Subgroups — #11770922

Order 1152.

371 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592624	C₂	1	1	yes	maximal, characteristic
2	C₃×#4936	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	#67713	—	3	1	no	maximal
5	#67631	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	#63657	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
12	C₃×C₃⋊#95	—	2	1	no	—
13	C₃×C₃⋊#95	C₄	1	2	yes	—
14	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
15	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
16	C₃×#423	—	3	1	no	—
17	C₃×#382	—	3	1	no	—
18	#4936	D₃	1	1	yes	characteristic
19	#4925	—	3	1	no	—
20	#4925	—	3	1	no	—
21	#4925	—	6	1	no	—
22	#4924	D₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	—	2	2	no	—
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	D₄	1	1	yes	characteristic
30	C₃²×C₄²	Q₈	1	1	yes	characteristic
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₃×C₄⋊Dic₃	—	4	1	no	—
33	C₃×C₄⋊Dic₃	—	4	1	no	—
34	C₃×#208	—	4	1	no	—
35	C₃×#208	—	4	1	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	#1286	—	9	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂×C₃×C₄⋊C₄	—	6	1	no	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₃	—	2	1	no	—
44	C₂×#208	—	2	1	no	—
45	C₃×C₄⋊C₈	—	3	2	no	—
46	C₃×C₄⋊C₈	—	6	1	no	—
47	C₃×#95	—	3	2	no	—
48	C₃×#95	—	6	1	no	—
49	C₃⋊#95	—	2	1	no	—
50	C₃⋊#95	Dic₃	1	2	yes	—
51	#830	—	6	2	no	—
52	#830	—	6	1	no	—
53	#830	—	6	1	no	—
54	C₄⋊(C₃⋊C₈)	—	2	1	no	—
55	C₄⋊(C₃⋊C₈)	—	6	2	no	—
56	C₄⋊(C₃⋊C₈)	—	6	1	no	—
57	C₄⋊(C₃⋊C₈)	—	6	1	no	—
58	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	complemented
59	C₄×C₃⋊C₈	—	3	2	no	—
60	C₄×C₃⋊C₈	—	3	2	no	—
61	C₄×C₃⋊C₈	—	6	2	no	—
62	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	4	1	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
73	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
74	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	4	2	no	—
76	C₂×C₃×Dic₃	—	4	2	no	—
77	C₂×C₃×Dic₃	—	4	2	no	—
78	C₃⋊(C₃⋊C₈)	—	4	2	no	—
79	C₃×C₃⋊C₈	—	4	2	no	—
80	#431	—	9	1	no	—
81	#423	—	3	1	no	—
82	#382	—	3	1	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	6	2	no	—
87	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	#67	1	1	yes	characteristic
91	C₂²×C₃×C₄	#67	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×Dic₃	—	2	2	no	—
94	C₃×M₁₆	—	12	1	no	—
95	C₃×M₁₆	—	12	1	no	—
96	C₂×C₃×C₈	—	6	2	no	—
97	C₃×C₄⋊C₄	—	12	1	no	—
98	C₃×C₄⋊C₄	—	12	1	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	Dic₆	1	1	yes	characteristic
104	C₃×C₄²	Dic₆	1	1	yes	characteristic
105	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
106	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	D₁₂	1	1	yes	characteristic
109	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₄⋊Dic₃	—	4	1	no	—
112	C₄⋊Dic₃	—	4	1	no	—
113	#208	—	4	1	no	—
114	#208	—	4	1	no	—
115	C₂×C₃⋊C₈	—	2	2	no	—
116	C₂×C₃⋊C₈	—	6	4	no	—
117	C₂×C₃⋊C₈	—	6	2	no	—
118	C₂×C₃⋊C₈	—	6	2	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	#95	1	1	yes	characteristic
122	C₂²×C₃²	#102	1	1	yes	characteristic
123	C₂²×C₃²	#103	1	1	yes	characteristic
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	#102	1	1	yes	characteristic
129	C₃²×C₄	#103	1	1	yes	characteristic
130	C₃²×C₄	#106	1	1	yes	characteristic
131	C₃²×C₄	#107	1	1	yes	characteristic
132	C₃×Dic₃	—	8	2	no	—
133	C₂×M₁₆	—	6	1	no	—
134	C₂×C₄⋊C₄	—	6	1	no	—
135	C₂×C₄²	D₃²	1	1	yes	characteristic
136	C₄⋊C₈	—	3	2	no	—
137	C₄⋊C₈	—	6	1	no	—
138	C₄⋊C₈	—	18	1	no	—
139	C₂²⋊C₈	—	18	1	no	—
140	C₄×C₈	—	9	2	no	—
141	#95	—	3	2	no	—
142	#95	—	6	1	no	—
143	C₂³×C₃	—	2	1	no	—
144	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
145	C₂³×C₃	#217	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	12	2	no	—
163	C₂×C₃×C₄	—	12	2	no	—
164	C₂×C₃×C₄	—	12	2	no	—
165	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	#213	1	1	yes	characteristic
177	C₂×C₃×C₄	#213	1	1	yes	characteristic
178	C₂×C₃×C₄	#214	1	1	yes	characteristic
179	C₂×C₃×C₄	#214	1	1	yes	characteristic
180	C₂×C₃×C₄	#215	1	1	yes	characteristic
181	C₂×C₃×C₄	#215	1	1	yes	characteristic
182	C₂×C₃×C₄	#216	1	1	yes	characteristic
183	C₂×C₃×C₄	#216	1	1	yes	characteristic
184	C₂×C₃×C₄	#217	1	1	yes	characteristic
185	C₂×Dic₃	—	4	2	no	—
186	C₂×Dic₃	—	4	2	no	—
187	C₂×Dic₃	—	4	2	no	—
188	C₃×C₈	—	12	2	no	—
189	C₃⋊C₈	—	4	2	no	—
190	C₃⋊C₈	—	12	4	no	—
191	C₃⋊C₈	—	12	2	no	—
192	C₃⋊C₈	—	12	2	no	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	H(3,ℤ/4)	1	1	yes	characteristic
196	C₂×C₃²	#340	1	1	yes	characteristic
197	C₂×C₃²	#344	1	1	yes	characteristic
198	C₂²×C₄	—	6	2	no	—
199	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	#625	1	1	yes	characteristic
201	C₂²×C₄	#626	1	1	yes	characteristic
202	M₁₆	—	12	1	no	—
203	M₁₆	—	12	1	no	—
204	C₂×C₈	—	6	2	no	—
205	C₂×C₈	—	18	2	no	—
206	C₄⋊C₄	—	12	1	no	—
207	C₄⋊C₄	—	12	1	no	—
208	C₄²	D₃×Dic₃	1	1	yes	characteristic
209	C₄²	#624	1	1	yes	characteristic
210	C₄²	#626	1	1	yes	characteristic
211	C₄²	#627	1	1	yes	characteristic
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	1	no	—
221	C₂²×C₃	#809	1	1	yes	characteristic
222	C₂²×C₃	#810	1	1	yes	characteristic
223	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
224	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
225	C₂²×C₃	#832	1	1	yes	characteristic
226	C₂²×C₃	#835	1	1	yes	characteristic
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	4	1	no	—
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	—	24	2	no	—
244	C₃×C₄	#807	1	1	yes	characteristic
245	C₃×C₄	#807	1	1	yes	characteristic
246	C₃×C₄	#808	1	1	yes	characteristic
247	C₃×C₄	#808	1	1	yes	characteristic
248	C₃×C₄	#809	1	1	yes	characteristic
249	C₃×C₄	#810	1	1	yes	characteristic
250	C₃×C₄	#832	1	1	yes	characteristic
251	C₃×C₄	#835	1	1	yes	characteristic
252	Dic₃	—	8	2	no	—
253	C₃²	#1286	1	1	yes	characteristic, complemented, Sylow
254	C₂³	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	12	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	—	12	2	no	—
260	C₂×C₄	#3713	1	1	yes	characteristic
261	C₂×C₄	#3715	1	1	yes	characteristic
262	C₂×C₄	#3715	1	1	yes	characteristic
263	C₂×C₄	#3718	1	1	yes	characteristic
264	C₂×C₄	Dic₃²	1	1	yes	characteristic
265	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
266	C₂×C₄	#3722	1	1	yes	characteristic
267	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
268	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
269	C₂×C₄	#3724	1	1	yes	characteristic
270	C₈	—	12	2	no	—
271	C₈	—	36	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	4	1	no	—
280	C₂×C₃	—	4	1	no	—
281	C₂×C₃	#4529	1	1	yes	characteristic
282	C₂×C₃	#4529	1	1	yes	characteristic
283	C₂×C₃	#4531	1	1	yes	characteristic
284	C₂×C₃	#4533	1	1	yes	characteristic
285	C₂×C₃	#4557	1	1	yes	characteristic
286	C₂×C₃	#4559	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	#63562	1	1	yes	characteristic
290	C₂²	#63565	1	1	yes	characteristic
291	C₂²	#63579	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	24	2	no	—
297	C₄	#63562	1	1	yes	characteristic
298	C₄	#63565	1	1	yes	characteristic
299	C₄	#63574	1	1	yes	characteristic
300	C₄	#63576	1	1	yes	characteristic
301	C₃	—	2	1	no	—
302	C₃	#67631	1	1	yes	characteristic, complemented
303	C₃	#67713	1	1	yes	characteristic, complemented
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	#10589158	1	1	yes	characteristic
307	C₂	#10589163	1	1	yes	characteristic
308	C₂	#10589170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63579
commutator	C₂×C₃²×C₄	C₄²
Frattini, hypercenter	C₂×C₄²	D₃²
Fitting, cosocle	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
hypocenter	C₃²	#1286
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
15. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
58. C₄⋊(C₃⋊C₈)	Dic₃	semidirect
253. C₃²	#1286	semidirect
302. C₃	#67631	semidirect
303. C₃	#67713	semidirect

Maximal subgroups

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

subgroup	index
#10592624	2
C₃×#4936	2
C₃×#4924	2
#67713	3
#67631	3