Subgroups — C₃×#10588515 #13142000

Order 1728.

326 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	C₃×#63431	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#63377	C₂	1	2	yes	maximal
5	C₃×#63376	C₂	1	2	yes	maximal
6	C₃×#4851	—	3	1	no	maximal
7	#10588515	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Dic₁₈	—	2	1	no	—
9	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊Dic₉	—	2	1	no	—
13	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₃×Dic₉⋊C₄	—	2	1	no	—
16	C₃×C₄×Dic₉	—	2	1	no	—
17	C₃×#869	—	3	1	no	—
18	C₃×C₄×Dic₆	—	3	1	no	—
19	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₃×#817	—	3	2	no	—
21	C₃×#816	—	3	2	no	—
22	#63431	C₂×C₃	1	1	yes	characteristic
23	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic
24	C₉×C₄⋊C₈	—	2	1	no	—
25	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
26	#63377	C₂×C₃	1	2	yes	—
27	#63376	C₂×C₃	1	2	yes	—
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃×Dic₉	—	2	1	no	—
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₂×C₃×Dic₉	—	2	1	no	—
34	C₃×Dic₁₈	—	2	1	no	—
35	C₃×Dic₁₈	—	4	2	no	—
36	C₃×Dic₁₈	Q₈	1	2	yes	complemented
37	C₃×C₈×C₉	—	2	2	no	—
38	C₃×#475	—	9	1	no	—
39	#4851	—	3	1	no	—
40	C₂×C₃×Dic₆	—	3	1	no	—
41	C₂×C₃×Dic₆	—	6	1	no	—
42	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
43	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₃×C₄⋊Dic₃	—	3	2	no	—
45	C₃×C₄⋊Dic₃	—	3	1	no	—
46	C₃×C₄⋊Dic₃	—	6	1	no	—
47	C₃×Dic₃⋊C₄	—	6	1	no	—
48	C₃×C₄×Dic₃	—	6	1	no	—
49	C₂×Dic₁₈	—	2	1	no	—
50	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₈×C₉	—	2	2	no	—
52	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
55	C₄⋊Dic₉	—	2	1	no	—
56	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
57	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
58	Dic₉⋊C₄	—	2	1	no	—
59	C₄×Dic₉	—	2	1	no	—
60	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
61	C₃×C₄×C₉	—	2	1	no	—
62	C₃×C₄×C₉	SD₁₆	1	2	yes	—
63	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
64	C₃×C₄×C₉	#41	1	1	yes	characteristic
65	C₃×Dic₉	—	2	2	no	—
66	C₃×Dic₉	—	4	2	no	—
67	C₃×Dic₉	—	4	1	no	—
68	C₃×C₄⋊Q₈	—	9	1	no	—
69	C₃×C₄×Q₈	—	9	1	no	—
70	#869	—	3	1	no	—
71	C₄×Dic₆	—	3	1	no	—
72	C₃×#106	—	9	2	no	—
73	C₃×C₄⋊C₈	—	2	1	no	—
74	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
76	C₃×#103	—	9	2	no	—
77	#817	—	3	2	no	—
78	#816	—	3	2	no	—
79	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
80	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	6	1	no	—
83	C₂×C₃×Dic₃	—	6	1	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₃×Dic₆	—	3	2	no	—
86	C₃×Dic₆	—	6	1	no	—
87	C₃×Dic₆	—	12	2	no	—
88	C₃²×C₈	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
95	C₂×Dic₉	—	2	1	no	—
96	C₂×Dic₉	—	2	1	no	—
97	C₂×Dic₉	—	2	2	no	—
98	Dic₁₈	—	2	1	no	—
99	Dic₁₈	—	4	2	no	—
100	Dic₁₈	C₃×Q₈	1	2	yes	complemented
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	—	4	2	no	—
103	#475	—	9	1	no	Sylow
104	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₃×C₉	#137	1	1	yes	characteristic
107	C₂×C₃×Q₈	—	9	1	no	—
108	C₂×C₃×Q₈	—	18	1	no	—
109	C₂×Dic₆	—	3	1	no	—
110	C₂×Dic₆	—	6	1	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
113	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
114	C₃×C₄⋊C₄	—	9	2	no	—
115	C₃×C₄⋊C₄	—	9	1	no	—
116	C₃×C₄⋊C₄	—	18	1	no	—
117	C₃×C₄⋊C₄	—	18	1	no	—
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	—	18	1	no	—
120	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
121	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
122	C₄⋊Dic₃	—	3	2	no	—
123	C₄⋊Dic₃	—	3	1	no	—
124	C₄⋊Dic₃	—	6	1	no	—
125	Dic₃⋊C₄	—	6	1	no	—
126	C₄×Dic₃	—	6	1	no	—
127	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	#204	1	2	yes	—
130	C₃²×C₄	#237	1	1	yes	characteristic
131	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
132	C₃×Dic₃	—	6	2	no	—
133	C₃×Dic₃	—	12	2	no	—
134	C₃×Dic₃	—	12	1	no	—
135	C₂²×C₉	—	2	1	no	—
136	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	1	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	4	1	no	—
142	C₄×C₉	C₃×SD₁₆	1	2	yes	—
143	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₄×C₉	C₃×#41	1	1	yes	characteristic
145	Dic₉	—	2	2	no	—
146	Dic₉	—	4	2	no	—
147	Dic₉	—	4	1	no	—
148	C₄⋊Q₈	—	9	1	no	—
149	C₄×Q₈	—	9	1	no	—
150	#106	—	9	2	no	—
151	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic
152	#103	—	9	2	no	—
153	C₃×C₉	#475	1	1	yes	characteristic, complemented, Sylow
154	C₃×Q₈	—	9	2	no	—
155	C₃×Q₈	—	18	1	no	—
156	C₃×Q₈	—	36	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	18	1	no	—
161	C₂×C₃×C₄	—	18	1	no	—
162	C₂×C₃×C₄	—	18	2	no	—
163	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
164	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
169	C₂×Dic₃	—	6	1	no	—
170	C₂×Dic₃	—	6	1	no	—
171	C₂×Dic₃	—	6	2	no	—
172	Dic₆	—	3	2	no	—
173	Dic₆	—	6	1	no	—
174	Dic₆	—	12	2	no	—
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	2	2	no	—
177	C₃×C₈	—	4	2	no	—
178	C₂×C₃²	#897	1	1	yes	characteristic
179	C₂×C₃²	C₂×#204	1	1	yes	characteristic
180	C₂×C₃²	#909	1	1	yes	characteristic
181	C₂×C₉	—	2	1	no	—
182	C₂×C₉	—	2	1	no	—
183	C₂×C₉	—	2	1	no	—
184	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
185	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
186	C₂×C₉	C₃×#137	1	1	yes	characteristic
187	C₂×Q₈	—	9	1	no	—
188	C₂×Q₈	—	18	1	no	—
189	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
190	C₄⋊C₄	—	9	2	no	—
191	C₄⋊C₄	—	9	1	no	—
192	C₄⋊C₄	—	18	1	no	—
193	C₄⋊C₄	—	18	1	no	—
194	C₄²	—	18	1	no	—
195	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
198	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	—	18	2	no	—
206	C₃×C₄	—	36	2	no	—
207	C₃×C₄	—	36	1	no	—
208	C₃×C₄	#3664	1	2	yes	—
209	C₃×C₄	#3699	1	1	yes	characteristic
210	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
211	C₃×C₄	C₃×#204	1	2	yes	—
212	C₃×C₄	C₃×#237	1	1	yes	characteristic
213	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
214	Dic₃	—	6	2	no	—
215	Dic₃	—	12	2	no	—
216	Dic₃	—	12	1	no	—
217	C₃²	#4851	1	1	yes	characteristic
218	C₉	—	2	1	no	—
219	C₉	C₃×#475	1	1	yes	characteristic, complemented
220	Q₈	—	9	2	no	—
221	Q₈	—	18	1	no	—
222	Q₈	—	36	2	no	—
223	C₂×C₄	—	18	1	no	—
224	C₂×C₄	—	18	1	no	—
225	C₂×C₄	—	18	2	no	—
226	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
227	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	#63459	1	1	yes	characteristic
234	C₂×C₃	C₂×#3664	1	1	yes	characteristic
235	C₂×C₃	#63471	1	1	yes	characteristic
236	C₂×C₃	C₃×#897	1	1	yes	characteristic
237	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
238	C₂×C₃	C₃×#909	1	1	yes	characteristic
239	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	18	2	no	—
242	C₄	—	36	2	no	—
243	C₄	—	36	1	no	—
244	C₄	C₃×#3664	1	2	yes	—
245	C₄	C₃×#3699	1	1	yes	characteristic
246	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	#10588515	1	1	yes	characteristic, direct product factor
249	C₃	C₃×#4851	1	1	yes	characteristic
250	C₂	C₃×#63459	1	1	yes	characteristic
251	C₂	C₂×C₃×#3664	1	1	yes	characteristic
252	C₂	C₃×#63471	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#475
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
7. #10588515	C₃	direct (N × H)
36. C₃×Dic₁₈	Q₈	semidirect
100. Dic₁₈	C₃×Q₈	semidirect
153. C₃×C₉	#475	semidirect
219. C₉	C₃×#475	semidirect
248. C₃	#10588515	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×#63431	2
C₃×C₄×Dic₁₈	2
C₃×C₉×C₄⋊C₈	2
C₃×#63377	2
C₃×#63376	2
C₃×#4851	3
#10588515	3

Subgroups — C3×#10588515 #13142000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588515 #13142000