Subgroups — #11784205

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 333 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×C₄²×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	#70441	—	3	1	no	maximal
7	#69391	—	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	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
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×C₄³	—	3	1	no	—
20	C₂×#815	—	3	2	no	—
21	C₄²×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×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
26	C₂×#803	—	3	2	no	—
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	2	yes	—
33	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
34	C₃²×C₄²	C₂×C₄	1	2	yes	—
35	C₃²×C₄²	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
39	C₃×C₄×Dic₃	—	2	4	no	—
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₃×C₄×Dic₃	—	2	4	no	—
43	C₃×C₄×Dic₃	D₄	1	2	yes	—
44	C₃×C₄×Dic₃	Q₈	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
48	#1767	—	9	1	no	Sylow
49	C₂²×C₃×C₈	—	3	2	no	—
50	C₂×C₃×C₄²	—	2	1	no	—
51	C₂×C₃×C₄²	—	3	1	no	—
52	C₂×C₃×C₄²	—	3	2	no	—
53	C₂×C₃×C₄²	—	3	2	no	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄×Dic₃	Dic₃	1	2	yes	—
58	C₂×C₄×Dic₃	Dic₃	1	2	yes	—
59	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	—	3	2	no	—
62	C₂²×C₃⋊C₈	—	3	2	no	—
63	C₂²×C₃⋊C₈	—	6	2	no	—
64	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
65	C₃×C₄⋊C₈	—	3	4	no	—
66	#815	—	6	4	no	—
67	C₄⋊(C₃⋊C₈)	—	3	4	no	—
68	C₄⋊(C₃⋊C₈)	—	3	4	no	—
69	C₄⋊(C₃⋊C₈)	—	6	4	no	—
70	C₄⋊(C₃⋊C₈)	Dic₃	1	4	yes	complemented
71	#803	—	6	4	no	—
72	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	C₄²	1	2	yes	—
75	C₂×C₃²×C₄	C₄²	1	2	yes	—
76	C₂×C₃²×C₄	M₁₆	1	4	yes	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	1	yes	characteristic
83	C₂×C₃×Dic₃	—	2	2	no	—
84	C₂×C₃×Dic₃	—	2	2	no	—
85	C₂×C₃×Dic₃	—	2	2	no	—
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
88	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
89	C₃⋊(C₃⋊C₈)	—	2	4	no	—
90	C₃×C₃⋊C₈	—	2	4	no	—
91	C₂×C₄⋊C₈	—	3	1	no	—
92	C₂×C₄⋊C₈	—	9	1	no	—
93	C₂×#97	—	9	2	no	—
94	C₄³	—	3	1	no	—
95	#336	—	9	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	3	2	no	—
100	C₂²×C₃×C₄	—	3	2	no	—
101	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
108	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
109	C₂×C₃×C₈	—	3	4	no	—
110	C₂×C₃×C₈	—	6	2	no	—
111	C₂×C₃×C₈	—	6	2	no	—
112	C₃×C₄²	—	2	2	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	—	3	2	no	—
115	C₃×C₄²	—	3	2	no	—
116	C₃×C₄²	—	6	4	no	—
117	C₃×C₄²	—	6	4	no	—
118	C₃×C₄²	—	6	1	no	—
119	C₃×C₄²	—	6	1	no	—
120	C₃×C₄²	C₄×D₃	1	2	yes	—
121	C₃×C₄²	C₄×D₃	1	2	yes	—
122	C₃×C₄²	C₄×D₃	1	2	yes	—
123	C₃×C₄²	C₂×Dic₃	1	2	yes	—
124	C₄×Dic₃	—	2	4	no	—
125	C₄×Dic₃	—	2	1	no	—
126	C₄×Dic₃	—	2	1	no	—
127	C₄×Dic₃	—	2	4	no	—
128	C₄×Dic₃	Dic₆	1	2	yes	—
129	C₄×Dic₃	D₁₂	1	2	yes	—
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	—	3	4	no	—
133	C₂×C₃⋊C₈	—	3	4	no	—
134	C₂×C₃⋊C₈	—	6	4	no	—
135	C₂×C₃⋊C₈	—	6	2	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	—	6	2	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	—	12	2	no	—
140	C₂×C₃⋊C₈	—	12	2	no	—
141	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
142	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
143	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₃²	#117	1	1	yes	characteristic
145	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
146	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
147	C₃²×C₄	—	2	2	no	—
148	C₃²×C₄	#97	1	4	yes	complemented
149	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
150	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
151	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
152	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
153	C₃×Dic₃	—	2	4	no	—
154	C₃×Dic₃	—	2	4	no	—
155	C₂²×C₈	—	3	2	no	—
156	C₂²×C₈	—	9	2	no	—
157	C₂×C₄²	—	3	1	no	—
158	C₂×C₄²	—	3	2	no	—
159	C₂×C₄²	—	3	2	no	—
160	C₂×C₄²	—	3	1	no	—
161	C₂×C₄²	D₃²	1	1	yes	characteristic
162	C₄⋊C₈	—	3	4	no	—
163	C₄⋊C₈	—	9	4	no	—
164	#97	—	18	4	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	3	4	no	—
180	C₂×C₃×C₄	—	3	4	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	6	2	no	—
183	C₂×C₃×C₄	—	6	2	no	—
184	C₂×C₃×C₄	—	6	2	no	—
185	C₂×C₃×C₄	—	6	2	no	—
186	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
187	C₂×C₃×C₄	#208	1	4	yes	—
188	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
189	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
190	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
191	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
192	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
197	C₂×C₃×C₄	#235	1	1	yes	characteristic
198	C₂×C₃×C₄	#235	1	1	yes	characteristic
199	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	#237	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
202	C₂×C₃×C₄	#239	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
204	C₂×Dic₃	—	2	2	no	—
205	C₂×Dic₃	—	2	2	no	—
206	C₂×Dic₃	—	2	2	no	—
207	C₂×Dic₃	—	2	2	no	—
208	C₂×Dic₃	#208	1	4	yes	—
209	C₂×Dic₃	C₄⋊Dic₃	1	4	yes	—
210	C₃×C₈	—	6	4	no	—
211	C₃⋊C₈	—	2	4	no	—
212	C₃⋊C₈	—	6	4	no	—
213	C₃⋊C₈	—	6	4	no	—
214	C₃⋊C₈	—	12	4	no	—
215	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₃²	C₂×#97	1	1	yes	characteristic
217	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
218	C₂×C₃²	#423	1	2	yes	—
219	C₂×C₃²	#432	1	2	yes	—
220	C₂²×C₄	—	3	2	no	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
223	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
225	C₂×C₈	—	3	4	no	—
226	C₂×C₈	—	6	2	no	—
227	C₂×C₈	—	6	2	no	—
228	C₂×C₈	—	9	4	no	—
229	C₂×C₈	—	18	2	no	—
230	C₂×C₈	—	18	2	no	—
231	C₄²	—	3	2	no	—
232	C₄²	—	3	2	no	—
233	C₄²	—	6	4	no	—
234	C₄²	—	6	4	no	—
235	C₄²	—	6	1	no	—
236	C₄²	—	6	1	no	—
237	C₄²	D₃×Dic₃	1	2	yes	—
238	C₄²	#624	1	2	yes	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
245	C₂²×C₃	#892	1	1	yes	characteristic
246	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
247	C₂²×C₃	D₃×M₁₆	1	2	yes	—
248	C₂²×C₃	C₂×#208	1	2	yes	—
249	C₂²×C₃	C₂×#208	1	2	yes	—
250	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
252	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
253	C₂²×C₃	#926	1	1	yes	characteristic
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	4	no	—
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	—	4	2	no	—
262	C₃×C₄	—	6	4	no	—
263	C₃×C₄	—	6	4	no	—
264	C₃×C₄	#803	1	4	yes	complemented
265	C₃×C₄	#815	1	4	yes	complemented
266	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
267	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
268	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
269	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
270	C₃×C₄	#887	1	1	yes	characteristic
271	C₃×C₄	#893	1	1	yes	characteristic
272	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
273	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
274	Dic₃	—	2	4	no	—
275	Dic₃	—	2	4	no	—
276	C₃²	#1767	1	1	yes	characteristic, complemented, Sylow
277	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	3	4	no	—
280	C₂×C₄	—	3	4	no	—
281	C₂×C₄	—	6	2	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	#3711	1	4	yes	—
286	C₂×C₄	Dic₃²	1	2	yes	—
287	C₂×C₄	Dic₃²	1	2	yes	—
288	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
289	C₂×C₄	#3795	1	1	yes	characteristic
290	C₂×C₄	#3799	1	1	yes	characteristic
291	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
292	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
293	C₂×C₄	C₂×#624	1	1	yes	characteristic
294	C₈	—	6	4	no	—
295	C₈	—	18	4	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	#4835	1	2	yes	—
302	C₂×C₃	#4837	1	2	yes	—
303	C₂×C₃	C₂×#803	1	1	yes	characteristic
304	C₂×C₃	C₄×#208	1	1	yes	characteristic
305	C₂×C₃	#4924	1	2	yes	—
306	C₂×C₃	#4926	1	2	yes	—
307	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
308	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
309	C₂×C₃	C₂×#815	1	1	yes	characteristic
310	C₂×C₃	Dic₃×M₁₆	1	1	yes	characteristic
311	C₂²	D₃×#208	1	2	yes	—
312	C₂²	C₂×#3711	1	2	yes	—
313	C₂²	#63841	1	1	yes	characteristic
314	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
315	C₂²	C₂×Dic₃²	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	6	4	no	—
318	C₄	—	6	4	no	—
319	C₄	#63554	1	4	yes	complemented
320	C₄	Dic₃×Dic₆	1	1	yes	characteristic
321	C₄	#63844	1	1	yes	characteristic
322	C₄	Dic₃×D₁₂	1	1	yes	characteristic
323	C₄	#63894	1	1	yes	characteristic
324	C₃	—	2	1	no	—
325	C₃	#69391	1	1	yes	characteristic, complemented
326	C₃	#70441	1	1	yes	characteristic, complemented
327	C₂	#10590263	1	2	yes	—
328	C₂	#10590336	1	2	yes	—
329	C₂	Dic₃×#208	1	1	yes	characteristic
330	C₂	C₂×#63554	1	1	yes	characteristic
331	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	#1767
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
15. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
16. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
70. C₄⋊(C₃⋊C₈)	Dic₃	semidirect
148. C₃²×C₄	#97	semidirect
264. C₃×C₄	#803	semidirect
265. C₃×C₄	#815	semidirect
276. C₃²	#1767	semidirect
319. C₄	#63554	semidirect
325. C₃	#69391	semidirect
326. C₃	#70441	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×C₄²×Dic₃	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×#63554	2
#10589155	2
#70441	3
#69391	3