Subgroups — #11797363

Order 1152.

309 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5101	C₂	1	1	yes	maximal, characteristic
2	#10589415	C₂	1	1	yes	maximal, characteristic
3	C₃×#4549	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#69883	—	3	1	no	maximal
9	#69094	—	3	1	no	maximal
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃×#926	C₄	1	1	yes	characteristic
12	C₂×C₃×#208	C₄	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₃×#815	C₄	1	1	yes	characteristic
16	C₃×#815	C₄	1	1	yes	characteristic
17	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₃×#405	—	3	1	no	—
22	#5101	D₃	1	1	yes	characteristic
23	C₃×C₂²⋊C₁₆	—	3	1	no	—
24	#4549	—	3	1	no	—
25	#4549	—	3	1	no	—
26	#4549	—	6	1	no	—
27	#4549	D₃	1	1	yes	characteristic
28	#4502	—	3	1	no	—
29	Dic₃⋊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	#4461	—	3	1	no	—
35	C₄×C₃⋊C₁₆	—	3	1	no	—
36	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	—	2	1	no	—
38	C₂×C₃²×C₈	—	2	1	no	—
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₃×#217	C₈	1	1	yes	characteristic
42	C₃×#217	C₈	1	1	yes	characteristic
43	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
44	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₃×#208	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₃⋊(C₃⋊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	#2164	—	9	1	no	Sylow
55	C₂×C₃×M₁₆	—	3	1	no	—
56	C₂²×C₃×C₈	—	2	1	no	—
57	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×#117	—	3	1	no	—
60	#926	Dic₃	1	1	yes	characteristic
61	C₂×#208	Dic₃	1	1	yes	characteristic
62	C₂×C₃×C₁₆	—	3	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₃×#97	—	3	1	no	—
65	C₃×#97	—	3	1	no	—
66	C₃×C₄×C₈	—	3	1	no	—
67	C₃×C₄×C₈	—	3	1	no	—
68	#815	Dic₃	1	1	yes	characteristic
69	#815	Dic₃	1	1	yes	characteristic
70	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
71	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃⋊C₁₆	—	3	1	no	—
73	C₂×C₃⋊C₁₆	—	3	1	no	—
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	—	6	1	no	—
78	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
79	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
80	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×C₄	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	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
89	C₃²×C₈	—	2	1	no	—
90	C₃²×C₈	#41	1	1	yes	characteristic
91	C₃²×C₈	#41	1	1	yes	characteristic
92	C₃×C₃⋊C₈	—	2	1	no	—
93	C₃×C₃⋊C₈	M₁₆	1	2	yes	—
94	#405	—	3	1	no	—
95	C₄⋊C₁₆	—	9	1	no	—
96	C₄⋊C₁₆	—	9	1	no	—
97	C₂²⋊C₁₆	—	3	1	no	—
98	C₂²⋊C₁₆	—	9	1	no	—
99	#346	—	9	1	no	—
100	C₄×C₁₆	—	9	1	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
104	C₃×M₁₆	—	6	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	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	—	4	1	no	—
114	C₂×C₃×C₈	—	4	1	no	—
115	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	3	1	no	—
120	C₃×C₄⋊C₄	—	3	1	no	—
121	C₃×C₂²⋊C₄	—	3	1	no	—
122	C₃×C₂²⋊C₄	—	3	1	no	—
123	C₃×C₄²	—	3	1	no	—
124	C₃×C₄²	—	3	1	no	—
125	#217	C₃⋊C₈	1	1	yes	characteristic
126	#217	C₃⋊C₈	1	1	yes	characteristic
127	C₄⋊Dic₃	C₃⋊C₈	1	1	yes	characteristic
128	C₄⋊Dic₃	C₃⋊C₈	1	1	yes	characteristic
129	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
130	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
131	#208	—	2	2	no	—
132	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
134	C₃×C₁₆	—	6	1	no	—
135	C₃×C₁₆	—	6	1	no	—
136	C₃⋊C₁₆	—	2	1	no	—
137	C₃⋊C₁₆	—	2	1	no	—
138	C₃⋊C₁₆	—	6	1	no	—
139	C₃⋊C₁₆	—	6	1	no	—
140	C₃⋊C₁₆	—	6	1	no	—
141	C₃⋊C₁₆	—	6	1	no	—
142	C₃⋊C₁₆	—	12	1	no	—
143	C₃⋊C₁₆	—	12	1	no	—
144	C₂²×C₃²	—	2	1	no	—
145	C₂²×C₃²	—	2	1	no	—
146	C₂²×C₃²	—	2	1	no	—
147	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	#117	1	1	yes	characteristic
150	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
151	C₃×Dic₃	—	2	1	no	—
152	C₃×Dic₃	—	2	1	no	—
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	—	2	1	no	—
155	C₂×M₁₆	—	3	1	no	—
156	C₂²×C₈	D₃²	1	1	yes	characteristic
157	#117	—	3	1	no	—
158	C₂×C₁₆	—	3	1	no	—
159	C₂×C₁₆	—	3	1	no	—
160	C₂×C₁₆	—	9	1	no	—
161	C₂×C₁₆	—	9	1	no	—
162	#97	—	3	1	no	—
163	#97	—	3	1	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	3	1	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
168	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
186	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
187	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
188	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	—	3	2	no	—
194	C₃×C₈	—	4	1	no	—
195	C₃×C₈	—	6	1	no	—
196	C₃×C₈	#235	1	1	yes	characteristic
197	C₃×C₈	#235	1	1	yes	characteristic
198	C₃×C₈	#237	1	1	yes	characteristic
199	C₃×C₈	#237	1	1	yes	characteristic
200	C₃⋊C₈	—	2	1	no	—
201	C₃⋊C₈	#208	1	2	yes	—
202	C₂×C₃²	—	4	1	no	—
203	C₂×C₃²	#431	1	1	yes	characteristic
204	C₂×C₃²	#504	1	1	yes	characteristic
205	C₂×C₃²	#504	1	1	yes	characteristic
206	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
207	M₁₆	—	6	2	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	3	1	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
214	C₄⋊C₄	—	3	1	no	—
215	C₄⋊C₄	—	3	1	no	—
216	C₂²⋊C₄	—	3	1	no	—
217	C₂²⋊C₄	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	—	3	1	no	—
220	C₁₆	—	6	1	no	—
221	C₁₆	—	6	1	no	—
222	C₁₆	—	18	1	no	—
223	C₁₆	—	18	1	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	4	1	no	—
233	C₂²×C₃	—	4	1	no	—
234	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
235	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	6	1	no	—
242	C₃×C₄	—	6	1	no	—
243	C₃×C₄	—	6	1	no	—
244	C₃×C₄	—	6	1	no	—
245	C₃×C₄	#877	1	1	yes	characteristic
246	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
247	C₃×C₄	C₂×#208	1	1	yes	characteristic
248	C₃×C₄	#926	1	1	yes	characteristic
249	Dic₃	—	2	1	no	—
250	Dic₃	—	2	1	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	2	1	no	—
253	C₃²	#2164	1	1	yes	characteristic, complemented, Sylow
254	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	3	2	no	—
265	C₈	—	6	1	no	—
266	C₈	#3804	1	1	yes	characteristic
267	C₈	#3804	1	1	yes	characteristic
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	#4718	1	1	yes	characteristic
275	C₂×C₃	#4901	1	1	yes	characteristic
276	C₂×C₃	#4907	1	1	yes	characteristic
277	C₂×C₃	#4925	1	1	yes	characteristic
278	C₂×C₃	#5140	1	1	yes	characteristic
279	C₂×C₃	#5140	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	1	no	—
289	C₄	D₃×#208	1	1	yes	characteristic
290	C₄	#63955	1	1	yes	characteristic
291	C₃	—	2	1	no	—
292	C₃	#69094	1	1	yes	characteristic, complemented
293	C₃	#69883	1	1	yes	characteristic, complemented
294	C₂	—	4	1	no	—
295	C₂	#10590621	1	1	yes	characteristic
296	C₂	#10590622	1	1	yes	characteristic
297	C₂	#10591053	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂²×C₃²×C₈	C₂²
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2164
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
5. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
6. #10589103	C₂	semidirect
7. Dic₃×C₃⋊C₁₆	C₂	semidirect
253. C₃²	#2164	semidirect
292. C₃	#69094	semidirect
293. C₃	#69883	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5101	2
#10589415	2
C₃×#4549	2
Dic₃⋊(C₃⋊C₁₆)	2
#10589103	2
Dic₃×C₃⋊C₁₆	2
#69883	3
#69094	3