Subgroups — C₃×C₃⋊D₃×#106 #13162897

Order 1728.

924 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₄×C₃⋊D₃	C₂	1	2	yes	maximal
3	C₃³×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63644	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63635	C₂	1	2	yes	maximal, complemented
6	C₃⋊D₃×#106	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×D₃×#106	—	3	4	no	maximal
8	C₂×C₃×C₄×C₃⋊D₃	—	2	2	no	—
9	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	C₃×#3750	—	2	2	no	—
14	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×C₈×C₃⋊D₃	C₄	1	4	yes	complemented
16	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
17	C₄⋊C₄×C₃⋊D₃	C₂×C₃	1	2	yes	—
18	C₂×C₃×D₃×C₈	—	3	4	no	—
19	C₃×D₃×C₄⋊C₄	—	3	8	no	—
20	C₃²×#106	—	2	4	no	—
21	C₃²×#106	D₃	1	4	yes	complemented
22	C₃²×#106	C₂×C₃	1	1	yes	characteristic, complemented
23	#63644	C₂×C₃	1	1	yes	characteristic, complemented
24	#63635	C₂×C₃	1	2	yes	complemented
25	C₃×#817	—	3	4	no	—
26	C₃×#808	—	3	8	no	—
27	C₂²×C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	—	2	2	no	—
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
31	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
32	C₃×C₄×C₃⋊D₃	—	4	4	no	—
33	C₃×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
34	C₃×C₄×C₃⋊D₃	Q₈	1	2	yes	—
35	C₃³×C₈	C₂×C₄	1	2	yes	complemented
36	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
37	C₂×C₃×#106	—	9	1	no	—
38	D₃×#106	—	3	4	no	—
39	C₂×C₄×C₃⋊D₃	—	2	2	no	—
40	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	3	4	no	—
42	C₂×C₃×C₄×D₃	—	6	8	no	—
43	C₂×C₃²×C₈	—	2	4	no	—
44	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	—	2	8	no	—
47	C₃²×C₄⋊C₄	C₂×D₃	1	8	yes	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
50	#3750	—	2	2	no	—
51	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
52	C₈×C₃⋊D₃	C₃×C₄	1	4	yes	complemented
53	C₃×C₄⋊Dic₃	—	3	8	no	—
54	C₃×Dic₃⋊C₄	—	6	8	no	—
55	C₂×C₃×C₃⋊C₈	—	3	4	no	—
56	C₃×D₃×C₈	—	3	16	no	—
57	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₃⋊D₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₃×C₃⋊D₃	SD₁₆	1	4	yes	—
60	C₃³×C₄	—	4	2	no	—
61	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₃⋊Dic₃	—	4	2	no	—
64	C₃×C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₃×C₈	—	9	1	no	—
66	C₂×C₃×C₄⋊C₄	—	9	2	no	—
67	C₂×D₃×C₈	—	3	4	no	—
68	D₃×C₄⋊C₄	—	3	8	no	—
69	C₃×#106	—	2	4	no	—
70	C₃×#106	—	9	1	no	—
71	C₃×#106	—	9	2	no	—
72	C₃×#106	C₃×D₃	1	4	yes	complemented
73	C₃×#106	C₃⋊D₃	1	1	yes	characteristic, direct product factor
74	#817	—	3	4	no	—
75	#808	—	3	8	no	—
76	C₂²×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	3	4	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	8	no	—
81	C₂×C₃²×C₄	—	4	8	no	—
82	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	—	2	2	no	—
85	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
86	C₄×C₃⋊D₃	—	4	4	no	—
87	C₄×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
88	C₄×C₃⋊D₃	C₃×Q₈	1	2	yes	—
89	C₂×C₃×Dic₃	—	3	4	no	—
90	C₂×C₃×Dic₃	—	6	8	no	—
91	C₃×C₄×D₃	—	3	8	no	—
92	C₃×C₄×D₃	—	3	8	no	—
93	C₃×C₄×D₃	—	12	16	no	—
94	C₃²×C₈	—	2	8	no	—
95	C₃²×C₈	C₄×D₃	1	8	yes	complemented
96	C₃²×C₈	C₂×C₃×C₄	1	2	yes	complemented
97	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	2	yes	complemented
98	C₃×C₃⋊C₈	—	3	8	no	—
99	C₂×#106	—	9	1	no	Sylow
100	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₃³	C₂×SD₁₆	1	2	yes	—
102	C₃×C₃⋊D₃	#106	1	4	yes	direct product factor
103	C₂²×C₃×C₄	—	9	1	no	—
104	C₂²×C₃×C₄	—	18	2	no	—
105	C₂×C₄×D₃	—	3	4	no	—
106	C₂×C₄×D₃	—	6	8	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	—	9	1	no	—
109	C₂×C₃×C₈	—	9	4	no	—
110	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
111	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	2	8	no	—
113	C₃×C₄⋊C₄	—	9	2	no	—
114	C₃×C₄⋊C₄	—	18	2	no	—
115	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
116	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
117	C₄⋊Dic₃	—	3	8	no	—
118	Dic₃⋊C₄	—	6	8	no	—
119	C₂×C₃⋊C₈	—	3	4	no	—
120	D₃×C₈	—	3	16	no	—
121	C₂²×C₃²	—	2	4	no	—
122	C₂²×C₃²	D₃×D₄	1	4	yes	—
123	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₃⋊D₃	C₃×C₄⋊C₄	1	2	yes	—
125	C₂×C₃⋊D₃	C₃×SD₁₆	1	4	yes	—
126	C₂×C₃×D₃	—	3	8	no	—
127	C₂×C₃×D₃	—	3	16	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	—	2	4	no	—
130	C₃²×C₄	—	4	2	no	—
131	C₃²×C₄	—	4	8	no	—
132	C₃²×C₄	—	8	8	no	—
133	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
134	C₃²×C₄	D₃×Q₈	1	4	yes	—
135	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
137	C₃⋊Dic₃	—	4	2	no	—
138	C₃⋊Dic₃	C₃×C₄⋊C₄	1	2	yes	—
139	C₃×Dic₃	—	3	8	no	—
140	C₃×Dic₃	—	12	8	no	—
141	C₂²×C₈	—	9	1	no	—
142	C₂×C₄⋊C₄	—	9	2	no	—
143	#106	—	9	1	no	—
144	#106	—	9	2	no	—
145	#106	C₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor
146	C₃³	C₂×#106	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₃	—	9	1	no	—
148	C₂²×D₃	—	3	4	no	—
149	C₂×C₃×C₄	—	2	8	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	4	8	no	—
153	C₂×C₃×C₄	—	9	1	no	—
154	C₂×C₃×C₄	—	9	2	no	—
155	C₂×C₃×C₄	—	9	2	no	—
156	C₂×C₃×C₄	—	18	2	no	—
157	C₂×C₃×C₄	—	36	4	no	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
159	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
160	C₂×Dic₃	—	3	4	no	—
161	C₂×Dic₃	—	6	8	no	—
162	C₄×D₃	—	3	8	no	—
163	C₄×D₃	—	3	8	no	—
164	C₄×D₃	—	12	16	no	—
165	C₃×C₈	—	2	8	no	—
166	C₃×C₈	—	9	2	no	—
167	C₃×C₈	C₃×C₄×D₃	1	8	yes	complemented
168	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	complemented
169	C₃⋊C₈	—	3	8	no	—
170	C₂×C₃²	—	2	4	no	—
171	C₂×C₃²	—	2	8	no	—
172	C₂×C₃²	D₃×C₄⋊C₄	1	4	yes	—
173	C₂×C₃²	D₃×SD₁₆	1	8	yes	—
174	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₃²	C₂×C₃×SD₁₆	1	2	yes	—
176	C₃⋊D₃	C₃×#106	1	4	yes	direct product factor
177	C₃×D₃	—	3	16	no	—
178	C₂²×C₄	—	9	1	no	—
179	C₂²×C₄	—	18	2	no	—
180	C₂×C₈	—	9	1	no	—
181	C₂×C₈	—	9	4	no	—
182	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
183	C₄⋊C₄	—	9	2	no	—
184	C₄⋊C₄	—	18	2	no	—
185	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	—	9	2	no	—
188	C₂²×C₃	—	9	4	no	—
189	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
190	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
191	C₂×D₃	—	3	8	no	—
192	C₂×D₃	—	3	16	no	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	4	8	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	8	8	no	—
198	C₃×C₄	—	9	2	no	—
199	C₃×C₄	—	36	2	no	—
200	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
201	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
202	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
203	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
204	Dic₃	—	3	8	no	—
205	Dic₃	—	12	8	no	—
206	C₃²	—	2	4	no	—
207	C₃²	D₃×#106	1	4	yes	complemented
208	C₃²	C₂×C₃×#106	1	1	yes	characteristic, complemented
209	C₂³	—	9	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	9	1	no	—
212	C₂×C₄	—	9	2	no	—
213	C₂×C₄	—	9	2	no	—
214	C₂×C₄	—	18	2	no	—
215	C₂×C₄	—	36	4	no	—
216	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
217	C₈	—	9	2	no	—
218	C₈	C₃×C₄×C₃⋊D₃	1	2	yes	complemented
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	2	8	no	—
221	C₂×C₃	—	9	4	no	—
222	C₂×C₃	C₃×D₃×C₄⋊C₄	1	4	yes	—
223	C₂×C₃	C₃×D₃×SD₁₆	1	8	yes	—
224	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
225	C₂×C₃	SD₁₆×C₃⋊D₃	1	2	yes	—
226	D₃	—	3	16	no	—
227	C₂²	—	9	2	no	—
228	C₂²	—	9	4	no	—
229	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
230	C₄	—	4	2	no	—
231	C₄	—	9	2	no	—
232	C₄	—	36	2	no	—
233	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
234	C₄	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
235	C₃	—	2	4	no	—
236	C₃	C₃×D₃×#106	1	4	yes	complemented
237	C₃	C₃⋊D₃×#106	1	1	yes	characteristic, direct product factor
238	C₂	—	9	4	no	—
239	C₂	C₃×C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
240	C₂	C₃×SD₁₆×C₃⋊D₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#106	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂²×C₃×C₃⋊D₃	D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#106	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×#106
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

Complements

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

N	H ≅ G/N	type
3. C₃³×#106	C₂	semidirect
4. C₃×#63644	C₂	semidirect
5. C₃×#63635	C₂	semidirect
6. C₃⋊D₃×#106	C₃	direct (N × H)
15. C₃×C₈×C₃⋊D₃	C₄	semidirect
21. C₃²×#106	D₃	semidirect
22. C₃²×#106	C₂×C₃	semidirect
23. #63644	C₂×C₃	semidirect
24. #63635	C₂×C₃	semidirect
35. C₃³×C₈	C₂×C₄	semidirect
36. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
52. C₈×C₃⋊D₃	C₃×C₄	semidirect
72. C₃×#106	C₃×D₃	semidirect
73. C₃×#106	C₃⋊D₃	direct (N × H)
95. C₃²×C₈	C₄×D₃	semidirect
96. C₃²×C₈	C₂×C₃×C₄	semidirect
97. C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	semidirect
102. C₃×C₃⋊D₃	#106	direct (N × H)
145. #106	C₃×C₃⋊D₃	direct (N × H)
146. C₃³	C₂×#106	semidirect
167. C₃×C₈	C₃×C₄×D₃	semidirect
168. C₃×C₈	C₄×C₃⋊D₃	semidirect
176. C₃⋊D₃	C₃×#106	direct (N × H)
207. C₃²	D₃×#106	semidirect
208. C₃²	C₂×C₃×#106	semidirect
218. C₈	C₃×C₄×C₃⋊D₃	semidirect
236. C₃	C₃×D₃×#106	semidirect
237. C₃	C₃⋊D₃×#106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×C₃⋊D₃	2
C₃×C₄⋊C₄×C₃⋊D₃	2
C₃³×#106	2
C₃×#63644	2
C₃×#63635	2
C₃⋊D₃×#106	3
C₃×D₃×#106	3

Subgroups — C3×C3⋊D3×#106 #13162897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊D₃×#106 #13162897