Subgroups — C₃²×#4894 #13162330

Order 1728.

744 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#444	—	3	1	no	maximal
7	C₃×#4894	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×#97	—	3	1	no	—
27	C₃×#825	C₂×C₃	1	8	yes	complemented
28	C₃×#815	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	2	yes	—
39	C₃×#444	—	3	4	no	—
40	#4894	C₃²	1	1	yes	characteristic, direct product factor
41	C₃²×#41	—	3	2	no	—
42	C₃×#235	C₂²×C₃	1	8	yes	—
43	C₃²×Q₁₆	—	3	1	no	—
44	C₃²×SD₁₆	—	3	2	no	—
45	C₃²×D₈	—	3	1	no	—
46	C₃²×M₁₆	—	2	8	no	—
47	C₃²×M₁₆	—	6	2	no	—
48	C₃²×M₁₆	C₂×D₃	1	2	yes	—
49	C₃²×M₁₆	C₂²×C₃	1	8	yes	—
50	C₂×C₃²×C₈	—	2	4	no	—
51	C₂×C₃²×C₈	—	3	1	no	—
52	C₂×C₃²×C₈	—	6	2	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
55	C₃²×C₄²	—	3	1	no	—
56	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
57	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
58	C₃×Dic₁₂	C₃×C₄	1	4	yes	complemented
59	C₃×D₂₄	C₃×C₄	1	4	yes	complemented
60	C₃×#204	C₃×C₄	1	8	yes	complemented
61	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
62	C₃×D₃×C₈	—	2	8	no	—
63	C₂²×C₃³	#41	1	1	yes	characteristic
64	C₂×C₃²×D₃	—	2	2	no	—
65	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃²×Dic₃	—	2	2	no	—
68	C₃²×Dic₃	—	4	1	no	—
69	C₃×#135	—	3	4	no	—
70	C₃×#131	—	3	8	no	—
71	#907	C₂×C₃²	1	2	yes	complemented
72	#904	C₂×C₃²	1	1	yes	characteristic
73	C₃×#108	—	2	4	no	—
74	C₃×#108	C₃×D₃	1	4	yes	complemented
75	C₃×#108	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₃×#104	—	3	8	no	—
77	C₃×#97	—	3	4	no	—
78	#825	C₂×C₃²	1	2	yes	complemented
79	#815	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₃²×Q₈	—	3	2	no	—
81	C₃²×D₄	—	3	2	no	—
82	C₃²×D₄	—	6	2	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	6	2	no	—
85	C₂×C₃²×C₄	—	6	1	no	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
88	C₃×#67	—	2	8	no	—
89	C₂×C₃×Dic₃	—	2	4	no	—
90	C₃×D₁₂	C₂×C₃×C₄	1	8	yes	—
91	C₃×C₄×D₃	—	2	8	no	—
92	C₃×Dic₆	C₂×C₃×C₄	1	8	yes	—
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	—	2	4	no	—
95	C₃²×C₈	—	2	4	no	—
96	C₃²×C₈	—	2	8	no	—
97	C₃²×C₈	—	3	2	no	—
98	C₃²×C₈	—	4	8	no	—
99	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
100	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
101	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
102	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
103	C₃×C₃⋊C₈	C₃×D₄	1	8	yes	—
104	#444	—	3	1	no	Sylow
105	C₂×C₃³	—	2	1	no	—
106	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
107	C₃²×D₃	—	4	2	no	—
108	C₃×#41	—	3	8	no	—
109	#235	C₂²×C₃²	1	2	yes	—
110	C₃×Q₁₆	—	3	4	no	—
111	C₃×SD₁₆	—	3	8	no	—
112	C₃×D₈	—	3	4	no	—
113	C₃×M₁₆	—	2	8	no	—
114	C₃×M₁₆	—	6	8	no	—
115	C₃×M₁₆	C₂×C₃×D₃	1	8	yes	—
116	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	6	8	no	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
121	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
122	C₃×C₄²	—	3	4	no	—
123	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
124	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
125	Dic₁₂	C₃²×C₄	1	1	yes	characteristic, complemented
126	D₂₄	C₃²×C₄	1	1	yes	characteristic, complemented
127	#204	C₃²×C₄	1	2	yes	complemented
128	ΓL(1,𝔽₂₅)	—	2	2	no	—
129	D₃×C₈	—	2	2	no	—
130	C₂²×C₃²	—	2	4	no	—
131	C₂²×C₃²	—	6	2	no	—
132	C₂²×C₃²	#239	1	1	yes	characteristic
133	C₂²×C₃²	C₃×#41	1	4	yes	—
134	C₂×C₃×D₃	—	2	8	no	—
135	C₃²×C₄	—	2	4	no	—
136	C₃²×C₄	—	2	4	no	—
137	C₃²×C₄	—	6	2	no	—
138	C₃²×C₄	—	12	1	no	—
139	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
140	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
141	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
142	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
143	C₃×Dic₃	—	2	8	no	—
144	C₃×Dic₃	—	4	4	no	—
145	#135	—	3	1	no	—
146	#131	—	3	2	no	—
147	#108	C₃²×D₃	1	1	yes	characteristic, complemented
148	#104	—	3	2	no	—
149	#97	—	3	1	no	—
150	C₃³	#444	1	1	yes	characteristic, complemented, Sylow
151	C₃×Q₈	—	3	8	no	—
152	C₃×D₄	—	3	8	no	—
153	C₃×D₄	—	6	8	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	6	8	no	—
156	C₂×C₃×C₄	—	6	4	no	—
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
158	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
159	#67	—	2	2	no	—
160	C₂×Dic₃	—	2	1	no	—
161	D₁₂	C₂×C₃²×C₄	1	2	yes	—
162	C₄×D₃	—	2	2	no	—
163	Dic₆	C₂×C₃²×C₄	1	2	yes	—
164	C₃×C₈	—	2	8	no	—
165	C₃×C₈	—	2	2	no	—
166	C₃×C₈	—	2	4	no	—
167	C₃×C₈	—	2	4	no	—
168	C₃×C₈	—	3	8	no	—
169	C₃×C₈	—	4	8	no	—
170	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
171	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
172	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
173	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
174	C₃⋊C₈	C₃²×D₄	1	2	yes	—
175	C₂×C₃²	—	2	1	no	—
176	C₂×C₃²	—	2	4	no	—
177	C₂×C₃²	—	2	4	no	—
178	C₂×C₃²	—	4	4	no	—
179	C₂×C₃²	—	12	2	no	—
180	C₂×C₃²	#893	1	1	yes	characteristic
181	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
182	C₃×D₃	—	4	8	no	—
183	#41	—	3	2	no	—
184	Q₁₆	—	3	1	no	—
185	SD₁₆	—	3	2	no	—
186	D₈	—	3	1	no	—
187	M₁₆	—	6	2	no	—
188	M₁₆	C₂×C₃²×D₃	1	2	yes	—
189	C₂×C₈	—	3	1	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
192	C₄²	—	3	1	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	6	8	no	—
195	C₂²×C₃	C₃×#239	1	4	yes	—
196	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
197	C₂×D₃	—	2	2	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	6	8	no	—
201	C₃×C₄	—	12	4	no	—
202	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
203	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
204	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
205	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
206	Dic₃	—	2	2	no	—
207	Dic₃	—	4	1	no	—
208	C₃²	—	2	4	no	—
209	C₃²	#4894	1	1	yes	characteristic, direct product factor
210	C₃²	C₃×#444	1	4	yes	complemented
211	Q₈	—	3	2	no	—
212	D₄	—	3	2	no	—
213	D₄	—	6	2	no	—
214	C₂×C₄	—	6	2	no	—
215	C₂×C₄	—	6	1	no	—
216	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	3	2	no	—
219	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
220	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
221	C₂×C₃	—	2	4	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	4	no	—
224	C₂×C₃	—	4	4	no	—
225	C₂×C₃	—	12	8	no	—
226	C₂×C₃	C₃×#893	1	4	yes	—
227	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
228	D₃	—	4	2	no	—
229	C₂²	—	6	2	no	—
230	C₂²	C₃²×#239	1	1	yes	characteristic
231	C₄	—	6	2	no	—
232	C₄	—	12	1	no	—
233	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
234	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
235	C₃	—	2	4	no	—
236	C₃	C₃×#4894	1	4	yes	direct product factor
237	C₃	C₃²×#444	1	1	yes	characteristic, complemented
238	C₂	—	2	1	no	—
239	C₂	—	12	2	no	—
240	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×#444
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×#815	C₂	semidirect
7. C₃×#4894	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×#815	C₂×C₃	semidirect
40. #4894	C₃²	direct (N × H)
58. C₃×Dic₁₂	C₃×C₄	semidirect
59. C₃×D₂₄	C₃×C₄	semidirect
60. C₃×#204	C₃×C₄	semidirect
71. #907	C₂×C₃²	semidirect
74. C₃×#108	C₃×D₃	semidirect
75. C₃×#108	C₂×C₃²	semidirect
78. #825	C₂×C₃²	semidirect
79. #815	C₂×C₃²	semidirect
125. Dic₁₂	C₃²×C₄	semidirect
126. D₂₄	C₃²×C₄	semidirect
127. #204	C₃²×C₄	semidirect
147. #108	C₃²×D₃	semidirect
150. C₃³	#444	semidirect
209. C₃²	#4894	direct (N × H)
210. C₃²	C₃×#444	semidirect
236. C₃	C₃×#4894	direct (N × H)
237. C₃	C₃²×#444	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×#815	2
C₃²×#444	3
C₃×#4894	3

Subgroups — C32×#4894 #13162330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4894 #13162330