Subgroups — #11797603

Order 1152.

316 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5144	C₂	1	1	yes	maximal, characteristic
2	C₃×#5140	C₂	1	1	yes	maximal, characteristic, complemented
3	#10590621	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589427	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589134	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589111	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
8	#71453	—	3	1	no	maximal
9	#70924	—	3	1	no	maximal
10	C₃²×#131	C₂²	1	1	yes	characteristic
11	C₃×#950	C₄	1	1	yes	characteristic
12	C₃×#901	C₂²	1	1	yes	characteristic
13	#63639	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₃×#837	C₄	1	1	yes	characteristic
16	C₃×#825	C₄	1	1	yes	characteristic
17	C₃×#822	C₄	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₃×#812	—	2	1	no	—
20	C₃×#812	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
22	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	#63543	—	2	1	no	—
24	D₃×C₃⋊C₁₆	—	2	1	no	—
25	C₃×#504	—	3	1	no	—
26	C₃×#443	—	3	1	no	—
27	#5144	D₃	1	1	yes	characteristic
28	#5140	—	3	1	no	—
29	#5140	D₃	1	1	yes	characteristic, complemented
30	#4901	—	3	1	no	—
31	#4561	—	3	1	no	—
32	#4561	—	3	1	no	—
33	#4561	—	3	1	no	—
34	#4561	—	6	1	no	—
35	#4515	—	3	1	no	—
36	#4511	—	3	1	no	—
37	#4502	—	3	1	no	—
38	#4463	—	3	1	no	—
39	#4461	—	3	1	no	—
40	C₃²×#41	C₂×C₄	1	1	yes	characteristic
41	C₃×#235	C₂×C₄	1	1	yes	characteristic
42	C₃²×M₁₆	—	2	1	no	—
43	C₃²×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	—	2	1	no	—
45	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₃×#216	C₈	1	1	yes	characteristic
47	C₃×#215	C₈	1	1	yes	characteristic
48	C₃×#214	C₈	1	1	yes	characteristic
49	C₃×#213	C₈	1	1	yes	characteristic
50	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×#208	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
54	C₃×D₃×C₈	—	2	1	no	—
55	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
56	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
57	C₃×C₃⋊C₁₆	—	2	1	no	—
58	C₃×C₃⋊C₁₆	D₄	1	1	yes	characteristic
59	C₃×C₃⋊C₁₆	D₄	1	1	yes	characteristic
60	#2171	—	9	1	no	Sylow
61	C₃×#135	—	3	1	no	—
62	C₃×#131	—	2	1	no	—
63	C₃×#131	—	3	1	no	—
64	C₃×#131	C₂×D₃	1	1	yes	characteristic
65	C₃×#131	C₂×D₃	1	1	yes	characteristic
66	#950	Dic₃	1	1	yes	characteristic
67	#901	C₂×D₃	1	1	yes	characteristic
68	C₃×M₃₂	—	3	1	no	—
69	C₃×M₃₂	—	6	1	no	—
70	C₂×C₃×C₁₆	—	3	1	no	—
71	C₂×C₃×C₁₆	—	6	1	no	—
72	C₃×#108	—	3	1	no	—
73	C₃×#104	—	3	1	no	—
74	C₃×#104	—	3	1	no	—
75	C₃×C₄×C₈	—	3	1	no	—
76	#837	Dic₃	1	1	yes	characteristic
77	#825	Dic₃	1	1	yes	characteristic
78	#822	Dic₃	1	1	yes	characteristic
79	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
80	#812	—	2	1	no	—
81	#812	—	3	1	no	—
82	#812	—	3	1	no	—
83	#812	—	6	1	no	—
84	#812	—	6	1	no	—
85	#812	C₂×D₃	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	—	2	1	no	—
87	C₂×C₃⋊C₁₆	—	3	1	no	—
88	C₂×C₃⋊C₁₆	—	3	1	no	—
89	C₂×C₃⋊C₁₆	—	6	1	no	—
90	C₂×C₃⋊C₁₆	—	6	1	no	—
91	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
92	#798	—	6	1	no	—
93	D₃×C₁₆	—	6	1	no	—
94	C₃²×Q₈	C₂×C₈	1	1	yes	characteristic
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	C₂×C₈	1	1	yes	characteristic
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₃×#67	—	2	1	no	—
100	C₂×C₃×Dic₃	—	2	1	no	—
101	C₃×D₁₂	C₂×C₈	1	1	yes	characteristic
102	C₃×C₄×D₃	—	2	1	no	—
103	C₃×Dic₆	C₂×C₈	1	1	yes	characteristic
104	C₃²×C₈	—	2	1	no	—
105	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₃²×C₈	#41	1	1	yes	characteristic
107	C₃×C₃⋊C₈	—	2	1	no	—
108	C₃×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
109	C₃×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
110	#504	—	3	1	no	—
111	#504	—	9	1	no	—
112	#443	—	3	1	no	—
113	#364	—	9	1	no	—
114	#350	—	9	1	no	—
115	#350	—	9	1	no	—
116	#346	—	9	1	no	—
117	C₃×#41	—	2	1	no	—
118	C₃×#41	—	3	1	no	—
119	C₃×#41	C₄×D₃	1	1	yes	characteristic
120	C₃×#41	C₂×Dic₃	1	1	yes	characteristic
121	#235	C₂×Dic₃	1	1	yes	characteristic
122	C₃×Q₁₆	—	3	1	no	—
123	C₃×SD₁₆	—	3	1	no	—
124	C₃×SD₁₆	—	3	1	no	—
125	C₃×D₈	—	3	1	no	—
126	C₃×M₁₆	—	2	1	no	—
127	C₃×M₁₆	—	2	1	no	—
128	C₃×M₁₆	—	2	1	no	—
129	C₃×M₁₆	—	3	1	no	—
130	C₃×M₁₆	—	4	1	no	—
131	C₃×M₁₆	—	6	1	no	—
132	C₃×M₁₆	C₄×D₃	1	1	yes	characteristic
133	C₃×M₁₆	C₂×Dic₃	1	1	yes	characteristic
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	3	1	no	—
138	C₂×C₃×C₈	—	4	1	no	—
139	C₂×C₃×C₈	—	6	1	no	—
140	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
142	C₃×C₄²	—	3	1	no	—
143	#216	C₃⋊C₈	1	1	yes	characteristic
144	#215	C₃⋊C₈	1	1	yes	characteristic
145	#214	C₃⋊C₈	1	1	yes	characteristic
146	#213	C₃⋊C₈	1	1	yes	characteristic
147	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
148	#208	C₂×Dic₃	1	1	yes	characteristic
149	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
150	ΓL(1,𝔽₂₅)	—	2	1	no	—
151	D₃×C₈	—	2	1	no	—
152	C₃×C₁₆	—	3	1	no	—
153	C₃×C₁₆	—	3	1	no	—
154	C₃×C₁₆	—	6	1	no	—
155	C₃⋊C₁₆	—	2	1	no	—
156	C₃⋊C₁₆	—	6	1	no	—
157	C₃⋊C₁₆	—	6	1	no	—
158	C₃⋊C₁₆	—	6	1	no	—
159	C₃⋊C₁₆	—	6	1	no	—
160	C₃⋊C₁₆	—	12	1	no	—
161	C₃⋊C₁₆	—	12	1	no	—
162	C₃⋊C₁₆	#67	1	1	yes	characteristic
163	C₃⋊C₁₆	#67	1	1	yes	characteristic
164	C₂²×C₃²	—	2	1	no	—
165	C₂²×C₃²	#131	1	1	yes	characteristic
166	C₂×C₃×D₃	—	2	1	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
169	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₃×Dic₃	—	2	1	no	—
171	C₃×Dic₃	—	4	1	no	—
172	#135	—	3	1	no	—
173	#131	—	3	1	no	—
174	#131	D₃²	1	1	yes	characteristic
175	M₃₂	—	3	1	no	—
176	M₃₂	—	6	1	no	—
177	M₃₂	—	9	1	no	—
178	M₃₂	—	18	1	no	—
179	C₂×C₁₆	—	3	1	no	—
180	C₂×C₁₆	—	6	1	no	—
181	C₂×C₁₆	—	9	1	no	—
182	C₂×C₁₆	—	18	1	no	—
183	#108	—	3	1	no	—
184	#104	—	3	1	no	—
185	#104	—	3	1	no	—
186	C₄×C₈	—	3	1	no	—
187	C₃×Q₈	—	2	1	no	—
188	C₃×Q₈	—	3	1	no	—
189	C₃×Q₈	D₃×C₈	1	1	yes	characteristic
190	C₃×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
191	C₃×D₄	—	2	1	no	—
192	C₃×D₄	—	2	1	no	—
193	C₃×D₄	—	2	1	no	—
194	C₃×D₄	—	3	1	no	—
195	C₃×D₄	—	4	1	no	—
196	C₃×D₄	—	6	1	no	—
197	C₃×D₄	D₃×C₈	1	1	yes	characteristic
198	C₃×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	—	6	1	no	—
204	C₂×C₃×C₄	—	6	1	no	—
205	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
207	#67	—	2	1	no	—
208	C₂×Dic₃	—	2	1	no	—
209	D₁₂	C₂×C₃⋊C₈	1	1	yes	characteristic
210	C₄×D₃	—	2	1	no	—
211	Dic₆	C₂×C₃⋊C₈	1	1	yes	characteristic
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	3	1	no	—
217	C₃×C₈	—	3	1	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	—	6	1	no	—
220	C₃×C₈	#235	1	1	yes	characteristic
221	C₃×C₈	D₃×D₄	1	1	yes	characteristic
222	C₃×C₈	#237	1	1	yes	characteristic
223	C₃×C₈	C₂×#67	1	1	yes	characteristic
224	C₃⋊C₈	—	2	1	no	—
225	C₃⋊C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
226	C₃⋊C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
227	C₂×C₃²	—	2	1	no	—
228	C₂×C₃²	—	4	1	no	—
229	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
230	C₃×D₃	—	4	1	no	—
231	#41	—	3	1	no	—
232	#41	D₃×Dic₃	1	1	yes	characteristic
233	Q₁₆	—	3	1	no	—
234	SD₁₆	—	3	1	no	—
235	SD₁₆	—	3	1	no	—
236	D₈	—	3	1	no	—
237	M₁₆	—	2	1	no	—
238	M₁₆	—	3	1	no	—
239	M₁₆	—	6	1	no	—
240	M₁₆	D₃×Dic₃	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
245	C₄²	—	3	1	no	—
246	C₁₆	—	3	1	no	—
247	C₁₆	—	3	1	no	—
248	C₁₆	—	6	1	no	—
249	C₁₆	—	18	1	no	—
250	C₁₆	—	18	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	6	1	no	—
256	C₂²×C₃	#901	1	1	yes	characteristic
257	C₂²×C₃	#948	1	1	yes	characteristic
258	C₂×D₃	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	6	1	no	—
265	C₃×C₄	—	12	1	no	—
266	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
267	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
268	C₃×C₄	C₄×#67	1	1	yes	characteristic
269	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
270	Dic₃	—	2	1	no	—
271	Dic₃	—	4	1	no	—
272	C₃²	#2171	1	1	yes	characteristic, complemented, Sylow
273	Q₈	—	3	1	no	—
274	Q₈	D₃×C₃⋊C₈	1	1	yes	characteristic
275	D₄	—	2	1	no	—
276	D₄	—	3	1	no	—
277	D₄	—	6	1	no	—
278	D₄	D₃×C₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	3	1	no	—
285	C₈	—	3	1	no	—
286	C₈	—	6	1	no	—
287	C₈	#3804	1	1	yes	characteristic
288	C₈	D₃×#67	1	1	yes	characteristic
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	4	1	no	—
293	C₂×C₃	—	4	1	no	—
294	C₂×C₃	—	4	1	no	—
295	C₂×C₃	—	8	1	no	—
296	C₂×C₃	—	12	1	no	—
297	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
298	C₂×C₃	C₈×#67	1	1	yes	characteristic
299	D₃	—	4	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	#63814	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	6	1	no	—
305	C₄	—	12	1	no	—
306	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
307	C₄	Dic₃×#67	1	1	yes	characteristic
308	C₃	—	2	1	no	—
309	C₃	#70924	1	1	yes	characteristic, complemented
310	C₃	#71453	1	1	yes	characteristic, complemented
311	C₂	—	2	1	no	—
312	C₂	—	4	1	no	—
313	C₂	—	12	1	no	—
314	C₂	C₃⋊C₈×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₃×#67
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×#131	C₂²
socle	C₂×C₃²	C₈×D₄
prime radical	C₃×#950	C₄
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#131	D₃²
hypocenter	C₃²	#2171
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₃×#5140	C₂	semidirect
3. #10590621	C₂	semidirect
4. #10589427	C₂	semidirect
5. #10589134	C₂	semidirect
6. #10589111	C₂	semidirect
7. #10589103	C₂	semidirect
29. #5140	D₃	semidirect
272. C₃²	#2171	semidirect
309. C₃	#70924	semidirect
310. C₃	#71453	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5144	2
C₃×#5140	2
#10590621	2
#10589427	2
#10589134	2
#10589111	2
#10589103	2
#71453	3
#70924	3