Subgroups — #13160387

Order 1728.

988 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₃×#64135	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#208	C₂	1	1	yes	maximal, characteristic
4	#11722051	C₂	1	4	yes	maximal, complemented
5	#10592818	—	3	1	no	maximal
6	#10591117	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
10	C₃×#3757	C₄	1	2	yes	—
11	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
12	C₃×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
13	C₃²×#208	—	2	2	no	—
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃²×M₁₆	—	3	1	no	—
16	#64135	D₃	1	1	yes	characteristic
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	4	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	—	6	4	no	—
20	C₃×#926	—	3	4	no	—
21	C₂×C₃×#208	D₃	1	4	yes	—
22	#63641	—	3	4	no	—
23	Dic₃⋊(C₃⋊C₈)	—	3	16	no	—
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	D₄	1	2	yes	—
27	C₂×C₃³×C₄	Q₈	1	2	yes	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
30	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
31	C₃²×C₃⋊C₈	—	2	2	no	—
32	#5119	—	9	4	no	—
33	#4999	—	9	1	no	—
34	C₂²×C₃²×C₄	—	2	4	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₃²×M₁₆	—	6	2	no	—
38	C₂×C₃²×C₈	—	3	2	no	—
39	#3757	Dic₃	1	2	yes	—
40	C₄⋊(C₃⋊Dic₃)	Dic₃	1	2	yes	—
41	C₄×C₃⋊Dic₃	C₂×D₃	1	2	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	6	16	no	—
46	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
48	C₃×#217	—	3	8	no	—
49	C₃×C₄⋊Dic₃	—	3	8	no	—
50	C₃×C₄×Dic₃	—	3	8	no	—
51	C₃×#208	—	2	8	no	—
52	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
53	C₂²×C₃³	—	2	2	no	—
54	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
55	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
56	C₃³×C₄	C₄⋊C₄	1	2	yes	—
57	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
58	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
59	C₃×C₃⋊Dic₃	—	2	4	no	—
60	C₂×C₃×M₁₆	—	3	4	no	—
61	C₃×#117	—	9	1	no	—
62	#926	—	3	4	no	—
63	C₂×#208	C₃⋊D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	—	9	4	no	—
65	C₂²×C₃⋊C₈	—	9	1	no	—
66	C₂²×C₃⋊C₈	—	18	4	no	—
67	Dic₃⋊C₈	—	9	16	no	—
68	C₄⋊(C₃⋊C₈)	—	9	4	no	—
69	C₂³×C₃²	—	2	4	no	—
70	C₂³×C₃²	C₄×D₃	1	4	yes	—
71	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	4	no	—
73	C₂×C₃²×C₄	—	2	8	no	—
74	C₂×C₃²×C₄	—	2	8	no	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	Dic₆	1	8	yes	—
77	C₂×C₃²×C₄	Dic₆	1	2	yes	—
78	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
79	C₂×C₃²×C₄	D₁₂	1	2	yes	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	#67	1	8	yes	—
82	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	C₂×Dic₃	1	4	yes	—
85	C₂×C₃×Dic₃	—	3	16	no	—
86	C₃²×C₈	—	6	2	no	—
87	C₃⋊(C₃⋊C₈)	—	6	16	no	—
88	C₃⋊(C₃⋊C₈)	—	6	2	no	—
89	C₃⋊(C₃⋊C₈)	—	6	8	no	—
90	C₃×C₃⋊C₈	—	2	8	no	—
91	#424	—	27	1	no	Sylow
92	C₂×C₃³	—	2	2	no	—
93	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃³	#131	1	2	yes	—
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	D₃²	1	4	yes	—
97	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₃×M₁₆	—	6	8	no	—
99	C₂×C₃×C₈	—	3	8	no	—
100	C₃×C₄⋊C₄	—	9	2	no	—
101	C₃×C₂²⋊C₄	—	9	2	no	—
102	C₃×C₄²	—	9	2	no	—
103	#217	—	3	8	no	—
104	C₄⋊Dic₃	—	3	8	no	—
105	C₄×Dic₃	—	3	8	no	—
106	#208	—	2	2	no	—
107	C₂×C₃⋊C₈	—	9	8	no	—
108	C₂×C₃⋊C₈	—	9	2	no	—
109	C₂×C₃⋊C₈	—	18	8	no	—
110	C₂×C₃⋊C₈	—	18	16	no	—
111	C₂×C₃⋊C₈	—	18	8	no	—
112	C₂×C₃⋊C₈	—	18	2	no	—
113	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
114	C₂²×C₃²	—	2	16	no	—
115	C₂²×C₃²	—	2	8	no	—
116	C₂²×C₃²	—	2	2	no	—
117	C₂²×C₃²	—	2	8	no	—
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	Dic₃⋊C₄	1	8	yes	—
120	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
121	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
122	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
123	C₃²×C₄	—	2	8	no	—
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	Dic₃⋊C₄	1	8	yes	—
127	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
128	C₃²×C₄	C₂×Dic₆	1	4	yes	—
129	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
130	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
131	C₃²×C₄	C₂×#67	1	4	yes	—
132	C₃⋊Dic₃	—	2	4	no	—
133	C₃×Dic₃	—	6	16	no	—
134	C₂×M₁₆	—	3	1	no	—
135	C₂²×C₈	—	27	1	no	—
136	#117	—	9	1	no	—
137	C₄⋊C₈	—	27	4	no	—
138	C₃³	#424	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	D₃×Dic₃	1	4	yes	—
141	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	8	no	—
144	C₂×C₃×C₄	—	2	8	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	9	4	no	—
147	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
148	C₂×C₃×C₄	#626	1	8	yes	—
149	C₂×C₃×C₄	#627	1	8	yes	—
150	C₂×C₃×C₄	#634	1	2	yes	—
151	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
152	C₂×C₃×C₄	#638	1	2	yes	—
153	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
154	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
155	C₂×Dic₃	—	3	16	no	—
156	C₃×C₈	—	6	8	no	—
157	C₃⋊C₈	—	2	2	no	—
158	C₃⋊C₈	—	18	16	no	—
159	C₃⋊C₈	—	18	8	no	—
160	C₃⋊C₈	—	18	2	no	—
161	C₂×C₃²	—	2	16	no	—
162	C₂×C₃²	—	2	2	no	—
163	C₂×C₃²	—	2	8	no	—
164	C₂×C₃²	—	2	8	no	—
165	C₂×C₃²	—	2	4	no	—
166	C₂×C₃²	#907	1	8	yes	—
167	C₂×C₃²	C₂×Dic₃⋊C₄	1	4	yes	—
168	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂×C₃²	#948	1	2	yes	—
170	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
171	M₁₆	—	6	2	no	—
172	C₂×C₈	—	3	2	no	—
173	C₂×C₈	—	27	2	no	—
174	C₂×C₈	—	54	2	no	—
175	C₄⋊C₄	—	9	2	no	—
176	C₂²⋊C₄	—	9	2	no	—
177	C₄²	—	9	2	no	—
178	C₂²×C₃	—	2	16	no	—
179	C₂²×C₃	—	2	8	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	—	2	8	no	—
183	C₂²×C₃	Dic₃⋊Dic₃	1	8	yes	—
184	C₂²×C₃	#3750	1	2	yes	—
185	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
186	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	8	no	—
190	C₃×C₄	—	18	4	no	—
191	C₃×C₄	Dic₃⋊Dic₃	1	8	yes	—
192	C₃×C₄	#3750	1	2	yes	—
193	C₃×C₄	C₂×#626	1	4	yes	—
194	C₃×C₄	C₂×#627	1	4	yes	—
195	C₃×C₄	C₂×#634	1	1	yes	characteristic
196	C₃×C₄	C₂×#638	1	1	yes	characteristic
197	Dic₃	—	6	16	no	—
198	C₃²	—	2	4	no	—
199	C₃²	#4999	1	1	yes	characteristic, complemented
200	C₃²	#5119	1	4	yes	complemented
201	C₂³	Dic₃×C₃⋊D₃	1	1	yes	characteristic
202	C₂×C₄	—	9	4	no	—
203	C₂×C₄	Dic₃×C₃⋊D₃	1	1	yes	characteristic
204	C₂×C₄	#6291	1	2	yes	—
205	C₂×C₄	#6293	1	2	yes	—
206	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
207	C₈	—	6	2	no	—
208	C₈	—	54	2	no	—
209	C₂×C₃	—	2	16	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	8	no	—
212	C₂×C₃	—	2	8	no	—
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	#63815	1	8	yes	—
215	C₂×C₃	C₂×Dic₃⋊Dic₃	1	4	yes	—
216	C₂×C₃	#64116	1	2	yes	—
217	C₂×C₃	C₂×#3750	1	1	yes	characteristic
218	C₂²	—	2	2	no	—
219	C₂²	#88807	1	2	yes	—
220	C₂²	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
221	C₄	—	18	4	no	—
222	C₄	#88807	1	2	yes	—
223	C₄	C₂×#6291	1	1	yes	characteristic
224	C₄	C₂×#6293	1	1	yes	characteristic
225	C₃	—	2	4	no	—
226	C₃	#10591117	1	4	yes	complemented
227	C₃	#10592818	1	1	yes	characteristic, complemented
228	C₂	—	2	2	no	—
229	C₂	#11723354	1	2	yes	—
230	C₂	C₂×#88807	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₂²×C₃³×C₄	C₂²
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	#424
LCS	G ▷ C₂²×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. #11722051	C₂	semidirect
138. C₃³	#424	semidirect
199. C₃²	#4999	semidirect
200. C₃²	#5119	semidirect
226. C₃	#10591117	semidirect
227. C₃	#10592818	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
C₃×#64135	2
C₂×C₃²×#208	2
#11722051	2
#10592818	3
#10591117	3