Subgroups — #13158914

Order 1728.

956 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×#88786	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63630	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722019	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592696	—	3	1	no	maximal
9	#10590319	—	3	4	no	maximal
10	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃²×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
17	C₃²×Dic₃⋊C₄	C₄	1	2	yes	—
18	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
19	#88786	C₄	1	4	yes	complemented
20	D₃×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
21	C₃²×#117	—	3	1	no	—
22	C₂×#3748	—	3	1	no	—
23	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
24	C₃×#872	D₃	1	4	yes	—
25	C₂×#3711	—	3	4	no	—
26	C₂×D₃×C₃⋊C₈	—	3	4	no	—
27	#63630	—	3	1	no	—
28	#63630	D₃	1	1	yes	characteristic, complemented
29	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
30	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
31	C₄×C₃⋊(C₃⋊C₈)	—	3	4	no	—
32	C₄×C₃⋊(C₃⋊C₈)	—	6	4	no	—
33	C₃×#803	—	3	4	no	—
34	#63554	—	3	4	no	—
35	Dic₃×C₃⋊C₈	—	3	4	no	—
36	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
38	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
43	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	—
44	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	—
45	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
46	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
47	#4997	—	9	4	no	—
48	#4708	—	9	1	no	—
49	C₂²×C₃²×C₄	—	3	1	no	—
50	C₂×C₃×C₄×D₃	C₂×D₃	1	4	yes	—
51	C₃²×C₄⋊C₄	—	3	2	no	—
52	C₃²×C₂²⋊C₄	—	3	2	no	—
53	C₃²×C₄²	—	2	4	no	—
54	C₃²×C₄²	—	3	1	no	—
55	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂×D₃	1	4	yes	—
57	#3748	—	3	4	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
59	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
61	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
62	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
63	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
64	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
65	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
67	C₃×(C₂×D₃)⋊C₄	Dic₃	1	8	yes	—
68	C₃×Dic₃⋊C₄	Dic₃	1	8	yes	—
69	C₃×C₄×Dic₃	C₂×D₃	1	4	yes	—
70	C₂×C₃×C₃⋊C₈	—	3	4	no	—
71	C₂×C₃×C₃⋊C₈	—	3	4	no	—
72	#3711	—	3	16	no	—
73	D₃×C₃⋊C₈	—	3	16	no	—
74	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×D₃	—	2	2	no	—
76	C₂×C₃²×D₃	C₄²	1	2	yes	—
77	C₃³×C₄	—	2	2	no	—
78	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃²×Dic₃	—	2	2	no	—
81	C₃²×Dic₃	C₄²	1	2	yes	—
82	C₃×#117	—	3	4	no	—
83	C₂×#208	—	9	4	no	—
84	C₂²×C₃⋊C₈	—	9	4	no	—
85	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
86	C₂×D₃×C₈	—	9	1	no	—
87	#872	C₃⋊D₃	1	1	yes	characteristic
88	C₃×#97	—	9	1	no	—
89	#815	—	9	1	no	—
90	C₈×Dic₃	—	9	1	no	—
91	#803	—	3	4	no	—
92	#803	—	9	4	no	—
93	C₄×C₃⋊C₈	—	9	4	no	—
94	C₄×C₃⋊C₈	—	9	4	no	—
95	C₄×C₃⋊C₈	—	9	1	no	—
96	C₄×C₃⋊C₈	—	18	4	no	—
97	C₂³×C₃²	—	3	1	no	—
98	C₂²×C₃×D₃	C₂×Dic₃	1	4	yes	—
99	C₂×C₃²×C₄	—	2	8	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	3	1	no	—
102	C₂×C₃²×C₄	—	3	2	no	—
103	C₂×C₃²×C₄	—	3	2	no	—
104	C₂×C₃²×C₄	—	3	2	no	—
105	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
106	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
107	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	C₂×Dic₃	1	8	yes	—
110	C₂×C₃×Dic₃	C₂×Dic₃	1	4	yes	—
111	C₃×C₄×D₃	C₄×D₃	1	8	yes	—
112	C₃×C₄×D₃	C₄×D₃	1	8	yes	—
113	C₃⋊(C₃⋊C₈)	—	3	2	no	—
114	C₃⋊(C₃⋊C₈)	—	3	2	no	—
115	C₃⋊(C₃⋊C₈)	—	3	8	no	—
116	C₃⋊(C₃⋊C₈)	—	3	8	no	—
117	C₃⋊(C₃⋊C₈)	—	6	8	no	—
118	C₃⋊(C₃⋊C₈)	—	6	8	no	—
119	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	—
120	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	—
121	C₃×C₃⋊C₈	—	3	8	no	—
122	C₃×C₃⋊C₈	—	3	8	no	—
123	#405	—	27	1	no	Sylow
124	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₃³	#131	1	2	yes	—
126	C₃²×D₃	—	2	2	no	—
127	C₂²×C₃×C₄	—	3	4	no	—
128	C₂×C₄×D₃	C₂×C₃⋊D₃	1	1	yes	characteristic
129	C₂×C₃×C₈	—	9	1	no	—
130	C₂×C₃×C₈	—	9	1	no	—
131	C₃×C₄⋊C₄	—	3	8	no	—
132	C₃×C₂²⋊C₄	—	3	8	no	—
133	C₃×C₄²	—	2	4	no	—
134	C₃×C₄²	—	3	4	no	—
135	C₃×C₄²	D₃²	1	4	yes	—
136	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
137	(C₂×D₃)⋊C₄	C₃⋊Dic₃	1	2	yes	—
138	Dic₃⋊C₄	C₃⋊Dic₃	1	2	yes	—
139	C₄×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
140	#208	—	9	16	no	—
141	C₂×C₃⋊C₈	—	3	4	no	—
142	C₂×C₃⋊C₈	—	3	4	no	—
143	C₂×C₃⋊C₈	—	9	16	no	—
144	C₂×C₃⋊C₈	—	9	4	no	—
145	C₂×C₃⋊C₈	—	9	4	no	—
146	C₂×C₃⋊C₈	—	9	1	no	—
147	C₂×C₃⋊C₈	—	9	1	no	—
148	C₂×C₃⋊C₈	—	18	4	no	—
149	C₂×C₃⋊C₈	—	18	4	no	—
150	ΓL(1,𝔽₂₅)	—	9	4	no	—
151	D₃×C₈	—	9	4	no	—
152	C₂²×C₃²	—	2	4	no	—
153	C₂²×C₃²	—	3	2	no	—
154	C₂²×C₃²	—	6	2	no	—
155	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
157	C₂×C₃×D₃	—	2	8	no	—
158	C₂×C₃×D₃	C₄×Dic₃	1	8	yes	—
159	C₃²×C₄	—	2	16	no	—
160	C₃²×C₄	—	2	2	no	—
161	C₃²×C₄	—	2	8	no	—
162	C₃²×C₄	—	2	4	no	—
163	C₃²×C₄	—	2	4	no	—
164	C₃²×C₄	—	3	2	no	—
165	C₃²×C₄	—	6	2	no	—
166	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
167	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
168	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₃×Dic₃	—	2	8	no	—
171	C₃×Dic₃	C₄×Dic₃	1	8	yes	—
172	C₂×M₁₆	—	27	1	no	—
173	C₂²×C₈	—	27	1	no	—
174	#117	—	3	1	no	—
175	#97	—	9	1	no	—
176	#97	—	27	1	no	—
177	C₄×C₈	—	27	1	no	—
178	C₄×C₈	—	27	1	no	—
179	C₃³	#405	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	3	4	no	—
181	C₂²×D₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	3	4	no	—
185	C₂×C₃×C₄	—	3	8	no	—
186	C₂×C₃×C₄	—	3	8	no	—
187	C₂×C₃×C₄	—	3	8	no	—
188	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
189	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
190	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
191	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
192	C₂×Dic₃	C₂×C₃⋊Dic₃	1	2	yes	—
193	C₂×Dic₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
194	C₄×D₃	C₄×C₃⋊D₃	1	2	yes	—
195	C₄×D₃	C₄×C₃⋊D₃	1	2	yes	—
196	C₃×C₈	—	9	2	no	—
197	C₃×C₈	—	9	2	no	—
198	C₃⋊C₈	—	3	8	no	—
199	C₃⋊C₈	—	3	8	no	—
200	C₃⋊C₈	—	9	8	no	—
201	C₃⋊C₈	—	9	8	no	—
202	C₃⋊C₈	—	9	2	no	—
203	C₃⋊C₈	—	9	2	no	—
204	C₃⋊C₈	—	18	8	no	—
205	C₃⋊C₈	—	18	8	no	—
206	C₂×C₃²	—	2	8	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	—	6	2	no	—
209	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
210	C₂×C₃²	#907	1	2	yes	—
211	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
212	C₂×C₃²	#948	1	8	yes	—
213	C₃×D₃	—	2	8	no	—
214	C₂²×C₄	—	3	1	no	—
215	M₁₆	—	27	4	no	—
216	C₂×C₈	—	9	1	no	—
217	C₂×C₈	—	9	1	no	—
218	C₂×C₈	—	27	4	no	—
219	C₂×C₈	—	27	1	no	—
220	C₂×C₈	—	27	1	no	—
221	C₄⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₄²	—	3	1	no	—
224	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	3	8	no	—
227	C₂²×C₃	—	6	8	no	—
228	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
229	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	C₄×C₃⋊Dic₃	1	2	yes	—
232	C₃×C₄	—	2	16	no	—
233	C₃×C₄	—	2	8	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	4	no	—
237	C₃×C₄	—	3	8	no	—
238	C₃×C₄	—	6	8	no	—
239	C₃×C₄	C₄×D₃²	1	4	yes	—
240	C₃×C₄	C₄×D₃²	1	4	yes	—
241	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
242	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
243	Dic₃	—	2	2	no	—
244	Dic₃	C₄×C₃⋊Dic₃	1	2	yes	—
245	C₃²	—	2	4	no	—
246	C₃²	#4708	1	1	yes	characteristic, complemented
247	C₃²	#4997	1	4	yes	complemented
248	C₂³	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	D₃×C₃⋊Dic₃	1	2	yes	—
254	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
255	C₈	—	9	2	no	—
256	C₈	—	9	2	no	—
257	C₈	—	27	2	no	—
258	C₈	—	27	2	no	—
259	C₂×C₃	—	2	8	no	—
260	C₂×C₃	—	2	4	no	—
261	C₂×C₃	—	6	8	no	—
262	C₂×C₃	#63815	1	8	yes	—
263	C₂×C₃	C₄×D₃×Dic₃	1	4	yes	—
264	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
265	C₂×C₃	#64157	1	2	yes	—
266	D₃	—	2	2	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	3	2	no	—
272	C₄	—	6	2	no	—
273	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
274	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
275	C₃	—	2	4	no	—
276	C₃	#10590319	1	4	yes	complemented
277	C₃	#10592696	1	1	yes	characteristic, complemented
278	C₂	—	6	2	no	—
279	C₂	#11723350	1	2	yes	—
280	C₂	C₄×D₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂×C₄²
Fitting	C₃³×C₄²	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃²×D₃	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄²	D₃×C₃⋊D₃
hypocenter	C₃³	#405
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
5. C₃×#63630	C₂	semidirect
6. #11722019	C₂	semidirect
7. Dic₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
19. #88786	C₄	semidirect
20. D₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
28. #63630	D₃	semidirect
179. C₃³	#405	semidirect
246. C₃²	#4708	semidirect
247. C₃²	#4997	semidirect
276. C₃	#10590319	semidirect
277. C₃	#10592696	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#872	2
C₂×#88786	2
C₂×D₃×C₃⋊(C₃⋊C₈)	2
C₄×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₃×#63630	2
#11722019	2
Dic₃×C₃⋊(C₃⋊C₈)	2
#10592696	3
#10590319	3