Subgroups — #11797132

Order 1152.

309 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	D₃×#819	C₂	1	1	yes	maximal, characteristic
3	#10590706	C₂	1	1	yes	maximal, characteristic
4	#10589379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4513	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589137	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589136	C₂	1	1	yes	maximal, characteristic, complemented
8	#71427	—	3	1	no	maximal
9	#69312	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	#63841	—	2	1	no	—
12	D₃×#208	—	2	1	no	—
13	C₃²×M₃₂	C₂²	1	1	yes	characteristic
14	#63646	C₂²	1	1	yes	characteristic
15	#63644	C₂²	1	1	yes	characteristic
16	C₃×#819	C₂²	1	1	yes	characteristic
17	C₃×#817	C₂²	1	1	yes	characteristic
18	C₃×#812	C₂²	1	1	yes	characteristic
19	C₃×#798	C₄	1	2	yes	complemented
20	C₃×D₃×C₁₆	C₄	1	2	yes	complemented
21	#63577	—	2	1	no	—
22	#63575	—	2	1	no	—
23	C₂×C₃×M₃₂	—	3	1	no	—
24	C₂×#819	—	3	1	no	—
25	#5106	—	3	1	no	—
26	D₃×M₃₂	D₃	1	1	yes	characteristic
27	D₃×#108	—	3	1	no	—
28	#4860	—	3	1	no	—
29	C₃×#365	—	3	1	no	—
30	#4513	—	3	1	no	—
31	#4513	—	3	1	no	—
32	#4513	—	3	1	no	—
33	#4513	—	3	1	no	—
34	#4513	—	6	1	no	—
35	#4513	D₃	1	1	yes	characteristic, complemented
36	#4494	—	3	1	no	—
37	#4487	—	3	1	no	—
38	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
41	#3748	—	2	1	no	—
42	C₃×C₄⋊Dic₃	—	2	1	no	—
43	C₃×#208	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
45	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
48	C₃×D₃×C₈	Q₈	1	1	yes	characteristic
49	(C₂×D₃)⋊Dic₃	—	4	1	no	—
50	Dic₃²	—	4	1	no	—
51	#3711	—	4	1	no	—
52	D₃×C₃⋊C₈	—	4	1	no	—
53	C₃²×C₁₆	C₂×C₄	1	2	yes	—
54	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
55	#2155	—	9	1	no	Sylow
56	C₂²×C₃×C₈	—	3	1	no	—
57	#926	—	6	1	no	—
58	C₂×#208	—	6	1	no	—
59	D₃×M₁₆	—	6	1	no	—
60	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
61	#892	—	6	1	no	—
62	C₃×M₃₂	—	2	1	no	—
63	C₃×M₃₂	—	3	1	no	—
64	C₃×M₃₂	—	3	2	no	—
65	C₃×M₃₂	C₂×D₃	1	1	yes	characteristic
66	C₃×M₃₂	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₁₆	—	3	2	no	—
68	C₃×#108	—	3	1	no	—
69	C₃×#106	—	3	1	no	—
70	#822	—	6	1	no	—
71	#819	—	3	1	no	—
72	#819	—	3	1	no	—
73	#819	—	6	1	no	—
74	#819	—	6	1	no	—
75	#819	C₂×D₃	1	1	yes	characteristic
76	#818	—	6	1	no	—
77	#817	—	3	1	no	—
78	#817	—	3	1	no	—
79	#817	—	6	1	no	—
80	#817	C₂×D₃	1	1	yes	characteristic
81	#812	C₂×D₃	1	1	yes	characteristic
82	#807	—	6	1	no	—
83	#798	Dic₃	1	2	yes	complemented
84	D₃×C₁₆	Dic₃	1	2	yes	complemented
85	C₂²×C₃×D₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	4	1	no	—
88	C₂×C₃×Dic₃	—	4	1	no	—
89	C₂×C₃×Dic₃	SD₁₆	1	1	yes	characteristic
90	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
91	C₃×C₄×D₃	C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
93	C₃×C₄×D₃	SD₁₆	1	1	yes	characteristic
94	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₃⋊(C₃⋊C₈)	—	4	1	no	—
97	C₃×C₃⋊C₈	—	4	1	no	—
98	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₃×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂×M₃₂	—	3	1	no	—
101	C₂×#108	—	9	1	no	—
102	#427	—	9	1	no	—
103	#365	—	3	1	no	—
104	#365	—	9	1	no	—
105	#365	—	9	1	no	—
106	#365	—	9	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	D₁₂	1	1	yes	characteristic
109	C₃×M₁₆	—	6	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	3	1	no	—
112	C₂×C₃×C₈	—	3	1	no	—
113	C₂×C₃×C₈	—	3	1	no	—
114	C₂×C₃×C₈	—	3	1	no	—
115	C₂×C₃×C₈	—	3	1	no	—
116	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	6	1	no	—
119	#217	—	12	1	no	—
120	(C₂×D₃)⋊C₄	—	12	1	no	—
121	C₄⋊Dic₃	—	2	1	no	—
122	C₄⋊Dic₃	—	6	1	no	—
123	C₄⋊Dic₃	—	6	1	no	—
124	C₄⋊Dic₃	—	12	1	no	—
125	C₄×Dic₃	—	12	1	no	—
126	C₄×Dic₃	—	12	1	no	—
127	#208	—	2	1	no	—
128	#208	—	6	1	no	—
129	#208	—	6	1	no	—
130	#208	—	12	1	no	—
131	#208	—	12	1	no	—
132	C₂×C₃⋊C₈	—	12	1	no	—
133	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
134	ΓL(1,𝔽₂₅)	—	12	1	no	—
135	D₃×C₈	—	12	1	no	—
136	D₃×C₈	Dic₆	1	1	yes	characteristic
137	D₃×C₈	Dic₆	1	1	yes	characteristic
138	D₃×C₈	C₂×Dic₃	1	1	yes	characteristic
139	D₃×C₈	C₂×Dic₃	1	1	yes	characteristic
140	C₃×C₁₆	—	2	2	no	—
141	C₃×C₁₆	—	3	2	no	—
142	C₃×C₁₆	C₄×D₃	1	2	yes	—
143	C₃×C₁₆	C₂×Dic₃	1	2	yes	—
144	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	—
145	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₃×D₃	—	4	1	no	—
147	C₂×C₃×D₃	#106	1	1	yes	characteristic
148	C₂×C₃×D₃	#106	1	1	yes	characteristic
149	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
151	C₃⋊Dic₃	—	8	1	no	—
152	C₃×Dic₃	—	8	1	no	—
153	C₃×Dic₃	#106	1	1	yes	characteristic
154	C₃×Dic₃	#106	1	1	yes	characteristic
155	C₂×M₁₆	—	18	1	no	—
156	C₂²×C₈	—	3	1	no	—
157	#117	—	18	1	no	—
158	M₃₂	—	3	1	no	—
159	M₃₂	—	3	2	no	—
160	M₃₂	D₃²	1	1	yes	characteristic
161	C₂×C₁₆	—	3	2	no	—
162	#108	—	3	1	no	—
163	#108	—	9	1	no	—
164	#108	—	18	1	no	—
165	#107	—	18	1	no	—
166	#106	—	3	1	no	—
167	#106	—	9	1	no	—
168	C₂³×C₃	—	3	1	no	—
169	C₂²×D₃	#204	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	12	1	no	—
177	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
179	C₂×Dic₃	—	4	1	no	—
180	C₂×Dic₃	—	12	1	no	—
181	C₂×Dic₃	—	12	1	no	—
182	C₂×Dic₃	—	24	1	no	—
183	C₂×Dic₃	#204	1	1	yes	characteristic
184	C₄×D₃	#204	1	1	yes	characteristic
185	C₄×D₃	#204	1	1	yes	characteristic
186	C₄×D₃	C₄⋊Dic₃	1	1	yes	characteristic
187	C₄×D₃	C₄⋊Dic₃	1	1	yes	characteristic
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	3	1	no	—
191	C₃×C₈	—	3	1	no	—
192	C₃×C₈	—	12	1	no	—
193	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
194	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
195	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
196	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
197	C₃⋊C₈	—	4	1	no	—
198	C₃⋊C₈	—	12	1	no	—
199	C₃⋊C₈	—	12	1	no	—
200	C₃⋊C₈	—	24	1	no	—
201	C₃⋊C₈	C₄⋊Dic₃	1	1	yes	characteristic
202	C₃⋊C₈	C₄⋊Dic₃	1	1	yes	characteristic
203	C₂×C₃²	—	2	1	no	—
204	C₂×C₃²	C₂×#106	1	1	yes	characteristic
205	C₃×D₃	—	2	1	no	—
206	C₃×D₃	—	2	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	M₁₆	—	6	1	no	—
209	M₁₆	—	18	1	no	—
210	M₁₆	—	36	1	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	3	1	no	—
214	C₂×C₈	—	3	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	36	1	no	—
217	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
218	C₄⋊C₄	—	6	1	no	—
219	C₄⋊C₄	—	18	1	no	—
220	C₂²⋊C₄	—	36	1	no	—
221	C₄²	—	36	1	no	—
222	C₁₆	—	3	2	no	—
223	C₁₆	D₃×Dic₃	1	2	yes	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	3	1	no	—
226	C₂²×C₃	—	3	1	no	—
227	C₂²×C₃	—	12	1	no	—
228	C₂²×C₃	C₂×#204	1	1	yes	characteristic
229	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
230	C₂×D₃	—	4	1	no	—
231	C₂×D₃	#817	1	1	yes	characteristic
232	C₂×D₃	#817	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	3	1	no	—
236	C₃×C₄	—	3	1	no	—
237	C₃×C₄	—	24	1	no	—
238	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
239	C₃×C₄	C₂×#204	1	1	yes	characteristic
240	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
241	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
242	Dic₃	—	8	1	no	—
243	Dic₃	—	24	1	no	—
244	Dic₃	—	24	1	no	—
245	Dic₃	—	48	1	no	—
246	Dic₃	#817	1	1	yes	characteristic
247	Dic₃	#817	1	1	yes	characteristic
248	C₃²	#2155	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	12	1	no	—
256	C₂×C₄	—	36	1	no	—
257	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
258	C₈	—	3	1	no	—
259	C₈	—	3	1	no	—
260	C₈	—	12	1	no	—
261	C₈	—	36	1	no	—
262	C₈	D₃×Dic₆	1	1	yes	characteristic
263	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	4	1	no	—
268	C₂×C₃	—	6	1	no	—
269	C₂×C₃	—	6	1	no	—
270	C₂×C₃	D₃×#106	1	1	yes	characteristic
271	C₂×C₃	C₂×#817	1	1	yes	characteristic
272	D₃	—	2	1	no	—
273	D₃	—	2	1	no	—
274	C₂²	—	3	1	no	—
275	C₂²	—	3	1	no	—
276	C₂²	—	12	1	no	—
277	C₂²	D₃×#204	1	1	yes	characteristic
278	C₄	—	3	1	no	—
279	C₄	—	3	1	no	—
280	C₄	—	24	1	no	—
281	C₄	—	72	1	no	—
282	C₄	D₃×#204	1	1	yes	characteristic
283	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
284	C₃	—	2	1	no	—
285	C₃	#69312	1	1	yes	characteristic, complemented
286	C₃	#71427	1	1	yes	characteristic, complemented
287	C₂	—	2	1	no	—
288	C₂	—	6	1	no	—
289	C₂	—	6	1	no	—
290	C₂	D₃×#817	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×#817
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×M₃₂	C₂²
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃×D₃	SD₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	M₃₂	D₃²
hypocenter	C₃²	#2155
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #10589379	C₂	semidirect
5. C₃×#4513	C₂	semidirect
6. #10589137	C₂	semidirect
7. #10589136	C₂	semidirect
19. C₃×#798	C₄	semidirect
20. C₃×D₃×C₁₆	C₄	semidirect
35. #4513	D₃	semidirect
83. #798	Dic₃	semidirect
84. D₃×C₁₆	Dic₃	semidirect
248. C₃²	#2155	semidirect
285. C₃	#69312	semidirect
286. C₃	#71427	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×M₃₂	2
D₃×#819	2
#10590706	2
#10589379	2
C₃×#4513	2
#10589137	2
#10589136	2
#71427	3
#69312	3