Subgroups — C₃²×#4688 #13162124

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#436	—	3	1	no	maximal
7	C₃×#4688	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
14	C₃²×C₄⋊Dic₃	C₄	1	1	yes	characteristic
15	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃²×ΓL(1,𝔽₂₅)	—	2	4	no	—
17	C₂×C₃²×M₁₆	—	3	2	no	—
18	C₃²×C₄×D₄	—	3	1	no	—
19	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	8	yes	complemented
20	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
21	C₃²×C₄⋊C₈	—	3	1	no	—
22	C₃²×C₂²⋊C₈	—	3	2	no	—
23	C₃²×C₄×C₈	—	2	4	no	—
24	C₃²×C₄×C₈	D₃	1	1	yes	characteristic, complemented
25	C₃²×C₄×C₈	C₂×C₃	1	4	yes	complemented
26	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	8	yes	complemented
27	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
33	C₃²×D₁₂	—	2	2	no	—
34	C₃²×C₄×D₃	—	2	2	no	—
35	C₃²×C₄×D₃	—	2	2	no	—
36	C₃³×C₈	—	2	1	no	—
37	C₃³×C₈	D₄	1	2	yes	complemented
38	C₃²×C₃⋊C₈	—	2	2	no	—
39	C₃×#436	—	3	4	no	—
40	#4688	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×D₄	—	3	1	no	—
42	C₂²×C₃²×C₄	—	3	2	no	—
43	C₂×C₃×D₁₂	C₃×C₄	1	4	yes	—
44	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
45	C₃²×M₁₆	—	6	4	no	—
46	C₂×C₃²×C₈	—	2	4	no	—
47	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	—	3	2	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
53	C₃²×C₄⋊C₄	—	3	1	no	—
54	C₃²×C₂²⋊C₄	—	3	2	no	—
55	C₃²×C₄²	—	2	4	no	—
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₄²	C₂²×C₃	1	4	yes	—
58	C₃×(C₂×D₃)⋊C₄	C₃×C₄	1	8	yes	—
59	C₃×C₄⋊Dic₃	C₃×C₄	1	4	yes	—
60	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
61	C₃×ΓL(1,𝔽₂₅)	—	2	16	no	—
62	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃²×D₃	—	2	2	no	—
64	C₂×C₃²×D₃	—	2	2	no	—
65	C₂×C₃²×D₃	—	2	2	no	—
66	C₃³×C₄	—	2	1	no	—
67	C₃³×C₄	M₁₆	1	2	yes	complemented
68	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
69	C₃³×C₄	#41	1	1	yes	characteristic
70	C₃²×Dic₃	—	2	2	no	—
71	C₂×C₃×M₁₆	—	3	8	no	—
72	C₃×C₄×D₄	—	3	4	no	—
73	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	2	yes	complemented
74	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	3	4	no	—
76	C₃×C₂²⋊C₈	—	3	8	no	—
77	C₃×C₄×C₈	—	2	4	no	—
78	C₃×C₄×C₈	C₃×D₃	1	4	yes	complemented
79	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
80	(C₂×D₃)⋊C₈	C₂×C₃²	1	2	yes	complemented
81	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
82	C₂³×C₃²	—	3	2	no	—
83	C₂²×C₃×D₃	C₂×C₃×C₄	1	8	yes	—
84	C₃²×D₄	—	6	2	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	3	2	no	—
89	C₂×C₃²×C₄	—	6	2	no	—
90	C₂×C₃²×C₄	—	6	2	no	—
91	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
97	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
98	C₃×D₁₂	—	2	8	no	—
99	C₃×C₄×D₃	—	2	8	no	—
100	C₃×C₄×D₃	—	2	8	no	—
101	C₃²×C₈	—	2	8	no	—
102	C₃²×C₈	—	2	1	no	—
103	C₃²×C₈	—	2	4	no	—
104	C₃²×C₈	—	2	8	no	—
105	C₃²×C₈	—	6	2	no	—
106	C₃²×C₈	D₁₂	1	2	yes	complemented
107	C₃²×C₈	C₃×D₄	1	8	yes	complemented
108	C₃×C₃⋊C₈	—	2	8	no	—
109	#436	—	3	1	no	Sylow
110	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₃³	#131	1	1	yes	characteristic
113	C₃²×D₃	—	4	2	no	—
114	C₂×C₃×D₄	—	3	4	no	—
115	C₂²×C₃×C₄	—	3	8	no	—
116	C₂×D₁₂	C₃²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
118	C₃×M₁₆	—	6	16	no	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	—	3	8	no	—
122	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
123	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
124	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	3	4	no	—
127	C₃×C₂²⋊C₄	—	3	8	no	—
128	C₃×C₄²	—	2	4	no	—
129	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
130	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
131	(C₂×D₃)⋊C₄	C₃²×C₄	1	2	yes	—
132	C₄⋊Dic₃	C₃²×C₄	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
134	ΓL(1,𝔽₂₅)	—	2	4	no	—
135	C₂²×C₃²	—	2	4	no	—
136	C₂²×C₃²	—	6	2	no	—
137	C₂²×C₃²	—	6	2	no	—
138	C₂²×C₃²	—	6	2	no	—
139	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
140	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
141	C₂×C₃×D₃	—	2	8	no	—
142	C₂×C₃×D₃	—	2	8	no	—
143	C₂×C₃×D₃	—	2	8	no	—
144	C₃²×C₄	—	2	8	no	—
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	2	8	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	6	2	no	—
151	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	complemented
152	C₃²×C₄	C₃×M₁₆	1	8	yes	complemented
153	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
154	C₃²×C₄	#235	1	1	yes	characteristic
155	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
156	C₃²×C₄	C₃×#41	1	4	yes	—
157	C₃×Dic₃	—	2	8	no	—
158	C₂×M₁₆	—	3	2	no	—
159	C₄×D₄	—	3	1	no	—
160	C₄⋊C₈	—	3	1	no	—
161	C₂²⋊C₈	—	3	2	no	—
162	C₄×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
163	C₃³	#436	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	3	8	no	—
165	C₂²×D₃	C₂×C₃²×C₄	1	2	yes	—
166	C₃×D₄	—	6	8	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	3	8	no	—
171	C₂×C₃×C₄	—	6	8	no	—
172	C₂×C₃×C₄	—	6	8	no	—
173	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
174	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
175	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
178	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
179	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
180	D₁₂	—	2	2	no	—
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	—	2	2	no	—
183	C₃×C₈	—	2	8	no	—
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	—	2	8	no	—
187	C₃×C₈	—	6	8	no	—
188	C₃×C₈	C₃×D₁₂	1	8	yes	complemented
189	C₃×C₈	C₃²×D₄	1	2	yes	complemented
190	C₃⋊C₈	—	2	2	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	—	2	4	no	—
194	C₂×C₃²	—	12	2	no	—
195	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
196	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
197	C₂×C₃²	#901	1	1	yes	characteristic
198	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
199	C₂×C₃²	C₂×C₃×M₁₆	1	4	yes	—
200	C₂×C₃²	C₃×#131	1	4	yes	—
201	C₃×D₃	—	4	8	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂²×C₄	—	3	2	no	—
204	M₁₆	—	6	4	no	—
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
208	C₄⋊C₄	—	3	1	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	6	8	no	—
213	C₂²×C₃	—	6	8	no	—
214	C₂²×C₃	—	6	8	no	—
215	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
216	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
217	C₂×D₃	—	2	2	no	—
218	C₂×D₃	—	2	2	no	—
219	C₂×D₃	—	2	2	no	—
220	C₃×C₄	—	2	8	no	—
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	—	6	8	no	—
227	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	8	yes	complemented
228	C₃×C₄	C₃²×M₁₆	1	2	yes	complemented
229	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
230	C₃×C₄	C₃×#235	1	4	yes	—
231	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
232	C₃×C₄	C₃²×#41	1	1	yes	characteristic
233	Dic₃	—	2	2	no	—
234	C₃²	—	2	4	no	—
235	C₃²	#4688	1	1	yes	characteristic, direct product factor
236	C₃²	C₃×#436	1	4	yes	complemented
237	C₂³	—	3	2	no	—
238	D₄	—	6	2	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
243	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	6	2	no	—
247	C₈	C₃²×D₁₂	1	2	yes	complemented
248	C₂×C₃	—	2	4	no	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	12	8	no	—
252	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
253	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	4	yes	—
254	C₂×C₃	C₃×#901	1	4	yes	—
255	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
257	C₂×C₃	C₃²×#131	1	1	yes	characteristic
258	D₃	—	4	2	no	—
259	C₂²	—	6	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	6	2	no	—
262	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	6	2	no	—
265	C₄	C₃²×ΓL(1,𝔽₂₅)	1	2	yes	complemented
266	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
267	C₄	C₃²×#235	1	1	yes	characteristic
268	C₃	—	2	4	no	—
269	C₃	C₃×#4688	1	4	yes	direct product factor
270	C₃	C₃²×#436	1	1	yes	characteristic, complemented
271	C₂	—	12	2	no	—
272	C₂	C₃²×C₄×D₁₂	1	1	yes	characteristic
273	C₂	C₂×C₃²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
274	C₂	C₃²×#901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄×C₈	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃²×D₁₂	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄×C₈	D₃
hypocenter	C₃	C₃²×#436
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	semidirect
3. C₃³×C₄×C₈	C₂	semidirect
4. C₃²×(C₂×D₃)⋊C₈	C₂	semidirect
5. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#4688	C₃	direct (N × H)
19. C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
24. C₃²×C₄×C₈	D₃	semidirect
25. C₃²×C₄×C₈	C₂×C₃	semidirect
26. C₃×(C₂×D₃)⋊C₈	C₂×C₃	semidirect
27. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
37. C₃³×C₈	D₄	semidirect
40. #4688	C₃²	direct (N × H)
67. C₃³×C₄	M₁₆	semidirect
73. C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	semidirect
78. C₃×C₄×C₈	C₃×D₃	semidirect
79. C₃×C₄×C₈	C₂×C₃²	semidirect
80. (C₂×D₃)⋊C₈	C₂×C₃²	semidirect
81. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
106. C₃²×C₈	D₁₂	semidirect
107. C₃²×C₈	C₃×D₄	semidirect
151. C₃²×C₄	ΓL(1,𝔽₂₅)	semidirect
152. C₃²×C₄	C₃×M₁₆	semidirect
162. C₄×C₈	C₃²×D₃	semidirect
163. C₃³	#436	semidirect
188. C₃×C₈	C₃×D₁₂	semidirect
189. C₃×C₈	C₃²×D₄	semidirect
227. C₃×C₄	C₃×ΓL(1,𝔽₂₅)	semidirect
228. C₃×C₄	C₃²×M₁₆	semidirect
235. C₃²	#4688	direct (N × H)
236. C₃²	C₃×#436	semidirect
247. C₈	C₃²×D₁₂	semidirect
265. C₄	C₃²×ΓL(1,𝔽₂₅)	semidirect
269. C₃	C₃×#4688	direct (N × H)
270. C₃	C₃²×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×ΓL(1,𝔽₂₅)	2
C₃²×C₄×D₁₂	2
C₃³×C₄×C₈	2
C₃²×(C₂×D₃)⋊C₈	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#436	3
C₃×#4688	3

Subgroups — C32×#4688 #13162124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4688 #13162124