Subgroups — #13144406

Order 1728.

600 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88543	C₂	1	2	yes	maximal
3	#11721706	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720527	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592718	—	3	1	no	maximal
7	#10588695	—	3	3	no	maximal
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×#6226	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	#88543	—	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×D₄	D₃	1	1	yes	characteristic
19	C₂×#3754	—	3	2	no	—
20	#64078	—	3	1	no	—
21	#63637	—	3	2	no	—
22	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
23	C₄×D₄×C₉	D₃	1	3	yes	—
24	C₂×#3672	—	3	6	no	—
25	#63431	—	3	3	no	—
26	#63368	—	3	6	no	—
27	C₄⋊(C₉⋊C₈)	—	3	3	no	—
28	C₂³×C₃×C₉	—	2	1	no	—
29	C₃×D₄×C₉	—	2	1	no	—
30	C₃×D₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	4	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×C₃⋊Dic₉	—	2	2	no	—
37	#6226	—	2	2	no	—
38	#6226	—	2	2	no	—
39	C₃⋊(C₉⋊C₈)	—	2	2	no	—
40	#5019	—	9	3	no	—
41	#5019	—	9	1	no	—
42	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	—	2	1	no	—
44	C₂×#634	—	3	2	no	—
45	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
46	C₃²×C₂²⋊C₄	—	2	1	no	—
47	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
48	#3754	—	6	4	no	—
49	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
50	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
52	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
53	C₂²×C₄×C₉	—	2	3	no	—
54	C₂×Dic₁₈	—	3	6	no	—
55	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
56	C₉×C₂²⋊C₄	—	2	3	no	—
57	C₄²×C₉	C₂×D₃	1	3	yes	—
58	#3672	—	6	12	no	—
59	C₄⋊Dic₉	—	6	3	no	—
60	C₄⋊Dic₉	—	6	3	no	—
61	C₂×C₉⋊C₈	—	3	6	no	—
62	C₂²×C₃×C₉	—	2	2	no	—
63	C₂²×C₃×C₉	—	4	1	no	—
64	C₂²×C₃×C₉	—	4	1	no	—
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	—	4	1	no	—
68	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₉	#41	1	1	yes	characteristic
71	C₃⋊Dic₉	—	4	2	no	—
72	C₃×C₄×D₄	D₉	1	3	yes	—
73	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
74	C₂×#214	—	9	6	no	—
75	C₂×#214	—	9	2	no	—
76	#869	—	9	3	no	—
77	#869	—	9	1	no	—
78	#810	—	9	6	no	—
79	#810	—	9	2	no	—
80	C₄⋊(C₃⋊C₈)	—	9	3	no	—
81	C₄⋊(C₃⋊C₈)	—	9	1	no	—
82	C₂³×C₃²	—	2	1	no	—
83	C₃²×D₄	—	2	1	no	—
84	C₃²×D₄	D₁₂	1	2	yes	—
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	—	4	2	no	—
87	C₂×C₃²×C₄	#67	1	1	yes	characteristic
88	C₂×C₃²×C₄	#67	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₃⋊Dic₃	—	6	2	no	—
91	#634	—	6	2	no	—
92	#634	—	6	2	no	—
93	C₂³×C₉	—	2	3	no	—
94	C₃⋊(C₃⋊C₈)	—	6	2	no	—
95	D₄×C₉	—	2	3	no	—
96	D₄×C₉	D₁₂	1	6	yes	—
97	C₂×C₄×C₉	—	2	3	no	—
98	C₂×C₄×C₉	—	4	6	no	—
99	C₂×C₄×C₉	#67	1	3	yes	—
100	C₂×C₄×C₉	#67	1	3	yes	—
101	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
102	C₂×Dic₉	—	6	6	no	—
103	Dic₁₈	—	6	6	no	—
104	Dic₁₈	—	6	6	no	—
105	C₉⋊C₈	—	6	6	no	—
106	#461	—	27	1	no	Sylow
107	C₂×C₃×C₉	—	4	2	no	—
108	C₂×C₃×C₉	#121	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₃×C₉	#137	1	1	yes	characteristic
111	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
112	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	3	no	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂×Dic₆	—	9	6	no	—
116	C₂×Dic₆	—	9	2	no	—
117	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
118	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
119	C₃×C₂²⋊C₄	—	2	3	no	—
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₄²	C₂×D₉	1	3	yes	—
122	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
123	#214	—	18	12	no	—
124	#214	—	18	4	no	—
125	C₄⋊Dic₃	—	18	3	no	—
126	C₄⋊Dic₃	—	18	3	no	—
127	C₄⋊Dic₃	—	18	1	no	—
128	C₄⋊Dic₃	—	18	1	no	—
129	C₂×C₃⋊C₈	—	9	6	no	—
130	C₂×C₃⋊C₈	—	9	2	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	4	1	no	—
133	C₂²×C₃²	—	4	1	no	—
134	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
135	C₃²×C₄	—	2	1	no	—
136	C₃²×C₄	—	4	1	no	—
137	C₃²×C₄	#214	1	2	yes	—
138	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₃²×C₄	#235	1	1	yes	characteristic
140	C₃⋊Dic₃	—	12	2	no	—
141	C₂²×C₉	—	2	6	no	—
142	C₂²×C₉	—	4	3	no	—
143	C₂²×C₉	—	4	3	no	—
144	C₂²×C₉	C₂×#67	1	3	yes	—
145	C₄×C₉	—	2	3	no	—
146	C₄×C₉	—	4	3	no	—
147	C₄×C₉	#214	1	6	yes	complemented
148	C₄×C₉	C₂×D₁₂	1	3	yes	—
149	C₄×C₉	#235	1	3	yes	—
150	Dic₉	—	12	6	no	—
151	C₂×SD₁₆	—	27	2	no	—
152	C₄⋊Q₈	—	27	1	no	—
153	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic
154	C₄⋊C₈	—	27	1	no	—
155	#103	—	27	2	no	—
156	C₃×C₉	#461	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	3	no	—
158	C₂³×C₃	—	2	1	no	—
159	C₃×D₄	—	2	3	no	—
160	C₃×D₄	—	2	1	no	—
161	C₃×D₄	D₃₆	1	6	yes	—
162	C₃×D₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
163	C₂×C₃×C₄	—	2	3	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	4	6	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	#611	1	3	yes	—
168	C₂×C₃×C₄	#611	1	3	yes	—
169	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
170	C₂×C₃×C₄	#638	1	1	yes	characteristic
171	C₂×C₃×C₄	#638	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
173	C₂×Dic₃	—	18	6	no	—
174	C₂×Dic₃	—	18	2	no	—
175	Dic₆	—	18	6	no	—
176	Dic₆	—	18	6	no	—
177	Dic₆	—	18	2	no	—
178	Dic₆	—	18	2	no	—
179	C₃⋊C₈	—	18	6	no	—
180	C₃⋊C₈	—	18	2	no	—
181	C₂×C₃²	—	4	2	no	—
182	C₂×C₃²	#930	1	1	yes	characteristic
183	C₂×C₃²	C₂×#214	1	1	yes	characteristic
184	C₂×C₃²	#951	1	1	yes	characteristic
185	C₂×C₉	—	4	6	no	—
186	C₂×C₉	#930	1	3	yes	—
187	C₂×C₉	C₂×#214	1	3	yes	—
188	C₂×C₉	#951	1	3	yes	—
189	C₂×Q₈	—	27	2	no	—
190	C₂×D₄	C₂×C₃⋊D₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	SD₁₆	—	54	4	no	—
193	C₂×C₈	—	27	2	no	—
194	C₄⋊C₄	—	54	1	no	—
195	C₄⋊C₄	—	54	1	no	—
196	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	1	no	—
198	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
199	C₂²×C₃	—	2	6	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	4	3	no	—
202	C₂²×C₃	—	4	3	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	C₂×#611	1	3	yes	—
206	C₂²×C₃	C₂×#638	1	1	yes	characteristic
207	C₃×C₄	—	2	3	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	4	3	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	#3672	1	6	yes	complemented
212	C₃×C₄	C₂×D₃₆	1	3	yes	—
213	C₃×C₄	#3697	1	3	yes	—
214	C₃×C₄	#3754	1	2	yes	—
215	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
216	C₃×C₄	#3828	1	1	yes	characteristic
217	Dic₃	—	36	6	no	—
218	Dic₃	—	36	2	no	—
219	C₃²	#5019	1	1	yes	characteristic
220	C₉	#5019	1	3	yes	complemented
221	C₂³	—	2	1	no	—
222	Q₈	—	54	2	no	—
223	Q₈	—	54	2	no	—
224	D₄	—	2	1	no	—
225	D₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	54	2	no	—
229	C₂×C₄	#6233	1	1	yes	characteristic
230	C₂×C₄	#6233	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
232	C₈	—	54	2	no	—
233	C₂×C₃	—	4	6	no	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	#63492	1	3	yes	—
236	C₂×C₃	C₂×#3672	1	3	yes	—
237	C₂×C₃	#63511	1	3	yes	—
238	C₂×C₃	#64139	1	1	yes	characteristic
239	C₂×C₃	C₂×#3754	1	1	yes	characteristic
240	C₂×C₃	#64160	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₂×#6233	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	108	2	no	—
248	C₄	#88543	1	2	yes	complemented
249	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
250	C₄	#88736	1	1	yes	characteristic
251	C₃	#10588695	1	3	yes	complemented
252	C₃	#10592718	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	#11721819	1	1	yes	characteristic
255	C₂	C₂×#88543	1	1	yes	characteristic
256	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×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	#461
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
3. #11721706	C₂	semidirect
4. #11720527	C₂	semidirect
5. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
68. C₃×C₄×C₉	SD₁₆	semidirect
147. C₄×C₉	#214	semidirect
156. C₃×C₉	#461	semidirect
211. C₃×C₄	#3672	semidirect
220. C₉	#5019	semidirect
248. C₄	#88543	semidirect
251. C₃	#10588695	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₂×#88543	2
#11721706	2
#11720527	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592718	3
#10588695	3