Subgroups — C₃×#10588508 #13141996

Order 1728.

344 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₉⋊C₈	C₂	1	2	yes	maximal
4	C₃×C₈×Dic₉	C₂	1	2	yes	maximal
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#4830	—	3	1	no	maximal
7	#10588508	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Dic₁₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊Dic₉	C₄	1	1	yes	characteristic
12	C₃×Dic₉⋊C₄	C₄	1	2	yes	—
13	C₃×C₄×Dic₉	C₂²	1	2	yes	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×Dic₆	—	3	1	no	—
17	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	—	3	2	no	—
19	C₃×C₈×Dic₃	—	3	2	no	—
20	C₃×C₄×C₃⋊C₈	—	3	1	no	—
21	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic
22	C₉×C₄⋊C₈	—	2	1	no	—
23	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
24	Dic₉⋊C₈	C₂×C₃	1	2	yes	—
25	C₈×Dic₉	C₂×C₃	1	2	yes	—
26	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
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×C₃×Dic₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
32	C₃×Dic₁₈	C₈	1	4	yes	complemented
33	C₃×C₈×C₉	—	2	2	no	—
34	C₃×C₉⋊C₈	—	2	1	no	—
35	C₃×C₉⋊C₈	Q₈	1	2	yes	complemented
36	C₃×C₈×Q₈	—	9	1	no	—
37	#4830	—	3	1	no	—
38	C₂×C₃×Dic₆	—	3	1	no	—
39	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
40	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	—	3	1	no	—
42	C₃×Dic₃⋊C₄	—	3	2	no	—
43	C₃×C₄×Dic₃	—	3	2	no	—
44	C₂×C₃×C₃⋊C₈	—	3	1	no	—
45	C₂×C₃×C₃⋊C₈	—	3	1	no	—
46	C₂×Dic₁₈	C₃×C₄	1	1	yes	characteristic
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
49	C₄²×C₉	—	2	1	no	—
50	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
51	C₄⋊Dic₉	C₃×C₄	1	1	yes	characteristic
52	Dic₉⋊C₄	C₃×C₄	1	2	yes	—
53	C₄×Dic₉	C₂²×C₃	1	2	yes	—
54	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	2	1	no	—
58	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
59	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₃×C₄×C₉	#41	1	1	yes	characteristic
61	C₃×Dic₉	—	2	2	no	—
62	C₃×Dic₉	C₂×C₈	1	4	yes	—
63	C₃×C₄×Q₈	—	9	1	no	—
64	C₄×Dic₆	—	3	1	no	—
65	C₃×C₄⋊C₈	—	2	1	no	—
66	C₃×C₄⋊C₈	—	9	2	no	—
67	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic
68	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
69	C₃×C₄×C₈	—	9	2	no	—
70	C₃×C₄×C₈	—	9	1	no	—
71	Dic₃⋊C₈	—	3	2	no	—
72	C₈×Dic₃	—	3	2	no	—
73	C₄×C₃⋊C₈	—	3	1	no	—
74	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃×Dic₃	—	3	2	no	—
78	C₂×C₃×Dic₃	—	3	2	no	—
79	C₃×Dic₆	—	3	4	no	—
80	C₃²×C₈	—	2	2	no	—
81	C₃×C₃⋊C₈	—	3	2	no	—
82	C₃×C₃⋊C₈	—	6	1	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
89	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
90	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
91	Dic₁₈	C₃×C₈	1	4	yes	complemented
92	C₈×C₉	—	2	2	no	—
93	C₈×C₉	—	4	2	no	—
94	C₉⋊C₈	—	2	1	no	—
95	C₉⋊C₈	C₃×Q₈	1	2	yes	complemented
96	C₈×Q₈	—	9	1	no	Sylow
97	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₃×C₉	#131	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	9	1	no	—
101	C₂×Dic₆	—	3	1	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	9	1	no	—
104	C₂×C₃×C₈	—	9	1	no	—
105	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
106	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
107	C₃×C₄⋊C₄	—	9	2	no	—
108	C₃×C₄⋊C₄	—	9	1	no	—
109	C₃×C₄²	—	2	1	no	—
110	C₃×C₄²	—	9	2	no	—
111	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
112	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
113	C₄⋊Dic₃	—	3	1	no	—
114	Dic₃⋊C₄	—	3	2	no	—
115	C₄×Dic₃	—	3	2	no	—
116	C₂×C₃⋊C₈	—	3	1	no	—
117	C₂×C₃⋊C₈	—	3	1	no	—
118	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	D₃×C₈	1	2	yes	—
121	C₃²×C₄	#237	1	1	yes	characteristic
122	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
123	C₃×Dic₃	—	3	4	no	—
124	C₃×Dic₃	—	6	2	no	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	1	no	—
131	C₄×C₉	—	4	1	no	—
132	C₄×C₉	C₂×C₃×C₈	1	2	yes	—
133	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
134	C₄×C₉	C₃×#41	1	1	yes	characteristic
135	Dic₉	—	2	2	no	—
136	Dic₉	C₂×C₃×C₈	1	4	yes	—
137	C₄×Q₈	—	9	1	no	—
138	C₄⋊C₈	—	9	2	no	—
139	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic
140	C₄×C₈	—	9	2	no	—
141	C₄×C₈	—	9	1	no	—
142	C₃×C₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
143	C₃×Q₈	—	9	4	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	9	2	no	—
148	C₂×C₃×C₄	—	9	2	no	—
149	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
152	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
155	C₂×Dic₃	—	3	2	no	—
156	C₂×Dic₃	—	3	2	no	—
157	Dic₆	—	3	4	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	—	4	2	no	—
161	C₃×C₈	—	9	2	no	—
162	C₃×C₈	—	18	1	no	—
163	C₃⋊C₈	—	3	2	no	—
164	C₃⋊C₈	—	6	1	no	—
165	C₂×C₃²	#887	1	1	yes	characteristic
166	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
167	C₂×C₃²	#907	1	1	yes	characteristic
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	C₃×C₄×Q₈	1	1	yes	characteristic
172	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
173	C₂×C₉	C₃×#131	1	1	yes	characteristic
174	C₂×Q₈	—	9	1	no	—
175	C₂×C₈	—	9	1	no	—
176	C₂×C₈	—	9	1	no	—
177	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
178	C₄⋊C₄	—	9	2	no	—
179	C₄⋊C₄	—	9	1	no	—
180	C₄²	—	9	2	no	—
181	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
184	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	4	1	no	—
191	C₃×C₄	—	9	4	no	—
192	C₃×C₄	—	18	2	no	—
193	C₃×C₄	C₈×D₉	1	2	yes	—
194	C₃×C₄	#3699	1	1	yes	characteristic
195	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
196	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
197	C₃×C₄	C₃×#237	1	1	yes	characteristic
198	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
199	Dic₃	—	3	4	no	—
200	Dic₃	—	6	2	no	—
201	C₃²	#4830	1	1	yes	characteristic
202	C₉	—	2	1	no	—
203	C₉	C₃×C₈×Q₈	1	1	yes	characteristic, complemented
204	Q₈	—	9	4	no	—
205	C₂×C₄	—	9	2	no	—
206	C₂×C₄	—	9	2	no	—
207	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
208	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	9	2	no	—
212	C₈	—	18	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	#63449	1	1	yes	characteristic
217	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
218	C₂×C₃	#63469	1	1	yes	characteristic
219	C₂×C₃	C₃×#887	1	1	yes	characteristic
220	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
221	C₂×C₃	C₃×#907	1	1	yes	characteristic
222	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	9	4	no	—
225	C₄	—	18	2	no	—
226	C₄	C₃×C₈×D₉	1	2	yes	—
227	C₄	C₃×#3699	1	1	yes	characteristic
228	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
229	C₃	—	2	1	no	—
230	C₃	#10588508	1	1	yes	characteristic, direct product factor
231	C₃	C₃×#4830	1	1	yes	characteristic
232	C₂	C₃×#63449	1	1	yes	characteristic
233	C₂	C₂×C₃×C₈×D₉	1	1	yes	characteristic
234	C₂	C₃×#63469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃²	C₂×C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×C₈×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
7. #10588508	C₃	direct (N × H)
32. C₃×Dic₁₈	C₈	semidirect
35. C₃×C₉⋊C₈	Q₈	semidirect
91. Dic₁₈	C₃×C₈	semidirect
95. C₉⋊C₈	C₃×Q₈	semidirect
142. C₃×C₉	C₈×Q₈	semidirect
203. C₉	C₃×C₈×Q₈	semidirect
230. C₃	#10588508	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×Dic₁₈	2
C₃×C₉×C₄⋊C₈	2
C₃×Dic₉⋊C₈	2
C₃×C₈×Dic₉	2
C₃×C₄×C₉⋊C₈	2
C₃×#4830	3
#10588508	3

Subgroups — C3×#10588508 #13141996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588508 #13141996