Subgroups — #72222543

Order 2304.

448 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#69085	C₂	1	1	yes	maximal, characteristic
2	#11797611	C₂	1	2	yes	maximal, complemented
3	C₃×#67514	C₂	1	1	yes	maximal, characteristic, complemented
4	#11771667	C₂	1	2	yes	maximal, complemented
5	#11771319	C₂	1	1	yes	maximal, characteristic, complemented
6	#10707077	—	3	1	no	maximal
7	#10707076	—	3	1	no	maximal
8	C₃×#4904	C₄	1	1	yes	characteristic
9	C₃×#4700	C₂²	1	2	yes	—
10	#10590622	—	2	2	no	—
11	C₃²×#346	C₂²	1	1	yes	characteristic
12	#10589327	C₂²	1	1	yes	characteristic
13	#10589321	C₂²	1	1	yes	characteristic
14	C₃×#4510	C₄	1	2	yes	—
15	C₃×#4506	C₄	1	1	yes	characteristic
16	C₃×#4463	C₂²	1	2	yes	—
17	#10589111	—	2	2	no	—
18	C₃×#2179	—	3	1	no	—
19	#70924	—	3	2	no	—
20	#69301	—	3	2	no	—
21	#69085	D₃	1	1	yes	characteristic
22	C₃×#1369	—	3	1	no	—
23	#67767	—	3	2	no	—
24	#67569	—	3	2	no	—
25	#67514	D₃	1	1	yes	characteristic, complemented
26	#67511	—	3	1	no	—
27	#67511	—	3	1	no	—
28	#67511	—	6	1	no	—
29	C₃×#904	C₂×C₄	1	2	yes	—
30	C₃×#901	—	2	2	no	—
31	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊(C₃⋊C₁₆)	D₄	1	2	yes	—
35	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	C₃×#819	C₂×C₄	1	2	yes	—
37	C₃×#812	—	2	2	no	—
38	C₃×#805	—	2	2	no	—
39	C₃×Dic₂₄	C₈	1	1	yes	characteristic, complemented
40	C₃×#800	C₈	1	2	yes	complemented
41	C₃×D₄₈	C₈	1	1	yes	characteristic, complemented
42	#63543	—	4	2	no	—
43	D₃×C₃⋊C₁₆	—	4	2	no	—
44	#12068	—	9	1	no	Sylow
45	C₃×#508	—	3	1	no	—
46	C₃×#443	—	3	2	no	—
47	#5140	—	6	2	no	—
48	#4907	—	6	2	no	—
49	#4904	Dic₃	1	1	yes	characteristic
50	#4700	C₂×D₃	1	2	yes	—
51	C₃×#368	—	3	1	no	—
52	C₃×#364	—	3	2	no	—
53	C₃×#359	—	3	2	no	—
54	C₃×#346	—	2	1	no	—
55	C₃×#346	C₂×D₃	1	1	yes	characteristic
56	C₃×#346	C₂×D₃	1	1	yes	characteristic
57	#4561	—	6	2	no	—
58	#4515	—	6	2	no	—
59	#4510	Dic₃	1	2	yes	—
60	#4506	Dic₃	1	1	yes	characteristic
61	#4463	C₂×D₃	1	2	yes	—
62	#4461	—	3	1	no	—
63	#4461	—	3	1	no	—
64	#4461	—	6	1	no	—
65	#4455	—	3	1	no	—
66	#4455	—	3	1	no	—
67	#4455	—	6	1	no	—
68	C₃×#235	—	2	2	no	—
69	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃²×C₄²	#41	1	1	yes	characteristic
72	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
73	C₃×#208	—	2	2	no	—
74	C₃×Dic₁₂	C₂×C₈	1	2	yes	—
75	C₃×D₂₄	C₂×C₈	1	2	yes	—
76	C₃×#204	—	2	2	no	—
77	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
78	C₃×D₃×C₈	—	4	2	no	—
79	C₃²×C₁₆	—	2	1	no	—
80	C₃²×C₁₆	C₂×C₈	1	1	yes	characteristic
81	C₃²×C₁₆	C₂×C₈	1	1	yes	characteristic
82	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
83	C₃×C₃⋊C₁₆	—	4	2	no	—
84	#2179	—	3	1	no	—
85	#2171	—	9	2	no	—
86	#1369	—	3	1	no	—
87	#1336	—	9	2	no	—
88	#1313	—	9	1	no	—
89	C₃×#135	—	3	2	no	—
90	C₃×#131	—	6	2	no	—
91	#904	C₂×Dic₃	1	2	yes	—
92	#901	—	2	2	no	—
93	C₃×Q₃₂	—	3	1	no	—
94	C₃×SD₃₂	—	3	2	no	—
95	C₃×D₁₆	—	3	1	no	—
96	C₃×M₃₂	—	6	2	no	—
97	C₂×C₃×C₁₆	—	2	1	no	—
98	C₂×C₃×C₁₆	—	2	1	no	—
99	C₂×C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₁₆	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₁₆	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₁₆	C₂×Dic₃	1	1	yes	characteristic
103	C₃×#108	—	3	2	no	—
104	C₃×#104	—	6	2	no	—
105	C₃×C₄×C₈	—	2	1	no	—
106	C₃×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₃×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
108	#819	C₂×Dic₃	1	2	yes	—
109	#812	—	2	2	no	—
110	#812	—	12	2	no	—
111	C₂×C₃⋊C₁₆	—	3	2	no	—
112	C₂×C₃⋊C₁₆	—	3	2	no	—
113	C₂×C₃⋊C₁₆	—	6	2	no	—
114	C₂×C₃⋊C₁₆	—	12	2	no	—
115	#805	—	2	2	no	—
116	C₄×C₃⋊C₈	—	6	2	no	—
117	C₄×C₃⋊C₈	—	6	1	no	—
118	C₄×C₃⋊C₈	—	6	1	no	—
119	Dic₂₄	C₃⋊C₈	1	1	yes	characteristic, complemented
120	#800	C₃⋊C₈	1	2	yes	complemented
121	D₄₈	C₃⋊C₈	1	1	yes	characteristic, complemented
122	#798	—	12	2	no	—
123	D₃×C₁₆	—	12	2	no	—
124	C₂×C₃²×C₄	—	2	1	no	—
125	C₂×C₃²×C₄	C₄×D₄	1	1	yes	characteristic
126	C₃×#67	—	4	2	no	—
127	C₃×D₁₂	—	2	2	no	—
128	C₃×C₄×D₃	—	4	2	no	—
129	C₃×Dic₆	—	2	2	no	—
130	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
131	C₃²×C₈	#131	1	1	yes	characteristic
132	C₃²×C₈	AGL(1,ℤ/8)	1	2	yes	—
133	C₃⋊(C₃⋊C₈)	—	8	1	no	—
134	C₃×C₃⋊C₈	—	4	2	no	—
135	#508	—	3	1	no	—
136	#504	—	18	2	no	—
137	#443	—	3	2	no	—
138	#368	—	3	1	no	—
139	#364	—	3	2	no	—
140	#359	—	3	2	no	—
141	#350	—	18	2	no	—
142	#346	—	9	1	no	—
143	#346	D₃²	1	1	yes	characteristic
144	#322	—	9	1	no	—
145	C₃×#41	—	6	2	no	—
146	#235	—	2	2	no	—
147	C₃×Q₁₆	—	3	2	no	—
148	C₃×SD₁₆	—	6	2	no	—
149	C₃×D₈	—	3	2	no	—
150	C₃×M₁₆	—	6	2	no	—
151	C₃×M₁₆	—	12	2	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	—	2	1	no	—
154	C₂×C₃×C₈	—	12	2	no	—
155	C₂×C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₃×C₈	D₃×D₄	1	1	yes	characteristic
157	C₂×C₃×C₈	D₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
159	C₃×C₄²	—	2	1	no	—
160	C₃×C₄²	#237	1	1	yes	characteristic
161	C₃×C₄²	#239	1	1	yes	characteristic
162	#208	—	2	2	no	—
163	C₂×C₃⋊C₈	—	12	2	no	—
164	C₂×C₃⋊C₈	—	12	1	no	—
165	C₂×C₃⋊C₈	—	12	1	no	—
166	Dic₁₂	C₂×C₃⋊C₈	1	2	yes	—
167	D₂₄	C₂×C₃⋊C₈	1	2	yes	—
168	#204	—	2	2	no	—
169	ΓL(1,𝔽₂₅)	—	4	2	no	—
170	D₃×C₈	—	4	2	no	—
171	C₃×C₁₆	—	2	2	no	—
172	C₃×C₁₆	—	2	1	no	—
173	C₃×C₁₆	—	2	1	no	—
174	C₃×C₁₆	—	2	1	no	—
175	C₃×C₁₆	—	2	1	no	—
176	C₃×C₁₆	—	12	2	no	—
177	C₃×C₁₆	D₃×C₈	1	1	yes	characteristic
178	C₃×C₁₆	D₃×C₈	1	1	yes	characteristic
179	C₃×C₁₆	C₂×C₃⋊C₈	1	1	yes	characteristic
180	C₃×C₁₆	C₂×C₃⋊C₈	1	1	yes	characteristic
181	C₃⋊C₁₆	—	4	2	no	—
182	C₃⋊C₁₆	—	6	2	no	—
183	C₃⋊C₁₆	—	6	2	no	—
184	C₃⋊C₁₆	—	12	2	no	—
185	C₂²×C₃²	#444	1	1	yes	characteristic
186	C₂×C₃×D₃	—	4	2	no	—
187	C₃²×C₄	—	4	1	no	—
188	C₃²×C₄	C₈×D₄	1	1	yes	characteristic
189	C₃²×C₄	#442	1	1	yes	characteristic
190	C₃×Dic₃	—	4	2	no	—
191	#135	—	3	2	no	—
192	#131	—	6	2	no	—
193	Q₃₂	—	3	1	no	—
194	SD₃₂	—	3	2	no	—
195	D₁₆	—	3	1	no	—
196	M₃₂	—	6	2	no	—
197	M₃₂	—	36	2	no	—
198	C₂×C₁₆	—	9	2	no	—
199	C₂×C₁₆	—	36	2	no	—
200	C₂×C₁₆	D₃×Dic₃	1	1	yes	characteristic
201	C₂×C₁₆	D₃×Dic₃	1	1	yes	characteristic
202	#108	—	3	2	no	—
203	#104	—	6	2	no	—
204	C₄×C₈	—	18	1	no	—
205	C₄×C₈	C₂×D₃²	1	1	yes	characteristic
206	C₃×Q₈	—	6	2	no	—
207	C₃×D₄	—	6	2	no	—
208	C₃×D₄	—	12	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	12	2	no	—
214	C₂×C₃×C₄	#893	1	1	yes	characteristic
215	C₂×C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
216	#67	—	4	2	no	—
217	D₁₂	—	2	2	no	—
218	C₄×D₃	—	4	2	no	—
219	Dic₆	—	2	2	no	—
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	12	2	no	—
224	C₃×C₈	C₂×D₃×C₈	1	1	yes	characteristic
225	C₃×C₈	#907	1	1	yes	characteristic
226	C₃×C₈	#911	1	2	yes	—
227	C₃×C₈	#914	1	2	yes	—
228	C₃×C₈	C₂²×C₃⋊C₈	1	1	yes	characteristic
229	C₃×C₈	#948	1	1	yes	characteristic
230	C₃⋊C₈	—	4	2	no	—
231	C₃⋊C₈	—	24	2	no	—
232	C₃⋊C₈	—	24	1	no	—
233	C₃⋊C₈	—	24	1	no	—
234	C₂×C₃²	—	2	1	no	—
235	C₂×C₃²	#1580	1	1	yes	characteristic
236	C₃×D₃	—	8	2	no	—
237	#41	—	6	2	no	—
238	Q₁₆	—	3	2	no	—
239	SD₁₆	—	6	2	no	—
240	D₈	—	3	2	no	—
241	M₁₆	—	6	2	no	—
242	M₁₆	—	12	2	no	—
243	C₂×C₈	—	12	2	no	—
244	C₂×C₈	—	36	1	no	—
245	C₂×C₈	#3802	1	1	yes	characteristic
246	C₂×C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
247	C₄²	#3796	1	1	yes	characteristic
248	C₁₆	—	2	1	no	—
249	C₁₆	—	12	2	no	—
250	C₁₆	—	18	2	no	—
251	C₁₆	D₃×C₃⋊C₈	1	1	yes	characteristic
252	C₁₆	D₃×C₃⋊C₈	1	1	yes	characteristic
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	12	2	no	—
255	C₂²×C₃	#4894	1	1	yes	characteristic
256	C₂²×C₃	#5197	1	1	yes	characteristic
257	C₂×D₃	—	4	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	12	2	no	—
264	C₃×C₄	#4834	1	1	yes	characteristic
265	C₃×C₄	#4869	1	1	yes	characteristic
266	C₃×C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
267	C₃×C₄	#5152	1	1	yes	characteristic
268	Dic₃	—	4	2	no	—
269	C₃²	#12068	1	1	yes	characteristic, complemented, Sylow
270	Q₈	—	6	2	no	—
271	D₄	—	6	2	no	—
272	D₄	—	12	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	12	2	no	—
275	C₂×C₄	#63898	1	1	yes	characteristic
276	C₈	—	12	2	no	—
277	C₈	—	72	1	no	—
278	C₈	#63798	1	2	yes	—
279	C₈	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
280	C₈	#63815	1	1	yes	characteristic
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	24	2	no	—
286	C₂×C₃	#68518	1	1	yes	characteristic
287	C₂×C₃	#70878	1	1	yes	characteristic
288	D₃	—	8	2	no	—
289	C₂²	—	12	2	no	—
290	C₂²	#10590920	1	1	yes	characteristic
291	C₄	—	4	1	no	—
292	C₄	—	12	2	no	—
293	C₄	#10590329	1	1	yes	characteristic
294	C₄	#10590781	1	1	yes	characteristic
295	C₃	—	2	1	no	—
296	C₃	#10707076	1	1	yes	characteristic, complemented
297	C₃	#10707077	1	1	yes	characteristic, complemented
298	C₂	—	2	1	no	—
299	C₂	—	24	2	no	—
300	C₂	#11777359	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10590781
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂×D₃²
Fitting	C₃²×#346	C₂²
socle	C₂×C₃²	#1580
prime radical	C₃×#4904	C₄
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	#346	D₃²
hypocenter	C₃²	#12068
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
2. #11797611	C₂	semidirect
3. C₃×#67514	C₂	semidirect
4. #11771667	C₂	semidirect
5. #11771319	C₂	semidirect
25. #67514	D₃	semidirect
39. C₃×Dic₂₄	C₈	semidirect
40. C₃×#800	C₈	semidirect
41. C₃×D₄₈	C₈	semidirect
119. Dic₂₄	C₃⋊C₈	semidirect
120. #800	C₃⋊C₈	semidirect
121. D₄₈	C₃⋊C₈	semidirect
269. C₃²	#12068	semidirect
296. C₃	#10707076	semidirect
297. C₃	#10707077	semidirect

Maximal subgroups

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

subgroup	index
C₃×#69085	2
#11797611	2
C₃×#67514	2
#11771667	2
#11771319	2
#10707077	3
#10707076	3