Subgroups — C₃²×#4895 #13162331

Order 1728.

756 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	C₃²×#907	C₂	1	2	yes	maximal, complemented
2	C₃²×#904	C₂	1	1	yes	maximal, characteristic
3	C₃³×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#825	C₂	1	2	yes	maximal, complemented
5	C₃²×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#443	—	3	1	no	maximal
7	C₃×#4895	C₃	1	4	yes	maximal, direct product factor
8	C₃²×#235	C₂²	1	2	yes	—
9	C₃³×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×Dic₁₂	C₄	1	1	yes	characteristic, complemented
14	C₃²×D₂₄	C₄	1	1	yes	characteristic, complemented
15	C₃²×#204	C₄	1	2	yes	complemented
16	C₃²×ΓL(1,𝔽₂₅)	—	2	2	no	—
17	C₃²×D₃×C₈	—	2	2	no	—
18	C₃²×#135	—	3	1	no	—
19	C₃²×#131	—	3	2	no	—
20	C₃×#907	C₂×C₃	1	8	yes	complemented
21	C₃×#904	C₂×C₃	1	4	yes	—
22	C₃²×#108	—	2	4	no	—
23	C₃²×#108	D₃	1	1	yes	characteristic, complemented
24	C₃²×#108	C₂×C₃	1	4	yes	complemented
25	C₃²×#104	—	3	2	no	—
26	C₃²×C₄×C₈	—	3	1	no	—
27	C₃×#825	C₂×C₃	1	8	yes	complemented
28	C₃×C₈×Dic₃	C₂×C₃	1	4	yes	complemented
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₃²×#67	—	2	2	no	—
31	C₂×C₃²×Dic₃	—	2	1	no	—
32	C₃²×D₁₂	C₂×C₄	1	2	yes	—
33	C₃²×C₄×D₃	—	2	2	no	—
34	C₃²×Dic₆	C₂×C₄	1	2	yes	—
35	C₃³×C₈	—	2	2	no	—
36	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
39	C₃²×C₃⋊C₈	D₄	1	1	yes	characteristic
40	C₃×#443	—	3	4	no	—
41	#4895	C₃²	1	1	yes	characteristic, direct product factor
42	C₃²×#41	—	3	2	no	—
43	C₃×#235	C₂²×C₃	1	8	yes	—
44	C₃²×Q₁₆	—	3	1	no	—
45	C₃²×SD₁₆	—	3	2	no	—
46	C₃²×D₈	—	3	1	no	—
47	C₃²×M₁₆	—	2	8	no	—
48	C₃²×M₁₆	—	6	2	no	—
49	C₃²×M₁₆	C₂×D₃	1	2	yes	—
50	C₃²×M₁₆	C₂²×C₃	1	8	yes	—
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	—	3	1	no	—
53	C₂×C₃²×C₈	—	6	2	no	—
54	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
56	C₃²×C₄²	—	3	1	no	—
57	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
58	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
59	C₃×Dic₁₂	C₃×C₄	1	4	yes	complemented
60	C₃×D₂₄	C₃×C₄	1	4	yes	complemented
61	C₃×#204	C₃×C₄	1	8	yes	complemented
62	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
63	C₃×D₃×C₈	—	2	8	no	—
64	C₂²×C₃³	#41	1	1	yes	characteristic
65	C₂×C₃²×D₃	—	2	2	no	—
66	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃²×Dic₃	—	2	2	no	—
69	C₃²×Dic₃	—	2	2	no	—
70	C₃×#135	—	3	4	no	—
71	C₃×#131	—	3	8	no	—
72	#907	C₂×C₃²	1	2	yes	complemented
73	#904	C₂×C₃²	1	1	yes	characteristic
74	C₃×#108	—	2	4	no	—
75	C₃×#108	C₃×D₃	1	4	yes	complemented
76	C₃×#108	C₂×C₃²	1	1	yes	characteristic, complemented
77	C₃×#104	—	3	8	no	—
78	C₃×C₄×C₈	—	3	4	no	—
79	#825	C₂×C₃²	1	2	yes	complemented
80	C₈×Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₃²×Q₈	—	3	2	no	—
82	C₃²×D₄	—	3	2	no	—
83	C₃²×D₄	—	6	2	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	6	1	no	—
86	C₂×C₃²×C₄	—	6	2	no	—
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
89	C₃×#67	—	2	8	no	—
90	C₂×C₃×Dic₃	—	2	4	no	—
91	C₃×D₁₂	C₂×C₃×C₄	1	8	yes	—
92	C₃×C₄×D₃	—	2	8	no	—
93	C₃×Dic₆	C₂×C₃×C₄	1	8	yes	—
94	C₃²×C₈	—	2	2	no	—
95	C₃²×C₈	—	2	4	no	—
96	C₃²×C₈	—	2	4	no	—
97	C₃²×C₈	—	2	8	no	—
98	C₃²×C₈	—	3	1	no	—
99	C₃²×C₈	—	3	1	no	—
100	C₃²×C₈	—	4	8	no	—
101	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
102	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
103	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
104	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
105	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
106	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
107	#443	—	3	1	no	Sylow
108	C₂×C₃³	—	2	1	no	—
109	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
110	C₃²×D₃	—	4	2	no	—
111	C₃×#41	—	3	8	no	—
112	#235	C₂²×C₃²	1	2	yes	—
113	C₃×Q₁₆	—	3	4	no	—
114	C₃×SD₁₆	—	3	8	no	—
115	C₃×D₈	—	3	4	no	—
116	C₃×M₁₆	—	2	8	no	—
117	C₃×M₁₆	—	6	8	no	—
118	C₃×M₁₆	C₂×C₃×D₃	1	8	yes	—
119	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	—	3	4	no	—
122	C₂×C₃×C₈	—	6	8	no	—
123	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
124	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
125	C₃×C₄²	—	3	4	no	—
126	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
127	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
128	Dic₁₂	C₃²×C₄	1	1	yes	characteristic, complemented
129	D₂₄	C₃²×C₄	1	1	yes	characteristic, complemented
130	#204	C₃²×C₄	1	2	yes	complemented
131	ΓL(1,𝔽₂₅)	—	2	2	no	—
132	D₃×C₈	—	2	2	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	6	2	no	—
135	C₂²×C₃²	#239	1	1	yes	characteristic
136	C₂²×C₃²	C₃×#41	1	4	yes	—
137	C₂×C₃×D₃	—	2	8	no	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	6	2	no	—
141	C₃²×C₄	—	6	2	no	—
142	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
143	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
144	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
145	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
146	C₃×Dic₃	—	2	8	no	—
147	C₃×Dic₃	—	2	8	no	—
148	#135	—	3	1	no	—
149	#131	—	3	2	no	—
150	#108	C₃²×D₃	1	1	yes	characteristic, complemented
151	#104	—	3	2	no	—
152	C₄×C₈	—	3	1	no	—
153	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
154	C₃×Q₈	—	3	8	no	—
155	C₃×D₄	—	3	8	no	—
156	C₃×D₄	—	6	8	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	6	4	no	—
159	C₂×C₃×C₄	—	6	8	no	—
160	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
161	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
162	#67	—	2	2	no	—
163	C₂×Dic₃	—	2	1	no	—
164	D₁₂	C₂×C₃²×C₄	1	2	yes	—
165	C₄×D₃	—	2	2	no	—
166	Dic₆	C₂×C₃²×C₄	1	2	yes	—
167	C₃×C₈	—	2	8	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	2	4	no	—
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	3	4	no	—
172	C₃×C₈	—	3	4	no	—
173	C₃×C₈	—	4	8	no	—
174	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
175	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
176	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
177	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
178	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
179	C₃⋊C₈	C₃²×D₄	1	1	yes	characteristic
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	4	4	no	—
184	C₂×C₃²	—	12	2	no	—
185	C₂×C₃²	#893	1	1	yes	characteristic
186	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
187	C₃×D₃	—	4	8	no	—
188	#41	—	3	2	no	—
189	Q₁₆	—	3	1	no	—
190	SD₁₆	—	3	2	no	—
191	D₈	—	3	1	no	—
192	M₁₆	—	6	2	no	—
193	M₁₆	C₂×C₃²×D₃	1	2	yes	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	6	2	no	—
196	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
197	C₄²	—	3	1	no	—
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	6	8	no	—
200	C₂²×C₃	C₃×#239	1	4	yes	—
201	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
202	C₂×D₃	—	2	2	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	6	8	no	—
206	C₃×C₄	—	6	8	no	—
207	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
208	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
209	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
210	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
211	Dic₃	—	2	2	no	—
212	Dic₃	—	2	2	no	—
213	C₃²	—	2	4	no	—
214	C₃²	#4895	1	1	yes	characteristic, direct product factor
215	C₃²	C₃×#443	1	4	yes	complemented
216	Q₈	—	3	2	no	—
217	D₄	—	3	2	no	—
218	D₄	—	6	2	no	—
219	C₂×C₄	—	6	1	no	—
220	C₂×C₄	—	6	2	no	—
221	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	3	1	no	—
224	C₈	—	3	1	no	—
225	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
226	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
227	C₂×C₃	—	2	4	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	4	4	no	—
231	C₂×C₃	—	12	8	no	—
232	C₂×C₃	C₃×#893	1	4	yes	—
233	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
234	D₃	—	4	2	no	—
235	C₂²	—	6	2	no	—
236	C₂²	C₃²×#239	1	1	yes	characteristic
237	C₄	—	6	2	no	—
238	C₄	—	6	2	no	—
239	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
240	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
241	C₃	—	2	4	no	—
242	C₃	C₃×#4895	1	4	yes	direct product factor
243	C₃	C₃²×#443	1	1	yes	characteristic, complemented
244	C₂	—	2	1	no	—
245	C₂	—	12	2	no	—
246	C₂	C₃²×#893	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#108	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃²×#904	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#108	D₃
hypocenter	C₃	C₃²×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#108

Complements

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

N	H ≅ G/N	type
1. C₃²×#907	C₂	semidirect
3. C₃³×#108	C₂	semidirect
4. C₃²×#825	C₂	semidirect
5. C₃²×C₈×Dic₃	C₂	semidirect
7. C₃×#4895	C₃	direct (N × H)
13. C₃²×Dic₁₂	C₄	semidirect
14. C₃²×D₂₄	C₄	semidirect
15. C₃²×#204	C₄	semidirect
20. C₃×#907	C₂×C₃	semidirect
23. C₃²×#108	D₃	semidirect
24. C₃²×#108	C₂×C₃	semidirect
27. C₃×#825	C₂×C₃	semidirect
28. C₃×C₈×Dic₃	C₂×C₃	semidirect
41. #4895	C₃²	direct (N × H)
59. C₃×Dic₁₂	C₃×C₄	semidirect
60. C₃×D₂₄	C₃×C₄	semidirect
61. C₃×#204	C₃×C₄	semidirect
72. #907	C₂×C₃²	semidirect
75. C₃×#108	C₃×D₃	semidirect
76. C₃×#108	C₂×C₃²	semidirect
79. #825	C₂×C₃²	semidirect
80. C₈×Dic₃	C₂×C₃²	semidirect
128. Dic₁₂	C₃²×C₄	semidirect
129. D₂₄	C₃²×C₄	semidirect
130. #204	C₃²×C₄	semidirect
150. #108	C₃²×D₃	semidirect
153. C₃³	#443	semidirect
214. C₃²	#4895	direct (N × H)
215. C₃²	C₃×#443	semidirect
242. C₃	C₃×#4895	direct (N × H)
243. C₃	C₃²×#443	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#907	2
C₃²×#904	2
C₃³×#108	2
C₃²×#825	2
C₃²×C₈×Dic₃	2
C₃²×#443	3
C₃×#4895	3

Subgroups — C32×#4895 #13162331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4895 #13162331