Subgroups — #13160282

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic
3	C₃²×#926	C₂	1	1	yes	maximal, characteristic
4	#11722019	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#10592696	—	3	1	no	maximal
7	#10591057	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
12	C₃×#3748	C₄	1	4	yes	complemented
13	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃²×#217	C₄	1	2	yes	—
15	C₃²×C₄⋊Dic₃	C₄	1	2	yes	—
16	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
17	C₃²×#117	—	3	1	no	—
18	C₂×#3748	D₃	1	1	yes	characteristic
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	4	no	—
21	C₂²×C₃⋊(C₃⋊C₈)	—	6	4	no	—
22	C₃×#926	D₃	1	4	yes	—
23	C₂×C₃×#208	—	3	4	no	—
24	#63630	—	3	2	no	—
25	C₄×C₃⋊(C₃⋊C₈)	—	3	2	no	—
26	#63554	—	3	8	no	—
27	Dic₃×C₃⋊C₈	—	3	8	no	—
28	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	C₂×C₄	1	4	yes	—
34	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	—
35	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	—
36	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
37	#5122	—	9	1	no	—
38	#4997	—	9	4	no	—
39	C₂²×C₃²×C₄	—	2	4	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₃²×C₄⋊C₄	—	3	2	no	—
43	C₃²×C₂²⋊C₄	—	3	2	no	—
44	C₃²×C₄²	—	3	2	no	—
45	#3748	Dic₃	1	4	yes	complemented
46	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	3	16	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	6	16	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
55	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
56	C₃×#217	Dic₃	1	8	yes	—
57	C₃×C₄⋊Dic₃	Dic₃	1	8	yes	—
58	C₃×C₄×Dic₃	C₂×D₃	1	8	yes	—
59	C₃×#208	—	3	16	no	—
60	C₂×C₃×C₃⋊C₈	—	3	8	no	—
61	C₂²×C₃³	—	2	2	no	—
62	C₂²×C₃³	C₄²	1	2	yes	—
63	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
64	C₃³×C₄	C₄²	1	2	yes	—
65	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃²×Dic₃	—	2	4	no	—
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₃×#117	—	3	4	no	—
70	#926	C₃⋊D₃	1	1	yes	characteristic
71	C₂×#208	—	3	4	no	—
72	C₂²×C₃⋊C₈	—	9	4	no	—
73	C₂²×C₃⋊C₈	—	9	1	no	—
74	C₂²×C₃⋊C₈	—	18	4	no	—
75	#815	—	9	2	no	—
76	C₈×Dic₃	—	9	2	no	—
77	#803	—	9	8	no	—
78	C₄×C₃⋊C₈	—	9	8	no	—
79	C₂³×C₃²	—	2	4	no	—
80	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
81	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	2	8	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	3	4	no	—
87	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
89	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
91	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
92	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
94	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₂×Dic₃	1	16	yes	—
96	C₃⋊(C₃⋊C₈)	—	3	2	no	—
97	C₃⋊(C₃⋊C₈)	—	3	2	no	—
98	C₃⋊(C₃⋊C₈)	—	3	8	no	—
99	C₃⋊(C₃⋊C₈)	—	3	8	no	—
100	C₃⋊(C₃⋊C₈)	—	6	8	no	—
101	C₃⋊(C₃⋊C₈)	—	6	8	no	—
102	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	4	yes	—
103	C₃×C₃⋊C₈	—	3	16	no	—
104	#405	—	27	1	no	Sylow
105	C₂×C₃³	—	2	2	no	—
106	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₃³	#131	1	2	yes	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	D₃²	1	4	yes	—
110	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₃×M₁₆	—	9	4	no	—
112	C₂×C₃×C₈	—	9	2	no	—
113	C₃×C₄⋊C₄	—	3	8	no	—
114	C₃×C₂²⋊C₄	—	3	8	no	—
115	C₃×C₄²	—	3	8	no	—
116	#217	C₃⋊Dic₃	1	2	yes	—
117	C₄⋊Dic₃	C₃⋊Dic₃	1	2	yes	—
118	C₄×Dic₃	C₂×C₃⋊D₃	1	2	yes	—
119	#208	—	3	16	no	—
120	C₂×C₃⋊C₈	—	3	8	no	—
121	C₂×C₃⋊C₈	—	9	16	no	—
122	C₂×C₃⋊C₈	—	9	4	no	—
123	C₂×C₃⋊C₈	—	9	4	no	—
124	C₂×C₃⋊C₈	—	9	4	no	—
125	C₂×C₃⋊C₈	—	9	1	no	—
126	C₂×C₃⋊C₈	—	9	1	no	—
127	C₂×C₃⋊C₈	—	18	16	no	—
128	C₂×C₃⋊C₈	—	18	4	no	—
129	C₂×C₃⋊C₈	—	18	4	no	—
130	C₂²×C₃²	—	2	16	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	2	8	no	—
133	C₂²×C₃²	—	2	8	no	—
134	C₂²×C₃²	—	2	4	no	—
135	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
136	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
137	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
139	C₃²×C₄	—	2	8	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	6	4	no	—
143	C₃²×C₄	C₄×Dic₃	1	8	yes	—
144	C₃²×C₄	C₄×Dic₃	1	2	yes	—
145	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
146	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
147	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
148	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
149	C₃×Dic₃	—	2	16	no	—
150	C₂×M₁₆	—	9	1	no	—
151	C₂²×C₈	—	27	1	no	—
152	#117	—	3	1	no	—
153	#97	—	27	2	no	—
154	C₄×C₈	—	27	2	no	—
155	C₃³	#405	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	4	no	—
157	C₂³×C₃	D₃×Dic₃	1	4	yes	—
158	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	3	16	no	—
164	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
165	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
166	C₂×C₃×C₄	#624	1	8	yes	—
167	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
168	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
171	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
172	C₂×Dic₃	C₂×C₃⋊Dic₃	1	4	yes	—
173	C₃×C₈	—	9	4	no	—
174	C₃⋊C₈	—	3	16	no	—
175	C₃⋊C₈	—	9	8	no	—
176	C₃⋊C₈	—	9	8	no	—
177	C₃⋊C₈	—	9	2	no	—
178	C₃⋊C₈	—	9	2	no	—
179	C₃⋊C₈	—	18	8	no	—
180	C₃⋊C₈	—	18	8	no	—
181	C₂×C₃²	—	2	16	no	—
182	C₂×C₃²	—	2	2	no	—
183	C₂×C₃²	—	2	8	no	—
184	C₂×C₃²	—	2	8	no	—
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	#907	1	2	yes	—
187	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
188	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₂×C₃²	#948	1	8	yes	—
190	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
191	M₁₆	—	9	4	no	—
192	C₂×C₈	—	9	2	no	—
193	C₂×C₈	—	27	4	no	—
194	C₂×C₈	—	27	1	no	—
195	C₂×C₈	—	27	1	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₂²⋊C₄	—	3	2	no	—
198	C₄²	—	3	2	no	—
199	C₂²×C₃	—	2	16	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	4	no	—
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	Dic₃²	1	8	yes	—
205	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
206	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
207	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	2	8	no	—
211	C₃×C₄	—	6	16	no	—
212	C₃×C₄	Dic₃²	1	8	yes	—
213	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
214	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
215	C₃×C₄	C₂×#624	1	4	yes	—
216	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
217	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
218	Dic₃	—	2	4	no	—
219	C₃²	—	2	4	no	—
220	C₃²	#4997	1	4	yes	complemented
221	C₃²	#5122	1	1	yes	characteristic, complemented
222	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
223	C₂×C₄	—	3	4	no	—
224	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
225	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
226	C₂×C₄	#6289	1	2	yes	—
227	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
228	C₈	—	9	4	no	—
229	C₈	—	27	2	no	—
230	C₈	—	27	2	no	—
231	C₂×C₃	—	2	16	no	—
232	C₂×C₃	—	2	8	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	2	8	no	—
236	C₂×C₃	#63815	1	8	yes	—
237	C₂×C₃	C₂×Dic₃²	1	4	yes	—
238	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
239	C₂×C₃	#64157	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
242	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
243	C₄	—	6	4	no	—
244	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
245	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
246	C₄	C₂×#6289	1	1	yes	characteristic
247	C₃	—	2	4	no	—
248	C₃	#10591057	1	4	yes	complemented
249	C₃	#10592696	1	1	yes	characteristic, complemented
250	C₂	—	2	2	no	—
251	C₂	#11723350	1	2	yes	—
252	C₂	C₂×Dic₃×C₃⋊Dic₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #11722019	C₂	semidirect
5. Dic₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
9. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
12. C₃×#3748	C₄	semidirect
45. #3748	Dic₃	semidirect
155. C₃³	#405	semidirect
220. C₃²	#4997	semidirect
221. C₃²	#5122	semidirect
248. C₃	#10591057	semidirect
249. C₃	#10592696	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₃×#3748	2
C₃²×#926	2
#11722019	2
Dic₃×C₃⋊(C₃⋊C₈)	2
#10592696	3
#10591057	3