characteristic, direct product factor
319	C₃	C₃×#1667	1	1	yes	characteristic, complemented
320	C₂	—	4	2	no	—
321	C₂	—	48	2	no	—
322	C₂	C₃×#5015	1	1	yes	characteristic
323	C₂	C₃×#5115	1	1	yes	characteristic
324	C₂	C₃×#5142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#930
commutator, cosocle	C₃×C₄²	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₈×D₄	C₂
socle	C₂²×C₃²	#121
hypercenter	C₃×C₈×D₄	D₃
hypocenter	C₃	C₃×#1667
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₃²×C₈×D₄	C₂	semidirect
2. C₃×#5015	C₂	semidirect
3. C₃×#4695	C₂	semidirect
4. C₃×#4485	C₂	semidirect
5. C₃×#4458	C₂	semidirect
7. #70847	C₃	direct (N × H)
22. C₃×C₈×D₄	D₃	semidirect
23. C₃×C₈×D₄	C₂×C₃	semidirect
24. #5015	C₂×C₃	semidirect
25. #4695	C₂×C₃	semidirect
28. #4485	C₂×C₃	semidirect
29. #4458	C₂×C₃	semidirect
77. C₃²×D₄	D₈	semidirect
91. C₃²×C₈	D₈	semidirect
95. C₈×D₄	C₃×D₃	semidirect
177. C₃×D₄	D₂₄	semidirect
178. C₃×D₄	C₃×D₈	semidirect
210. C₃×C₈	#213	semidirect
211. C₃×C₈	C₃×D₈	semidirect
270. C₃²	#1667	semidirect
279. D₄	C₃×D₂₄	semidirect
289. C₈	C₃×#213	semidirect
318. C₃	#70847	direct (N × H)
319. C₃	C₃×#1667	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×D₄	2
C₃×#5015	2
C₃×#4695	2
C₃×#4485	2
C₃×#4458	2
C₃×#1667	3
#70847	3

Subgroups — C3×#70847 #11780884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70847 #11780884