Subgroups — #72222535

Order 2304.

544 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11797054	C₂	1	2	yes	maximal
2	C₃⋊D₃×#346	C₂	1	1	yes	maximal, characteristic
3	#11772198	C₂	1	2	yes	maximal, complemented
4	C₃×#67514	C₂	1	2	yes	maximal, complemented
5	D₃×#1369	—	3	2	no	maximal
6	C₂×C₁₆×C₃⋊D₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₃⋊D₃	C₄	1	1	yes	characteristic
8	C₄×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	#10590625	—	2	2	no	—
10	C₃²×#346	C₂²	1	1	yes	characteristic
11	#10589368	C₄	1	1	yes	characteristic
12	#10589368	C₄	1	1	yes	characteristic
13	#10589327	C₂²	1	1	yes	characteristic
14	C₃×#4463	C₂²	1	4	yes	—
15	#10589135	—	2	2	no	—
16	D₃×#364	—	3	4	no	—
17	D₃×#346	—	3	2	no	—
18	D₃×#346	—	6	1	no	—
19	C₃×#1369	—	3	2	no	—
20	#67567	—	3	4	no	—
21	#67514	D₃	1	2	yes	complemented
22	C₂×C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
24	C₄²×C₃⋊D₃	Q₈	1	1	yes	characteristic
25	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₈×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₈×C₃⋊Dic₃	D₄	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)	Q₈	1	1	yes	characteristic
33	C₁₆×C₃⋊D₃	—	2	1	no	—
34	C₁₆×C₃⋊D₃	—	2	1	no	—
35	C₁₆×C₃⋊D₃	C₈	1	2	yes	—
36	C₁₆×C₃⋊D₃	C₈	1	2	yes	—
37	C₃×#812	—	2	4	no	—
38	#63544	—	4	2	no	—
39	#63542	—	4	2	no	—
40	C₂×#1369	—	9	1	no	Sylow
41	D₃×M₃₂	—	6	4	no	—
42	C₂×D₃×C₁₆	—	3	2	no	—
43	C₂×D₃×C₁₆	—	3	2	no	—
44	C₂×D₃×C₁₆	—	6	1	no	—
45	C₂×D₃×C₁₆	—	6	1	no	—
46	C₄×D₃×C₈	—	3	2	no	—
47	C₄×D₃×C₈	—	6	1	no	—
48	C₃×#364	—	3	4	no	—
49	C₃×#346	—	2	1	no	—
50	C₃×#346	C₂×D₃	1	2	yes	—
51	#4511	—	6	4	no	—
52	#4502	—	3	2	no	—
53	#4502	—	3	2	no	—
54	#4502	—	6	1	no	—
55	#4502	—	6	1	no	—
56	#4463	C₂×D₃	1	4	yes	—
57	#4461	—	3	2	no	—
58	#4461	—	6	1	no	—
59	C₂×C₄×C₃⋊D₃	—	2	1	no	—
60	C₂×C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₃²×C₄²	C₂×Q₈	1	1	yes	characteristic
64	C₄×C₃⋊Dic₃	—	2	1	no	—
65	C₄×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	1	yes	characteristic
68	C₈×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
69	C₈×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
70	C₈×C₃⋊D₃	M₁₆	1	1	yes	characteristic
71	C₈×C₃⋊D₃	M₁₆	1	1	yes	characteristic
72	C₈×C₃⋊D₃	SD₁₆	1	2	yes	—
73	C₈×C₃⋊D₃	SD₁₆	1	2	yes	—
74	C₃²×C₁₆	—	2	1	no	—
75	C₃²×C₁₆	C₂×C₈	1	2	yes	—
76	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
77	C₃⋊(C₃⋊C₁₆)	C₂×C₈	1	2	yes	—
78	C₃×C₃⋊C₁₆	—	4	4	no	—
79	C₂×#364	—	9	2	no	—
80	C₂×#346	—	9	1	no	—
81	#1369	—	3	2	no	—
82	#1369	—	9	2	no	—
83	C₂×D₃×C₈	—	3	2	no	—
84	C₂×D₃×C₈	—	3	2	no	—
85	C₂×D₃×C₈	—	6	1	no	—
86	C₂×D₃×C₈	—	6	1	no	—
87	C₄²×D₃	—	3	2	no	—
88	C₄²×D₃	—	6	1	no	—
89	C₃×M₃₂	—	6	4	no	—
90	C₂×C₃×C₁₆	—	2	1	no	—
91	C₂×C₃×C₁₆	—	2	1	no	—
92	C₂×C₃×C₁₆	C₄×D₃	1	2	yes	—
93	C₂×C₃×C₁₆	C₄×D₃	1	2	yes	—
94	C₃×C₄×C₈	—	2	1	no	—
95	C₃×C₄×C₈	C₂²×D₃	1	2	yes	—
96	C₈×Dic₃	—	3	2	no	—
97	C₈×Dic₃	—	3	2	no	—
98	C₈×Dic₃	—	6	1	no	—
99	C₈×Dic₃	—	6	1	no	—
100	#812	—	2	4	no	—
101	C₂×C₃⋊C₁₆	—	3	2	no	—
102	C₂×C₃⋊C₁₆	—	3	2	no	—
103	C₂×C₃⋊C₁₆	—	6	1	no	—
104	C₂×C₃⋊C₁₆	—	6	1	no	—
105	C₄×C₃⋊C₈	—	3	2	no	—
106	C₄×C₃⋊C₈	—	6	1	no	—
107	#798	—	12	4	no	—
108	D₃×C₁₆	—	3	4	no	—
109	D₃×C₁₆	—	3	4	no	—
110	D₃×C₁₆	—	6	2	no	—
111	D₃×C₁₆	—	6	2	no	—
112	D₃×C₁₆	—	6	2	no	—
113	D₃×C₁₆	—	6	2	no	—
114	D₃×C₁₆	—	6	2	no	—
115	D₃×C₁₆	—	6	2	no	—
116	D₃×C₁₆	—	12	4	no	—
117	C₂²×C₃⋊D₃	#108	1	1	yes	characteristic
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃⋊Dic₃	—	2	1	no	—
121	C₂×C₃⋊Dic₃	#108	1	1	yes	characteristic
122	C₄×C₃⋊D₃	—	4	2	no	—
123	C₄×C₃⋊D₃	C₄⋊C₈	1	1	yes	characteristic
124	C₄×C₃⋊D₃	C₄⋊C₈	1	1	yes	characteristic
125	C₄×C₃⋊D₃	#106	1	1	yes	characteristic
126	C₄×C₃⋊D₃	#106	1	1	yes	characteristic
127	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
128	C₃²×C₈	C₂×M₁₆	1	1	yes	characteristic
129	C₃²×C₈	C₂×SD₁₆	1	2	yes	—
130	C₃⋊(C₃⋊C₈)	C₄⋊C₈	1	1	yes	characteristic
131	C₃⋊(C₃⋊C₈)	C₄⋊C₈	1	1	yes	characteristic
132	C₃⋊(C₃⋊C₈)	#106	1	2	yes	—
133	C₂×M₃₂	—	18	2	no	—
134	C₂²×C₁₆	—	9	1	no	—
135	C₂²×C₁₆	—	9	1	no	—
136	C₂×C₄×C₈	—	9	1	no	—
137	#364	—	3	4	no	—
138	#364	—	18	2	no	—
139	#346	—	9	1	no	—
140	#346	—	9	1	no	—
141	#346	—	9	1	no	—
142	#346	D₃²	1	1	yes	characteristic
143	C₂×C₄×D₃	—	3	2	no	—
144	C₂×C₄×D₃	—	6	1	no	—
145	C₂×C₄×D₃	—	6	2	no	—
146	C₂×C₄×D₃	—	6	2	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	C₂×C₄×D₃	1	2	yes	—
150	C₂×C₃×C₈	D₃×D₄	1	2	yes	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	D₃×Q₈	1	2	yes	—
153	C₄×Dic₃	—	3	2	no	—
154	C₄×Dic₃	—	6	1	no	—
155	C₄×Dic₃	—	6	2	no	—
156	C₄×Dic₃	—	6	2	no	—
157	C₂×C₃⋊C₈	—	3	2	no	—
158	C₂×C₃⋊C₈	—	3	2	no	—
159	C₂×C₃⋊C₈	—	6	1	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	D₃×C₈	—	3	2	no	—
162	D₃×C₈	—	3	2	no	—
163	D₃×C₈	—	3	2	no	—
164	D₃×C₈	—	3	2	no	—
165	D₃×C₈	—	3	4	no	—
166	D₃×C₈	—	3	4	no	—
167	D₃×C₈	—	6	1	no	—
168	D₃×C₈	—	6	1	no	—
169	D₃×C₈	—	6	1	no	—
170	D₃×C₈	—	6	1	no	—
171	D₃×C₈	—	6	2	no	—
172	D₃×C₈	—	6	2	no	—
173	C₃×C₁₆	—	2	2	no	—
174	C₃×C₁₆	—	2	2	no	—
175	C₃×C₁₆	—	2	2	no	—
176	C₃×C₁₆	—	12	4	no	—
177	C₃×C₁₆	D₃×C₈	1	4	yes	—
178	C₃⋊C₁₆	—	3	4	no	—
179	C₃⋊C₁₆	—	4	4	no	—
180	C₃⋊C₁₆	—	6	2	no	—
181	C₃⋊C₁₆	—	6	2	no	—
182	C₃⋊C₁₆	—	6	2	no	—
183	C₂²×C₃²	C₂×#108	1	1	yes	characteristic
184	C₂×C₃⋊D₃	—	2	2	no	—
185	C₂×C₃⋊D₃	#334	1	1	yes	characteristic
186	C₂×C₃⋊D₃	#334	1	1	yes	characteristic
187	C₃²×C₄	—	4	1	no	—
188	C₃²×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
189	C₃²×C₄	C₂×#106	1	1	yes	characteristic
190	C₃⋊Dic₃	—	4	1	no	—
191	C₃⋊Dic₃	#334	1	1	yes	characteristic
192	C₃⋊Dic₃	#334	1	1	yes	characteristic
193	C₂²×C₈	—	9	1	no	—
194	C₂²×C₈	—	9	1	no	—
195	C₂×C₄²	—	9	1	no	—
196	M₃₂	—	6	4	no	—
197	M₃₂	—	36	2	no	—
198	C₂×C₁₆	—	9	1	no	—
199	C₂×C₁₆	—	9	2	no	—
200	C₂×C₁₆	—	9	2	no	—
201	C₂×C₁₆	—	9	1	no	—
202	C₂×C₁₆	—	18	1	no	—
203	C₂×C₁₆	—	18	1	no	—
204	C₂×C₁₆	—	36	2	no	—
205	C₂×C₁₆	#624	1	1	yes	characteristic
206	C₂×C₁₆	#624	1	1	yes	characteristic
207	C₄×C₈	—	9	1	no	—
208	C₄×C₈	—	9	1	no	—
209	C₄×C₈	—	9	1	no	—
210	C₄×C₈	C₂×D₃²	1	1	yes	characteristic
211	C₂²×D₃	—	3	2	no	—
212	C₂²×D₃	—	6	1	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	D₃×C₄⋊C₄	1	2	yes	—
217	C₂×Dic₃	—	3	2	no	—
218	C₂×Dic₃	—	6	1	no	—
219	C₂×Dic₃	—	6	2	no	—
220	C₂×Dic₃	—	6	2	no	—
221	C₄×D₃	—	3	2	no	—
222	C₄×D₃	—	3	2	no	—
223	C₄×D₃	—	3	2	no	—
224	C₄×D₃	—	3	2	no	—
225	C₄×D₃	—	6	1	no	—
226	C₄×D₃	—	6	1	no	—
227	C₄×D₃	—	6	1	no	—
228	C₄×D₃	—	6	1	no	—
229	C₄×D₃	—	12	4	no	—
230	C₄×D₃	—	12	4	no	—
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	C₂×D₃×C₈	1	2	yes	—
235	C₃×C₈	D₃×M₁₆	1	2	yes	—
236	C₃×C₈	D₃×SD₁₆	1	4	yes	—
237	C₃⋊C₈	—	3	2	no	—
238	C₃⋊C₈	—	3	2	no	—
239	C₃⋊C₈	—	3	4	no	—
240	C₃⋊C₈	—	6	1	no	—
241	C₃⋊C₈	—	6	1	no	—
242	C₃⋊C₈	—	6	2	no	—
243	C₂×C₃²	—	2	1	no	—
244	C₂×C₃²	C₂×#334	1	1	yes	characteristic
245	C₃⋊D₃	—	2	1	no	—
246	C₃⋊D₃	#1369	1	2	yes	complemented
247	C₂²×C₄	—	9	1	no	—
248	C₂²×C₄	—	18	1	no	—
249	C₂×C₈	—	9	1	no	—
250	C₂×C₈	—	9	1	no	—
251	C₂×C₈	—	9	2	no	—
252	C₂×C₈	—	9	2	no	—
253	C₂×C₈	—	9	1	no	—
254	C₂×C₈	—	9	1	no	—
255	C₂×C₈	—	9	1	no	—
256	C₂×C₈	—	9	1	no	—
257	C₂×C₈	#3802	1	1	yes	characteristic
258	C₂×C₈	C₂×#624	1	1	yes	characteristic
259	C₄²	—	9	1	no	—
260	C₄²	—	18	1	no	—
261	C₄²	#3797	1	1	yes	characteristic
262	C₁₆	—	2	1	no	—
263	C₁₆	—	9	2	no	—
264	C₁₆	—	12	4	no	—
265	C₁₆	—	18	1	no	—
266	C₁₆	#3710	1	2	yes	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	D₃×#108	1	2	yes	—
269	C₂×D₃	—	3	2	no	—
270	C₂×D₃	—	3	2	no	—
271	C₂×D₃	—	6	1	no	—
272	C₂×D₃	—	6	1	no	—
273	C₂×D₃	—	6	4	no	—
274	C₂×D₃	—	6	4	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	4	2	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	D₃×C₄⋊C₈	1	2	yes	—
280	C₃×C₄	D₃×#106	1	2	yes	—
281	Dic₃	—	3	2	no	—
282	Dic₃	—	3	2	no	—
283	Dic₃	—	6	1	no	—
284	Dic₃	—	6	1	no	—
285	Dic₃	—	12	2	no	—
286	Dic₃	—	12	2	no	—
287	C₃²	C₂×#1369	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	9	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	9	1	no	—
291	C₂×C₄	—	9	1	no	—
292	C₂×C₄	—	9	1	no	—
293	C₂×C₄	—	9	1	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	18	1	no	—
296	C₂×C₄	—	36	2	no	—
297	C₂×C₄	#63900	1	1	yes	characteristic
298	C₈	—	9	1	no	—
299	C₈	—	9	1	no	—
300	C₈	—	9	2	no	—
301	C₈	#63796	1	2	yes	—
302	C₈	C₂×#3710	1	1	yes	characteristic
303	C₈	#63818	1	1	yes	characteristic
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	D₃×#334	1	2	yes	—
308	D₃	—	3	4	no	—
309	D₃	—	6	2	no	—
310	D₃	—	6	2	no	—
311	D₃	—	6	2	no	—
312	C₂²	—	9	1	no	—
313	C₂²	—	9	1	no	—
314	C₂²	—	18	2	no	—
315	C₂²	#10590913	1	1	yes	characteristic
316	C₄	—	4	1	no	—
317	C₄	—	9	1	no	—
318	C₄	—	9	1	no	—
319	C₄	—	36	1	no	—
320	C₄	#10590331	1	1	yes	characteristic
321	C₄	#10590685	1	1	yes	characteristic
322	C₃	—	2	1	no	—
323	C₃	D₃×#1369	1	2	yes	complemented
324	C₂	—	2	1	no	—
325	C₂	—	9	2	no	—
326	C₂	—	18	1	no	—
327	C₂	#11776763	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10590685
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂×D₃²
Fitting	C₃²×#346	C₂²
socle	C₂×C₃²	C₂×#334
prime radical	C₂²×C₃⋊D₃	#108
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	#346	D₃²
hypocenter	C₃²	C₂×#1369
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. #11772198	C₂	semidirect
4. C₃×#67514	C₂	semidirect
21. #67514	D₃	semidirect
246. C₃⋊D₃	#1369	semidirect
287. C₃²	C₂×#1369	semidirect
323. C₃	D₃×#1369	semidirect

Maximal subgroups

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

subgroup	index
#11797054	2
C₃⋊D₃×#346	2
#11772198	2
C₃×#67514	2
D₃×#1369	3