Subgroups — #11797028

Order 1152.

345 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4689	C₂	1	1	yes	maximal, characteristic
2	C₂×#63543	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	#69882	—	3	1	no	maximal
9	#69285	—	3	1	no	maximal
10	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×#872	C₄	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×#815	C₄	1	1	yes	characteristic
17	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₃×#803	C₄	1	1	yes	characteristic
21	#63543	—	2	2	no	—
22	D₃×C₃⋊C₁₆	—	2	2	no	—
23	C₃×#405	—	3	1	no	—
24	C₂×#812	—	3	1	no	—
25	C₂²×C₃⋊C₁₆	—	3	1	no	—
26	C₂×#798	—	3	1	no	—
27	C₂×D₃×C₁₆	—	3	1	no	—
28	#4689	D₃	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	—	3	1	no	—
30	Dic₃⋊C₁₆	—	3	1	no	—
31	Dic₃⋊C₁₆	—	3	1	no	—
32	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
33	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
34	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
35	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
36	C₄⋊(C₃⋊C₁₆)	—	6	1	no	—
37	C₄⋊(C₃⋊C₁₆)	D₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
43	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
44	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
45	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
50	C₃×D₃×C₈	D₄	1	2	yes	—
51	C₃×D₃×C₈	Q₈	1	2	yes	—
52	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
53	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
54	C₃×C₃⋊C₁₆	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	#2151	—	9	1	no	Sylow
57	C₂×C₃×M₁₆	—	3	1	no	—
58	C₂²×C₃×C₈	—	3	1	no	—
59	C₃×#117	—	3	1	no	—
60	C₂×ΓL(1,𝔽₂₅)	Dic₃	1	1	yes	characteristic
61	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
62	#872	Dic₃	1	1	yes	characteristic
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	—	3	1	no	—
65	C₃×#97	—	3	1	no	—
66	C₃×#97	—	3	1	no	—
67	C₃×C₄×C₈	—	2	1	no	—
68	C₃×C₄×C₈	—	3	1	no	—
69	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	#815	Dic₃	1	1	yes	characteristic
72	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
73	#812	—	6	2	no	—
74	C₂×C₃⋊C₁₆	—	3	1	no	—
75	C₂×C₃⋊C₁₆	—	3	1	no	—
76	C₂×C₃⋊C₁₆	—	3	1	no	—
77	C₂×C₃⋊C₁₆	—	3	1	no	—
78	C₂×C₃⋊C₁₆	—	6	1	no	—
79	C₂×C₃⋊C₁₆	—	6	1	no	—
80	C₂×C₃⋊C₁₆	—	6	2	no	—
81	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
83	#803	Dic₃	1	1	yes	characteristic
84	#798	—	6	2	no	—
85	D₃×C₁₆	—	6	2	no	—
86	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
93	C₃×C₄×D₃	C₄⋊C₄	1	2	yes	—
94	C₃×C₄×D₃	M₁₆	1	2	yes	—
95	C₃²×C₈	—	2	1	no	—
96	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₃×C₃⋊C₈	—	2	1	no	—
99	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
100	C₂×M₃₂	—	9	1	no	—
101	C₂²×C₁₆	—	9	1	no	—
102	#405	—	3	1	no	—
103	C₄⋊C₁₆	—	3	1	no	—
104	C₄⋊C₁₆	—	9	1	no	—
105	C₄⋊C₁₆	—	9	1	no	—
106	C₄⋊C₁₆	—	9	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
109	C₃×M₁₆	—	6	2	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	3	1	no	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₂×C₃×C₈	—	3	2	no	—
115	C₂×C₃×C₈	—	3	1	no	—
116	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	3	1	no	—
121	C₃×C₄⋊C₄	—	3	1	no	—
122	C₃×C₂²⋊C₄	—	3	1	no	—
123	C₃×C₂²⋊C₄	—	3	1	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	3	1	no	—
126	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
127	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
128	(C₂×D₃)⋊C₄	C₃⋊C₈	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	C₃⋊C₈	1	1	yes	characteristic
130	Dic₃⋊C₄	C₃⋊C₈	1	1	yes	characteristic
131	Dic₃⋊C₄	C₃⋊C₈	1	1	yes	characteristic
132	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
134	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
135	ΓL(1,𝔽₂₅)	—	2	2	no	—
136	D₃×C₈	Dic₆	1	2	yes	—
137	D₃×C₈	D₁₂	1	2	yes	—
138	C₃×C₁₆	—	6	1	no	—
139	C₃×C₁₆	—	6	1	no	—
140	C₃⋊C₁₆	—	2	1	no	—
141	C₃⋊C₁₆	—	2	1	no	—
142	C₃⋊C₁₆	—	6	1	no	—
143	C₃⋊C₁₆	—	6	1	no	—
144	C₃⋊C₁₆	—	6	1	no	—
145	C₃⋊C₁₆	—	6	1	no	—
146	C₃⋊C₁₆	—	12	1	no	—
147	C₃⋊C₁₆	—	12	1	no	—
148	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₃×D₃	—	2	1	no	—
150	C₂×C₃×D₃	—	2	1	no	—
151	C₂×C₃×D₃	C₄⋊C₈	1	2	yes	—
152	C₃²×C₄	—	2	1	no	—
153	C₃²×C₄	—	2	1	no	—
154	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
156	C₃×Dic₃	—	2	1	no	—
157	C₃×Dic₃	—	2	1	no	—
158	C₃×Dic₃	C₄⋊C₈	1	2	yes	—
159	C₂×M₁₆	—	3	1	no	—
160	C₂²×C₈	—	3	1	no	—
161	#117	—	3	1	no	—
162	M₃₂	—	18	2	no	—
163	C₂×C₁₆	—	3	1	no	—
164	C₂×C₁₆	—	3	1	no	—
165	C₂×C₁₆	—	9	1	no	—
166	C₂×C₁₆	—	9	1	no	—
167	C₂×C₁₆	—	18	2	no	—
168	#97	—	3	1	no	—
169	#97	—	3	1	no	—
170	C₄×C₈	—	3	1	no	—
171	C₄×C₈	D₃²	1	1	yes	characteristic
172	C₂³×C₃	—	3	1	no	—
173	C₂²×D₃	C₂×C₃⋊C₈	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	2	no	—
179	C₂×C₃×C₄	—	3	2	no	—
180	C₂×C₃×C₄	—	3	1	no	—
181	C₂×C₃×C₄	—	3	1	no	—
182	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
183	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
189	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
190	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
191	C₄×D₃	#208	1	2	yes	—
192	C₄×D₃	C₄⋊Dic₃	1	2	yes	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	3	2	no	—
198	C₃×C₈	—	4	1	no	—
199	C₃×C₈	—	6	1	no	—
200	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
201	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
202	C₃×C₈	D₃×D₄	1	1	yes	characteristic
203	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
204	C₃⋊C₈	—	2	1	no	—
205	C₃⋊C₈	C₄⋊Dic₃	1	2	yes	—
206	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
207	C₂×C₃²	#504	1	1	yes	characteristic
208	C₂×C₃²	#504	1	1	yes	characteristic
209	C₃×D₃	—	2	2	no	—
210	C₂²×C₄	—	3	1	no	—
211	M₁₆	—	6	2	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	3	2	no	—
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
217	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₂²⋊C₄	—	3	1	no	—
221	C₂²⋊C₄	—	3	1	no	—
222	C₄²	—	3	1	no	—
223	C₄²	D₃×Dic₃	1	1	yes	characteristic
224	C₁₆	—	6	1	no	—
225	C₁₆	—	6	1	no	—
226	C₁₆	—	18	1	no	—
227	C₁₆	—	18	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	3	2	no	—
230	C₂²×C₃	—	6	1	no	—
231	C₂²×C₃	—	6	1	no	—
232	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
233	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	C₄⋊(C₃⋊C₈)	1	2	yes	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	3	2	no	—
245	C₃×C₄	—	4	1	no	—
246	C₃×C₄	—	6	1	no	—
247	C₃×C₄	—	6	1	no	—
248	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
249	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
250	C₃×C₄	C₂×#208	1	1	yes	characteristic
251	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
252	Dic₃	—	2	1	no	—
253	Dic₃	—	2	1	no	—
254	Dic₃	C₄⋊(C₃⋊C₈)	1	2	yes	—
255	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	3	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
263	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	3	2	no	—
267	C₈	—	6	1	no	—
268	C₈	D₃×Dic₆	1	1	yes	characteristic
269	C₈	D₃×D₁₂	1	1	yes	characteristic
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	6	2	no	—
274	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
275	C₂×C₃	#4901	1	1	yes	characteristic
276	C₂×C₃	#4907	1	1	yes	characteristic
277	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
278	C₂×C₃	#5140	1	1	yes	characteristic
279	C₂×C₃	#5140	1	1	yes	characteristic
280	D₃	—	2	2	no	—
281	C₂²	—	3	2	no	—
282	C₂²	—	6	1	no	—
283	C₂²	—	6	1	no	—
284	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	3	2	no	—
288	C₄	—	6	1	no	—
289	C₄	—	6	1	no	—
290	C₄	D₃×#208	1	1	yes	characteristic
291	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
292	C₃	—	2	1	no	—
293	C₃	#69285	1	1	yes	characteristic, complemented
294	C₃	#69882	1	1	yes	characteristic, complemented
295	C₂	—	6	2	no	—
296	C₂	D₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
297	C₂	#10590621	1	1	yes	characteristic
298	C₂	#10590622	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄×C₈	C₂²
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	semidirect
5. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
6. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
7. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
37. C₄⋊(C₃⋊C₁₆)	D₃	semidirect
255. C₃²	#2151	semidirect
293. C₃	#69285	semidirect
294. C₃	#69882	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4689	2
C₂×#63543	2
C₂×D₃×C₃⋊C₁₆	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
C₃×C₄⋊(C₃⋊C₁₆)	2
Dic₃⋊(C₃⋊C₁₆)	2
#69882	3
#69285	3