Subgroups — C₃²×#4893 #13162329

Order 1728.

732 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₃³×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#822	C₂	1	2	yes	maximal, complemented
5	C₃²×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#430	—	3	1	no	maximal
7	C₃×#4893	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₃³×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃²×#208	C₂²	1	2	yes	—
12	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×ΓL(1,𝔽₂₅)	—	2	2	no	—
14	C₃²×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
15	C₃²×D₃×C₈	—	2	2	no	—
16	C₂×C₃²×M₁₆	—	3	1	no	—
17	C₂×C₃²×M₁₆	—	3	2	no	—
18	C₃×D₃×M₁₆	C₂×C₃	1	8	yes	—
19	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	4	yes	—
20	C₃²×#108	—	2	4	no	—
21	C₃²×#108	—	3	2	no	—
22	C₃²×#108	—	3	1	no	—
23	C₃²×#108	D₃	1	1	yes	characteristic, complemented
24	C₃²×#108	C₂×C₃	1	4	yes	complemented
25	C₃×#822	C₂×C₃	1	8	yes	complemented
26	C₃×#819	C₂×C₃	1	4	yes	complemented
27	C₂²×C₃²×D₃	Q₈	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃²×Dic₃	Q₈	1	1	yes	characteristic
30	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
31	C₃²×C₄×D₃	D₄	1	2	yes	—
32	C₃³×C₈	—	2	2	no	—
33	C₃³×C₈	C₂×C₄	1	2	yes	—
34	C₃²×C₃⋊C₈	—	2	2	no	—
35	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×#430	—	3	4	no	—
37	#4893	C₃²	1	1	yes	characteristic, direct product factor
38	C₂²×C₃²×C₄	—	3	1	no	—
39	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
40	C₃²×M₁₆	—	2	8	no	—
41	C₃²×M₁₆	—	3	4	no	—
42	C₃²×M₁₆	—	3	2	no	—
43	C₃²×M₁₆	—	6	2	no	—
44	C₃²×M₁₆	C₂×D₃	1	2	yes	—
45	C₃²×M₁₆	C₂²×C₃	1	8	yes	—
46	C₂×C₃²×C₈	—	2	4	no	—
47	C₂×C₃²×C₈	—	3	1	no	—
48	C₂×C₃²×C₈	—	6	2	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
51	C₃×#208	C₂²×C₃	1	8	yes	—
52	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
53	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
54	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	16	yes	—
55	C₃×D₃×C₈	—	2	8	no	—
56	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃²×D₃	—	4	1	no	—
58	C₂×C₃²×D₃	C₄⋊C₄	1	2	yes	—
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
61	C₃²×Dic₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₃×M₁₆	—	3	4	no	—
63	C₂×C₃×M₁₆	—	3	8	no	—
64	D₃×M₁₆	C₂×C₃²	1	2	yes	—
65	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	1	yes	characteristic
66	C₃×#108	—	2	4	no	—
67	C₃×#108	—	3	8	no	—
68	C₃×#108	—	3	4	no	—
69	C₃×#108	C₃×D₃	1	4	yes	complemented
70	C₃×#108	C₂×C₃²	1	1	yes	characteristic, complemented
71	#822	C₂×C₃²	1	2	yes	complemented
72	#819	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂³×C₃²	—	3	1	no	—
74	C₂²×C₃×D₃	C₃×Q₈	1	4	yes	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	—	3	1	no	—
77	C₂×C₃²×C₄	—	3	2	no	—
78	C₂×C₃²×C₄	—	3	2	no	—
79	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
81	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
82	C₃×C₄×D₃	C₂×C₃×C₄	1	8	yes	—
83	C₃×C₄×D₃	C₃×D₄	1	8	yes	—
84	C₃²×C₈	—	2	2	no	—
85	C₃²×C₈	—	2	8	no	—
86	C₃²×C₈	—	2	8	no	—
87	C₃²×C₈	—	3	2	no	—
88	C₃²×C₈	—	4	8	no	—
89	C₃²×C₈	—	6	2	no	—
90	C₃²×C₈	C₄×D₃	1	2	yes	—
91	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
92	C₃×C₃⋊C₈	—	2	8	no	—
93	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
94	#430	—	3	1	no	Sylow
95	C₂×C₃³	—	2	1	no	—
96	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃²×D₃	—	2	2	no	—
98	C₂²×C₃×C₄	—	3	4	no	—
99	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
100	C₃×M₁₆	—	2	8	no	—
101	C₃×M₁₆	—	3	16	no	—
102	C₃×M₁₆	—	3	8	no	—
103	C₃×M₁₆	—	6	8	no	—
104	C₃×M₁₆	C₂×C₃×D₃	1	8	yes	—
105	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	—	3	4	no	—
108	C₂×C₃×C₈	—	6	8	no	—
109	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
110	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
111	#208	C₂²×C₃²	1	2	yes	—
112	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
113	ΓL(1,𝔽₂₅)	—	2	2	no	—
114	ΓL(1,𝔽₂₅)	C₃²×C₄	1	4	yes	—
115	D₃×C₈	—	2	2	no	—
116	C₂²×C₃²	—	2	4	no	—
117	C₂²×C₃²	—	3	2	no	—
118	C₂²×C₃²	—	12	1	no	—
119	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
120	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	—
121	C₂×C₃×D₃	—	4	4	no	—
122	C₂×C₃×D₃	C₃×C₄⋊C₄	1	8	yes	—
123	C₃²×C₄	—	2	4	no	—
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	—	3	2	no	—
126	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
127	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
128	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
129	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
130	C₃×Dic₃	C₃×C₄⋊C₄	1	8	yes	—
131	C₂×M₁₆	—	3	1	no	—
132	C₂×M₁₆	—	3	2	no	—
133	#108	—	3	2	no	—
134	#108	—	3	1	no	—
135	#108	C₃²×D₃	1	1	yes	characteristic, complemented
136	C₃³	#430	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₃	—	3	4	no	—
138	C₂²×D₃	C₃²×Q₈	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	3	4	no	—
141	C₂×C₃×C₄	—	3	8	no	—
142	C₂×C₃×C₄	—	3	8	no	—
143	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
144	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
145	C₂×Dic₃	C₃²×Q₈	1	1	yes	characteristic
146	C₄×D₃	C₂×C₃²×C₄	1	2	yes	—
147	C₄×D₃	C₃²×D₄	1	2	yes	—
148	C₃×C₈	—	2	8	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	—	2	8	no	—
151	C₃×C₈	—	3	8	no	—
152	C₃×C₈	—	4	8	no	—
153	C₃×C₈	—	6	8	no	—
154	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
155	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
156	C₃⋊C₈	—	2	2	no	—
157	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	—	2	4	no	—
161	C₂×C₃²	—	4	4	no	—
162	C₂×C₃²	—	6	2	no	—
163	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
165	C₃×D₃	—	2	8	no	—
166	C₂²×C₄	—	3	1	no	—
167	M₁₆	—	3	4	no	—
168	M₁₆	—	3	2	no	—
169	M₁₆	—	6	2	no	—
170	M₁₆	C₂×C₃²×D₃	1	2	yes	—
171	C₂×C₈	—	3	1	no	—
172	C₂×C₈	—	6	2	no	—
173	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	4	no	—
175	C₂²×C₃	—	3	8	no	—
176	C₂²×C₃	—	12	4	no	—
177	C₂²×C₃	C₃×D₃×Q₈	1	4	yes	—
178	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
179	C₂×D₃	—	4	1	no	—
180	C₂×D₃	C₃²×C₄⋊C₄	1	2	yes	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	—	3	8	no	—
184	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
185	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
186	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
187	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
188	Dic₃	C₃²×C₄⋊C₄	1	2	yes	—
189	C₃²	—	2	4	no	—
190	C₃²	#4893	1	1	yes	characteristic, direct product factor
191	C₃²	C₃×#430	1	4	yes	complemented
192	C₂³	—	3	1	no	—
193	C₂×C₄	—	3	1	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	3	2	no	—
199	C₈	—	6	2	no	—
200	C₈	C₃²×C₄×D₃	1	2	yes	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	—	6	8	no	—
206	C₂×C₃	C₃×D₃×C₄⋊C₄	1	4	yes	—
207	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
208	D₃	—	2	2	no	—
209	C₂²	—	3	2	no	—
210	C₂²	—	12	1	no	—
211	C₂²	C₃²×D₃×Q₈	1	1	yes	characteristic
212	C₄	—	3	2	no	—
213	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
214	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
215	C₃	—	2	4	no	—
216	C₃	C₃×#4893	1	4	yes	direct product factor
217	C₃	C₃²×#430	1	1	yes	characteristic, complemented
218	C₂	—	2	1	no	—
219	C₂	—	6	2	no	—
220	C₂	C₃²×D₃×C₄⋊C₄	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	C₂×C₄⋊C₄
prime radical	C₂²×C₃²×D₃	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#108	D₃
hypocenter	C₃	C₃²×#430
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
3. C₃³×#108	C₂	semidirect
4. C₃²×#822	C₂	semidirect
5. C₃²×#819	C₂	semidirect
7. C₃×#4893	C₃	direct (N × H)
23. C₃²×#108	D₃	semidirect
24. C₃²×#108	C₂×C₃	semidirect
25. C₃×#822	C₂×C₃	semidirect
26. C₃×#819	C₂×C₃	semidirect
37. #4893	C₃²	direct (N × H)
69. C₃×#108	C₃×D₃	semidirect
70. C₃×#108	C₂×C₃²	semidirect
71. #822	C₂×C₃²	semidirect
72. #819	C₂×C₃²	semidirect
135. #108	C₃²×D₃	semidirect
136. C₃³	#430	semidirect
190. C₃²	#4893	direct (N × H)
191. C₃²	C₃×#430	semidirect
216. C₃	C₃×#4893	direct (N × H)
217. C₃	C₃²×#430	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₃×M₁₆	2
C₂×C₃²×ΓL(1,𝔽₂₅)	2
C₃³×#108	2
C₃²×#822	2
C₃²×#819	2
C₃²×#430	3
C₃×#4893	3

Subgroups — C32×#4893 #13162329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4893 #13162329