Subgroups — #13144431

Order 1728.

540 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88547	C₂	1	2	yes	maximal
3	#11721706	C₂	1	1	yes	maximal, characteristic
4	#11720527	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	#10592730	—	3	1	no	maximal
7	#10588707	—	3	3	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×#6226	C₂²	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	#88547	—	2	4	no	—
15	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
16	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
19	C₂×#3756	—	3	2	no	—
20	#64078	—	3	1	no	—
21	#63637	—	3	2	no	—
22	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
23	C₄×Q₈×C₉	D₃	1	3	yes	—
24	C₂×#3674	—	3	6	no	—
25	#63431	—	3	3	no	—
26	#63368	—	3	6	no	—
27	C₄⋊(C₉⋊C₈)	—	3	3	no	—
28	C₃×Q₈×C₉	—	2	1	no	—
29	C₃×Q₈×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊Dic₉	—	2	2	no	—
36	#6226	—	2	2	no	—
37	#6226	—	2	2	no	—
38	C₃⋊(C₉⋊C₈)	—	2	2	no	—
39	#5031	—	9	3	no	—
40	#5031	—	9	1	no	—
41	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₂×#634	—	3	2	no	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
45	C₃²×C₄²	—	2	1	no	—
46	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
47	#3756	—	6	4	no	—
48	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
49	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
52	C₂×Dic₁₈	—	3	6	no	—
53	C₉×C₄⋊C₄	—	2	3	no	—
54	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
55	C₄²×C₉	—	2	3	no	—
56	C₄²×C₉	C₂×D₃	1	3	yes	—
57	#3674	—	6	12	no	—
58	C₄⋊Dic₉	—	6	3	no	—
59	C₄⋊Dic₉	—	6	3	no	—
60	C₂×C₉⋊C₈	—	3	6	no	—
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	—	2	1	no	—
64	C₃×C₄×C₉	—	4	1	no	—
65	C₃×C₄×C₉	Q₁₆	1	2	yes	complemented
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	#41	1	1	yes	characteristic
68	C₃⋊Dic₉	—	4	2	no	—
69	C₃×C₄×Q₈	D₉	1	3	yes	—
70	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
71	C₂×#216	—	9	6	no	—
72	C₂×#216	—	9	2	no	—
73	#869	—	9	3	no	—
74	#869	—	9	1	no	—
75	#810	—	9	6	no	—
76	#810	—	9	2	no	—
77	C₄⋊(C₃⋊C₈)	—	9	3	no	—
78	C₄⋊(C₃⋊C₈)	—	9	1	no	—
79	C₃²×Q₈	—	2	1	no	—
80	C₃²×Q₈	D₁₂	1	2	yes	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	#67	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	—	6	2	no	—
87	#634	—	6	2	no	—
88	#634	—	6	2	no	—
89	C₃⋊(C₃⋊C₈)	—	6	2	no	—
90	Q₈×C₉	—	2	3	no	—
91	Q₈×C₉	D₁₂	1	6	yes	—
92	C₂×C₄×C₉	—	2	3	no	—
93	C₂×C₄×C₉	—	2	3	no	—
94	C₂×C₄×C₉	#67	1	3	yes	—
95	C₂×C₄×C₉	#67	1	3	yes	—
96	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
97	C₂×Dic₉	—	6	6	no	—
98	Dic₁₈	—	6	6	no	—
99	Dic₁₈	—	6	6	no	—
100	C₉⋊C₈	—	6	6	no	—
101	#462	—	27	1	no	Sylow
102	C₂×C₃×C₉	#121	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₃×C₉	#137	1	1	yes	characteristic
105	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
106	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂×Dic₆	—	9	6	no	—
108	C₂×Dic₆	—	9	2	no	—
109	C₃×C₄⋊C₄	—	2	3	no	—
110	C₃×C₄⋊C₄	—	2	1	no	—
111	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
112	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₃×C₄²	—	2	3	no	—
114	C₃×C₄²	—	2	1	no	—
115	C₃×C₄²	C₂×D₉	1	3	yes	—
116	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
117	#216	—	18	12	no	—
118	#216	—	18	4	no	—
119	C₄⋊Dic₃	—	18	3	no	—
120	C₄⋊Dic₃	—	18	3	no	—
121	C₄⋊Dic₃	—	18	1	no	—
122	C₄⋊Dic₃	—	18	1	no	—
123	C₂×C₃⋊C₈	—	9	6	no	—
124	C₂×C₃⋊C₈	—	9	2	no	—
125	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	4	1	no	—
129	C₃²×C₄	#216	1	2	yes	—
130	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
131	C₃²×C₄	#235	1	1	yes	characteristic
132	C₃⋊Dic₃	—	12	2	no	—
133	C₂²×C₉	C₂×#67	1	3	yes	—
134	C₄×C₉	—	2	6	no	—
135	C₄×C₉	—	2	3	no	—
136	C₄×C₉	—	4	3	no	—
137	C₄×C₉	#216	1	6	yes	complemented
138	C₄×C₉	C₂×D₁₂	1	3	yes	—
139	C₄×C₉	#235	1	3	yes	—
140	Dic₉	—	12	6	no	—
141	C₂×Q₁₆	—	27	2	no	—
142	C₄⋊Q₈	—	27	1	no	—
143	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
144	C₄⋊C₈	—	27	1	no	—
145	#103	—	27	2	no	—
146	C₃×C₉	#462	1	1	yes	characteristic, complemented, Sylow
147	C₃×Q₈	—	2	3	no	—
148	C₃×Q₈	—	2	1	no	—
149	C₃×Q₈	D₃₆	1	6	yes	—
150	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
151	C₂×C₃×C₄	—	2	3	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	3	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	#611	1	3	yes	—
156	C₂×C₃×C₄	#611	1	3	yes	—
157	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
158	C₂×C₃×C₄	#638	1	1	yes	characteristic
159	C₂×C₃×C₄	#638	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
161	C₂×Dic₃	—	18	6	no	—
162	C₂×Dic₃	—	18	2	no	—
163	Dic₆	—	18	6	no	—
164	Dic₆	—	18	6	no	—
165	Dic₆	—	18	2	no	—
166	Dic₆	—	18	2	no	—
167	C₃⋊C₈	—	18	6	no	—
168	C₃⋊C₈	—	18	2	no	—
169	C₂×C₃²	#930	1	1	yes	characteristic
170	C₂×C₃²	C₂×#216	1	1	yes	characteristic
171	C₂×C₃²	#951	1	1	yes	characteristic
172	C₂×C₉	#930	1	3	yes	—
173	C₂×C₉	C₂×#216	1	3	yes	—
174	C₂×C₉	#951	1	3	yes	—
175	C₂×Q₈	—	27	2	no	—
176	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
177	Q₁₆	—	54	4	no	—
178	C₂×C₈	—	27	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	54	1	no	—
181	C₄⋊C₄	—	54	1	no	—
182	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
183	C₄²	—	2	1	no	—
184	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
185	C₂²×C₃	C₂×#611	1	3	yes	—
186	C₂²×C₃	C₂×#638	1	1	yes	characteristic
187	C₃×C₄	—	2	6	no	—
188	C₃×C₄	—	2	3	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	3	no	—
192	C₃×C₄	—	4	1	no	—
193	C₃×C₄	#3674	1	6	yes	complemented
194	C₃×C₄	C₂×D₃₆	1	3	yes	—
195	C₃×C₄	#3697	1	3	yes	—
196	C₃×C₄	#3756	1	2	yes	—
197	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
198	C₃×C₄	#3828	1	1	yes	characteristic
199	Dic₃	—	36	6	no	—
200	Dic₃	—	36	2	no	—
201	C₃²	#5031	1	1	yes	characteristic
202	C₉	#5031	1	3	yes	complemented
203	Q₈	—	2	1	no	—
204	Q₈	—	54	2	no	—
205	Q₈	—	54	2	no	—
206	Q₈	C₄⋊(C₃⋊D₉)	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	54	2	no	—
210	C₂×C₄	#6233	1	1	yes	characteristic
211	C₂×C₄	#6233	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
213	C₈	—	54	2	no	—
214	C₂×C₃	#63492	1	3	yes	—
215	C₂×C₃	C₂×#3674	1	3	yes	—
216	C₂×C₃	#63511	1	3	yes	—
217	C₂×C₃	#64139	1	1	yes	characteristic
218	C₂×C₃	C₂×#3756	1	1	yes	characteristic
219	C₂×C₃	#64160	1	1	yes	characteristic
220	C₂²	C₂×#6233	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	108	2	no	—
225	C₄	#88547	1	2	yes	complemented
226	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
227	C₄	#88736	1	1	yes	characteristic
228	C₃	#10588707	1	3	yes	complemented
229	C₃	#10592730	1	1	yes	characteristic
230	C₂	#11721819	1	1	yes	characteristic
231	C₂	C₂×#88547	1	1	yes	characteristic
232	C₂	#11721866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#462
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
65. C₃×C₄×C₉	Q₁₆	semidirect
137. C₄×C₉	#216	semidirect
146. C₃×C₉	#462	semidirect
193. C₃×C₄	#3674	semidirect
202. C₉	#5031	semidirect
225. C₄	#88547	semidirect
228. C₃	#10588707	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₂×#88547	2
#11721706	2
#11720527	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592730	3
#10588707	3