Subgroups — C₃×#10592895 #13163231

Order 1728.

884 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#135	C₂	1	1	yes	maximal, characteristic
2	C₃×#64157	C₂	1	2	yes	maximal, complemented
3	C₃×#63664	C₂	1	2	yes	maximal, complemented
4	C₃×#63646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592895	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5196	—	3	4	no	maximal
8	C₃³×#41	C₂²	1	2	yes	—
9	C₃³×Q₁₆	C₄	1	1	yes	characteristic, complemented
10	C₃³×SD₁₆	C₄	1	2	yes	complemented
11	C₃³×D₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₃×#3748	—	2	2	no	—
15	C₃×#3748	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃²×#135	—	2	4	no	—
19	C₃²×#135	D₃	1	4	yes	—
20	C₃²×#135	C₂×C₃	1	1	yes	characteristic
21	#64157	C₂×C₃	1	2	yes	complemented
22	C₃×#948	—	3	8	no	—
23	#63664	C₂×C₃	1	2	yes	complemented
24	#63646	C₂×C₃	1	1	yes	characteristic, complemented
25	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#837	—	3	8	no	—
27	C₃×#819	—	3	4	no	—
28	C₃×C₈×Dic₃	—	3	4	no	—
29	C₃³×Q₈	C₂×C₄	1	2	yes	—
30	C₃³×D₄	—	2	2	no	—
31	C₃³×D₄	C₂×C₄	1	2	yes	—
32	C₂×C₃³×C₄	—	2	2	no	—
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
35	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
39	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
40	C₃×#443	—	9	1	no	—
41	#5196	—	3	4	no	—
42	C₃²×#41	—	2	8	no	—
43	C₃²×#41	C₂×D₃	1	8	yes	—
44	C₃²×#41	C₂²×C₃	1	2	yes	—
45	C₃²×Q₁₆	—	2	4	no	—
46	C₃²×Q₁₆	Dic₃	1	4	yes	complemented
47	C₃²×Q₁₆	C₃×C₄	1	1	yes	characteristic, complemented
48	C₃²×SD₁₆	—	2	8	no	—
49	C₃²×SD₁₆	Dic₃	1	8	yes	complemented
50	C₃²×SD₁₆	C₃×C₄	1	2	yes	complemented
51	C₃²×D₈	—	2	4	no	—
52	C₃²×D₈	Dic₃	1	4	yes	complemented
53	C₃²×D₈	C₃×C₄	1	1	yes	characteristic, complemented
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
56	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
58	#3748	—	2	2	no	—
59	#3748	C₂²×C₃	1	2	yes	—
60	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
61	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
62	C₃×C₄×Dic₃	—	3	4	no	—
63	C₃×#208	—	3	8	no	—
64	C₃×#208	—	6	8	no	—
65	C₂×C₃×C₃⋊C₈	—	3	4	no	—
66	C₂×C₃×C₃⋊C₈	—	6	8	no	—
67	C₂²×C₃³	—	2	2	no	—
68	C₂²×C₃³	#41	1	1	yes	characteristic
69	C₃³×C₄	—	2	2	no	—
70	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃×C₃⋊Dic₃	—	2	2	no	—
73	C₃×#135	—	2	4	no	—
74	C₃×#135	C₃×D₃	1	4	yes	—
75	C₃×#135	C₃⋊D₃	1	1	yes	characteristic
76	C₃×#131	—	9	2	no	—
77	#948	—	3	8	no	—
78	C₃×#108	—	9	1	no	—
79	C₃×#104	—	9	2	no	—
80	C₃×C₄×C₈	—	9	1	no	—
81	#837	—	3	8	no	—
82	#819	—	3	4	no	—
83	C₈×Dic₃	—	3	4	no	—
84	C₃²×Q₈	—	2	8	no	—
85	C₃²×Q₈	C₂×Dic₃	1	8	yes	—
86	C₃²×Q₈	C₂×C₃×C₄	1	2	yes	—
87	C₃²×D₄	—	2	2	no	—
88	C₃²×D₄	—	2	8	no	—
89	C₃²×D₄	—	2	8	no	—
90	C₃²×D₄	—	4	8	no	—
91	C₃²×D₄	C₂×Dic₃	1	8	yes	—
92	C₃²×D₄	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	8	no	—
96	C₂×C₃²×C₄	—	4	8	no	—
97	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	—	2	1	no	—
100	C₂×C₃×Dic₃	—	6	4	no	—
101	C₃²×C₈	—	2	4	no	—
102	C₃²×C₈	—	2	4	no	—
103	C₃²×C₈	C₂×Dic₃	1	4	yes	—
104	C₃²×C₈	C₂×Dic₃	1	4	yes	—
105	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₃⋊(C₃⋊C₈)	—	2	2	no	—
108	C₃⋊(C₃⋊C₈)	C₃×D₄	1	1	yes	characteristic
109	C₃⋊(C₃⋊C₈)	C₃×D₄	1	1	yes	characteristic
110	C₃×C₃⋊C₈	—	3	4	no	—
111	C₃×C₃⋊C₈	—	3	4	no	—
112	C₃×C₃⋊C₈	—	6	8	no	—
113	#443	—	9	1	no	Sylow
114	C₂×C₃³	—	2	1	no	—
115	C₂×C₃³	—	4	2	no	—
116	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
117	C₃×#41	—	2	8	no	—
118	C₃×#41	C₂×C₃×D₃	1	8	yes	—
119	C₃×#41	C₂×C₃⋊D₃	1	2	yes	—
120	C₃×Q₁₆	—	2	4	no	—
121	C₃×Q₁₆	C₃×Dic₃	1	4	yes	complemented
122	C₃×Q₁₆	C₃⋊Dic₃	1	1	yes	characteristic, complemented
123	C₃×SD₁₆	—	2	8	no	—
124	C₃×SD₁₆	C₃×Dic₃	1	8	yes	complemented
125	C₃×SD₁₆	C₃⋊Dic₃	1	2	yes	complemented
126	C₃×D₈	—	2	4	no	—
127	C₃×D₈	C₃×Dic₃	1	4	yes	complemented
128	C₃×D₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
129	C₃×M₁₆	—	9	2	no	—
130	C₃×M₁₆	—	18	2	no	—
131	C₂×C₃×C₈	—	2	4	no	—
132	C₂×C₃×C₈	—	9	1	no	—
133	C₂×C₃×C₈	—	18	2	no	—
134	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
135	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₃×C₄²	—	9	1	no	—
137	C₄×Dic₃	—	3	4	no	—
138	#208	—	3	8	no	—
139	#208	—	6	8	no	—
140	C₂×C₃⋊C₈	—	3	4	no	—
141	C₂×C₃⋊C₈	—	6	8	no	—
142	C₂²×C₃²	—	2	2	no	—
143	C₂²×C₃²	—	2	4	no	—
144	C₂²×C₃²	—	2	8	no	—
145	C₂²×C₃²	—	4	8	no	—
146	C₂²×C₃²	#237	1	4	yes	—
147	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
148	C₃²×C₄	—	2	2	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	2	4	no	—
151	C₃²×C₄	—	2	8	no	—
152	C₃²×C₄	—	4	8	no	—
153	C₃²×C₄	D₃×D₄	1	4	yes	—
154	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
155	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₃⋊Dic₃	—	2	2	no	—
158	C₃×Dic₃	—	6	8	no	—
159	#135	C₃×C₃⋊D₃	1	1	yes	characteristic
160	#131	—	9	2	no	—
161	#108	—	9	1	no	—
162	#104	—	9	2	no	—
163	C₄×C₈	—	9	1	no	—
164	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
165	C₃×Q₈	—	2	8	no	—
166	C₃×Q₈	C₂×C₃×Dic₃	1	8	yes	—
167	C₃×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
168	C₃×D₄	—	2	8	no	—
169	C₃×D₄	—	2	2	no	—
170	C₃×D₄	—	2	8	no	—
171	C₃×D₄	—	4	8	no	—
172	C₃×D₄	C₂×C₃×Dic₃	1	8	yes	—
173	C₃×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
174	C₂×C₃×C₄	—	2	8	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	4	8	no	—
178	C₂×C₃×C₄	—	18	1	no	—
179	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
180	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
181	C₂×Dic₃	—	6	4	no	—
182	C₃×C₈	—	2	4	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	9	1	no	—
185	C₃×C₈	—	9	1	no	—
186	C₃×C₈	—	18	2	no	—
187	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
188	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
189	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
190	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
191	C₃⋊C₈	—	3	4	no	—
192	C₃⋊C₈	—	3	4	no	—
193	C₃⋊C₈	—	6	8	no	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	—	2	4	no	—
196	C₂×C₃²	—	2	4	no	—
197	C₂×C₃²	—	4	4	no	—
198	C₂×C₃²	—	4	2	no	—
199	C₂×C₃²	—	4	8	no	—
200	C₂×C₃²	—	8	8	no	—
201	C₂×C₃²	D₄×Dic₃	1	4	yes	—
202	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
203	#41	C₂×C₃×C₃⋊D₃	1	2	yes	—
204	Q₁₆	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
205	SD₁₆	C₃×C₃⋊Dic₃	1	2	yes	complemented
206	D₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
207	M₁₆	—	9	2	no	—
208	M₁₆	—	18	2	no	—
209	C₂×C₈	—	9	1	no	—
210	C₂×C₈	—	18	2	no	—
211	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
212	C₄²	—	9	1	no	—
213	C₂²×C₃	—	2	8	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	4	no	—
216	C₂²×C₃	—	4	8	no	—
217	C₂²×C₃	C₃×#237	1	4	yes	—
218	C₂²×C₃	#3830	1	1	yes	characteristic
219	C₃×C₄	—	2	8	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	4	8	no	—
224	C₃×C₄	—	18	2	no	—
225	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
226	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
227	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
228	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
229	Dic₃	—	6	8	no	—
230	C₃²	—	2	4	no	—
231	C₃²	#5196	1	4	yes	complemented
232	C₃²	C₃×#443	1	1	yes	characteristic, complemented
233	Q₈	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
234	D₄	—	2	2	no	—
235	D₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	18	1	no	—
238	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
239	C₈	—	9	1	no	—
240	C₈	—	9	1	no	—
241	C₈	—	18	2	no	—
242	C₈	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
243	C₈	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	—	4	4	no	—
248	C₂×C₃	—	4	8	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	—	8	8	no	—
251	C₂×C₃	C₃×D₄×Dic₃	1	4	yes	—
252	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
253	C₂²	—	2	2	no	—
254	C₂²	C₃×#3830	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	18	2	no	—
257	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
258	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
259	C₃	—	2	4	no	—
260	C₃	C₃×#5196	1	4	yes	complemented
261	C₃	#10592895	1	1	yes	characteristic, direct product factor
262	C₂	—	2	1	no	—
263	C₂	—	4	2	no	—
264	C₂	C₃×D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting, prime radical	C₃³×#135	C₂
socle	C₂×C₃³	C₄×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#135	C₃⋊D₃
hypocenter	C₃²	C₃×#443
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
2. C₃×#64157	C₂	semidirect
3. C₃×#63664	C₂	semidirect
4. C₃×#63646	C₂	semidirect
5. C₃×C₈×C₃⋊Dic₃	C₂	semidirect
6. #10592895	C₃	direct (N × H)
9. C₃³×Q₁₆	C₄	semidirect
10. C₃³×SD₁₆	C₄	semidirect
11. C₃³×D₈	C₄	semidirect
21. #64157	C₂×C₃	semidirect
23. #63664	C₂×C₃	semidirect
24. #63646	C₂×C₃	semidirect
25. C₈×C₃⋊Dic₃	C₂×C₃	semidirect
46. C₃²×Q₁₆	Dic₃	semidirect
47. C₃²×Q₁₆	C₃×C₄	semidirect
49. C₃²×SD₁₆	Dic₃	semidirect
50. C₃²×SD₁₆	C₃×C₄	semidirect
52. C₃²×D₈	Dic₃	semidirect
53. C₃²×D₈	C₃×C₄	semidirect
121. C₃×Q₁₆	C₃×Dic₃	semidirect
122. C₃×Q₁₆	C₃⋊Dic₃	semidirect
124. C₃×SD₁₆	C₃×Dic₃	semidirect
125. C₃×SD₁₆	C₃⋊Dic₃	semidirect
127. C₃×D₈	C₃×Dic₃	semidirect
128. C₃×D₈	C₃⋊Dic₃	semidirect
164. C₃³	#443	semidirect
204. Q₁₆	C₃×C₃⋊Dic₃	semidirect
205. SD₁₆	C₃×C₃⋊Dic₃	semidirect
206. D₈	C₃×C₃⋊Dic₃	semidirect
231. C₃²	#5196	semidirect
232. C₃²	C₃×#443	semidirect
260. C₃	C₃×#5196	semidirect
261. C₃	#10592895	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#135	2
C₃×#64157	2
C₃×#63664	2
C₃×#63646	2
C₃×C₈×C₃⋊Dic₃	2
#10592895	3
C₃×#5196	3

Subgroups — C3×#10592895 #13163231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592895 #13163231