Subgroups — #72222480

Order 2304.

544 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11800468	C₂	1	2	yes	maximal
2	C₃⋊D₃×#346	C₂	1	1	yes	maximal, characteristic
3	#11772681	C₂	1	2	yes	maximal, complemented
4	C₃×#67519	C₂	1	2	yes	maximal, complemented
5	D₃×#1398	—	3	2	no	maximal
6	C₂×C₁₆×C₃⋊D₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₃⋊D₃	C₄	1	1	yes	characteristic
9	C₃²×#346	C₂²	1	1	yes	characteristic
10	#10589368	C₄	1	1	yes	characteristic
11	#10589368	C₂²	1	1	yes	characteristic
12	#10589327	C₄	1	1	yes	characteristic
13	C₃×#4499	C₂²	1	4	yes	—
14	#10589069	C₄	1	4	yes	complemented
15	#10589067	C₄	1	4	yes	complemented
16	D₃×M₆₄	—	3	4	no	—
17	D₃×#346	—	3	2	no	—
18	D₃×#346	—	6	1	no	—
19	C₃×#1398	—	3	2	no	—
20	#67600	—	3	4	no	—
21	#67519	D₃	1	2	yes	complemented
22	C₂×C₈×C₃⋊D₃	C₈	1	1	yes	characteristic
23	C₂×C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₃⋊D₃	C₈	1	1	yes	characteristic
25	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₁₆	C₂³	1	1	yes	characteristic
27	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₈×C₃⋊Dic₃	C₈	1	1	yes	characteristic
29	C₈×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	1	yes	characteristic
32	C₄×C₃⋊(C₃⋊C₈)	C₈	1	1	yes	characteristic
33	C₁₆×C₃⋊D₃	C₈	1	2	yes	—
34	C₁₆×C₃⋊D₃	C₈	1	2	yes	—
35	C₁₆×C₃⋊D₃	C₂×C₄	1	2	yes	—
36	C₁₆×C₃⋊D₃	C₂×C₄	1	2	yes	—
37	C₃×C₃⋊C₃₂	C₂×C₄	1	8	yes	complemented
38	C₂×#1398	—	9	1	no	Sylow
39	C₂×D₃×C₁₆	—	3	2	no	—
40	C₂×D₃×C₁₆	—	3	2	no	—
41	C₂×D₃×C₁₆	—	6	1	no	—
42	C₂×D₃×C₁₆	—	6	1	no	—
43	C₄×D₃×C₈	—	3	2	no	—
44	C₄×D₃×C₈	—	6	1	no	—
45	C₃×M₆₄	—	3	4	no	—
46	C₃×#346	—	2	1	no	—
47	C₃×#346	C₂×D₃	1	2	yes	—
48	#4502	—	3	2	no	—
49	#4502	—	3	2	no	—
50	#4502	—	6	1	no	—
51	#4502	—	6	1	no	—
52	#4499	C₂×D₃	1	4	yes	—
53	#4461	—	3	2	no	—
54	#4461	—	6	1	no	—
55	#4447	—	3	8	no	—
56	D₃×C₃₂	—	3	8	no	—
57	C₂×C₄×C₃⋊D₃	—	2	1	no	—
58	C₂×C₄×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₃²×C₄²	C₂×C₈	1	1	yes	characteristic
62	C₄×C₃⋊Dic₃	—	2	1	no	—
63	C₄×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃⋊(C₃⋊C₈)	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃⋊(C₃⋊C₈)	C₂×C₈	1	1	yes	characteristic
66	C₈×C₃⋊D₃	—	2	1	no	—
67	C₈×C₃⋊D₃	—	2	1	no	—
68	C₈×C₃⋊D₃	C₄²	1	1	yes	characteristic
69	C₈×C₃⋊D₃	C₄²	1	1	yes	characteristic
70	C₈×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
71	C₈×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
72	C₃²×C₁₆	C₂×C₈	1	2	yes	—
73	C₃²×C₁₆	C₂²×C₄	1	2	yes	—
74	C₃⋊(C₃⋊C₁₆)	C₄²	1	2	yes	—
75	C₃⋊(C₃⋊C₁₆)	C₂×C₈	1	2	yes	—
76	C₂×M₆₄	—	9	2	no	—
77	C₂×#346	—	9	1	no	—
78	#1398	—	3	2	no	—
79	#1398	—	9	2	no	—
80	C₂×D₃×C₈	—	3	2	no	—
81	C₂×D₃×C₈	—	3	2	no	—
82	C₂×D₃×C₈	—	6	1	no	—
83	C₂×D₃×C₈	—	6	1	no	—
84	C₄²×D₃	—	3	2	no	—
85	C₄²×D₃	—	6	1	no	—
86	C₂×C₃×C₁₆	—	2	1	no	—
87	C₂×C₃×C₁₆	—	2	1	no	—
88	C₂×C₃×C₁₆	C₄×D₃	1	2	yes	—
89	C₂×C₃×C₁₆	C₂²×D₃	1	2	yes	—
90	C₃×C₄×C₈	—	2	1	no	—
91	C₃×C₄×C₈	C₄×D₃	1	2	yes	—
92	C₈×Dic₃	—	3	2	no	—
93	C₈×Dic₃	—	3	2	no	—
94	C₈×Dic₃	—	6	1	no	—
95	C₈×Dic₃	—	6	1	no	—
96	C₂×C₃⋊C₁₆	—	3	2	no	—
97	C₂×C₃⋊C₁₆	—	3	2	no	—
98	C₂×C₃⋊C₁₆	—	6	1	no	—
99	C₂×C₃⋊C₁₆	—	6	1	no	—
100	C₄×C₃⋊C₈	—	3	2	no	—
101	C₄×C₃⋊C₈	—	6	1	no	—
102	D₃×C₁₆	—	3	4	no	—
103	D₃×C₁₆	—	3	4	no	—
104	D₃×C₁₆	—	3	4	no	—
105	D₃×C₁₆	—	3	4	no	—
106	D₃×C₁₆	—	6	2	no	—
107	D₃×C₁₆	—	6	2	no	—
108	D₃×C₁₆	—	6	2	no	—
109	D₃×C₁₆	—	6	2	no	—
110	C₃×C₃₂	—	3	8	no	—
111	C₃⋊C₃₂	C₄×D₃	1	8	yes	complemented
112	C₂²×C₃⋊D₃	M₃₂	1	1	yes	characteristic
113	C₂×C₃²×C₄	—	2	1	no	—
114	C₂×C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₃⋊Dic₃	—	2	1	no	—
116	C₂×C₃⋊Dic₃	M₃₂	1	1	yes	characteristic
117	C₄×C₃⋊D₃	—	4	2	no	—
118	C₄×C₃⋊D₃	C₄×C₈	1	1	yes	characteristic
119	C₄×C₃⋊D₃	C₄×C₈	1	1	yes	characteristic
120	C₄×C₃⋊D₃	M₃₂	1	1	yes	characteristic
121	C₄×C₃⋊D₃	M₃₂	1	1	yes	characteristic
122	C₃²×C₈	—	2	1	no	—
123	C₃²×C₈	C₂×C₄²	1	1	yes	characteristic
124	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
125	C₃⋊(C₃⋊C₈)	—	2	1	no	—
126	C₃⋊(C₃⋊C₈)	C₄×C₈	1	1	yes	characteristic
127	C₃⋊(C₃⋊C₈)	C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₆	—	9	1	no	—
129	C₂²×C₁₆	—	9	1	no	—
130	C₂×C₄×C₈	—	9	1	no	—
131	M₆₄	—	3	4	no	—
132	M₆₄	—	9	4	no	—
133	C₂×C₃₂	—	9	4	no	—
134	#346	—	9	1	no	—
135	#346	—	9	1	no	—
136	#346	—	9	1	no	—
137	#346	D₃²	1	1	yes	characteristic
138	C₂×C₄×D₃	—	3	2	no	—
139	C₂×C₄×D₃	—	6	1	no	—
140	C₂×C₄×D₃	—	6	2	no	—
141	C₂×C₄×D₃	—	6	2	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	D₃×C₈	1	2	yes	—
145	C₂×C₃×C₈	C₂×C₄×D₃	1	2	yes	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	D₃×C₈	1	2	yes	—
148	C₄×Dic₃	—	3	2	no	—
149	C₄×Dic₃	—	6	1	no	—
150	C₄×Dic₃	—	6	2	no	—
151	C₄×Dic₃	—	6	2	no	—
152	C₂×C₃⋊C₈	—	3	2	no	—
153	C₂×C₃⋊C₈	—	3	2	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	D₃×C₈	—	3	2	no	—
157	D₃×C₈	—	3	2	no	—
158	D₃×C₈	—	3	2	no	—
159	D₃×C₈	—	3	2	no	—
160	D₃×C₈	—	6	1	no	—
161	D₃×C₈	—	6	1	no	—
162	D₃×C₈	—	6	1	no	—
163	D₃×C₈	—	6	1	no	—
164	D₃×C₈	—	6	2	no	—
165	D₃×C₈	—	6	2	no	—
166	D₃×C₈	—	6	2	no	—
167	D₃×C₈	—	6	2	no	—
168	C₃×C₁₆	—	2	2	no	—
169	C₃×C₁₆	—	2	2	no	—
170	C₃×C₁₆	D₃×C₈	1	4	yes	—
171	C₃×C₁₆	C₂×C₄×D₃	1	4	yes	—
172	C₃⋊C₁₆	—	3	4	no	—
173	C₃⋊C₁₆	—	3	4	no	—
174	C₃⋊C₁₆	—	6	2	no	—
175	C₃⋊C₁₆	—	6	2	no	—
176	C₂²×C₃²	C₂×M₃₂	1	1	yes	characteristic
177	C₂×C₃⋊D₃	—	2	2	no	—
178	C₂×C₃⋊D₃	#346	1	1	yes	characteristic
179	C₂×C₃⋊D₃	#346	1	1	yes	characteristic
180	C₃²×C₄	—	4	1	no	—
181	C₃²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
182	C₃²×C₄	C₂×M₃₂	1	1	yes	characteristic
183	C₃⋊Dic₃	—	4	1	no	—
184	C₃⋊Dic₃	#346	1	1	yes	characteristic
185	C₃⋊Dic₃	#346	1	1	yes	characteristic
186	C₂²×C₈	—	9	1	no	—
187	C₂²×C₈	—	9	1	no	—
188	C₂×C₄²	—	9	1	no	—
189	C₂×C₁₆	—	9	1	no	—
190	C₂×C₁₆	—	9	1	no	—
191	C₂×C₁₆	—	9	2	no	—
192	C₂×C₁₆	—	9	2	no	—
193	C₂×C₁₆	—	9	2	no	—
194	C₂×C₁₆	—	9	2	no	—
195	C₂×C₁₆	#624	1	1	yes	characteristic
196	C₂×C₁₆	C₂×D₃²	1	1	yes	characteristic
197	C₄×C₈	—	9	1	no	—
198	C₄×C₈	—	9	1	no	—
199	C₄×C₈	—	9	1	no	—
200	C₄×C₈	#624	1	1	yes	characteristic
201	C₃₂	—	3	8	no	—
202	C₂²×D₃	—	3	2	no	—
203	C₂²×D₃	—	6	1	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	C₂×D₃×C₈	1	2	yes	—
208	C₂×Dic₃	—	3	2	no	—
209	C₂×Dic₃	—	6	1	no	—
210	C₂×Dic₃	—	6	2	no	—
211	C₂×Dic₃	—	6	2	no	—
212	C₄×D₃	—	3	2	no	—
213	C₄×D₃	—	3	2	no	—
214	C₄×D₃	—	3	2	no	—
215	C₄×D₃	—	3	2	no	—
216	C₄×D₃	—	6	1	no	—
217	C₄×D₃	—	6	1	no	—
218	C₄×D₃	—	6	1	no	—
219	C₄×D₃	—	6	1	no	—
220	C₄×D₃	—	12	4	no	—
221	C₄×D₃	—	12	4	no	—
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	—	2	2	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	—	2	1	no	—
226	C₃×C₈	C₄²×D₃	1	2	yes	—
227	C₃×C₈	C₂×D₃×C₈	1	2	yes	—
228	C₃⋊C₈	—	3	2	no	—
229	C₃⋊C₈	—	3	2	no	—
230	C₃⋊C₈	—	6	1	no	—
231	C₃⋊C₈	—	6	1	no	—
232	C₃⋊C₈	—	6	2	no	—
233	C₃⋊C₈	—	6	2	no	—
234	C₂×C₃²	—	2	1	no	—
235	C₂×C₃²	C₂×#346	1	1	yes	characteristic
236	C₃⋊D₃	—	2	1	no	—
237	C₃⋊D₃	#1398	1	2	yes	complemented
238	C₂²×C₄	—	9	1	no	—
239	C₂²×C₄	—	18	1	no	—
240	C₂×C₈	—	9	1	no	—
241	C₂×C₈	—	9	1	no	—
242	C₂×C₈	—	9	1	no	—
243	C₂×C₈	—	9	1	no	—
244	C₂×C₈	—	9	1	no	—
245	C₂×C₈	—	9	1	no	—
246	C₂×C₈	—	18	1	no	—
247	C₂×C₈	—	18	1	no	—
248	C₂×C₈	#3710	1	1	yes	characteristic
249	C₂×C₈	C₂×#624	1	1	yes	characteristic
250	C₄²	—	9	1	no	—
251	C₄²	—	18	1	no	—
252	C₄²	#3710	1	1	yes	characteristic
253	C₁₆	—	9	2	no	—
254	C₁₆	—	9	2	no	—
255	C₁₆	#3710	1	2	yes	—
256	C₁₆	C₄×D₃²	1	2	yes	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	D₃×M₃₂	1	2	yes	—
259	C₂×D₃	—	3	2	no	—
260	C₂×D₃	—	3	2	no	—
261	C₂×D₃	—	6	1	no	—
262	C₂×D₃	—	6	1	no	—
263	C₂×D₃	—	6	4	no	—
264	C₂×D₃	—	6	4	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	4	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	C₄×D₃×C₈	1	2	yes	—
270	C₃×C₄	D₃×M₃₂	1	2	yes	—
271	Dic₃	—	3	2	no	—
272	Dic₃	—	3	2	no	—
273	Dic₃	—	6	1	no	—
274	Dic₃	—	6	1	no	—
275	Dic₃	—	12	2	no	—
276	Dic₃	—	12	2	no	—
277	C₃²	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	9	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	9	1	no	—
281	C₂×C₄	—	9	1	no	—
282	C₂×C₄	—	9	1	no	—
283	C₂×C₄	—	9	1	no	—
284	C₂×C₄	—	9	1	no	—
285	C₂×C₄	—	18	1	no	—
286	C₂×C₄	—	36	2	no	—
287	C₂×C₄	C₂×#3710	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	9	1	no	—
290	C₈	—	9	1	no	—
291	C₈	—	18	1	no	—
292	C₈	C₂×#3710	1	1	yes	characteristic
293	C₈	C₄×#624	1	1	yes	characteristic
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	D₃×#346	1	2	yes	—
298	D₃	—	3	4	no	—
299	D₃	—	6	2	no	—
300	D₃	—	6	2	no	—
301	D₃	—	6	2	no	—
302	C₂²	—	9	1	no	—
303	C₂²	—	9	1	no	—
304	C₂²	—	18	2	no	—
305	C₂²	#10590625	1	1	yes	characteristic
306	C₄	—	4	1	no	—
307	C₄	—	9	1	no	—
308	C₄	—	9	1	no	—
309	C₄	—	36	1	no	—
310	C₄	C₄×#3710	1	1	yes	characteristic
311	C₄	#10590625	1	1	yes	characteristic
312	C₃	—	2	1	no	—
313	C₃	D₃×#1398	1	2	yes	complemented
314	C₂	—	2	1	no	—
315	C₂	—	9	2	no	—
316	C₂	—	18	1	no	—
317	C₂	#11795837	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×#624
commutator	C₃²×C₄	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂×D₃²
Fitting	C₃²×#346	C₂²
socle	C₂×C₃²	C₂×#346
prime radical	C₂²×C₃⋊D₃	M₃₂
cosocle	C₂×C₃²×C₁₆	C₂³
hypercenter	#346	D₃²
hypocenter	C₃²	C₂×#1398
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. #11772681	C₂	semidirect
4. C₃×#67519	C₂	semidirect
14. #10589069	C₄	semidirect
15. #10589067	C₄	semidirect
21. #67519	D₃	semidirect
37. C₃×C₃⋊C₃₂	C₂×C₄	semidirect
111. C₃⋊C₃₂	C₄×D₃	semidirect
237. C₃⋊D₃	#1398	semidirect
277. C₃²	C₂×#1398	semidirect
313. C₃	D₃×#1398	semidirect

Maximal subgroups

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

subgroup	index
#11800468	2
C₃⋊D₃×#346	2
#11772681	2
C₃×#67519	2
D₃×#1398	3