Subgroups — C₃×#10592399 #13162738

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64113	C₂	1	1	yes	maximal, characteristic
2	C₃×#64110	C₂	1	2	yes	maximal, complemented
3	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63632	C₂	1	2	yes	maximal, complemented
6	#10592399	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4700	—	3	4	no	maximal
8	C₃×#3828	C₂²	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃×#3748	C₂²	1	2	yes	—
13	C₃×#3746	C₄	1	1	yes	characteristic, complemented
14	C₃×#3745	C₄	1	1	yes	characteristic, complemented
15	C₃×#3744	C₄	1	2	yes	complemented
16	C₃×#3743	—	2	2	no	—
17	C₃×C₈×C₃⋊D₃	—	2	2	no	—
18	#64113	C₂×C₃	1	1	yes	characteristic
19	#64110	C₂×C₃	1	2	yes	complemented
20	C₃×#904	—	3	4	no	—
21	C₃×#901	—	3	8	no	—
22	C₃²×C₄×C₈	—	2	4	no	—
23	C₃²×C₄×C₈	D₃	1	4	yes	complemented
24	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#63646	C₂×C₃	1	1	yes	characteristic, complemented
26	#63632	C₂×C₃	1	2	yes	complemented
27	C₃×#819	—	3	4	no	—
28	C₃×#805	—	3	8	no	—
29	C₂×C₃³×C₄	—	2	1	no	—
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₃×#638	—	2	2	no	—
32	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
33	C₃×C₄×C₃⋊D₃	—	2	2	no	—
34	C₃×#634	C₂×C₄	1	2	yes	—
35	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃³×C₈	D₄	1	1	yes	characteristic
38	C₃³×C₈	D₄	1	1	yes	characteristic
39	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃×#443	—	9	1	no	—
41	#4700	—	3	4	no	—
42	#3828	C₂²×C₃	1	2	yes	—
43	C₃×#235	—	3	8	no	—
44	C₂×C₃²×C₈	—	2	4	no	—
45	C₂×C₃²×C₈	—	2	4	no	—
46	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
47	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
48	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₃²×C₄²	—	2	4	no	—
51	C₃²×C₄²	C₂×D₃	1	4	yes	—
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	#3748	C₂²×C₃	1	2	yes	—
54	#3746	C₃×C₄	1	1	yes	characteristic, complemented
55	#3745	C₃×C₄	1	1	yes	characteristic, complemented
56	#3744	C₃×C₄	1	2	yes	complemented
57	#3743	—	2	2	no	—
58	C₈×C₃⋊D₃	—	2	2	no	—
59	C₃×#208	—	3	8	no	—
60	C₃×Dic₁₂	—	3	4	no	—
61	C₃×D₂₄	—	3	4	no	—
62	C₃×#204	—	3	8	no	—
63	C₃×ΓL(1,𝔽₂₅)	—	6	8	no	—
64	C₃×D₃×C₈	—	6	8	no	—
65	C₂²×C₃³	#41	1	1	yes	characteristic
66	C₂×C₃×C₃⋊D₃	—	2	2	no	—
67	C₃³×C₄	—	2	2	no	—
68	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
70	C₃×C₃⋊Dic₃	—	2	2	no	—
71	C₃×#135	—	9	1	no	—
72	C₃×#131	—	9	2	no	—
73	#904	—	3	4	no	—
74	#901	—	3	8	no	—
75	C₃×#108	—	9	1	no	—
76	C₃×#104	—	9	2	no	—
77	C₃×C₄×C₈	—	2	4	no	—
78	C₃×C₄×C₈	C₃×D₃	1	4	yes	complemented
79	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
80	#819	—	3	4	no	—
81	#805	—	3	8	no	—
82	C₂×C₃²×C₄	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	#638	—	2	2	no	—
89	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	2	yes	—
90	C₄×C₃⋊D₃	—	2	2	no	—
91	#634	C₂×C₃×C₄	1	2	yes	—
92	C₃×#67	—	6	8	no	—
93	C₃×D₁₂	—	3	8	no	—
94	C₃×C₄×D₃	—	6	8	no	—
95	C₃×Dic₆	—	3	8	no	—
96	C₃²×C₈	—	2	4	no	—
97	C₃²×C₈	—	2	4	no	—
98	C₃²×C₈	—	2	4	no	—
99	C₃²×C₈	—	2	4	no	—
100	C₃²×C₈	C₄×D₃	1	4	yes	—
101	C₃²×C₈	C₄×D₃	1	4	yes	—
102	C₃²×C₈	D₁₂	1	4	yes	—
103	C₃²×C₈	D₁₂	1	4	yes	—
104	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
107	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
108	C₃⋊(C₃⋊C₈)	—	2	2	no	—
109	C₃×C₃⋊C₈	—	6	8	no	—
110	#443	—	9	1	no	Sylow
111	C₂×C₃³	—	2	1	no	—
112	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
113	C₃×C₃⋊D₃	—	4	2	no	—
114	C₃×#41	—	9	2	no	—
115	#235	—	3	8	no	—
116	C₃×Q₁₆	—	9	1	no	—
117	C₃×SD₁₆	—	9	2	no	—
118	C₃×D₈	—	9	1	no	—
119	C₃×M₁₆	—	9	2	no	—
120	C₃×M₁₆	—	18	2	no	—
121	C₂×C₃×C₈	—	2	4	no	—
122	C₂×C₃×C₈	—	2	4	no	—
123	C₂×C₃×C₈	—	18	2	no	—
124	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
125	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
126	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
128	C₃×C₄²	—	2	4	no	—
129	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
130	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
131	#208	—	3	8	no	—
132	Dic₁₂	—	3	4	no	—
133	D₂₄	—	3	4	no	—
134	#204	—	3	8	no	—
135	ΓL(1,𝔽₂₅)	—	6	8	no	—
136	D₃×C₈	—	6	8	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	#235	1	4	yes	—
139	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
140	C₂×C₃⋊D₃	—	2	2	no	—
141	C₂×C₃×D₃	—	6	8	no	—
142	C₃²×C₄	—	2	16	no	—
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	2	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
148	C₃²×C₄	C₂×D₁₂	1	4	yes	—
149	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₃⋊Dic₃	—	2	2	no	—
152	C₃×Dic₃	—	6	8	no	—
153	#135	—	9	1	no	—
154	#131	—	9	2	no	—
155	#108	—	9	1	no	—
156	#104	—	9	2	no	—
157	C₄×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
158	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	9	2	no	—
160	C₃×D₄	—	9	2	no	—
161	C₃×D₄	—	18	2	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	18	2	no	—
167	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
168	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
169	#67	—	6	8	no	—
170	D₁₂	—	3	8	no	—
171	C₄×D₃	—	6	8	no	—
172	Dic₆	—	3	8	no	—
173	C₃×C₈	—	2	4	no	—
174	C₃×C₈	—	2	4	no	—
175	C₃×C₈	—	2	4	no	—
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	18	2	no	—
178	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
179	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
180	C₃×C₈	C₃×D₁₂	1	4	yes	—
181	C₃×C₈	C₃×D₁₂	1	4	yes	—
182	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
183	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
184	C₃×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
185	C₃×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
186	C₃⋊C₈	—	6	8	no	—
187	C₂×C₃²	—	2	8	no	—
188	C₂×C₃²	—	2	4	no	—
189	C₂×C₃²	—	2	1	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	C₄×D₁₂	1	4	yes	—
192	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
193	C₃⋊D₃	—	4	2	no	—
194	C₃×D₃	—	12	8	no	—
195	#41	—	9	2	no	—
196	Q₁₆	—	9	1	no	—
197	SD₁₆	—	9	2	no	—
198	D₈	—	9	1	no	—
199	M₁₆	—	9	2	no	—
200	M₁₆	—	18	2	no	—
201	C₂×C₈	—	18	2	no	—
202	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
204	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	18	2	no	—
207	C₂²×C₃	C₃×#235	1	4	yes	—
208	C₂²×C₃	#3828	1	1	yes	characteristic
209	C₂×D₃	—	6	8	no	—
210	C₃×C₄	—	2	16	no	—
211	C₃×C₄	—	2	8	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	18	2	no	—
216	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
217	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
218	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
219	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
220	Dic₃	—	6	8	no	—
221	C₃²	—	2	4	no	—
222	C₃²	#4700	1	4	yes	complemented
223	C₃²	C₃×#443	1	1	yes	characteristic, complemented
224	Q₈	—	9	2	no	—
225	D₄	—	9	2	no	—
226	D₄	—	18	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	18	2	no	—
229	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
230	C₈	—	18	2	no	—
231	C₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
232	C₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
233	C₈	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
234	C₈	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
235	C₂×C₃	—	2	8	no	—
236	C₂×C₃	—	2	4	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	36	2	no	—
240	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
241	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
242	D₃	—	12	8	no	—
243	C₂²	—	18	2	no	—
244	C₂²	C₃×#3828	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	18	2	no	—
247	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
248	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
249	C₃	—	2	4	no	—
250	C₃	C₃×#4700	1	4	yes	complemented
251	C₃	#10592399	1	1	yes	characteristic, direct product factor
252	C₂	—	2	1	no	—
253	C₂	—	36	2	no	—
254	C₂	C₃×C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×C₈	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃×#64113	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#443
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#64110	C₂	semidirect
3. C₃³×C₄×C₈	C₂	semidirect
4. C₃×#63646	C₂	semidirect
5. C₃×#63632	C₂	semidirect
6. #10592399	C₃	direct (N × H)
13. C₃×#3746	C₄	semidirect
14. C₃×#3745	C₄	semidirect
15. C₃×#3744	C₄	semidirect
19. #64110	C₂×C₃	semidirect
23. C₃²×C₄×C₈	D₃	semidirect
24. C₃²×C₄×C₈	C₂×C₃	semidirect
25. #63646	C₂×C₃	semidirect
26. #63632	C₂×C₃	semidirect
54. #3746	C₃×C₄	semidirect
55. #3745	C₃×C₄	semidirect
56. #3744	C₃×C₄	semidirect
78. C₃×C₄×C₈	C₃×D₃	semidirect
79. C₃×C₄×C₈	C₃⋊D₃	semidirect
157. C₄×C₈	C₃×C₃⋊D₃	semidirect
158. C₃³	#443	semidirect
222. C₃²	#4700	semidirect
223. C₃²	C₃×#443	semidirect
250. C₃	C₃×#4700	semidirect
251. C₃	#10592399	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×#64113	2
C₃×#64110	2
C₃³×C₄×C₈	2
C₃×#63646	2
C₃×#63632	2
#10592399	3
C₃×#4700	3

Subgroups — C3×#10592399 #13162738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592399 #13162738