Subgroups — C₃²×#4837 #13162265

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 302 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
2	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#423	—	3	1	no	maximal
7	C₃×#4837	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
15	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃²×ΓL(1,𝔽₂₅)	—	2	4	no	—
18	C₂×C₃²×M₁₆	—	3	2	no	—
19	C₂×C₃²×C₄²	—	3	1	no	—
20	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	8	yes	—
21	C₃×C₄²×D₃	C₂×C₃	1	4	yes	—
22	C₃²×C₄⋊C₈	—	2	4	no	—
23	C₃²×C₄⋊C₈	—	3	2	no	—
24	C₃²×C₄⋊C₈	—	3	1	no	—
25	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
27	C₃×Dic₃⋊C₈	C₂×C₃	1	8	yes	complemented
28	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	complemented
29	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄×D₃	—	2	2	no	—
37	C₃²×C₄×D₃	—	2	2	no	—
38	C₃²×C₄×D₃	D₄	1	2	yes	—
39	C₃²×C₄×D₃	Q₈	1	2	yes	—
40	C₃³×C₈	—	2	2	no	—
41	C₃²×C₃⋊C₈	—	2	2	no	—
42	C₃×#423	—	3	4	no	—
43	#4837	C₃²	1	1	yes	characteristic, direct product factor
44	C₂²×C₃²×C₄	—	3	1	no	—
45	C₂²×C₃²×C₄	—	3	1	no	—
46	C₂²×C₃²×C₄	—	3	1	no	—
47	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
48	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
49	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
50	C₃²×M₁₆	—	6	4	no	—
51	C₂×C₃²×C₈	—	2	8	no	—
52	C₂×C₃²×C₈	—	3	2	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
54	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
55	C₃²×C₄²	—	2	4	no	—
56	C₃²×C₄²	—	3	1	no	—
57	C₃²×C₄²	—	3	1	no	—
58	C₃²×C₄²	—	3	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	4	yes	—
61	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
62	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
63	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
64	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
65	C₃×ΓL(1,𝔽₂₅)	—	2	16	no	—
66	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃²×D₃	—	2	1	no	—
68	C₂×C₃²×D₃	—	2	1	no	—
69	C₂×C₃²×D₃	C₄⋊C₄	1	2	yes	—
70	C₃³×C₄	—	2	1	no	—
71	C₃³×C₄	M₁₆	1	2	yes	complemented
72	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
73	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
74	C₃²×Dic₃	—	2	1	no	—
75	C₃²×Dic₃	C₄⋊C₄	1	2	yes	—
76	C₃²×Dic₃	M₁₆	1	2	yes	complemented
77	C₂×C₃×M₁₆	—	3	8	no	—
78	C₂×C₃×C₄²	—	3	4	no	—
79	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	2	yes	—
80	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic
81	C₃×C₄⋊C₈	—	2	4	no	—
82	C₃×C₄⋊C₈	—	3	8	no	—
83	C₃×C₄⋊C₈	—	3	4	no	—
84	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
85	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
86	Dic₃⋊C₈	C₂×C₃²	1	2	yes	complemented
87	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
88	C₂³×C₃²	—	3	1	no	—
89	C₂²×C₃×D₃	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	3	1	no	—
94	C₂×C₃²×C₄	—	3	1	no	—
95	C₂×C₃²×C₄	—	3	2	no	—
96	C₂×C₃²×C₄	—	3	2	no	—
97	C₂×C₃²×C₄	—	3	1	no	—
98	C₂×C₃²×C₄	—	6	2	no	—
99	C₂×C₃²×C₄	—	6	2	no	—
100	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
103	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
106	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
107	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
108	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
109	C₃×C₄×D₃	—	2	8	no	—
110	C₃×C₄×D₃	—	2	8	no	—
111	C₃×C₄×D₃	C₃×D₄	1	8	yes	—
112	C₃×C₄×D₃	C₃×Q₈	1	8	yes	—
113	C₃²×C₈	—	2	2	no	—
114	C₃²×C₈	—	2	8	no	—
115	C₃²×C₈	—	4	8	no	—
116	C₃²×C₈	—	6	2	no	—
117	C₃×C₃⋊C₈	—	2	8	no	—
118	#423	—	3	1	no	Sylow
119	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
122	C₃²×D₃	—	2	2	no	—
123	C₂²×C₃×C₄	—	3	4	no	—
124	C₂²×C₃×C₄	—	3	4	no	—
125	C₂²×C₃×C₄	—	3	4	no	—
126	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
127	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
129	C₃×M₁₆	—	6	16	no	—
130	C₂×C₃×C₈	—	2	8	no	—
131	C₂×C₃×C₈	—	3	8	no	—
132	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
133	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
134	C₃×C₄²	—	2	4	no	—
135	C₃×C₄²	—	3	4	no	—
136	C₃×C₄²	—	3	4	no	—
137	C₃×C₄²	—	3	4	no	—
138	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
139	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
140	C₄×Dic₃	C₃²×C₄	1	1	yes	characteristic
141	C₄×Dic₃	C₃²×C₄	1	1	yes	characteristic
142	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
143	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
144	ΓL(1,𝔽₂₅)	—	2	4	no	—
145	C₂²×C₃²	—	2	4	no	—
146	C₂²×C₃²	—	3	2	no	—
147	C₂²×C₃²	—	6	1	no	—
148	C₂²×C₃²	—	6	1	no	—
149	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
151	C₂×C₃×D₃	—	2	4	no	—
152	C₂×C₃×D₃	—	2	4	no	—
153	C₂×C₃×D₃	C₃×C₄⋊C₄	1	8	yes	—
154	C₃²×C₄	—	2	1	no	—
155	C₃²×C₄	—	2	8	no	—
156	C₃²×C₄	—	2	4	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	—	2	4	no	—
159	C₃²×C₄	—	3	2	no	—
160	C₃²×C₄	—	3	2	no	—
161	C₃²×C₄	—	4	4	no	—
162	C₃²×C₄	—	6	1	no	—
163	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	complemented
164	C₃²×C₄	C₃×M₁₆	1	8	yes	complemented
165	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
166	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
167	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
168	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
169	C₃×Dic₃	—	2	4	no	—
170	C₃×Dic₃	C₃×C₄⋊C₄	1	8	yes	—
171	C₃×Dic₃	C₃×M₁₆	1	8	yes	complemented
172	C₂×M₁₆	—	3	2	no	—
173	C₂×C₄²	—	3	1	no	—
174	C₄⋊C₈	—	3	2	no	—
175	C₄⋊C₈	—	3	1	no	—
176	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
177	C₃³	#423	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	3	4	no	—
179	C₂²×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	3	4	no	—
184	C₂×C₃×C₄	—	3	4	no	—
185	C₂×C₃×C₄	—	3	8	no	—
186	C₂×C₃×C₄	—	3	8	no	—
187	C₂×C₃×C₄	—	3	4	no	—
188	C₂×C₃×C₄	—	6	8	no	—
189	C₂×C₃×C₄	—	6	8	no	—
190	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
191	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
192	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
195	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
196	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
197	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
198	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	2	2	no	—
201	C₄×D₃	C₃²×D₄	1	2	yes	—
202	C₄×D₃	C₃²×Q₈	1	2	yes	—
203	C₃×C₈	—	2	8	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	4	8	no	—
206	C₃×C₈	—	6	8	no	—
207	C₃⋊C₈	—	2	2	no	—
208	C₂×C₃²	—	2	4	no	—
209	C₂×C₃²	—	2	4	no	—
210	C₂×C₃²	—	2	4	no	—
211	C₂×C₃²	—	6	2	no	—
212	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
214	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
215	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
216	C₂×C₃²	C₂×C₃×M₁₆	1	4	yes	—
217	C₂×C₃²	C₂×C₃×M₁₆	1	4	yes	—
218	C₃×D₃	—	2	8	no	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	1	no	—
221	C₂²×C₄	—	3	1	no	—
222	M₁₆	—	6	4	no	—
223	C₂×C₈	—	3	2	no	—
224	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
225	C₄²	—	3	1	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
229	C₂²×C₃	—	2	4	no	—
230	C₂²×C₃	—	3	8	no	—
231	C₂²×C₃	—	6	4	no	—
232	C₂²×C₃	—	6	4	no	—
233	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
234	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	C₃²×C₄⋊C₄	1	2	yes	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	4	no	—
242	C₃×C₄	—	2	8	no	—
243	C₃×C₄	—	3	8	no	—
244	C₃×C₄	—	3	8	no	—
245	C₃×C₄	—	4	4	no	—
246	C₃×C₄	—	6	4	no	—
247	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	8	yes	complemented
248	C₃×C₄	C₃²×M₁₆	1	2	yes	complemented
249	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
250	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
251	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
252	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
253	Dic₃	—	2	1	no	—
254	Dic₃	C₃²×C₄⋊C₄	1	2	yes	—
255	Dic₃	C₃²×M₁₆	1	2	yes	complemented
256	C₃²	—	2	4	no	—
257	C₃²	#4837	1	1	yes	characteristic, direct product factor
258	C₃²	C₃×#423	1	4	yes	complemented
259	C₂³	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
268	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	6	2	no	—
272	C₂×C₃	—	2	4	no	—
273	C₂×C₃	—	2	4	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	6	8	no	—
276	C₂×C₃	C₃×D₃×C₄⋊C₄	1	4	yes	—
277	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	4	yes	—
278	C₂×C₃	C₃×D₃×M₁₆	1	4	yes	—
279	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
281	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
282	D₃	—	2	2	no	—
283	C₂²	—	3	2	no	—
284	C₂²	—	6	1	no	—
285	C₂²	—	6	1	no	—
286	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	3	2	no	—
289	C₄	—	3	2	no	—
290	C₄	—	6	1	no	—
291	C₄	C₃²×ΓL(1,𝔽₂₅)	1	2	yes	complemented
292	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
293	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
294	C₃	—	2	4	no	—
295	C₃	C₃×#4837	1	4	yes	direct product factor
296	C₃	C₃²×#423	1	1	yes	characteristic, complemented
297	C₂	—	6	2	no	—
298	C₂	C₃²×D₃×C₄⋊C₄	1	1	yes	characteristic
299	C₂	C₂×C₃²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
300	C₂	C₃²×D₃×M₁₆	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	C₂²×C₃²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#423
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
3. C₃³×C₄⋊C₈	C₂	semidirect
4. C₃²×Dic₃⋊C₈	C₂	semidirect
5. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#4837	C₃	direct (N × H)
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. C₃×Dic₃⋊C₈	C₂×C₃	semidirect
28. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
43. #4837	C₃²	direct (N × H)
71. C₃³×C₄	M₁₆	semidirect
76. C₃²×Dic₃	M₁₆	semidirect
84. C₃×C₄⋊C₈	C₃×D₃	semidirect
85. C₃×C₄⋊C₈	C₂×C₃²	semidirect
86. Dic₃⋊C₈	C₂×C₃²	semidirect
87. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
163. C₃²×C₄	ΓL(1,𝔽₂₅)	semidirect
164. C₃²×C₄	C₃×M₁₆	semidirect
171. C₃×Dic₃	C₃×M₁₆	semidirect
176. C₄⋊C₈	C₃²×D₃	semidirect
177. C₃³	#423	semidirect
247. C₃×C₄	C₃×ΓL(1,𝔽₂₅)	semidirect
248. C₃×C₄	C₃²×M₁₆	semidirect
255. Dic₃	C₃²×M₁₆	semidirect
257. C₃²	#4837	direct (N × H)
258. C₃²	C₃×#423	semidirect
291. C₄	C₃²×ΓL(1,𝔽₂₅)	semidirect
295. C₃	C₃×#4837	direct (N × H)
296. C₃	C₃²×#423	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×C₄²×D₃	2
C₃³×C₄⋊C₈	2
C₃²×Dic₃⋊C₈	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#423	3
C₃×#4837	3

Subgroups — C32×#4837 #13162265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4837 #13162265