Subgroups — C₃×#10592896 #13163232

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 260 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×#63642	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592896	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5197	—	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	#63642	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#837	—	3	8	no	—
27	C₃×#819	—	3	4	no	—
28	C₃×#815	—	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	2	yes	—
39	C₃×#444	—	9	1	no	—
40	#5197	—	3	4	no	—
41	C₃²×#41	—	2	8	no	—
42	C₃²×#41	C₂×D₃	1	8	yes	—
43	C₃²×#41	C₂²×C₃	1	2	yes	—
44	C₃²×Q₁₆	—	2	4	no	—
45	C₃²×Q₁₆	Dic₃	1	4	yes	complemented
46	C₃²×Q₁₆	C₃×C₄	1	1	yes	characteristic, complemented
47	C₃²×SD₁₆	—	2	8	no	—
48	C₃²×SD₁₆	Dic₃	1	8	yes	complemented
49	C₃²×SD₁₆	C₃×C₄	1	2	yes	complemented
50	C₃²×D₈	—	2	4	no	—
51	C₃²×D₈	Dic₃	1	4	yes	complemented
52	C₃²×D₈	C₃×C₄	1	1	yes	characteristic, complemented
53	C₂×C₃²×C₈	—	2	4	no	—
54	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
57	#3748	—	2	2	no	—
58	#3748	C₂²×C₃	1	2	yes	—
59	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
60	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄×Dic₃	—	3	4	no	—
62	C₃×#208	—	3	8	no	—
63	C₃×#208	—	6	8	no	—
64	C₂×C₃×C₃⋊C₈	—	3	4	no	—
65	C₂×C₃×C₃⋊C₈	—	6	8	no	—
66	C₂²×C₃³	—	2	2	no	—
67	C₂²×C₃³	#41	1	1	yes	characteristic
68	C₃³×C₄	—	2	2	no	—
69	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₃×C₃⋊Dic₃	—	4	1	no	—
72	C₃×#135	—	2	4	no	—
73	C₃×#135	C₃×D₃	1	4	yes	—
74	C₃×#135	C₃⋊D₃	1	1	yes	characteristic
75	C₃×#131	—	9	2	no	—
76	#948	—	3	8	no	—
77	C₃×#108	—	9	1	no	—
78	C₃×#104	—	9	2	no	—
79	C₃×#97	—	9	1	no	—
80	#837	—	3	8	no	—
81	#819	—	3	4	no	—
82	#815	—	3	4	no	—
83	C₃²×Q₈	—	2	8	no	—
84	C₃²×Q₈	C₂×Dic₃	1	8	yes	—
85	C₃²×Q₈	C₂×C₃×C₄	1	2	yes	—
86	C₃²×D₄	—	2	2	no	—
87	C₃²×D₄	—	2	8	no	—
88	C₃²×D₄	—	2	8	no	—
89	C₃²×D₄	—	4	8	no	—
90	C₃²×D₄	C₂×Dic₃	1	8	yes	—
91	C₃²×D₄	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	—	2	8	no	—
95	C₂×C₃²×C₄	—	4	8	no	—
96	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
97	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	1	no	—
99	C₂×C₃×Dic₃	—	6	4	no	—
100	C₃²×C₈	—	2	4	no	—
101	C₃²×C₈	—	2	4	no	—
102	C₃²×C₈	C₂×Dic₃	1	4	yes	—
103	C₃²×C₈	C₂×Dic₃	1	4	yes	—
104	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₃⋊(C₃⋊C₈)	—	2	2	no	—
107	C₃⋊(C₃⋊C₈)	C₃×D₄	1	2	yes	—
108	C₃×C₃⋊C₈	—	3	8	no	—
109	C₃×C₃⋊C₈	—	6	8	no	—
110	#444	—	9	1	no	Sylow
111	C₂×C₃³	—	2	1	no	—
112	C₂×C₃³	—	4	2	no	—
113	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
114	C₃×#41	—	2	8	no	—
115	C₃×#41	C₂×C₃×D₃	1	8	yes	—
116	C₃×#41	C₂×C₃⋊D₃	1	2	yes	—
117	C₃×Q₁₆	—	2	4	no	—
118	C₃×Q₁₆	C₃×Dic₃	1	4	yes	complemented
119	C₃×Q₁₆	C₃⋊Dic₃	1	1	yes	characteristic, complemented
120	C₃×SD₁₆	—	2	8	no	—
121	C₃×SD₁₆	C₃×Dic₃	1	8	yes	complemented
122	C₃×SD₁₆	C₃⋊Dic₃	1	2	yes	complemented
123	C₃×D₈	—	2	4	no	—
124	C₃×D₈	C₃×Dic₃	1	4	yes	complemented
125	C₃×D₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
126	C₃×M₁₆	—	9	2	no	—
127	C₃×M₁₆	—	18	2	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	—	9	1	no	—
130	C₂×C₃×C₈	—	18	2	no	—
131	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
132	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
133	C₃×C₄²	—	9	1	no	—
134	C₄×Dic₃	—	3	4	no	—
135	#208	—	3	8	no	—
136	#208	—	6	8	no	—
137	C₂×C₃⋊C₈	—	3	4	no	—
138	C₂×C₃⋊C₈	—	6	8	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	—	2	4	no	—
141	C₂²×C₃²	—	2	8	no	—
142	C₂²×C₃²	—	4	8	no	—
143	C₂²×C₃²	#237	1	4	yes	—
144	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
145	C₃²×C₄	—	2	2	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	2	4	no	—
148	C₃²×C₄	—	2	8	no	—
149	C₃²×C₄	—	4	8	no	—
150	C₃²×C₄	D₃×D₄	1	4	yes	—
151	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
152	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	C₃⋊Dic₃	—	4	1	no	—
155	C₃×Dic₃	—	12	4	no	—
156	#135	C₃×C₃⋊D₃	1	1	yes	characteristic
157	#131	—	9	2	no	—
158	#108	—	9	1	no	—
159	#104	—	9	2	no	—
160	#97	—	9	1	no	—
161	C₃³	#444	1	1	yes	characteristic, complemented, Sylow
162	C₃×Q₈	—	2	8	no	—
163	C₃×Q₈	C₂×C₃×Dic₃	1	8	yes	—
164	C₃×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
165	C₃×D₄	—	2	8	no	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	2	8	no	—
168	C₃×D₄	—	4	8	no	—
169	C₃×D₄	C₂×C₃×Dic₃	1	8	yes	—
170	C₃×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	4	8	no	—
175	C₂×C₃×C₄	—	18	1	no	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
177	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
178	C₂×Dic₃	—	6	4	no	—
179	C₃×C₈	—	2	4	no	—
180	C₃×C₈	—	2	4	no	—
181	C₃×C₈	—	9	2	no	—
182	C₃×C₈	—	18	2	no	—
183	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
184	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
185	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
186	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
187	C₃⋊C₈	—	3	8	no	—
188	C₃⋊C₈	—	6	8	no	—
189	C₂×C₃²	—	2	1	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	4	4	no	—
193	C₂×C₃²	—	4	2	no	—
194	C₂×C₃²	—	4	8	no	—
195	C₂×C₃²	—	8	8	no	—
196	C₂×C₃²	D₄×Dic₃	1	4	yes	—
197	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
198	#41	C₂×C₃×C₃⋊D₃	1	2	yes	—
199	Q₁₆	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
200	SD₁₆	C₃×C₃⋊Dic₃	1	2	yes	complemented
201	D₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
202	M₁₆	—	9	2	no	—
203	M₁₆	—	18	2	no	—
204	C₂×C₈	—	9	1	no	—
205	C₂×C₈	—	18	2	no	—
206	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
207	C₄²	—	9	1	no	—
208	C₂²×C₃	—	2	8	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	4	8	no	—
212	C₂²×C₃	C₃×#237	1	4	yes	—
213	C₂²×C₃	#3830	1	1	yes	characteristic
214	C₃×C₄	—	2	8	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	4	8	no	—
219	C₃×C₄	—	36	1	no	—
220	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
221	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
222	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
223	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
224	Dic₃	—	12	4	no	—
225	C₃²	—	2	4	no	—
226	C₃²	#5197	1	4	yes	complemented
227	C₃²	C₃×#444	1	1	yes	characteristic, complemented
228	Q₈	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	18	1	no	—
233	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
234	C₈	—	9	2	no	—
235	C₈	—	18	2	no	—
236	C₈	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
237	C₈	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	—	4	4	no	—
242	C₂×C₃	—	4	8	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	—	8	8	no	—
245	C₂×C₃	C₃×D₄×Dic₃	1	4	yes	—
246	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	C₃×#3830	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	36	1	no	—
251	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
252	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
253	C₃	—	2	4	no	—
254	C₃	C₃×#5197	1	4	yes	complemented
255	C₃	#10592896	1	1	yes	characteristic, direct product factor
256	C₂	—	2	1	no	—
257	C₂	—	4	2	no	—
258	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×#444
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×#63642	C₂	semidirect
6. #10592896	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. #63642	C₂×C₃	semidirect
45. C₃²×Q₁₆	Dic₃	semidirect
46. C₃²×Q₁₆	C₃×C₄	semidirect
48. C₃²×SD₁₆	Dic₃	semidirect
49. C₃²×SD₁₆	C₃×C₄	semidirect
51. C₃²×D₈	Dic₃	semidirect
52. C₃²×D₈	C₃×C₄	semidirect
118. C₃×Q₁₆	C₃×Dic₃	semidirect
119. C₃×Q₁₆	C₃⋊Dic₃	semidirect
121. C₃×SD₁₆	C₃×Dic₃	semidirect
122. C₃×SD₁₆	C₃⋊Dic₃	semidirect
124. C₃×D₈	C₃×Dic₃	semidirect
125. C₃×D₈	C₃⋊Dic₃	semidirect
161. C₃³	#444	semidirect
199. Q₁₆	C₃×C₃⋊Dic₃	semidirect
200. SD₁₆	C₃×C₃⋊Dic₃	semidirect
201. D₈	C₃×C₃⋊Dic₃	semidirect
226. C₃²	#5197	semidirect
227. C₃²	C₃×#444	semidirect
254. C₃	C₃×#5197	semidirect
255. C₃	#10592896	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×#63642	2
#10592896	3
C₃×#5197	3

Subgroups — C3×#10592896 #13163232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592896 #13163232