Subgroups — #13163651

Order 1728.

1736 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#88855	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	#11722200	C₂	1	1	yes	maximal, characteristic, complemented
5	#11722200	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722173	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592710	—	3	13	no	maximal
9	C₂×C₃³×D₄	C₄	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₃³×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₃³×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₃³×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	#88855	—	2	2	no	—
17	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
18	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
22	C₃²×C₄×D₄	D₃	1	13	yes	—
23	C₂×#3748	—	3	13	no	—
24	C₂²×C₃⋊(C₃⋊C₈)	—	3	13	no	—
25	#63657	—	3	13	no	—
26	#63657	—	3	13	no	—
27	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	13	no	—
28	#63630	—	3	13	no	—
29	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
31	C₃³×D₄	—	2	2	no	—
32	C₂×C₃³×C₄	—	2	1	no	—
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	—	2	1	no	—
35	C₂×C₃³×C₄	—	2	1	no	—
36	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
42	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
43	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
44	C₃⋊(C₃⋊(C₃⋊C₈))	D₄	1	2	yes	complemented
45	#5011	—	9	13	no	—
46	C₂×C₃²×D₄	Dic₃	1	13	yes	—
47	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
48	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
49	C₃²×C₄⋊C₄	Dic₃	1	13	yes	—
50	C₃²×C₂²⋊C₄	Dic₃	1	13	yes	—
51	C₃²×C₂²⋊C₄	Dic₃	1	13	yes	—
52	C₃²×C₄²	C₂×D₃	1	13	yes	—
53	#3748	—	6	26	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
55	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
56	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
57	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
59	C₂²×C₃³	—	2	1	no	—
60	C₂²×C₃³	—	2	1	no	—
61	C₂²×C₃³	—	2	1	no	—
62	C₂²×C₃³	—	2	1	no	—
63	C₂²×C₃³	—	2	1	no	—
64	C₂²×C₃³	M₁₆	1	2	yes	complemented
65	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	—	2	1	no	—
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	—	2	1	no	—
69	C₃³×C₄	—	2	1	no	—
70	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃³×C₄	#41	1	1	yes	characteristic
72	C₃×C₄×D₄	C₃⋊D₃	1	13	yes	—
73	C₂×#208	—	9	13	no	—
74	C₂²×C₃⋊C₈	—	9	13	no	—
75	#830	—	9	13	no	—
76	#830	—	9	13	no	—
77	C₄⋊(C₃⋊C₈)	—	9	13	no	—
78	#803	—	9	13	no	—
79	C₂³×C₃²	C₂×Dic₃	1	13	yes	—
80	C₂³×C₃²	C₂×Dic₃	1	13	yes	—
81	C₃²×D₄	—	2	26	no	—
82	C₂×C₃²×C₄	—	2	13	no	—
83	C₂×C₃²×C₄	—	2	13	no	—
84	C₂×C₃²×C₄	—	2	13	no	—
85	C₂×C₃²×C₄	—	2	13	no	—
86	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
87	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
90	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
91	C₃⋊(C₃⋊C₈)	—	3	26	no	—
92	C₃⋊(C₃⋊C₈)	—	6	13	no	—
93	C₃⋊(C₃⋊C₈)	—	6	13	no	—
94	C₃⋊(C₃⋊C₈)	—	6	13	no	—
95	#435	—	27	1	no	Sylow
96	C₂×C₃³	—	2	2	no	—
97	C₂×C₃³	—	4	1	no	—
98	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
99	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₃³	#131	1	1	yes	characteristic
101	C₂×C₃×D₄	C₃⋊Dic₃	1	13	yes	—
102	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
103	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
104	C₃×C₄⋊C₄	C₃⋊Dic₃	1	13	yes	—
105	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	13	yes	—
106	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	13	yes	—
107	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
108	#208	—	18	26	no	—
109	C₂×C₃⋊C₈	—	9	13	no	—
110	C₂×C₃⋊C₈	—	9	13	no	—
111	C₂×C₃⋊C₈	—	9	13	no	—
112	C₂×C₃⋊C₈	—	9	13	no	—
113	C₂×C₃⋊C₈	—	18	26	no	—
114	C₂²×C₃²	—	2	13	no	—
115	C₂²×C₃²	—	2	13	no	—
116	C₂²×C₃²	—	2	13	no	—
117	C₂²×C₃²	—	2	13	no	—
118	C₂²×C₃²	—	2	13	no	—
119	C₂²×C₃²	#208	1	26	yes	complemented
120	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
121	C₃²×C₄	—	2	13	no	—
122	C₃²×C₄	—	2	13	no	—
123	C₃²×C₄	—	2	13	no	—
124	C₃²×C₄	—	2	13	no	—
125	C₃²×C₄	D₃×D₄	1	13	yes	—
126	C₃²×C₄	#237	1	13	yes	—
127	C₂×M₁₆	—	27	1	no	—
128	C₂²×C₈	—	27	1	no	—
129	C₄×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
130	C₄⋊C₈	—	27	1	no	—
131	C₂²⋊C₈	—	27	1	no	—
132	C₂²⋊C₈	—	27	1	no	—
133	#97	—	27	1	no	—
134	C₃³	#435	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₃	C₂×C₃⋊Dic₃	1	13	yes	—
136	C₂³×C₃	C₂×C₃⋊Dic₃	1	13	yes	—
137	C₃×D₄	—	2	26	no	—
138	C₂×C₃×C₄	—	2	13	no	—
139	C₂×C₃×C₄	—	2	13	no	—
140	C₂×C₃×C₄	—	2	13	no	—
141	C₂×C₃×C₄	—	2	13	no	—
142	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
143	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
144	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
145	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
146	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
147	C₃⋊C₈	—	9	26	no	—
148	C₃⋊C₈	—	18	13	no	—
149	C₃⋊C₈	—	18	13	no	—
150	C₃⋊C₈	—	18	13	no	—
151	C₂×C₃²	—	2	26	no	—
152	C₂×C₃²	—	4	13	no	—
153	C₂×C₃²	C₂×#208	1	13	yes	—
154	C₂×C₃²	D₄×Dic₃	1	13	yes	—
155	C₂×C₃²	#948	1	13	yes	—
156	C₂×D₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
159	M₁₆	—	54	2	no	—
160	C₂×C₈	—	27	1	no	—
161	C₂×C₈	—	27	1	no	—
162	C₂×C₈	—	27	1	no	—
163	C₂×C₈	—	27	1	no	—
164	C₂×C₈	—	54	2	no	—
165	C₄⋊C₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
166	C₂²⋊C₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
167	C₂²⋊C₄	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
168	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
169	C₂²×C₃	—	2	13	no	—
170	C₂²×C₃	—	2	13	no	—
171	C₂²×C₃	—	2	13	no	—
172	C₂²×C₃	—	2	13	no	—
173	C₂²×C₃	—	2	13	no	—
174	C₂²×C₃	#3748	1	26	yes	complemented
175	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
176	C₃×C₄	—	2	13	no	—
177	C₃×C₄	—	2	13	no	—
178	C₃×C₄	—	2	13	no	—
179	C₃×C₄	—	2	13	no	—
180	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
181	C₃×C₄	#3830	1	13	yes	—
182	C₃²	#5011	1	13	yes	complemented
183	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
184	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
195	C₈	—	27	2	no	—
196	C₈	—	54	1	no	—
197	C₈	—	54	1	no	—
198	C₈	—	54	1	no	—
199	C₂×C₃	—	2	26	no	—
200	C₂×C₃	—	4	13	no	—
201	C₂×C₃	C₂×#3748	1	13	yes	—
202	C₂×C₃	D₄×C₃⋊Dic₃	1	13	yes	—
203	C₂×C₃	#64157	1	13	yes	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#88855	1	2	yes	complemented
210	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
216	C₄	#89078	1	1	yes	characteristic
217	C₃	#10592710	1	13	yes	complemented
218	C₂	—	2	2	no	—
219	C₂	—	4	1	no	—
220	C₂	C₂×#88855	1	1	yes	characteristic
221	C₂	D₄×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
222	C₂	#11723989	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₃³×C₄×D₄	C₂
prime radical	C₂×C₃³×D₄	C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#435
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88855	C₂	semidirect
3. C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
4. #11722200	C₂	semidirect
5. #11722200	C₂	semidirect
6. C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	semidirect
7. #11722173	C₂	semidirect
44. C₃⋊(C₃⋊(C₃⋊C₈))	D₄	semidirect
64. C₂²×C₃³	M₁₆	semidirect
119. C₂²×C₃²	#208	semidirect
134. C₃³	#435	semidirect
174. C₂²×C₃	#3748	semidirect
182. C₃²	#5011	semidirect
209. C₂²	#88855	semidirect
217. C₃	#10592710	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×D₄	2
C₂×#88855	2
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
#11722200	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#11722173	2
#10592710	3