Subgroups — #13137024

Order 1728.

326 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
5	#11720258	C₂	1	2	yes	maximal
6	#10590259	—	3	1	no	maximal
7	#10588701	—	3	1	no	maximal
8	C₂×C₉×Dic₆	C₄	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₉×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₄×C₉×Dic₃	C₂²	1	2	yes	—
15	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
16	C₃×C₄×Dic₆	D₃	1	1	yes	characteristic
17	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
19	Dic₃⋊(C₃⋊C₈)	—	3	2	no	—
20	#63554	—	3	2	no	—
21	C₄×Q₈×C₉	—	3	1	no	—
22	C₄⋊(C₉⋊C₈)	—	3	2	no	—
23	C₄⋊(C₉⋊C₈)	D₃	1	1	yes	characteristic
24	#63362	—	3	2	no	—
25	C₄×C₉⋊C₈	—	3	1	no	—
26	C₄×C₉⋊C₈	—	6	1	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
32	C₉×Dic₆	—	2	2	no	—
33	C₃⋊(C₉⋊C₈)	—	2	1	no	—
34	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	complemented
35	C₃×C₉⋊C₈	—	2	2	no	—
36	#5024	—	9	1	no	—
37	#4831	—	9	1	no	—
38	C₂×C₃×Dic₆	Dic₃	1	1	yes	characteristic
39	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
42	C₃×C₄⋊Dic₃	Dic₃	1	1	yes	characteristic
43	C₃×Dic₃⋊C₄	Dic₃	1	2	yes	—
44	C₃×C₄×Dic₃	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	—	3	2	no	—
46	C₂×Q₈×C₉	—	3	1	no	—
47	C₉×C₄⋊C₄	—	3	2	no	—
48	C₉×C₄⋊C₄	—	3	1	no	—
49	C₄²×C₉	—	2	1	no	—
50	C₄²×C₉	—	3	2	no	—
51	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂×C₉⋊C₈	—	3	1	no	—
53	C₂×C₉⋊C₈	—	3	1	no	—
54	C₂×C₉⋊C₈	—	6	1	no	—
55	C₂×C₉⋊C₈	—	6	1	no	—
56	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	2	1	no	—
59	C₃×C₄×C₉	M₁₆	1	2	yes	—
60	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₄×C₉	#41	1	1	yes	characteristic
62	C₉×Dic₃	—	2	2	no	—
63	C₉×Dic₃	—	2	2	no	—
64	C₃×C₄×Q₈	—	3	1	no	—
65	C₄×Dic₆	D₉	1	1	yes	characteristic
66	C₃×C₄⋊C₈	—	9	1	no	—
67	#815	—	9	2	no	—
68	Dic₃⋊C₈	—	9	2	no	—
69	C₄⋊(C₃⋊C₈)	—	3	1	no	—
70	C₄⋊(C₃⋊C₈)	—	9	2	no	—
71	#803	—	9	2	no	—
72	C₄×C₃⋊C₈	—	9	1	no	—
73	C₄×C₃⋊C₈	—	9	1	no	—
74	C₄×C₃⋊C₈	—	18	1	no	—
75	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
78	C₂×C₃×Dic₃	C₂×Dic₃	1	2	yes	—
79	C₂×C₃×Dic₃	C₂×Dic₃	1	2	yes	—
80	C₃×Dic₆	—	2	2	no	—
81	C₃⋊(C₃⋊C₈)	—	3	2	no	—
82	C₃⋊(C₃⋊C₈)	—	6	1	no	—
83	C₃×C₃⋊C₈	—	6	2	no	—
84	Q₈×C₉	—	6	2	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	3	2	no	—
89	C₂×C₄×C₉	—	3	2	no	—
90	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₉⋊C₈	—	2	2	no	—
94	C₉⋊C₈	—	3	2	no	—
95	C₉⋊C₈	—	6	2	no	—
96	C₉⋊C₈	—	6	2	no	—
97	C₉⋊C₈	—	6	1	no	—
98	#446	—	27	1	no	Sylow
99	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₃×C₉	#131	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	3	1	no	—
103	C₂×Dic₆	Dic₉	1	1	yes	characteristic
104	C₂×C₃×C₈	—	9	2	no	—
105	C₃×C₄⋊C₄	—	3	2	no	—
106	C₃×C₄⋊C₄	—	3	1	no	—
107	C₃×C₄²	—	2	1	no	—
108	C₃×C₄²	—	3	2	no	—
109	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
110	C₃×C₄²	D₃²	1	1	yes	characteristic
111	C₄⋊Dic₃	Dic₉	1	1	yes	characteristic
112	Dic₃⋊C₄	Dic₉	1	2	yes	—
113	C₄×Dic₃	C₂×D₉	1	2	yes	—
114	C₂×C₃⋊C₈	—	3	2	no	—
115	C₂×C₃⋊C₈	—	9	1	no	—
116	C₂×C₃⋊C₈	—	9	1	no	—
117	C₂×C₃⋊C₈	—	9	1	no	—
118	C₂×C₃⋊C₈	—	9	1	no	—
119	C₂×C₃⋊C₈	—	18	1	no	—
120	C₂×C₃⋊C₈	—	18	1	no	—
121	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	#208	1	2	yes	—
124	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
125	C₃²×C₄	#239	1	1	yes	characteristic
126	C₃×Dic₃	—	2	2	no	—
127	C₃×Dic₃	—	2	2	no	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	1	no	—
134	C₄×C₉	—	2	1	no	—
135	C₄×C₉	—	6	2	no	—
136	C₄×C₉	—	6	2	no	—
137	C₄×C₉	ΓL(1,𝔽₂₅)	1	2	yes	—
138	C₄×C₉	#237	1	1	yes	characteristic
139	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
140	C₄×Q₈	—	3	1	no	—
141	C₄⋊C₈	—	9	1	no	—
142	C₄⋊C₈	—	27	2	no	—
143	#97	—	27	2	no	—
144	C₄×C₈	—	27	1	no	—
145	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
146	C₃×Q₈	—	6	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	3	2	no	—
151	C₂×C₃×C₄	—	3	2	no	—
152	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
155	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
158	C₂×Dic₃	C₂×Dic₉	1	2	yes	—
159	C₂×Dic₃	C₂×Dic₉	1	2	yes	—
160	Dic₆	—	2	2	no	—
161	C₃×C₈	—	18	2	no	—
162	C₃⋊C₈	—	6	2	no	—
163	C₃⋊C₈	—	9	2	no	—
164	C₃⋊C₈	—	9	2	no	—
165	C₃⋊C₈	—	18	2	no	—
166	C₃⋊C₈	—	18	2	no	—
167	C₃⋊C₈	—	18	1	no	—
168	C₃⋊C₈	—	18	1	no	—
169	C₂×C₃²	C₂×#208	1	1	yes	characteristic
170	C₂×C₃²	Q₈×Dic₃	1	1	yes	characteristic
171	C₂×C₃²	#948	1	1	yes	characteristic
172	C₂×C₉	—	2	1	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	—	2	1	no	—
175	C₂×C₉	#887	1	1	yes	characteristic
176	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
177	C₂×C₉	#907	1	1	yes	characteristic
178	C₂×Q₈	—	3	1	no	—
179	C₂×C₈	—	9	2	no	—
180	C₂×C₈	—	27	1	no	—
181	C₂×C₈	—	27	1	no	—
182	C₄⋊C₄	—	3	2	no	—
183	C₄⋊C₄	—	3	1	no	—
184	C₄²	—	3	2	no	—
185	C₄²	D₃×D₉	1	1	yes	characteristic
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
188	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	6	2	no	—
196	C₃×C₄	—	6	2	no	—
197	C₃×C₄	#3667	1	2	yes	—
198	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
199	C₃×C₄	#3701	1	1	yes	characteristic
200	C₃×C₄	#3711	1	2	yes	—
201	C₃×C₄	#3796	1	1	yes	characteristic
202	C₃×C₄	#3797	1	1	yes	characteristic
203	Dic₃	—	2	2	no	—
204	Dic₃	—	2	2	no	—
205	C₃²	#5024	1	1	yes	characteristic
206	C₉	—	2	1	no	—
207	C₉	#4831	1	1	yes	characteristic, complemented
208	Q₈	—	6	2	no	—
209	C₂×C₄	—	3	2	no	—
210	C₂×C₄	—	3	2	no	—
211	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
212	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
213	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
214	C₈	—	18	2	no	—
215	C₈	—	27	2	no	—
216	C₈	—	54	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	C₂×#3667	1	1	yes	characteristic
221	C₂×C₃	Q₈×Dic₉	1	1	yes	characteristic
222	C₂×C₃	#63510	1	1	yes	characteristic
223	C₂×C₃	#63815	1	1	yes	characteristic
224	C₂×C₃	C₂×#3711	1	1	yes	characteristic
225	C₂×C₃	#63843	1	1	yes	characteristic
226	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	6	2	no	—
229	C₄	—	6	2	no	—
230	C₄	#88420	1	2	yes	—
231	C₄	#88634	1	1	yes	characteristic
232	C₄	(C₃⋊D₉)⋊Q₈	1	1	yes	characteristic
233	C₃	—	2	1	no	—
234	C₃	#10588701	1	1	yes	characteristic, complemented
235	C₃	#10590259	1	1	yes	characteristic
236	C₂	#11720878	1	1	yes	characteristic
237	C₂	C₂×#88420	1	1	yes	characteristic
238	C₂	#11720911	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle, prime radical	C₂²×C₃²	C₂²×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#446
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
34. C₃⋊(C₉⋊C₈)	Q₈	semidirect
145. C₃×C₉	#446	semidirect
207. C₉	#4831	semidirect
234. C₃	#10588701	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×Dic₆	2
C₄×C₃⋊(C₉⋊C₈)	2
C₃×C₄⋊(C₉⋊C₈)	2
Dic₃⋊(C₉⋊C₈)	2
#11720258	2
#10590259	3
#10588701	3