Subgroups — #11777080

Order 1152.

347 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#10590263	C₂	1	1	yes	maximal, characteristic
3	#10590261	C₂	1	1	yes	maximal, characteristic
4	#10589345	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589099	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
8	#70423	—	3	1	no	maximal
9	#68534	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	—	2	1	no	—
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×Dic₃⋊C₈	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	Dic₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
20	Dic₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
21	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
22	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
23	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
24	#63554	—	2	1	no	—
25	C₂×C₃×C₄⋊C₈	—	3	1	no	—
26	#4926	—	3	1	no	—
27	#4925	—	3	1	no	—
28	#4835	—	3	1	no	—
29	D₃×C₄⋊C₈	D₃	1	1	yes	characteristic
30	#4685	—	3	1	no	—
31	C₃×#334	—	3	1	no	—
32	#4480	—	3	1	no	—
33	#4480	—	3	1	no	—
34	#4480	D₃	1	1	yes	characteristic, complemented
35	#4479	—	3	1	no	—
36	#4479	—	3	1	no	—
37	#4479	—	3	1	no	—
38	#4479	—	6	1	no	—
39	#4458	—	3	1	no	—
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×D₃×C₈	—	4	2	no	—
56	#1573	—	9	1	no	Sylow
57	C₂²×C₃×C₈	—	6	1	no	—
58	C₂×C₃×C₄²	—	3	1	no	—
59	C₂×D₃×C₈	—	2	1	no	—
60	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
61	C₃×C₄⋊C₈	—	2	1	no	—
62	C₃×C₄⋊C₈	—	3	1	no	—
63	C₃×C₄⋊C₈	—	3	1	no	—
64	C₃×C₄⋊C₈	—	6	1	no	—
65	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄×C₈	—	3	1	no	—
68	#830	—	3	1	no	—
69	#830	—	3	1	no	—
70	#830	—	6	1	no	—
71	(C₂×D₃)⋊C₈	—	3	1	no	—
72	(C₂×D₃)⋊C₈	—	3	1	no	—
73	(C₂×D₃)⋊C₈	—	6	1	no	—
74	#815	—	6	1	no	—
75	Dic₃⋊C₈	—	2	1	no	—
76	Dic₃⋊C₈	—	3	1	no	—
77	Dic₃⋊C₈	—	3	1	no	—
78	C₄⋊(C₃⋊C₈)	—	3	1	no	—
79	C₄⋊(C₃⋊C₈)	—	3	1	no	—
80	C₄⋊(C₃⋊C₈)	—	3	1	no	—
81	C₄⋊(C₃⋊C₈)	—	3	1	no	—
82	C₄⋊(C₃⋊C₈)	—	6	1	no	—
83	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
84	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
85	#803	—	6	1	no	—
86	C₄×C₃⋊C₈	—	3	1	no	—
87	C₄×C₃⋊C₈	—	3	1	no	—
88	C₄×C₃⋊C₈	—	6	1	no	—
89	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
90	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
97	C₃×C₄×D₃	—	2	2	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃×C₄×D₃	—	2	1	no	—
100	C₃×C₄×D₃	—	2	1	no	—
101	C₃×C₄×D₃	—	2	1	no	—
102	C₃²×C₈	—	4	1	no	—
103	C₃⋊(C₃⋊C₈)	—	2	1	no	—
104	C₃⋊(C₃⋊C₈)	—	2	1	no	—
105	C₃⋊(C₃⋊C₈)	—	4	1	no	—
106	C₃×C₃⋊C₈	—	2	1	no	—
107	C₃×C₃⋊C₈	—	2	1	no	—
108	C₃×C₃⋊C₈	—	4	1	no	—
109	C₃×C₃⋊C₈	—	4	1	no	—
110	#432	—	9	1	no	—
111	#431	—	9	1	no	—
112	C₂×C₄⋊C₈	—	3	1	no	—
113	#335	—	9	1	no	—
114	#335	—	9	1	no	—
115	#334	—	3	1	no	—
116	#334	—	9	1	no	—
117	C₂²×C₃×C₄	—	3	1	no	—
118	C₂²×C₃×C₄	—	3	1	no	—
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂×C₄×D₃	Dic₆	1	1	yes	characteristic
121	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
122	C₂×C₄×D₃	#67	1	1	yes	characteristic
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	3	1	no	—
126	C₂×C₃×C₈	—	3	1	no	—
127	C₂×C₃×C₈	—	4	1	no	—
128	C₂×C₃×C₈	—	6	1	no	—
129	C₂×C₃×C₈	—	6	1	no	—
130	C₂×C₃×C₈	—	12	2	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	3	1	no	—
133	C₃×C₄²	—	3	1	no	—
134	C₃×C₄²	—	3	1	no	—
135	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
136	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
137	C₄×Dic₃	Dic₆	1	1	yes	characteristic
138	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
139	C₄×Dic₃	#67	1	1	yes	characteristic
140	C₂×C₃⋊C₈	—	2	1	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	3	1	no	—
143	C₂×C₃⋊C₈	—	3	1	no	—
144	C₂×C₃⋊C₈	—	3	1	no	—
145	C₂×C₃⋊C₈	—	3	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂×C₃⋊C₈	—	6	1	no	—
148	C₂×C₃⋊C₈	—	6	1	no	—
149	C₂×C₃⋊C₈	—	6	1	no	—
150	C₂×C₃⋊C₈	—	12	1	no	—
151	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
152	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
153	D₃×C₈	—	4	2	no	—
154	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₃×D₃	—	2	1	no	—
156	C₂×C₃×D₃	—	2	1	no	—
157	C₂×C₃×D₃	#108	1	2	yes	—
158	C₃²×C₄	—	2	1	no	—
159	C₃²×C₄	#131	1	1	yes	characteristic
160	C₃²×C₄	#131	1	1	yes	characteristic
161	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₃²×C₄	#137	1	1	yes	characteristic
163	C₃×Dic₃	—	2	1	no	—
164	C₃×Dic₃	—	2	1	no	—
165	C₃×Dic₃	—	2	1	no	—
166	C₂²×C₈	—	6	1	no	—
167	C₂×C₄²	—	3	1	no	—
168	C₄⋊C₈	—	3	1	no	—
169	C₄⋊C₈	—	3	1	no	—
170	C₄⋊C₈	—	6	1	no	—
171	C₄⋊C₈	—	9	1	no	—
172	C₄⋊C₈	—	9	1	no	—
173	C₄⋊C₈	—	9	1	no	—
174	C₄⋊C₈	D₃²	1	1	yes	characteristic
175	C₂²⋊C₈	—	9	1	no	—
176	C₂²⋊C₈	—	9	1	no	—
177	C₂²⋊C₈	—	18	1	no	—
178	#97	—	18	1	no	—
179	C₄×C₈	—	3	1	no	—
180	C₄×C₈	—	9	1	no	—
181	C₂³×C₃	—	3	1	no	—
182	C₂²×D₃	Dic₃⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	3	1	no	—
187	C₂×C₃×C₄	—	3	1	no	—
188	C₂×C₃×C₄	—	3	1	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	—	6	1	no	—
191	C₂×C₃×C₄	—	6	1	no	—
192	C₂×C₃×C₄	—	6	1	no	—
193	C₂×C₃×C₄	—	6	1	no	—
194	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
200	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
201	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
202	C₂×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
203	C₄×D₃	—	2	2	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	2	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	4	1	no	—
210	C₃×C₈	—	6	1	no	—
211	C₃×C₈	—	6	1	no	—
212	C₃×C₈	—	8	1	no	—
213	C₃×C₈	—	12	1	no	—
214	C₃×C₈	—	12	1	no	—
215	C₃⋊C₈	—	2	1	no	—
216	C₃⋊C₈	—	2	1	no	—
217	C₃⋊C₈	—	4	1	no	—
218	C₃⋊C₈	—	4	1	no	—
219	C₃⋊C₈	—	6	2	no	—
220	C₃⋊C₈	—	6	1	no	—
221	C₃⋊C₈	—	6	1	no	—
222	C₃⋊C₈	—	6	1	no	—
223	C₃⋊C₈	—	6	1	no	—
224	C₃⋊C₈	—	12	1	no	—
225	C₃⋊C₈	—	12	1	no	—
226	C₃⋊C₈	—	12	1	no	—
227	C₃⋊C₈	—	24	1	no	—
228	C₂×C₃²	#424	1	1	yes	characteristic
229	C₂×C₃²	#428	1	1	yes	characteristic
230	C₂×C₃²	C₂×#108	1	1	yes	characteristic
231	C₃×D₃	—	2	2	no	—
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	3	1	no	—
234	C₂²×C₄	—	3	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	3	1	no	—
238	C₂×C₈	—	6	1	no	—
239	C₂×C₈	—	6	1	no	—
240	C₂×C₈	—	9	1	no	—
241	C₂×C₈	—	9	1	no	—
242	C₂×C₈	—	12	2	no	—
243	C₂×C₈	—	18	1	no	—
244	C₄²	—	3	1	no	—
245	C₄²	—	3	1	no	—
246	C₄²	—	3	1	no	—
247	C₄²	C₂×D₃²	1	1	yes	characteristic
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	3	2	no	—
250	C₂²×C₃	—	6	1	no	—
251	C₂²×C₃	—	6	1	no	—
252	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
253	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	#822	1	2	yes	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	6	1	no	—
265	C₃×C₄	—	6	1	no	—
266	C₃×C₄	—	6	1	no	—
267	C₃×C₄	#901	1	1	yes	characteristic
268	C₃×C₄	#901	1	1	yes	characteristic
269	C₃×C₄	#901	1	1	yes	characteristic
270	C₃×C₄	#907	1	1	yes	characteristic
271	C₃×C₄	#914	1	1	yes	characteristic
272	C₃×C₄	#915	1	1	yes	characteristic
273	C₃×C₄	#932	1	1	yes	characteristic
274	C₃×C₄	#942	1	1	yes	characteristic
275	Dic₃	—	2	1	no	—
276	Dic₃	—	2	1	no	—
277	Dic₃	—	2	1	no	—
278	C₃²	#1573	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	6	1	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
289	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
290	C₂×C₄	D₃×#67	1	1	yes	characteristic
291	C₈	—	4	1	no	—
292	C₈	—	6	1	no	—
293	C₈	—	6	1	no	—
294	C₈	—	12	1	no	—
295	C₈	—	12	1	no	—
296	C₈	—	18	1	no	—
297	C₈	—	18	1	no	—
298	C₈	—	36	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	6	2	no	—
303	C₂×C₃	#4839	1	1	yes	characteristic
304	C₂×C₃	#4861	1	1	yes	characteristic
305	C₂×C₃	D₃×#108	1	1	yes	characteristic
306	C₂×C₃	#4964	1	1	yes	characteristic
307	C₂×C₃	#5102	1	1	yes	characteristic
308	C₂×C₃	C₂×#822	1	1	yes	characteristic
309	D₃	—	2	2	no	—
310	C₂²	—	3	2	no	—
311	C₂²	—	6	1	no	—
312	C₂²	—	6	1	no	—
313	C₂²	D₃×Dic₃⋊C₄	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	6	1	no	—
316	C₄	—	6	1	no	—
317	C₄	—	6	1	no	—
318	C₄	#63803	1	1	yes	characteristic
319	C₄	#63804	1	1	yes	characteristic
320	C₄	#63934	1	1	yes	characteristic
321	C₄	#63946	1	1	yes	characteristic
322	C₃	—	2	1	no	—
323	C₃	#68534	1	1	yes	characteristic, complemented
324	C₃	#70423	1	1	yes	characteristic, complemented
325	C₂	—	6	2	no	—
326	C₂	#10590581	1	1	yes	characteristic
327	C₂	#10590694	1	1	yes	characteristic
328	C₂	D₃×#822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×#67
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄⋊C₈	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄⋊C₈	D₃²
hypocenter	C₃²	#1573
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #10589345	C₂	semidirect
5. C₃×#4480	C₂	semidirect
6. #10589099	C₂	semidirect
7. #10589098	C₂	semidirect
34. #4480	D₃	semidirect
278. C₃²	#1573	semidirect
323. C₃	#68534	semidirect
324. C₃	#70423	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×C₄⋊C₈	2
#10590263	2
#10590261	2
#10589345	2
C₃×#4480	2
#10589099	2
#10589098	2
#70423	3
#68534	3