Subgroups — #11784307

Order 1152.

613 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#63554	C₂	1	2	yes	maximal
5	#10589155	C₂	1	2	yes	maximal
6	#69757	—	3	1	no	maximal
7	#69477	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
15	C₃×C₃⋊#95	C₄	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
17	#63554	—	2	4	no	—
18	C₂×C₃×C₄×C₈	—	3	1	no	—
19	C₃×C₄×C₄⋊C₄	—	3	1	no	—
20	C₂×#815	—	3	2	no	—
21	C₄×C₄⋊Dic₃	D₃	1	1	yes	characteristic
22	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
25	C₂×#803	—	3	2	no	—
26	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
27	#4550	—	3	2	no	—
28	#4524	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
33	C₃²×C₄²	C₂×C₄	1	2	yes	—
34	C₃²×C₄²	C₂×C₄	1	2	yes	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
37	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
38	C₃×C₄⋊Dic₃	—	2	4	no	—
39	C₃×C₄×Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
46	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
47	#1770	—	9	1	no	Sylow
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	—	3	2	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄⋊Dic₃	Dic₃	1	2	yes	—
56	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
57	C₂²×C₃⋊C₈	—	3	2	no	—
58	C₂²×C₃⋊C₈	—	3	2	no	—
59	C₂²×C₃⋊C₈	—	6	2	no	—
60	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄×C₈	—	3	4	no	—
63	C₃×#95	—	3	2	no	—
64	C₃⋊#95	Dic₃	1	2	yes	—
65	#815	—	6	4	no	—
66	C₄⋊(C₃⋊C₈)	—	3	4	no	—
67	C₄⋊(C₃⋊C₈)	—	3	4	no	—
68	C₄⋊(C₃⋊C₈)	—	6	4	no	—
69	#803	—	6	4	no	—
70	C₄×C₃⋊C₈	Dic₃	1	4	yes	complemented
71	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₄²	1	2	yes	—
74	C₂×C₃²×C₄	C₄²	1	2	yes	—
75	C₂×C₃²×C₄	M₁₆	1	4	yes	—
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	2	4	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	—	2	4	no	—
85	C₃⋊(C₃⋊C₈)	—	2	4	no	—
86	C₃×C₃⋊C₈	—	2	2	no	—
87	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	complemented
88	C₂×C₄⋊C₈	—	9	1	no	—
89	C₂×#97	—	9	2	no	—
90	C₂×C₄×C₈	—	3	1	no	—
91	C₄×C₄⋊C₄	—	3	1	no	—
92	#336	—	9	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	4	no	—
97	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×Dic₃	C₂×Dic₃	1	4	yes	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	3	2	no	—
108	C₂×C₃×C₈	—	6	1	no	—
109	C₂×C₃×C₈	—	6	1	no	—
110	C₃×C₄⋊C₄	—	6	4	no	—
111	C₃×C₄²	—	2	2	no	—
112	C₃×C₄²	—	2	2	no	—
113	C₃×C₄²	—	6	2	no	—
114	C₃×C₄²	—	6	2	no	—
115	C₃×C₄²	C₄×D₃	1	2	yes	—
116	C₃×C₄²	C₄×D₃	1	2	yes	—
117	C₃×C₄²	C₄×D₃	1	2	yes	—
118	C₃×C₄²	C₂×Dic₃	1	2	yes	—
119	C₄⋊Dic₃	—	2	4	no	—
120	C₄×Dic₃	—	2	2	no	—
121	C₄×Dic₃	—	2	2	no	—
122	C₂×C₃⋊C₈	—	2	1	no	—
123	C₂×C₃⋊C₈	—	2	1	no	—
124	C₂×C₃⋊C₈	—	3	4	no	—
125	C₂×C₃⋊C₈	—	3	4	no	—
126	C₂×C₃⋊C₈	—	6	4	no	—
127	C₂×C₃⋊C₈	—	6	2	no	—
128	C₂×C₃⋊C₈	—	6	2	no	—
129	C₂×C₃⋊C₈	—	6	2	no	—
130	C₂×C₃⋊C₈	—	6	2	no	—
131	C₂×C₃⋊C₈	—	12	2	no	—
132	C₂×C₃⋊C₈	—	12	2	no	—
133	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
134	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
136	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
137	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
138	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
141	C₂²×C₃²	#131	1	2	yes	—
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	#97	1	4	yes	complemented
144	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	#117	1	1	yes	characteristic
146	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₃×Dic₃	—	4	4	no	—
149	C₂²×C₈	—	3	1	no	—
150	C₂²×C₈	—	3	1	no	—
151	C₂²×C₈	—	9	2	no	—
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄²	—	3	2	no	—
154	C₂×C₄²	D₃²	1	1	yes	characteristic
155	C₄⋊C₈	—	9	4	no	—
156	#97	—	18	4	no	—
157	C₄×C₈	—	3	4	no	—
158	#95	—	3	2	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	—	6	4	no	—
176	C₂×C₃×C₄	—	6	4	no	—
177	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
178	C₂×C₃×C₄	#208	1	4	yes	—
179	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
180	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
181	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
182	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
183	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
188	C₂×C₃×C₄	#235	1	1	yes	characteristic
189	C₂×C₃×C₄	#235	1	1	yes	characteristic
190	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	#237	1	1	yes	characteristic
192	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
193	C₂×C₃×C₄	#239	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
195	C₂×Dic₃	—	2	4	no	—
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	—	2	4	no	—
198	C₃×C₈	—	3	4	no	—
199	C₃×C₈	—	6	2	no	—
200	C₃⋊C₈	—	2	2	no	—
201	C₃⋊C₈	—	6	4	no	—
202	C₃⋊C₈	—	6	4	no	—
203	C₃⋊C₈	—	12	4	no	—
204	C₃⋊C₈	C₄⋊Dic₃	1	4	yes	complemented
205	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₃²	C₂×#97	1	1	yes	characteristic
207	C₂×C₃²	#405	1	1	yes	characteristic
208	C₂×C₃²	#436	1	2	yes	—
209	C₂×C₃²	#446	1	2	yes	—
210	C₂²×C₄	—	3	4	no	—
211	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
212	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	2	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	3	2	no	—
218	C₂×C₈	—	6	1	no	—
219	C₂×C₈	—	6	1	no	—
220	C₂×C₈	—	9	4	no	—
221	C₂×C₈	—	18	2	no	—
222	C₂×C₈	—	18	2	no	—
223	C₄⋊C₄	—	6	4	no	—
224	C₄²	—	6	2	no	—
225	C₄²	—	6	2	no	—
226	C₄²	D₃×Dic₃	1	2	yes	—
227	C₄²	#624	1	2	yes	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
234	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
235	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
236	C₂²×C₃	#901	1	2	yes	—
237	C₂²×C₃	C₂×#208	1	2	yes	—
238	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
240	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
241	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
242	C₂²×C₃	#948	1	2	yes	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	12	4	no	—
252	C₃×C₄	#803	1	4	yes	complemented
253	C₃×C₄	#815	1	4	yes	complemented
254	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
255	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
256	C₃×C₄	#887	1	1	yes	characteristic
257	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
258	C₃×C₄	#892	1	1	yes	characteristic
259	C₃×C₄	#893	1	1	yes	characteristic
260	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
261	C₃×C₄	#926	1	1	yes	characteristic
262	Dic₃	—	4	4	no	—
263	C₃²	#1770	1	1	yes	characteristic, complemented, Sylow
264	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	6	4	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	#3711	1	4	yes	—
270	C₂×C₄	Dic₃²	1	2	yes	—
271	C₂×C₄	Dic₃²	1	2	yes	—
272	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
273	C₂×C₄	#3795	1	1	yes	characteristic
274	C₂×C₄	#3799	1	1	yes	characteristic
275	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
276	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#624	1	1	yes	characteristic
278	C₈	—	3	4	no	—
279	C₈	—	6	2	no	—
280	C₈	—	18	4	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	#4679	1	2	yes	—
287	C₂×C₃	#4688	1	2	yes	—
288	C₂×C₃	C₂×#803	1	1	yes	characteristic
289	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
290	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₃	#4997	1	1	yes	characteristic
292	C₂×C₃	#5012	1	2	yes	—
293	C₂×C₃	#5024	1	2	yes	—
294	C₂×C₃	C₂×#815	1	1	yes	characteristic
295	C₂×C₃	#5101	1	1	yes	characteristic
296	C₂²	#63814	1	2	yes	—
297	C₂²	C₂×#3711	1	2	yes	—
298	C₂²	Dic₃×Dic₆	1	1	yes	characteristic
299	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
300	C₂²	C₂×Dic₃²	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	12	4	no	—
303	C₄	#63554	1	4	yes	complemented
304	C₄	#63841	1	1	yes	characteristic
305	C₄	#63844	1	1	yes	characteristic
306	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
307	C₄	#63894	1	1	yes	characteristic
308	C₃	—	2	1	no	—
309	C₃	#69477	1	1	yes	characteristic, complemented
310	C₃	#69757	1	1	yes	characteristic, complemented
311	C₂	#10590258	1	2	yes	—
312	C₂	#10590337	1	2	yes	—
313	C₂	#10591055	1	1	yes	characteristic
314	C₂	C₂×#63554	1	1	yes	characteristic
315	C₂	Dic₃×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator	C₂²×C₃²	C₂×C₄²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1770
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
14. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
16. C₃×C₄×C₃⋊C₈	C₄	semidirect
70. C₄×C₃⋊C₈	Dic₃	semidirect
87. C₃×C₃⋊C₈	C₄⋊C₄	semidirect
143. C₃²×C₄	#97	semidirect
204. C₃⋊C₈	C₄⋊Dic₃	semidirect
252. C₃×C₄	#803	semidirect
253. C₃×C₄	#815	semidirect
263. C₃²	#1770	semidirect
303. C₄	#63554	semidirect
309. C₃	#69477	semidirect
310. C₃	#69757	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×#63554	2
#10589155	2
#69757	3
#69477	3