Subgroups — #11773686

Order 1152.

389 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4685	C₂	1	1	yes	maximal, characteristic
2	#10590261	C₂	1	1	yes	maximal, characteristic
3	#10590261	C₂	1	1	yes	maximal, characteristic
4	#10589320	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589078	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	#70744	—	3	1	no	maximal
9	#68074	—	3	1	no	maximal
10	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
15	C₃×Dic₃⋊C₈	C₄	1	2	yes	—
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	2	yes	—
20	Dic₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
21	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	2	yes	—
22	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	2	yes	—
23	C₃×#431	—	3	1	no	—
24	#4925	—	3	1	no	—
25	#4925	—	3	1	no	—
26	#4685	—	3	1	no	—
27	#4685	—	3	1	no	—
28	#4685	D₃	1	1	yes	characteristic
29	C₃×#322	—	3	1	no	—
30	#4455	—	3	1	no	—
31	#4455	D₃	1	1	yes	characteristic, complemented
32	#4454	—	3	1	no	—
33	#4454	—	3	1	no	—
34	#4454	—	3	1	no	—
35	#4454	—	6	1	no	—
36	C₈×C₃⋊C₈	—	3	1	no	—
37	C₈×C₃⋊C₈	—	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×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₃²×C₄²	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
44	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	#1454	—	9	1	no	Sylow
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
54	C₃×C₄⋊C₈	—	3	2	no	—
55	C₃×C₂²⋊C₈	—	3	2	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	—	3	1	no	—
58	C₃×C₄×C₈	—	3	1	no	—
59	C₃×C₄×C₈	—	3	1	no	—
60	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	#830	—	3	2	no	—
63	#830	—	3	2	no	—
64	(C₂×D₃)⋊C₈	—	3	2	no	—
65	(C₂×D₃)⋊C₈	—	3	2	no	—
66	(C₂×D₃)⋊C₈	Dic₃	1	2	yes	—
67	Dic₃⋊C₈	—	3	2	no	—
68	Dic₃⋊C₈	—	3	2	no	—
69	Dic₃⋊C₈	Dic₃	1	2	yes	—
70	C₄⋊(C₃⋊C₈)	—	3	2	no	—
71	C₄⋊(C₃⋊C₈)	—	3	2	no	—
72	C₄×C₃⋊C₈	—	3	1	no	—
73	C₄×C₃⋊C₈	—	3	1	no	—
74	C₄×C₃⋊C₈	—	3	1	no	—
75	C₄×C₃⋊C₈	—	3	1	no	—
76	C₄×C₃⋊C₈	—	6	1	no	—
77	C₄×C₃⋊C₈	—	6	1	no	—
78	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
79	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
80	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
88	C₃×C₄×D₃	—	2	2	no	—
89	C₃×C₄×D₃	—	2	2	no	—
90	C₃×C₄×D₃	—	2	2	no	—
91	C₃²×C₈	—	2	2	no	—
92	C₃⋊(C₃⋊C₈)	—	2	2	no	—
93	C₃⋊(C₃⋊C₈)	—	2	2	no	—
94	C₃×C₃⋊C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	#431	—	3	1	no	—
98	#431	—	9	1	no	—
99	#431	—	9	1	no	—
100	#322	—	3	1	no	—
101	#322	—	9	1	no	—
102	#322	—	9	1	no	—
103	C₈²	—	9	1	no	—
104	C₂²×C₃×C₄	—	3	1	no	—
105	C₂²×C₃×C₄	—	3	1	no	—
106	C₂²×C₃×C₄	—	3	1	no	—
107	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
108	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
109	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	—	3	1	no	—
118	C₃×C₄²	—	3	1	no	—
119	C₃×C₄²	—	3	1	no	—
120	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
121	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
122	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
123	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
124	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
125	C₂×C₃⋊C₈	—	3	2	no	—
126	C₂×C₃⋊C₈	—	3	2	no	—
127	C₂×C₃⋊C₈	—	3	2	no	—
128	C₂×C₃⋊C₈	—	3	2	no	—
129	C₂×C₃⋊C₈	—	6	2	no	—
130	C₂×C₃⋊C₈	—	6	2	no	—
131	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
132	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
133	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
134	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₃×D₃	—	2	1	no	—
136	C₂×C₃×D₃	—	2	1	no	—
137	C₂×C₃×D₃	—	2	1	no	—
138	C₃²×C₄	#131	1	2	yes	—
139	C₃²×C₄	#131	1	2	yes	—
140	C₃²×C₄	#131	1	2	yes	—
141	C₃×Dic₃	—	2	1	no	—
142	C₃×Dic₃	—	2	1	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₂×C₄²	—	3	1	no	—
145	C₄⋊C₈	—	3	2	no	—
146	C₄⋊C₈	—	9	2	no	—
147	C₄⋊C₈	—	9	2	no	—
148	C₂²⋊C₈	—	3	2	no	—
149	C₂²⋊C₈	—	9	2	no	—
150	C₂²⋊C₈	—	9	2	no	—
151	C₄×C₈	—	3	1	no	—
152	C₄×C₈	—	3	1	no	—
153	C₄×C₈	—	3	1	no	—
154	C₄×C₈	—	9	1	no	—
155	C₄×C₈	—	9	1	no	—
156	C₄×C₈	D₃²	1	1	yes	characteristic
157	C₂³×C₃	—	3	1	no	—
158	C₂²×D₃	C₄×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	C₂×C₃×C₄	—	6	2	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
174	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
175	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
176	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
177	C₄×D₃	—	2	2	no	—
178	C₄×D₃	—	2	2	no	—
179	C₄×D₃	—	2	2	no	—
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	4	2	no	—
183	C₃×C₈	—	6	2	no	—
184	C₃×C₈	—	6	2	no	—
185	C₃×C₈	—	6	2	no	—
186	C₃⋊C₈	—	2	2	no	—
187	C₃⋊C₈	—	2	2	no	—
188	C₃⋊C₈	—	2	2	no	—
189	C₃⋊C₈	—	6	4	no	—
190	C₃⋊C₈	—	6	2	no	—
191	C₃⋊C₈	—	6	2	no	—
192	C₃⋊C₈	—	6	2	no	—
193	C₃⋊C₈	—	6	2	no	—
194	C₃⋊C₈	—	12	2	no	—
195	C₂×C₃²	#405	1	1	yes	characteristic
196	C₂×C₃²	#405	1	1	yes	characteristic
197	C₂×C₃²	#405	1	1	yes	characteristic
198	C₃×D₃	—	4	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	3	1	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	3	2	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	—	9	2	no	—
206	C₂×C₈	—	9	2	no	—
207	C₂×C₈	D₃×Dic₃	1	2	yes	—
208	C₄²	—	3	1	no	—
209	C₄²	—	3	1	no	—
210	C₄²	—	3	1	no	—
211	C₄²	C₂×D₃²	1	1	yes	characteristic
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	6	1	no	—
214	C₂²×C₃	—	6	1	no	—
215	C₂²×C₃	—	6	1	no	—
216	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
217	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
218	C₂×D₃	—	2	1	no	—
219	C₂×D₃	—	2	1	no	—
220	C₂×D₃	—	2	1	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	6	1	no	—
225	C₃×C₄	—	6	1	no	—
226	C₃×C₄	—	6	1	no	—
227	C₃×C₄	#901	1	2	yes	—
228	C₃×C₄	#901	1	2	yes	—
229	C₃×C₄	#907	1	2	yes	—
230	C₃×C₄	#907	1	2	yes	—
231	C₃×C₄	#907	1	2	yes	—
232	C₃×C₄	#948	1	2	yes	—
233	Dic₃	—	2	1	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	—	2	1	no	—
236	C₃²	#1454	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	3	1	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
245	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
246	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	6	2	no	—
249	C₈	—	6	2	no	—
250	C₈	—	6	2	no	—
251	C₈	—	18	2	no	—
252	C₈	—	18	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	12	1	no	—
257	C₂×C₃	#4689	1	1	yes	characteristic
258	C₂×C₃	#4708	1	1	yes	characteristic
259	C₂×C₃	#4708	1	1	yes	characteristic
260	C₂×C₃	#4997	1	1	yes	characteristic
261	C₂×C₃	#5101	1	1	yes	characteristic
262	C₂×C₃	#5122	1	1	yes	characteristic
263	D₃	—	4	1	no	—
264	C₂²	—	6	1	no	—
265	C₂²	—	6	1	no	—
266	C₂²	—	6	1	no	—
267	C₂²	C₄×D₃×Dic₃	1	1	yes	characteristic
268	C₄	—	6	1	no	—
269	C₄	—	6	1	no	—
270	C₄	—	6	1	no	—
271	C₄	#63804	1	2	yes	—
272	C₄	#63805	1	2	yes	—
273	C₄	#63814	1	2	yes	—
274	C₃	—	2	1	no	—
275	C₃	#68074	1	1	yes	characteristic, complemented
276	C₃	#70744	1	1	yes	characteristic, complemented
277	C₂	—	12	1	no	—
278	C₂	#10590318	1	1	yes	characteristic
279	C₂	#10590693	1	1	yes	characteristic
280	C₂	#10590909	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂×C₄²
Fitting	C₃²×C₄×C₈	C₂²
prime radical	C₂²×C₃×D₃	C₄²
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1454
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10589320	C₂	semidirect
5. C₃×#4455	C₂	semidirect
6. #10589078	C₂	semidirect
7. C₃⋊C₈²	C₂	semidirect
31. #4455	D₃	semidirect
236. C₃²	#1454	semidirect
275. C₃	#68074	semidirect
276. C₃	#70744	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4685	2
#10590261	2
#10589320	2
C₃×#4455	2
#10589078	2
C₃⋊C₈²	2
#70744	3
#68074	3