Subgroups — #11797445

Order 1152.

373 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#625	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×#67	—	3	1	no	maximal
9	#67⋊C₁₆	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₄×#625	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₃×D₃×C₁₆	—	2	2	no	—
19	#63572	C₄	1	1	yes	characteristic
20	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
21	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
22	D₃×C₃⋊C₁₆	—	2	2	no	—
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	C₈×#67	—	3	1	no	—
25	C₈×#67	—	3	1	no	—
26	C₂²×C₃⋊C₁₆	—	3	1	no	—
27	C₂×D₃×C₁₆	—	3	1	no	—
28	C₂×D₃×C₁₆	D₃	1	1	yes	characteristic, complemented
29	C₃×C₂²⋊C₁₆	—	3	1	no	—
30	#4549	—	3	1	no	—
31	(C₂×D₃)⋊C₁₆	—	3	1	no	—
32	(C₂×D₃)⋊C₁₆	D₃	1	1	yes	characteristic, complemented
33	Dic₃⋊C₁₆	—	3	1	no	—
34	Dic₃⋊C₁₆	—	3	1	no	—
35	Dic₃⋊C₁₆	—	6	1	no	—
36	Dic₃×C₁₆	—	3	1	no	—
37	Dic₃×C₁₆	—	3	1	no	—
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#625	C₈	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
42	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×D₃×C₈	—	2	1	no	—
47	C₃×D₃×C₈	—	2	1	no	—
48	C₃×D₃×C₈	—	2	1	no	—
49	C₃×D₃×C₈	—	2	1	no	—
50	#3724	C₈	1	1	yes	characteristic
51	(C₂×D₃)⋊Dic₃	C₈	1	1	yes	characteristic
52	(C₂×D₃)⋊Dic₃	C₈	1	1	yes	characteristic
53	C₃²×C₁₆	—	2	1	no	—
54	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	C₃×C₃⋊C₁₆	D₄	1	2	yes	complemented
57	D₄×C₁₆	—	9	1	no	Sylow
58	C₂²×C₃×C₈	—	3	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₄×#67	—	3	1	no	—
61	C₄×#67	—	3	1	no	—
62	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
63	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×C₁₆	—	2	1	no	—
65	C₂×C₃×C₁₆	—	3	1	no	—
66	C₂×C₃×C₁₆	—	3	1	no	—
67	C₂×C₃×C₁₆	—	6	2	no	—
68	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
70	#830	—	3	1	no	—
71	#830	—	3	1	no	—
72	(C₂×D₃)⋊C₈	—	3	1	no	—
73	(C₂×D₃)⋊C₈	—	3	1	no	—
74	Dic₃⋊C₈	—	3	1	no	—
75	Dic₃⋊C₈	—	3	1	no	—
76	C₈×Dic₃	—	3	1	no	—
77	C₈×Dic₃	—	3	1	no	—
78	C₈×Dic₃	—	6	1	no	—
79	C₂×C₃⋊C₁₆	—	3	1	no	—
80	C₂×C₃⋊C₁₆	—	3	1	no	—
81	C₂×C₃⋊C₁₆	—	6	1	no	—
82	C₂×C₃⋊C₁₆	—	6	2	no	—
83	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
84	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
85	D₃×C₁₆	—	2	2	no	—
86	D₃×C₁₆	—	6	2	no	—
87	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
94	C₃×C₄×D₃	—	2	1	no	—
95	C₃×C₄×D₃	—	2	1	no	—
96	C₃×C₄×D₃	—	2	1	no	—
97	C₃×C₄×D₃	—	2	1	no	—
98	#625	C₁₆	1	4	yes	complemented
99	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃²×C₈	#41	1	1	yes	characteristic
101	C₃⋊(C₃⋊C₈)	—	2	1	no	—
102	C₃×C₃⋊C₈	—	2	1	no	—
103	C₃×C₃⋊C₈	—	2	1	no	—
104	C₂²×C₁₆	—	3	1	no	—
105	C₂²×C₁₆	—	9	1	no	—
106	C₈×D₄	—	9	1	no	—
107	C₄⋊C₁₆	—	9	1	no	—
108	C₂²⋊C₁₆	—	3	1	no	—
109	C₂²⋊C₁₆	—	9	1	no	—
110	C₄×C₁₆	—	9	1	no	—
111	C₂²×C₃×C₄	—	3	1	no	—
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂×#67	—	3	1	no	—
114	C₂×#67	—	3	1	no	—
115	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
116	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	6	1	no	—
121	C₂×C₃×C₈	—	6	1	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	1	no	—
124	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
126	#217	—	3	1	no	—
127	#217	—	3	1	no	—
128	(C₂×D₃)⋊C₄	—	3	1	no	—
129	(C₂×D₃)⋊C₄	—	3	1	no	—
130	Dic₃⋊C₄	—	3	1	no	—
131	Dic₃⋊C₄	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₄×Dic₃	—	3	1	no	—
134	C₄×Dic₃	—	6	1	no	—
135	C₂×C₃⋊C₈	—	3	1	no	—
136	C₂×C₃⋊C₈	—	3	1	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
140	D₃×C₈	—	2	1	no	—
141	D₃×C₈	—	2	1	no	—
142	D₃×C₈	—	2	1	no	—
143	D₃×C₈	—	2	1	no	—
144	C₃×C₁₆	—	2	1	no	—
145	C₃×C₁₆	—	2	1	no	—
146	C₃×C₁₆	—	3	2	no	—
147	C₃×C₁₆	—	4	1	no	—
148	C₃×C₁₆	—	6	1	no	—
149	C₃⋊C₁₆	—	2	1	no	—
150	C₃⋊C₁₆	—	6	1	no	—
151	C₃⋊C₁₆	—	6	1	no	—
152	C₃⋊C₁₆	—	12	1	no	—
153	C₃⋊C₁₆	#67	1	2	yes	complemented
154	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₃×D₃	—	2	1	no	—
156	C₂×C₃×D₃	—	2	1	no	—
157	C₂×C₃×D₃	—	2	1	no	—
158	C₂×C₃×D₃	—	2	1	no	—
159	C₂×C₃×D₃	C₂×C₁₆	1	2	yes	complemented
160	C₂×C₃×D₃	C₂×C₁₆	1	2	yes	complemented
161	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
162	C₃²×C₄	#131	1	1	yes	characteristic
163	C₃⋊Dic₃	—	2	1	no	—
164	C₃⋊Dic₃	C₂×C₁₆	1	2	yes	complemented
165	C₃×Dic₃	—	2	1	no	—
166	C₃×Dic₃	—	2	1	no	—
167	C₂²×C₈	—	3	1	no	—
168	C₂²×C₈	—	3	1	no	—
169	C₄×D₄	—	9	1	no	—
170	C₂×C₁₆	—	3	1	no	—
171	C₂×C₁₆	—	3	1	no	—
172	C₂×C₁₆	—	6	2	no	—
173	C₂×C₁₆	—	9	1	no	—
174	C₂×C₁₆	—	18	2	no	—
175	C₂×C₁₆	D₃²	1	1	yes	characteristic
176	C₄⋊C₈	—	9	1	no	—
177	C₂²⋊C₈	—	9	1	no	—
178	C₂²⋊C₈	—	9	1	no	—
179	C₄×C₈	—	9	1	no	—
180	C₂³×C₃	—	3	1	no	—
181	C₂³×C₃	—	3	1	no	—
182	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
183	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	3	1	no	—
186	C₂×C₃×C₄	—	3	1	no	—
187	C₂×C₃×C₄	—	6	1	no	—
188	C₂×C₃×C₄	—	6	1	no	—
189	C₂×C₃×C₄	—	6	1	no	—
190	C₂×C₃×C₄	—	6	1	no	—
191	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	#67	—	3	4	no	—
194	#67	—	3	4	no	—
195	C₂×Dic₃	—	3	1	no	—
196	C₂×Dic₃	—	3	1	no	—
197	C₂×Dic₃	—	3	1	no	—
198	C₂×Dic₃	—	3	1	no	—
199	C₂×Dic₃	—	6	1	no	—
200	C₂×Dic₃	—	6	1	no	—
201	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
202	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	2	1	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	6	1	no	—
210	C₃×C₈	—	6	1	no	—
211	C₃×C₈	#235	1	1	yes	characteristic
212	C₃×C₈	D₃×D₄	1	1	yes	characteristic
213	C₃×C₈	#237	1	1	yes	characteristic
214	C₃×C₈	C₂×#67	1	1	yes	characteristic
215	C₃⋊C₈	—	2	1	no	—
216	C₃⋊C₈	—	2	1	no	—
217	C₃⋊C₈	—	6	1	no	—
218	C₃⋊C₈	—	6	1	no	—
219	C₃⋊C₈	—	12	1	no	—
220	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
221	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
222	C₂×C₃²	#504	1	1	yes	characteristic
223	C₃×D₃	—	2	2	no	—
224	C₃×D₃	—	2	2	no	—
225	C₂×D₄	—	9	1	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	3	1	no	—
230	C₂×C₈	—	6	1	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	—	9	1	no	—
235	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
236	C₄⋊C₄	—	9	1	no	—
237	C₂²⋊C₄	—	9	1	no	—
238	C₂²⋊C₄	—	9	1	no	—
239	C₄²	—	9	1	no	—
240	C₁₆	—	2	1	no	—
241	C₁₆	—	3	2	no	—
242	C₁₆	—	6	1	no	—
243	C₁₆	—	18	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	3	2	no	—
246	C₂²×C₃	—	3	2	no	—
247	C₂²×C₃	—	6	1	no	—
248	C₂²×C₃	—	6	1	no	—
249	C₂²×C₃	—	6	1	no	—
250	C₂²×C₃	—	6	1	no	—
251	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
252	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	D₃×C₁₆	1	2	yes	complemented
258	C₂×D₃	D₃×C₁₆	1	2	yes	complemented
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	6	1	no	—
262	C₃×C₄	—	6	1	no	—
263	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
264	C₃×C₄	#901	1	1	yes	characteristic
265	C₃×C₄	#907	1	1	yes	characteristic
266	C₃×C₄	C₄×#67	1	1	yes	characteristic
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	1	no	—
269	Dic₃	—	3	2	no	—
270	Dic₃	—	3	2	no	—
271	Dic₃	—	6	2	no	—
272	Dic₃	—	6	1	no	—
273	Dic₃	—	6	1	no	—
274	Dic₃	—	12	1	no	—
275	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	3	1	no	—
277	C₂³	—	3	1	no	—
278	D₄	—	9	4	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	9	1	no	—
286	C₂×C₄	—	9	1	no	—
287	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
288	C₈	—	6	1	no	—
289	C₈	—	6	1	no	—
290	C₈	—	18	1	no	—
291	C₈	#3795	1	1	yes	characteristic
292	C₈	D₃×#67	1	1	yes	characteristic
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	6	2	no	—
297	C₂×C₃	—	6	2	no	—
298	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
299	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
300	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
301	C₂×C₃	#4901	1	1	yes	characteristic
302	C₂×C₃	#4907	1	1	yes	characteristic
303	C₂×C₃	C₈×#67	1	1	yes	characteristic
304	D₃	—	2	2	no	—
305	D₃	—	2	2	no	—
306	C₂²	—	3	2	no	—
307	C₂²	—	3	2	no	—
308	C₂²	—	6	1	no	—
309	C₂²	—	6	1	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	6	1	no	—
312	C₂²	D₃²×C₈	1	1	yes	characteristic
313	C₄	—	6	1	no	—
314	C₄	—	6	1	no	—
315	C₄	—	9	2	no	—
316	C₄	—	18	1	no	—
317	C₄	#63804	1	1	yes	characteristic
318	C₄	#63879	1	1	yes	characteristic
319	C₃	—	2	1	no	—
320	C₃	#67⋊C₁₆	1	1	yes	characteristic, complemented
321	C₃	C₁₆×#67	1	1	yes	characteristic, complemented
322	C₂	—	6	2	no	—
323	C₂	—	6	2	no	—
324	C₂	D₃²×C₁₆	1	1	yes	characteristic
325	C₂	#10590228	1	1	yes	characteristic
326	C₂	#10590717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₂×C₃²×C₁₆	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×#625	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	D₄×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₃×C₁₆	C₂	semidirect
3. C₂×D₃×C₃⋊C₁₆	C₂	semidirect
4. #10589367	C₂	semidirect
5. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. (C₂×D₃)⋊(C₃⋊C₁₆)	C₂	semidirect
7. #10589102	C₂	semidirect
28. C₂×D₃×C₁₆	D₃	semidirect
32. (C₂×D₃)⋊C₁₆	D₃	semidirect
56. C₃×C₃⋊C₁₆	D₄	semidirect
98. #625	C₁₆	semidirect
153. C₃⋊C₁₆	#67	semidirect
159. C₂×C₃×D₃	C₂×C₁₆	semidirect
160. C₂×C₃×D₃	C₂×C₁₆	semidirect
164. C₃⋊Dic₃	C₂×C₁₆	semidirect
257. C₂×D₃	D₃×C₁₆	semidirect
258. C₂×D₃	D₃×C₁₆	semidirect
275. C₃²	D₄×C₁₆	semidirect
320. C₃	#67⋊C₁₆	semidirect
321. C₃	C₁₆×#67	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×C₁₆	2
C₈×#625	2
C₂×D₃×C₃⋊C₁₆	2
#10589367	2
C₃×(C₂×D₃)⋊C₁₆	2
(C₂×D₃)⋊(C₃⋊C₁₆)	2
#10589102	2
C₁₆×#67	3
#67⋊C₁₆	3