Subgroups — C₃×#10592416 #13162755

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#97	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	#10592416	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4717	—	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×#97	—	2	4	no	—
23	C₃²×#97	D₃	1	4	yes	complemented
24	C₃²×#97	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	2	yes	—
38	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₃×#444	—	9	1	no	—
40	#4717	—	3	4	no	—
41	#3828	C₂²×C₃	1	2	yes	—
42	C₃×#235	—	3	8	no	—
43	C₂×C₃²×C₈	—	2	4	no	—
44	C₂×C₃²×C₈	—	2	4	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
46	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
47	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₃²×C₄²	—	2	4	no	—
50	C₃²×C₄²	C₂×D₃	1	4	yes	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	#3748	C₂²×C₃	1	2	yes	—
53	#3746	C₃×C₄	1	1	yes	characteristic, complemented
54	#3745	C₃×C₄	1	1	yes	characteristic, complemented
55	#3744	C₃×C₄	1	2	yes	complemented
56	#3743	—	2	2	no	—
57	C₈×C₃⋊D₃	—	2	2	no	—
58	C₃×#208	—	3	8	no	—
59	C₃×Dic₁₂	—	3	4	no	—
60	C₃×D₂₄	—	3	4	no	—
61	C₃×#204	—	3	8	no	—
62	C₃×ΓL(1,𝔽₂₅)	—	6	8	no	—
63	C₃×D₃×C₈	—	6	8	no	—
64	C₂²×C₃³	#41	1	1	yes	characteristic
65	C₂×C₃×C₃⋊D₃	—	2	2	no	—
66	C₃³×C₄	—	4	1	no	—
67	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
69	C₃×C₃⋊Dic₃	—	2	2	no	—
70	C₃×#135	—	9	1	no	—
71	C₃×#131	—	9	2	no	—
72	#904	—	3	4	no	—
73	#901	—	3	8	no	—
74	C₃×#108	—	9	1	no	—
75	C₃×#104	—	9	2	no	—
76	C₃×#97	—	2	4	no	—
77	C₃×#97	C₃×D₃	1	4	yes	complemented
78	C₃×#97	C₃⋊D₃	1	1	yes	characteristic, complemented
79	#819	—	3	4	no	—
80	#805	—	3	8	no	—
81	C₂×C₃²×C₄	—	2	8	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	1	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
87	#638	—	2	2	no	—
88	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	2	yes	—
89	C₄×C₃⋊D₃	—	2	2	no	—
90	#634	C₂×C₃×C₄	1	2	yes	—
91	C₃×#67	—	6	8	no	—
92	C₃×D₁₂	—	3	8	no	—
93	C₃×C₄×D₃	—	6	8	no	—
94	C₃×Dic₆	—	3	8	no	—
95	C₃²×C₈	—	2	4	no	—
96	C₃²×C₈	—	2	4	no	—
97	C₃²×C₈	—	2	8	no	—
98	C₃²×C₈	C₄×D₃	1	4	yes	—
99	C₃²×C₈	C₄×D₃	1	4	yes	—
100	C₃²×C₈	D₁₂	1	8	yes	—
101	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
102	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
103	C₃²×C₈	C₃×D₄	1	2	yes	—
104	C₃⋊(C₃⋊C₈)	—	2	2	no	—
105	C₃×C₃⋊C₈	—	6	8	no	—
106	#444	—	9	1	no	Sylow
107	C₂×C₃³	—	2	1	no	—
108	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
109	C₃×C₃⋊D₃	—	4	2	no	—
110	C₃×#41	—	9	2	no	—
111	#235	—	3	8	no	—
112	C₃×Q₁₆	—	9	1	no	—
113	C₃×SD₁₆	—	9	2	no	—
114	C₃×D₈	—	9	1	no	—
115	C₃×M₁₆	—	9	2	no	—
116	C₃×M₁₆	—	18	2	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	—	18	2	no	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
121	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
122	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
126	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
127	#208	—	3	8	no	—
128	Dic₁₂	—	3	4	no	—
129	D₂₄	—	3	4	no	—
130	#204	—	3	8	no	—
131	ΓL(1,𝔽₂₅)	—	6	8	no	—
132	D₃×C₈	—	6	8	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	#235	1	4	yes	—
135	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
136	C₂×C₃⋊D₃	—	2	2	no	—
137	C₂×C₃×D₃	—	6	8	no	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	4	8	no	—
141	C₃²×C₄	—	4	4	no	—
142	C₃²×C₄	—	4	1	no	—
143	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
144	C₃²×C₄	C₂×D₁₂	1	4	yes	—
145	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
147	C₃⋊Dic₃	—	2	2	no	—
148	C₃×Dic₃	—	6	8	no	—
149	#135	—	9	1	no	—
150	#131	—	9	2	no	—
151	#108	—	9	1	no	—
152	#104	—	9	2	no	—
153	#97	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
154	C₃³	#444	1	1	yes	characteristic, complemented, Sylow
155	C₃×Q₈	—	9	2	no	—
156	C₃×D₄	—	9	2	no	—
157	C₃×D₄	—	18	2	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	18	2	no	—
163	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
164	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
165	#67	—	6	8	no	—
166	D₁₂	—	3	8	no	—
167	C₄×D₃	—	6	8	no	—
168	Dic₆	—	3	8	no	—
169	C₃×C₈	—	2	4	no	—
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	2	8	no	—
172	C₃×C₈	—	18	2	no	—
173	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
174	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
175	C₃×C₈	C₃×D₁₂	1	8	yes	—
176	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
177	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
178	C₃×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
179	C₃⋊C₈	—	6	8	no	—
180	C₂×C₃²	—	2	8	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	1	no	—
183	C₂×C₃²	—	2	4	no	—
184	C₂×C₃²	C₄×D₁₂	1	4	yes	—
185	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
186	C₃⋊D₃	—	4	2	no	—
187	C₃×D₃	—	12	8	no	—
188	#41	—	9	2	no	—
189	Q₁₆	—	9	1	no	—
190	SD₁₆	—	9	2	no	—
191	D₈	—	9	1	no	—
192	M₁₆	—	9	2	no	—
193	M₁₆	—	18	2	no	—
194	C₂×C₈	—	18	2	no	—
195	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
197	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	18	2	no	—
200	C₂²×C₃	C₃×#235	1	4	yes	—
201	C₂²×C₃	#3828	1	1	yes	characteristic
202	C₂×D₃	—	6	8	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	4	8	no	—
206	C₃×C₄	—	4	4	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	18	2	no	—
209	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
210	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
211	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
212	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
213	Dic₃	—	6	8	no	—
214	C₃²	—	2	4	no	—
215	C₃²	#4717	1	4	yes	complemented
216	C₃²	C₃×#444	1	1	yes	characteristic, complemented
217	Q₈	—	9	2	no	—
218	D₄	—	9	2	no	—
219	D₄	—	18	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	18	2	no	—
222	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
223	C₈	—	18	2	no	—
224	C₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
225	C₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
226	C₈	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	—
227	C₂×C₃	—	2	8	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	36	2	no	—
232	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
233	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
234	D₃	—	12	8	no	—
235	C₂²	—	18	2	no	—
236	C₂²	C₃×#3828	1	1	yes	characteristic
237	C₄	—	4	1	no	—
238	C₄	—	18	2	no	—
239	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
240	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
241	C₃	—	2	4	no	—
242	C₃	C₃×#4717	1	4	yes	complemented
243	C₃	#10592416	1	1	yes	characteristic, direct product factor
244	C₂	—	2	1	no	—
245	C₂	—	36	2	no	—
246	C₂	C₃×C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#97	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃×#64113	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	C₃²	C₃×#444
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₃×#64110	C₂	semidirect
3. C₃³×#97	C₂	semidirect
4. C₃×#63646	C₂	semidirect
5. C₃×#63632	C₂	semidirect
6. #10592416	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×#97	D₃	semidirect
24. C₃²×#97	C₂×C₃	semidirect
25. #63646	C₂×C₃	semidirect
26. #63632	C₂×C₃	semidirect
53. #3746	C₃×C₄	semidirect
54. #3745	C₃×C₄	semidirect
55. #3744	C₃×C₄	semidirect
77. C₃×#97	C₃×D₃	semidirect
78. C₃×#97	C₃⋊D₃	semidirect
153. #97	C₃×C₃⋊D₃	semidirect
154. C₃³	#444	semidirect
215. C₃²	#4717	semidirect
216. C₃²	C₃×#444	semidirect
242. C₃	C₃×#4717	semidirect
243. C₃	#10592416	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×#97	2
C₃×#63646	2
C₃×#63632	2
#10592416	3
C₃×#4717	3

Subgroups — C3×#10592416 #13162755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592416 #13162755