Subgroups — #13139296

Order 1728.

466 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⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₉×#926	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720258	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₉⋊C₈	C₂	1	2	yes	maximal, complemented
6	#10591057	—	3	1	no	maximal
7	#10588675	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
10	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₉×#217	C₄	1	2	yes	—
13	C₉×C₄⋊Dic₃	C₄	1	2	yes	—
14	C₄×C₉×Dic₃	C₂²	1	2	yes	—
15	C₃×#3667	C₄	1	4	yes	complemented
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₃×#926	D₃	1	1	yes	characteristic
19	C₂×C₃×#208	—	3	1	no	—
20	#63554	—	3	2	no	—
21	Dic₃×C₃⋊C₈	—	3	2	no	—
22	C₉×#117	—	3	1	no	—
23	C₂×#3667	D₃	1	1	yes	characteristic
24	C₂²×C₉⋊C₈	—	3	1	no	—
25	C₂²×C₉⋊C₈	—	6	1	no	—
26	#63362	—	3	2	no	—
27	C₄×C₉⋊C₈	—	3	2	no	—
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
34	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
35	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
36	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	—
37	#5122	—	9	1	no	—
38	#4997	—	9	1	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
43	C₃×#217	Dic₃	1	2	yes	—
44	C₃×C₄⋊Dic₃	Dic₃	1	2	yes	—
45	C₃×C₄×Dic₃	C₂×D₃	1	2	yes	—
46	C₃×#208	—	3	4	no	—
47	C₂×C₃×C₃⋊C₈	—	3	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
50	C₉×C₄⋊C₄	—	3	2	no	—
51	C₉×C₂²⋊C₄	—	3	2	no	—
52	C₄²×C₉	—	3	2	no	—
53	#3667	Dic₃	1	4	yes	complemented
54	C₂×C₉⋊C₈	—	3	4	no	—
55	C₂×C₉⋊C₈	—	3	1	no	—
56	C₂×C₉⋊C₈	—	3	1	no	—
57	C₂×C₉⋊C₈	—	6	4	no	—
58	C₂×C₉⋊C₈	—	6	1	no	—
59	C₂×C₉⋊C₈	—	6	1	no	—
60	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	C₄²	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₄²	1	2	yes	—
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₉×Dic₃	—	2	4	no	—
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₃×#117	—	3	1	no	—
70	#926	D₉	1	1	yes	characteristic
71	C₂×#208	—	3	1	no	—
72	C₂²×C₃⋊C₈	—	9	1	no	—
73	C₂²×C₃⋊C₈	—	9	1	no	—
74	C₂²×C₃⋊C₈	—	18	1	no	—
75	#815	—	9	2	no	—
76	C₈×Dic₃	—	9	2	no	—
77	#803	—	9	2	no	—
78	C₄×C₃⋊C₈	—	9	2	no	—
79	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
81	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
82	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂×Dic₃	1	4	yes	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
87	C₃⋊(C₃⋊C₈)	—	3	2	no	—
88	C₃⋊(C₃⋊C₈)	—	3	2	no	—
89	C₃×C₃⋊C₈	—	3	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	3	4	no	—
95	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
96	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
97	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₉⋊C₈	—	3	2	no	—
100	C₉⋊C₈	—	3	2	no	—
101	C₉⋊C₈	—	6	2	no	—
102	C₉⋊C₈	—	6	2	no	—
103	C₉⋊C₈	C₂×Dic₃	1	4	yes	—
104	#405	—	27	1	no	Sylow
105	C₂×C₃×C₉	—	2	2	no	—
106	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₃×C₉	#131	1	2	yes	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
111	C₃×M₁₆	—	9	4	no	—
112	C₂×C₃×C₈	—	9	2	no	—
113	C₃×C₄⋊C₄	—	3	2	no	—
114	C₃×C₂²⋊C₄	—	3	2	no	—
115	C₃×C₄²	—	3	2	no	—
116	#217	Dic₉	1	2	yes	—
117	C₄⋊Dic₃	Dic₉	1	2	yes	—
118	C₄×Dic₃	C₂×D₉	1	2	yes	—
119	#208	—	3	4	no	—
120	C₂×C₃⋊C₈	—	3	2	no	—
121	C₂×C₃⋊C₈	—	9	4	no	—
122	C₂×C₃⋊C₈	—	9	1	no	—
123	C₂×C₃⋊C₈	—	9	1	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	4	no	—
128	C₂×C₃⋊C₈	—	18	1	no	—
129	C₂×C₃⋊C₈	—	18	1	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
132	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
133	C₃²×C₄	C₄×Dic₃	1	2	yes	—
134	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₃×Dic₃	—	2	4	no	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₄×Dic₃	1	2	yes	—
142	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	6	4	no	—
147	C₄×C₉	C₄×Dic₃	1	2	yes	—
148	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
149	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
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	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
158	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	3	4	no	—
164	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
165	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
166	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
168	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
169	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	#624	1	2	yes	—
171	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
172	C₂×Dic₃	C₂×Dic₉	1	4	yes	—
173	C₃×C₈	—	9	4	no	—
174	C₃⋊C₈	—	3	4	no	—
175	C₃⋊C₈	—	9	2	no	—
176	C₃⋊C₈	—	9	2	no	—
177	C₃⋊C₈	—	9	2	no	—
178	C₃⋊C₈	—	9	2	no	—
179	C₃⋊C₈	—	18	2	no	—
180	C₃⋊C₈	—	18	2	no	—
181	C₂×C₃²	—	2	2	no	—
182	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
183	C₂×C₃²	#948	1	2	yes	—
184	C₂×C₉	—	2	4	no	—
185	C₂×C₉	—	2	2	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	#907	1	2	yes	—
189	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
190	C₂²×C₄	D₃×D₉	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	4	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	C₄×Dic₉	1	2	yes	—
205	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
206	C₂²×C₃	Dic₃²	1	2	yes	—
207	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	6	4	no	—
212	C₃×C₄	C₄×Dic₉	1	2	yes	—
213	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
214	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
215	C₃×C₄	Dic₃²	1	2	yes	—
216	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
217	C₃×C₄	C₂×#624	1	1	yes	characteristic
218	Dic₃	—	2	4	no	—
219	C₃²	#4997	1	1	yes	characteristic
220	C₉	—	2	1	no	—
221	C₉	#5122	1	1	yes	characteristic, complemented
222	C₂³	D₃×Dic₉	1	1	yes	characteristic
223	C₂×C₄	—	3	4	no	—
224	C₂×C₄	Dic₃×D₉	1	2	yes	—
225	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
226	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
227	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
228	C₈	—	9	4	no	—
229	C₈	—	27	2	no	—
230	C₈	—	27	2	no	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
237	C₂×C₃	#63510	1	2	yes	—
238	C₂×C₃	#63815	1	2	yes	—
239	C₂×C₃	C₂×Dic₃²	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	Dic₃×Dic₉	1	2	yes	—
242	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
243	C₄	—	6	4	no	—
244	C₄	Dic₃×Dic₉	1	2	yes	—
245	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
246	C₄	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	#10588675	1	1	yes	characteristic, complemented
249	C₃	#10591057	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	#11720878	1	2	yes	—
252	C₂	C₂×Dic₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#405
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720258	C₂	semidirect
5. Dic₃×C₉⋊C₈	C₂	semidirect
9. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
15. C₃×#3667	C₄	semidirect
53. #3667	Dic₃	semidirect
155. C₃×C₉	#405	semidirect
221. C₉	#5122	semidirect
248. C₃	#10588675	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₉⋊C₈)	2
C₉×#926	2
C₂×C₃×#3667	2
#11720258	2
Dic₃×C₉⋊C₈	2
#10591057	3
#10588675	3