Subgroups — #11797329

Order 1152.

331 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	D₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	#10589368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
7	#10589104	C₂	1	1	yes	maximal, characteristic, complemented
8	#69275	—	3	1	no	maximal
9	#69093	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₄×D₃×Dic₃	C₄	1	1	yes	characteristic
12	C₂×D₃×C₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	#63553	C₄	1	1	yes	characteristic
20	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₃	—	3	1	no	—
22	C₄×D₃×C₈	—	3	1	no	—
23	C₃×C₄⋊C₁₆	—	3	1	no	—
24	C₃×C₂²⋊C₁₆	—	3	1	no	—
25	#4549	—	3	1	no	—
26	(C₂×D₃)⋊C₁₆	—	3	1	no	—
27	(C₂×D₃)⋊C₁₆	D₃	1	1	yes	characteristic
28	#4502	—	3	1	no	—
29	#4502	—	3	1	no	—
30	#4502	—	3	1	no	—
31	#4502	—	3	1	no	—
32	#4502	—	6	1	no	—
33	Dic₃⋊C₁₆	—	3	1	no	—
34	Dic₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
35	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
36	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₃×Dic₃	C₈	1	1	yes	characteristic
38	C₂×D₃×Dic₃	C₈	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×D₃×C₈	—	2	1	no	—
46	C₃×D₃×C₈	—	2	1	no	—
47	Dic₃²	C₈	1	1	yes	characteristic
48	Dic₃²	C₈	1	1	yes	characteristic
49	D₃×C₃⋊C₈	—	2	2	no	—
50	C₃²×C₁₆	—	2	1	no	—
51	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
52	C₃×C₃⋊C₁₆	—	2	1	no	—
53	C₃×C₃⋊C₁₆	—	2	1	no	—
54	#2163	—	9	1	no	Sylow
55	C₂²×C₃×C₈	—	3	1	no	—
56	C₂×C₄×Dic₃	—	3	1	no	—
57	C₂²×C₃⋊C₈	—	3	1	no	—
58	C₂×D₃×C₈	—	3	1	no	—
59	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄²×D₃	—	3	1	no	—
61	C₂×C₃×C₁₆	—	2	1	no	—
62	C₂×C₃×C₁₆	—	3	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₃×C₄×C₈	—	3	1	no	—
67	C₈×Dic₃	—	3	1	no	—
68	C₈×Dic₃	—	3	1	no	—
69	C₈×Dic₃	—	3	1	no	—
70	C₈×Dic₃	—	3	1	no	—
71	C₈×Dic₃	—	3	1	no	—
72	C₈×Dic₃	—	6	1	no	—
73	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃⋊C₁₆	—	3	1	no	—
75	C₂×C₃⋊C₁₆	—	3	1	no	—
76	C₂×C₃⋊C₁₆	—	6	1	no	—
77	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
79	C₄×C₃⋊C₈	—	3	1	no	—
80	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
87	C₃×C₄×D₃	—	2	1	no	—
88	C₃×C₄×D₃	—	2	1	no	—
89	D₃×Dic₃	—	2	2	no	—
90	D₃×Dic₃	—	2	2	no	—
91	C₃²×C₈	#41	1	1	yes	characteristic
92	C₃²×C₈	#41	1	1	yes	characteristic
93	C₃⋊(C₃⋊C₈)	M₁₆	1	2	yes	—
94	C₃×C₃⋊C₈	—	2	1	no	—
95	C₃×C₃⋊C₈	—	2	1	no	—
96	C₂×C₄×C₈	—	9	1	no	—
97	C₄⋊C₁₆	—	3	1	no	—
98	C₄⋊C₁₆	—	9	1	no	—
99	C₂²⋊C₁₆	—	3	1	no	—
100	C₂²⋊C₁₆	—	9	1	no	—
101	#346	—	9	1	no	—
102	#346	—	9	1	no	—
103	C₂²×C₃×C₄	—	3	1	no	—
104	C₂²×Dic₃	—	3	1	no	—
105	C₂²×Dic₃	—	3	1	no	—
106	C₂×C₄×D₃	—	3	1	no	—
107	C₂×C₄×D₃	—	3	1	no	—
108	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	3	1	no	—
111	C₂×C₃×C₈	—	3	1	no	—
112	C₂×C₃×C₈	—	6	1	no	—
113	C₂×C₃×C₈	—	6	1	no	—
114	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
116	C₃×C₄²	—	3	1	no	—
117	C₄×Dic₃	—	3	1	no	—
118	C₄×Dic₃	—	3	1	no	—
119	C₄×Dic₃	—	3	1	no	—
120	C₄×Dic₃	—	3	1	no	—
121	C₄×Dic₃	—	3	1	no	—
122	C₄×Dic₃	—	3	1	no	—
123	C₄×Dic₃	—	6	1	no	—
124	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃⋊C₈	—	3	1	no	—
126	C₂×C₃⋊C₈	—	3	1	no	—
127	C₂×C₃⋊C₈	—	6	1	no	—
128	C₂×C₃⋊C₈	—	6	2	no	—
129	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
130	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
131	D₃×C₈	—	2	1	no	—
132	D₃×C₈	—	2	1	no	—
133	D₃×C₈	—	6	2	no	—
134	C₃×C₁₆	—	2	2	no	—
135	C₃×C₁₆	—	2	1	no	—
136	C₃×C₁₆	—	2	1	no	—
137	C₃×C₁₆	—	6	1	no	—
138	C₃×C₁₆	—	6	1	no	—
139	C₃⋊C₁₆	—	2	1	no	—
140	C₃⋊C₁₆	—	2	1	no	—
141	C₃⋊C₁₆	—	6	2	no	—
142	C₃⋊C₁₆	—	6	1	no	—
143	C₃⋊C₁₆	—	6	1	no	—
144	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₃×D₃	—	2	1	no	—
146	C₂×C₃×D₃	—	2	1	no	—
147	C₂×C₃×D₃	M₃₂	1	2	yes	—
148	C₃²×C₄	#117	1	1	yes	characteristic
149	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
150	C₃⋊Dic₃	—	2	1	no	—
151	C₃⋊Dic₃	—	2	1	no	—
152	C₃×Dic₃	—	2	1	no	—
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	M₃₂	1	2	yes	—
155	C₂²×C₈	—	3	1	no	—
156	C₂²×C₈	—	9	1	no	—
157	C₂×C₄²	—	9	1	no	—
158	C₂×C₁₆	—	3	1	no	—
159	C₂×C₁₆	—	3	1	no	—
160	C₂×C₁₆	—	9	1	no	—
161	C₂×C₁₆	D₃²	1	1	yes	characteristic
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	9	1	no	—
164	C₄×C₈	—	9	1	no	—
165	C₄×C₈	—	9	1	no	—
166	C₂³×C₃	—	3	1	no	—
167	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	3	1	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	6	1	no	—
173	C₂×C₃×C₄	—	6	1	no	—
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×Dic₃	—	3	1	no	—
177	C₂×Dic₃	—	3	1	no	—
178	C₂×Dic₃	—	3	1	no	—
179	C₂×Dic₃	—	3	1	no	—
180	C₂×Dic₃	—	6	1	no	—
181	C₂×Dic₃	—	6	1	no	—
182	C₂×Dic₃	—	6	2	no	—
183	C₂×Dic₃	—	6	2	no	—
184	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
185	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
186	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
187	C₄×D₃	—	2	1	no	—
188	C₄×D₃	—	2	1	no	—
189	C₄×D₃	—	6	2	no	—
190	C₄×D₃	—	6	2	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	—	6	1	no	—
194	C₃×C₈	—	6	1	no	—
195	C₃×C₈	#237	1	1	yes	characteristic
196	C₃×C₈	#237	1	1	yes	characteristic
197	C₃×C₈	#237	1	1	yes	characteristic
198	C₃×C₈	#239	1	1	yes	characteristic
199	C₃⋊C₈	—	2	1	no	—
200	C₃⋊C₈	—	2	1	no	—
201	C₃⋊C₈	—	3	2	no	—
202	C₃⋊C₈	—	3	2	no	—
203	C₃⋊C₈	—	6	2	no	—
204	C₂×C₃²	#431	1	1	yes	characteristic
205	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
206	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
207	C₃×D₃	—	2	2	no	—
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	9	1	no	—
210	C₂²×C₄	—	9	1	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	6	1	no	—
214	C₂×C₈	—	6	1	no	—
215	C₂×C₈	—	9	1	no	—
216	C₂×C₈	—	18	2	no	—
217	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
218	C₄²	—	3	1	no	—
219	C₄²	—	9	1	no	—
220	C₄²	—	9	1	no	—
221	C₄²	—	9	1	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	—	6	1	no	—
224	C₁₆	—	6	1	no	—
225	C₁₆	—	18	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	3	2	no	—
228	C₂²×C₃	—	6	1	no	—
229	C₂²×C₃	—	6	1	no	—
230	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
231	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
232	C₂×D₃	—	2	1	no	—
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	#798	1	2	yes	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	3	2	no	—
238	C₃×C₄	—	6	1	no	—
239	C₃×C₄	—	6	1	no	—
240	C₃×C₄	#877	1	1	yes	characteristic
241	C₃×C₄	#892	1	1	yes	characteristic
242	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
243	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	6	1	no	—
247	Dic₃	—	6	1	no	—
248	Dic₃	—	6	1	no	—
249	Dic₃	—	6	1	no	—
250	Dic₃	—	12	1	no	—
251	Dic₃	—	12	1	no	—
252	Dic₃	#798	1	2	yes	—
253	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	9	1	no	—
261	C₂×C₄	—	9	1	no	—
262	C₂×C₄	—	18	2	no	—
263	C₂×C₄	—	18	2	no	—
264	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
265	C₈	—	6	1	no	—
266	C₈	—	6	1	no	—
267	C₈	—	9	2	no	—
268	C₈	#3796	1	1	yes	characteristic
269	C₈	#3805	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	#4718	1	1	yes	characteristic
275	C₂×C₃	#4833	1	1	yes	characteristic
276	C₂×C₃	C₂×#798	1	1	yes	characteristic
277	C₂×C₃	C₂×#798	1	1	yes	characteristic
278	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
279	C₂×C₃	D₃×M₃₂	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	D₃²×C₈	1	1	yes	characteristic
285	C₄	—	3	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	18	1	no	—
289	C₄	—	18	1	no	—
290	C₄	#63791	1	1	yes	characteristic
291	C₄	#63878	1	1	yes	characteristic
292	C₃	—	2	1	no	—
293	C₃	#69093	1	1	yes	characteristic, complemented
294	C₃	#69275	1	1	yes	characteristic, complemented
295	C₂	—	6	2	no	—
296	C₂	D₃×#798	1	1	yes	characteristic
297	C₂	D₃×#798	1	1	yes	characteristic
298	C₂	#10590656	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₁₆	C₂²
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	#2163
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #10589368	C₂	semidirect
4. C₃×Dic₃⋊C₁₆	C₂	semidirect
5. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
7. #10589104	C₂	semidirect
34. Dic₃⋊C₁₆	D₃	semidirect
253. C₃²	#2163	semidirect
293. C₃	#69093	semidirect
294. C₃	#69275	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₈×Dic₃	2
#10589368	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×Dic₃⋊C₁₆	2
Dic₃⋊(C₃⋊C₁₆)	2
(C₂×D₃)⋊(C₃⋊C₁₆)	2
#10589104	2
#69275	3
#69093	3