Subgroups — #11861355

Order 1152.

791 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5470	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10591116	C₂	1	2	yes	maximal
5	#10591055	C₂	1	2	yes	maximal
6	#10590258	C₂	1	4	yes	maximal, complemented
7	C₃⋊C₈×Dic₆	C₂	1	4	yes	maximal, complemented
8	#83328	—	3	1	no	maximal
9	#78445	—	3	1	no	maximal
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#3748	C₂²	1	2	yes	—
12	C₃×#926	C₂²	1	2	yes	—
13	C₃×#924	C₄	1	4	yes	—
14	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#870	C₄	1	2	yes	—
17	C₃×#869	C₄	1	2	yes	—
18	C₃×C₄×Dic₆	C₂²	1	4	yes	—
19	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	4	yes	—
20	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
21	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
22	Dic₃⋊(C₃⋊C₈)	C₂²	1	8	yes	—
23	#63554	C₂²	1	4	yes	—
24	Dic₃×C₃⋊C₈	C₂²	1	4	yes	—
25	C₃×#533	—	3	1	no	—
26	#5470	D₃	1	1	yes	characteristic
27	C₂×C₃×C₄×C₈	—	3	1	no	—
28	#5102	—	3	2	no	—
29	#5101	—	3	2	no	—
30	#5024	—	3	4	no	—
31	Q₈×C₃⋊C₈	—	3	4	no	—
32	#4999	—	3	2	no	—
33	#4997	—	3	2	no	—
34	#4924	—	3	1	no	—
35	#4924	—	3	1	no	—
36	#4924	—	6	1	no	—
37	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
38	#4679	—	3	4	no	—
39	C₈×Dic₆	—	3	4	no	—
40	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
43	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂³	1	2	yes	—
46	#3748	—	2	4	no	—
47	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
48	C₃×#217	C₂×C₄	1	4	yes	—
49	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
50	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
51	C₃×Dic₃⋊C₄	C₂×C₄	1	8	yes	—
52	C₃×C₄×Dic₃	C₂³	1	4	yes	—
53	C₂×C₃×C₃⋊C₈	—	2	2	no	—
54	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	complemented
55	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
58	#2960	—	9	1	no	Sylow
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₂²×C₃×C₈	—	3	1	no	—
61	C₃×C₄⋊Q₈	—	3	2	no	—
62	C₃×#125	—	3	2	no	—
63	C₃×C₂²⋊Q₈	—	3	4	no	—
64	C₃×C₄×Q₈	—	3	4	no	—
65	C₃×#117	—	3	2	no	—
66	C₂×C₃×C₄²	—	2	1	no	—
67	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
68	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
69	#926	C₂×D₃	1	2	yes	—
70	#924	Dic₃	1	4	yes	—
71	C₂×#208	—	3	2	no	—
72	C₂×#208	—	3	2	no	—
73	C₂×#208	—	6	2	no	—
74	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
76	#870	Dic₃	1	2	yes	—
77	#869	Dic₃	1	2	yes	—
78	C₄×Dic₆	C₂×D₃	1	4	yes	—
79	C₃×C₄×C₈	—	3	2	no	—
80	C₃×C₄×C₈	—	3	2	no	—
81	#815	—	3	4	no	—
82	Dic₃⋊C₈	—	3	8	no	—
83	C₈×Dic₃	—	3	4	no	—
84	C₄⋊(C₃⋊C₈)	—	3	4	no	—
85	C₄⋊(C₃⋊C₈)	—	3	4	no	—
86	C₄⋊(C₃⋊C₈)	—	3	8	no	—
87	C₄⋊(C₃⋊C₈)	—	6	4	no	—
88	#803	—	3	4	no	—
89	C₄×C₃⋊C₈	—	3	4	no	—
90	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
91	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
92	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
97	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
98	C₂×C₃²×C₄	#41	1	2	yes	—
99	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×Dic₃	C₂²×C₄	1	8	yes	—
101	C₃×Dic₆	—	2	8	no	—
102	C₃⋊(C₃⋊C₈)	—	2	4	no	—
103	C₃×C₃⋊C₈	—	2	2	no	—
104	C₃×C₃⋊C₈	C₂×Q₈	1	2	yes	—
105	C₃×C₃⋊C₈	C₂×Q₈	1	2	yes	—
106	#533	—	3	1	no	—
107	#446	—	9	4	no	—
108	C₈×Q₈	—	9	4	no	—
109	#424	—	9	2	no	—
110	#423	—	9	1	no	—
111	#405	—	9	2	no	—
112	C₂×C₄×C₈	—	3	1	no	—
113	C₂×C₃×Q₈	—	3	4	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
117	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
118	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂×Dic₆	C₂×Dic₃	1	4	yes	—
121	C₂×C₃×C₈	—	3	4	no	—
122	C₂×C₃×C₈	—	3	1	no	—
123	C₂×C₃×C₈	—	3	1	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₂×C₃×C₈	—	6	2	no	—
126	C₃×C₄⋊C₄	—	3	8	no	—
127	C₃×C₄⋊C₄	—	3	4	no	—
128	C₃×C₄⋊C₄	—	3	4	no	—
129	C₃×C₂²⋊C₄	—	3	4	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	2	2	no	—
133	C₃×C₄²	—	3	4	no	—
134	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
135	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
136	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
137	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
138	C₃×C₄²	C₂²×D₃	1	2	yes	—
139	C₃×C₄²	C₂²×D₃	1	2	yes	—
140	#217	C₂×Dic₃	1	4	yes	—
141	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
142	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
143	Dic₃⋊C₄	C₂×Dic₃	1	8	yes	—
144	C₄×Dic₃	C₂²×D₃	1	4	yes	—
145	#208	—	6	4	no	—
146	#208	—	6	4	no	—
147	#208	—	12	4	no	—
148	C₂×C₃⋊C₈	—	2	2	no	—
149	C₂×C₃⋊C₈	—	3	4	no	—
150	C₂×C₃⋊C₈	—	3	4	no	—
151	C₂×C₃⋊C₈	—	6	4	no	—
152	C₂×C₃⋊C₈	Dic₆	1	4	yes	complemented
153	C₂×C₃⋊C₈	C₂²×D₃	1	2	yes	—
154	C₂×C₃⋊C₈	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₃⋊C₈	C₂²×D₃	1	1	yes	characteristic
156	C₂²×C₃²	—	2	2	no	—
157	C₂²×C₃²	C₄×Q₈	1	2	yes	—
158	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
159	C₃²×C₄	—	2	2	no	—
160	C₃²×C₄	C₄×Q₈	1	2	yes	—
161	C₃²×C₄	#131	1	4	yes	—
162	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
163	C₃²×C₄	C₂×#41	1	1	yes	characteristic
164	C₃×Dic₃	—	2	8	no	—
165	C₂×M₁₆	—	9	2	no	—
166	C₂²×C₈	—	3	1	no	—
167	C₂²×C₈	—	3	1	no	—
168	C₄⋊Q₈	—	3	2	no	—
169	#125	—	3	2	no	—
170	C₂²⋊Q₈	—	3	4	no	—
171	C₄×Q₈	—	3	4	no	—
172	#117	—	3	2	no	—
173	C₂×C₄²	D₃²	1	1	yes	characteristic
174	C₄⋊C₈	—	9	8	no	—
175	C₄⋊C₈	—	9	4	no	—
176	#97	—	9	4	no	—
177	C₄×C₈	—	3	2	no	—
178	C₄×C₈	—	3	2	no	—
179	C₄×C₈	—	9	4	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
183	C₃×Q₈	—	6	8	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	3	8	no	—
193	C₂×C₃×C₄	—	4	4	no	—
194	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
195	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
198	C₂×C₃×C₄	#235	1	2	yes	—
199	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
200	C₂×C₃×C₄	#239	1	2	yes	—
201	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
202	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
204	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
206	C₂×Dic₃	C₂²×Dic₃	1	8	yes	—
207	Dic₆	—	2	8	no	—
208	C₃×C₈	—	3	2	no	—
209	C₃×C₈	—	3	2	no	—
210	C₃×C₈	—	6	2	no	—
211	C₃⋊C₈	—	2	2	no	—
212	C₃⋊C₈	—	6	4	no	—
213	C₃⋊C₈	—	6	4	no	—
214	C₃⋊C₈	—	12	4	no	—
215	C₃⋊C₈	C₂×Dic₆	1	2	yes	—
216	C₃⋊C₈	C₂×Dic₆	1	2	yes	—
217	C₂×C₃²	—	2	2	no	—
218	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
219	C₂×C₃²	C₂×#131	1	2	yes	—
220	C₂×Q₈	—	3	4	no	—
221	C₂²×C₄	D₃×Dic₃	1	2	yes	—
222	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
223	M₁₆	—	18	4	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	3	1	no	—
226	C₂×C₈	—	3	4	no	—
227	C₂×C₈	—	3	2	no	—
228	C₂×C₈	—	6	2	no	—
229	C₂×C₈	—	9	4	no	—
230	C₄⋊C₄	—	3	4	no	—
231	C₄⋊C₄	—	3	8	no	—
232	C₄⋊C₄	—	3	4	no	—
233	C₂²⋊C₄	—	3	4	no	—
234	C₄²	—	3	4	no	—
235	C₄²	D₃×Dic₃	1	1	yes	characteristic
236	C₄²	D₃×Dic₃	1	1	yes	characteristic
237	C₄²	C₂×D₃²	1	2	yes	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	4	2	no	—
243	C₂²×C₃	C₄×Dic₆	1	2	yes	—
244	C₂²×C₃	Q₈×Dic₃	1	2	yes	—
245	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
246	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	4	2	no	—
254	C₃×C₄	—	6	8	no	—
255	C₃×C₄	C₄×Dic₆	1	2	yes	—
256	C₃×C₄	#901	1	4	yes	—
257	C₃×C₄	Q₈×Dic₃	1	2	yes	—
258	C₃×C₄	#948	1	4	yes	—
259	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
260	C₃×C₄	C₂×#235	1	1	yes	characteristic
261	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
262	C₃×C₄	C₂×#239	1	1	yes	characteristic
263	Dic₃	—	2	8	no	—
264	C₃²	#2960	1	1	yes	characteristic, complemented, Sylow
265	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
266	Q₈	—	6	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	3	8	no	—
269	C₂×C₄	D₃×Dic₆	1	2	yes	—
270	C₂×C₄	#3799	1	2	yes	—
271	C₂×C₄	C₂×D₃×Dic₃	1	2	yes	—
272	C₂×C₄	C₂×D₃×Dic₃	1	2	yes	—
273	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₃²	1	1	yes	characteristic
275	C₈	—	3	2	no	—
276	C₈	—	3	2	no	—
277	C₈	—	6	2	no	—
278	C₈	—	18	4	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	4	2	no	—
284	C₂×C₃	C₂×C₄×Dic₆	1	1	yes	characteristic
285	C₂×C₃	C₂×#901	1	2	yes	—
286	C₂×C₃	C₂×Q₈×Dic₃	1	1	yes	characteristic
287	C₂×C₃	C₂×#948	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	Dic₃×Dic₆	1	2	yes	—
290	C₂²	C₂²×D₃×Dic₃	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	6	8	no	—
293	C₄	#63814	1	4	yes	—
294	C₄	Dic₃×Dic₆	1	2	yes	—
295	C₄	C₂×D₃×Dic₆	1	1	yes	characteristic
296	C₄	C₂×#3799	1	1	yes	characteristic
297	C₃	—	2	1	no	—
298	C₃	#78445	1	1	yes	characteristic, complemented
299	C₃	#83328	1	1	yes	characteristic, complemented
300	C₂	—	2	2	no	—
301	C₂	C₂×#63814	1	2	yes	—
302	C₂	C₂×Dic₃×Dic₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃²
commutator, socle	C₂²×C₃²	C₂³×C₄
Fitting	C₂×C₃²×C₄²	C₂²
prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#2960
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. #10590258	C₂	semidirect
7. C₃⋊C₈×Dic₆	C₂	semidirect
54. C₂×C₃×C₃⋊C₈	Q₈	semidirect
152. C₂×C₃⋊C₈	Dic₆	semidirect
264. C₃²	#2960	semidirect
298. C₃	#78445	semidirect
299. C₃	#83328	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5470	2
#10592623	2
C₂×C₃×C₄×C₃⋊C₈	2
#10591116	2
#10591055	2
#10590258	2
C₃⋊C₈×Dic₆	2
#83328	3
#78445	3