Subgroups — C₃×#10588701 #13142765

Order 1728.

352 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₃×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₃×#63362	C₂	1	2	yes	maximal
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#5024	—	3	1	no	maximal
7	#10588701	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	2	yes	—
13	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	—	3	2	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
19	C₃×#803	—	3	2	no	—
20	C₃×C₄×C₃⋊C₈	—	3	1	no	—
21	C₄×Q₈×C₉	—	2	1	no	—
22	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
23	C₄⋊(C₉⋊C₈)	C₂×C₃	1	2	yes	—
24	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic
25	#63362	C₂×C₃	1	2	yes	—
26	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
27	C₃×Q₈×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₃×C₉⋊C₈	—	2	1	no	—
34	C₃×C₉⋊C₈	—	2	2	no	—
35	C₃×C₉⋊C₈	Q₈	1	2	yes	complemented
36	C₃×#446	—	9	1	no	—
37	#5024	—	3	1	no	—
38	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
39	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
40	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
41	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂×D₃	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	—	3	1	no	—
44	C₂×C₃×C₃⋊C₈	—	3	1	no	—
45	C₂×C₃×C₃⋊C₈	—	3	2	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	C₃×C₄	1	1	yes	characteristic
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	C₃×C₄	1	2	yes	—
51	C₉×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	—	2	2	no	—
54	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
55	C₄²×C₉	C₂²×C₃	1	2	yes	—
56	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
58	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	2	no	—
61	C₃×C₄×C₉	—	2	1	no	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	M₁₆	1	2	yes	—
64	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
65	C₃×C₄×C₉	#41	1	1	yes	characteristic
66	C₃×C₄×Q₈	—	2	1	no	—
67	C₃×C₄×Q₈	D₉	1	1	yes	characteristic
68	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	9	1	no	—
70	C₃×C₄⋊C₈	—	9	2	no	—
71	C₃×#97	—	9	2	no	—
72	C₃×C₄×C₈	—	9	1	no	—
73	C₄⋊(C₃⋊C₈)	—	3	2	no	—
74	C₄⋊(C₃⋊C₈)	—	3	1	no	—
75	#803	—	3	2	no	—
76	C₄×C₃⋊C₈	—	3	1	no	—
77	C₃²×Q₈	—	2	2	no	—
78	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
81	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₃×C₃⋊C₈	—	3	2	no	—
84	C₃×C₃⋊C₈	—	6	1	no	—
85	C₃×C₃⋊C₈	—	6	2	no	—
86	Q₈×C₉	—	2	4	no	—
87	Q₈×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
98	C₉⋊C₈	—	2	1	no	—
99	C₉⋊C₈	—	2	2	no	—
100	C₉⋊C₈	C₃×Q₈	1	2	yes	complemented
101	#446	—	9	1	no	Sylow
102	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₃×C₉	#131	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	2	1	no	—
106	C₂×C₃×Q₈	Dic₉	1	1	yes	characteristic
107	C₂×C₃×Q₈	C₃×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	—	9	1	no	—
109	C₂×C₃×C₈	—	9	1	no	—
110	C₂×C₃×C₈	—	9	2	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	Dic₉	1	2	yes	—
114	C₃×C₄⋊C₄	Dic₉	1	1	yes	characteristic
115	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
116	C₃×C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
120	C₃×C₄²	C₂×D₉	1	2	yes	—
121	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
122	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
123	C₂×C₃⋊C₈	—	3	1	no	—
124	C₂×C₃⋊C₈	—	3	1	no	—
125	C₂×C₃⋊C₈	—	3	2	no	—
126	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
127	C₃²×C₄	—	2	2	no	—
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	#208	1	2	yes	—
131	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
132	C₃²×C₄	#239	1	1	yes	characteristic
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₄×C₉	—	2	4	no	—
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	4	1	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	C₃×M₁₆	1	2	yes	—
145	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₄×C₉	C₃×#41	1	1	yes	characteristic
147	C₄×Q₈	C₃×D₉	1	1	yes	characteristic
148	C₄⋊C₈	—	9	1	no	—
149	C₄⋊C₈	—	9	2	no	—
150	#97	—	9	2	no	—
151	C₄×C₈	—	9	1	no	—
152	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
153	C₃×Q₈	—	2	4	no	—
154	C₃×Q₈	—	2	2	no	—
155	C₃×Q₈	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
163	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
164	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₃×C₈	—	9	2	no	—
172	C₃×C₈	—	18	1	no	—
173	C₃×C₈	—	18	2	no	—
174	C₃⋊C₈	—	3	2	no	—
175	C₃⋊C₈	—	6	1	no	—
176	C₃⋊C₈	—	6	2	no	—
177	C₂×C₃²	C₂×#208	1	1	yes	characteristic
178	C₂×C₃²	Q₈×Dic₃	1	1	yes	characteristic
179	C₂×C₃²	#948	1	1	yes	characteristic
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	2	1	no	—
182	C₂×C₉	—	2	1	no	—
183	C₂×C₉	C₃×C₄×Q₈	1	1	yes	characteristic
184	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
185	C₂×C₉	C₃×#131	1	1	yes	characteristic
186	C₂×Q₈	C₃×Dic₉	1	1	yes	characteristic
187	C₂×C₈	—	9	1	no	—
188	C₂×C₈	—	9	1	no	—
189	C₂×C₈	—	9	2	no	—
190	C₄⋊C₄	C₃×Dic₉	1	2	yes	—
191	C₄⋊C₄	C₃×Dic₉	1	1	yes	characteristic
192	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
193	C₄²	C₂×C₃×D₉	1	2	yes	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
196	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	#3667	1	2	yes	—
210	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
211	C₃×C₄	#3701	1	1	yes	characteristic
212	C₃×C₄	C₃×#208	1	2	yes	—
213	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
214	C₃×C₄	C₃×#239	1	1	yes	characteristic
215	C₃²	#5024	1	1	yes	characteristic
216	C₉	—	2	1	no	—
217	C₉	C₃×#446	1	1	yes	characteristic, complemented
218	Q₈	—	2	2	no	—
219	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
221	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
222	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
224	C₈	—	9	2	no	—
225	C₈	—	18	1	no	—
226	C₈	—	18	2	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	C₂×#3667	1	1	yes	characteristic
231	C₂×C₃	Q₈×Dic₉	1	1	yes	characteristic
232	C₂×C₃	#63510	1	1	yes	characteristic
233	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
234	C₂×C₃	C₃×Q₈×Dic₃	1	1	yes	characteristic
235	C₂×C₃	C₃×#948	1	1	yes	characteristic
236	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	C₃×#3667	1	2	yes	—
241	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
242	C₄	C₃×#3701	1	1	yes	characteristic
243	C₃	—	2	1	no	—
244	C₃	#10588701	1	1	yes	characteristic, direct product factor
245	C₃	C₃×#5024	1	1	yes	characteristic
246	C₂	C₂×C₃×#3667	1	1	yes	characteristic
247	C₂	C₃×Q₈×Dic₉	1	1	yes	characteristic
248	C₂	C₃×#63510	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂²×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×#446
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
7. #10588701	C₃	direct (N × H)
35. C₃×C₉⋊C₈	Q₈	semidirect
100. C₉⋊C₈	C₃×Q₈	semidirect
152. C₃×C₉	#446	semidirect
217. C₉	C₃×#446	semidirect
244. C₃	#10588701	direct (N × H)

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
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#63362	2
C₃×C₄×C₉⋊C₈	2
C₃×#5024	3
#10588701	3

Subgroups — C3×#10588701 #13142765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588701 #13142765