Subgroups — #13159529

Order 1728.

1028 conjugacy classes of subgroups.

The subgroup lattice has 288 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	1	yes	maximal, characteristic
2	C₄×D₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#88786	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722020	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722018	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×C₃⋊Dic₃	—	3	1	no	maximal
9	#10590667	—	3	4	no	maximal
10	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×D₃×C₃⋊Dic₃	C₄	1	2	yes	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃²×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
19	Dic₃×C₃⋊Dic₃	C₄	1	2	yes	—
20	#88786	C₄	1	4	yes	complemented
21	C₂×C₃²×M₁₆	—	3	1	no	—
22	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
23	C₂×#3748	—	3	1	no	—
24	C₂×C₃×ΓL(1,𝔽₂₅)	D₃	1	4	yes	—
25	C₄×D₃×Dic₃	—	3	4	no	—
26	C₂×#3711	—	3	4	no	—
27	#63642	—	3	1	no	—
28	#63642	D₃	1	1	yes	characteristic, complemented
29	C₈×C₃⋊Dic₃	—	3	1	no	—
30	C₈×C₃⋊Dic₃	—	3	1	no	—
31	C₈×C₃⋊Dic₃	—	3	4	no	—
32	C₈×C₃⋊Dic₃	—	6	4	no	—
33	C₃×#815	—	3	4	no	—
34	#63554	—	3	4	no	—
35	#63553	—	3	4	no	—
36	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
43	D₃×C₃⋊Dic₃	—	2	4	no	—
44	C₃³×C₈	C₂×C₄	1	2	yes	complemented
45	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	complemented
46	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
47	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
48	Dic₃×M₁₆	—	9	4	no	—
49	#4707	—	9	1	no	—
50	C₂²×C₃²×C₄	—	3	1	no	—
51	C₂²×C₃⋊Dic₃	—	3	2	no	—
52	C₂×C₃×C₄×D₃	C₂×D₃	1	4	yes	—
53	C₂×D₃×Dic₃	—	3	8	no	—
54	C₃²×M₁₆	—	3	4	no	—
55	C₂×C₃²×C₈	—	2	4	no	—
56	C₂×C₃²×C₈	—	3	1	no	—
57	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
59	C₄×C₃⋊Dic₃	—	3	1	no	—
60	C₄×C₃⋊Dic₃	—	3	4	no	—
61	C₄×C₃⋊Dic₃	—	3	2	no	—
62	C₄×C₃⋊Dic₃	—	6	4	no	—
63	C₄×C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
64	#3748	—	3	4	no	—
65	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
67	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
68	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄×Dic₃	—	3	4	no	—
70	C₂×C₃×C₃⋊C₈	—	3	4	no	—
71	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
72	C₃×ΓL(1,𝔽₂₅)	Dic₃	1	16	yes	complemented
73	Dic₃²	—	3	8	no	—
74	#3711	—	3	16	no	—
75	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×D₃	—	2	2	no	—
77	C₂×C₃²×D₃	C₄²	1	2	yes	—
78	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃⋊(C₃⋊Dic₃)	M₁₆	1	4	yes	complemented
81	C₃×C₃⋊Dic₃	—	2	2	no	—
82	C₃²×Dic₃	C₄²	1	2	yes	—
83	C₂×C₃×M₁₆	—	3	4	no	—
84	C₂×C₄×Dic₃	—	9	4	no	—
85	C₂×#208	—	9	4	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
87	C₂×ΓL(1,𝔽₂₅)	C₃⋊D₃	1	1	yes	characteristic
88	C₄²×D₃	—	9	1	no	—
89	C₃×#97	—	9	1	no	—
90	#815	—	3	4	no	—
91	#815	—	9	4	no	—
92	C₈×Dic₃	—	9	4	no	—
93	C₈×Dic₃	—	9	4	no	—
94	C₈×Dic₃	—	9	1	no	—
95	C₈×Dic₃	—	9	1	no	—
96	C₈×Dic₃	—	18	4	no	—
97	#803	—	9	1	no	—
98	C₂³×C₃²	—	3	1	no	—
99	C₂²×C₃×D₃	C₄×D₃	1	4	yes	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	3	1	no	—
102	C₂×C₃²×C₄	—	3	2	no	—
103	C₂×C₃²×C₄	—	3	2	no	—
104	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	3	2	no	—
107	C₂×C₃⋊Dic₃	—	3	8	no	—
108	C₂×C₃⋊Dic₃	—	6	8	no	—
109	C₂×C₃⋊Dic₃	—	6	4	no	—
110	C₂×C₃⋊Dic₃	C₄×D₃	1	2	yes	—
111	C₂×C₃×Dic₃	—	3	8	no	—
112	C₂×C₃×Dic₃	C₄×D₃	1	4	yes	—
113	C₃×C₄×D₃	C₄×D₃	1	8	yes	—
114	C₃×C₄×D₃	C₂×Dic₃	1	8	yes	—
115	D₃×Dic₃	—	6	16	no	—
116	C₃²×C₈	—	2	8	no	—
117	C₃²×C₈	—	3	2	no	—
118	C₃²×C₈	C₄×D₃	1	2	yes	complemented
119	C₃²×C₈	C₂×Dic₃	1	8	yes	complemented
120	C₃⋊(C₃⋊C₈)	—	3	2	no	—
121	C₃⋊(C₃⋊C₈)	—	3	8	no	—
122	C₃⋊(C₃⋊C₈)	—	6	8	no	—
123	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	complemented
124	C₃×C₃⋊C₈	—	3	8	no	—
125	C₃×C₃⋊C₈	C₂×Dic₃	1	8	yes	complemented
126	C₄×M₁₆	—	27	1	no	Sylow
127	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₃³	C₂×M₁₆	1	2	yes	—
129	C₃²×D₃	—	2	2	no	—
130	C₂²×C₃×C₄	—	3	4	no	—
131	C₂²×Dic₃	—	9	8	no	—
132	C₂×C₄×D₃	—	9	2	no	—
133	C₂×C₄×D₃	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₃×M₁₆	—	3	16	no	—
135	C₂×C₃×C₈	—	2	4	no	—
136	C₂×C₃×C₈	—	3	4	no	—
137	C₂×C₃×C₈	—	9	1	no	—
138	C₂×C₃×C₈	D₃²	1	4	yes	—
139	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₃×C₄²	—	9	1	no	—
141	C₄×Dic₃	—	3	4	no	—
142	C₄×Dic₃	—	9	8	no	—
143	C₄×Dic₃	—	9	4	no	—
144	C₄×Dic₃	—	9	1	no	—
145	C₄×Dic₃	—	9	2	no	—
146	C₄×Dic₃	—	18	4	no	—
147	#208	—	9	16	no	—
148	C₂×C₃⋊C₈	—	3	4	no	—
149	C₂×C₃⋊C₈	—	9	4	no	—
150	C₂×C₃⋊C₈	—	9	1	no	—
151	C₂×C₃⋊C₈	—	18	4	no	—
152	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
153	ΓL(1,𝔽₂₅)	—	9	4	no	—
154	ΓL(1,𝔽₂₅)	C₃⋊Dic₃	1	4	yes	complemented
155	C₂²×C₃²	—	2	4	no	—
156	C₂²×C₃²	—	3	2	no	—
157	C₂²×C₃²	—	6	2	no	—
158	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
159	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	8	no	—
161	C₂×C₃×D₃	C₄×Dic₃	1	8	yes	—
162	C₃²×C₄	—	2	4	no	—
163	C₃²×C₄	—	2	4	no	—
164	C₃²×C₄	—	3	2	no	—
165	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
166	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
168	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
169	C₃⋊Dic₃	—	2	2	no	—
170	C₃⋊Dic₃	—	3	4	no	—
171	C₃⋊Dic₃	—	3	16	no	—
172	C₃⋊Dic₃	—	6	16	no	—
173	C₃×Dic₃	—	6	8	no	—
174	C₃×Dic₃	C₄×Dic₃	1	8	yes	—
175	C₂×M₁₆	—	3	1	no	—
176	C₂×M₁₆	—	27	1	no	—
177	C₂×C₄²	—	27	1	no	—
178	#97	—	9	1	no	—
179	#97	—	27	1	no	—
180	C₄×C₈	—	27	1	no	—
181	C₄×C₈	—	27	1	no	—
182	C₃³	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₃	—	3	4	no	—
184	C₂²×D₃	C₄×C₃⋊D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	3	4	no	—
187	C₂×C₃×C₄	—	3	8	no	—
188	C₂×C₃×C₄	—	3	8	no	—
189	C₂×C₃×C₄	—	9	2	no	—
190	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
191	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
192	C₂×Dic₃	—	3	8	no	—
193	C₂×Dic₃	—	9	8	no	—
194	C₂×Dic₃	—	9	2	no	—
195	C₂×Dic₃	—	18	8	no	—
196	C₂×Dic₃	—	18	16	no	—
197	C₂×Dic₃	C₄×C₃⋊D₃	1	1	yes	characteristic
198	C₄×D₃	—	18	4	no	—
199	C₄×D₃	C₄×C₃⋊D₃	1	2	yes	—
200	C₄×D₃	C₂×C₃⋊Dic₃	1	2	yes	—
201	C₃×C₈	—	2	8	no	—
202	C₃×C₈	—	3	8	no	—
203	C₃×C₈	—	9	2	no	—
204	C₃×C₈	D₃×Dic₃	1	8	yes	complemented
205	C₃×C₈	C₂×C₃⋊Dic₃	1	2	yes	complemented
206	C₃⋊C₈	—	3	8	no	—
207	C₃⋊C₈	—	9	8	no	—
208	C₃⋊C₈	—	9	2	no	—
209	C₃⋊C₈	—	18	8	no	—
210	C₃⋊C₈	C₂×C₃⋊Dic₃	1	2	yes	complemented
211	C₂×C₃²	—	2	8	no	—
212	C₂×C₃²	—	2	4	no	—
213	C₂×C₃²	—	6	2	no	—
214	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
215	C₂×C₃²	D₃×M₁₆	1	8	yes	—
216	C₂×C₃²	D₃×M₁₆	1	2	yes	—
217	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
218	C₃×D₃	—	2	8	no	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	27	2	no	—
221	M₁₆	—	3	4	no	—
222	M₁₆	—	27	4	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	9	1	no	—
225	C₂×C₈	—	27	1	no	—
226	C₂×C₈	D₃×C₃⋊D₃	1	1	yes	characteristic
227	C₄²	—	9	1	no	—
228	C₄²	—	27	2	no	—
229	C₄²	—	27	1	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	3	8	no	—
232	C₂²×C₃	—	6	8	no	—
233	C₂²×C₃	C₄×D₃²	1	4	yes	—
234	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
235	C₂×D₃	—	2	2	no	—
236	C₂×D₃	C₄×C₃⋊Dic₃	1	2	yes	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	3	8	no	—
240	C₃×C₄	—	18	2	no	—
241	C₃×C₄	C₄×D₃²	1	4	yes	—
242	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
243	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
244	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
245	Dic₃	—	6	8	no	—
246	Dic₃	—	9	16	no	—
247	Dic₃	—	9	4	no	—
248	Dic₃	—	18	16	no	—
249	Dic₃	C₄×C₃⋊Dic₃	1	2	yes	—
250	C₃²	—	2	4	no	—
251	C₃²	#4707	1	1	yes	characteristic, complemented
252	C₃²	Dic₃×M₁₆	1	4	yes	complemented
253	C₂³	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	9	2	no	—
258	C₂×C₄	—	27	2	no	—
259	C₂×C₄	—	54	4	no	—
260	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
261	C₈	—	3	2	no	—
262	C₈	—	9	2	no	—
263	C₈	—	27	2	no	—
264	C₈	D₃×C₃⋊Dic₃	1	2	yes	complemented
265	C₂×C₃	—	2	8	no	—
266	C₂×C₃	—	2	4	no	—
267	C₂×C₃	—	6	8	no	—
268	C₂×C₃	#63791	1	8	yes	—
269	C₂×C₃	C₄×D₃×Dic₃	1	4	yes	—
270	C₂×C₃	M₁₆×C₃⋊D₃	1	2	yes	—
271	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
272	D₃	—	2	2	no	—
273	C₂²	—	3	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
276	C₄	—	3	2	no	—
277	C₄	—	18	2	no	—
278	C₄	—	27	4	no	—
279	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
280	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
281	C₃	—	2	4	no	—
282	C₃	#10590667	1	4	yes	complemented
283	C₃	M₁₆×C₃⋊Dic₃	1	1	yes	characteristic, complemented
284	C₂	—	6	2	no	—
285	C₂	#11723312	1	2	yes	—
286	C₂	C₄×D₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂×C₄²
Fitting	C₂×C₃³×C₈	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃²×D₃	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×C₈	D₃×C₃⋊D₃
hypocenter	C₃³	C₄×M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
5. C₃×#63642	C₂	semidirect
6. #11722020	C₂	semidirect
7. #11722018	C₂	semidirect
18. C₃²×ΓL(1,𝔽₂₅)	C₄	semidirect
20. #88786	C₄	semidirect
28. #63642	D₃	semidirect
44. C₃³×C₈	C₂×C₄	semidirect
45. C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	semidirect
46. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
47. C₃²×C₃⋊C₈	C₂×C₄	semidirect
72. C₃×ΓL(1,𝔽₂₅)	Dic₃	semidirect
80. C₃⋊(C₃⋊Dic₃)	M₁₆	semidirect
118. C₃²×C₈	C₄×D₃	semidirect
119. C₃²×C₈	C₂×Dic₃	semidirect
123. C₃⋊(C₃⋊C₈)	C₄×D₃	semidirect
125. C₃×C₃⋊C₈	C₂×Dic₃	semidirect
154. ΓL(1,𝔽₂₅)	C₃⋊Dic₃	semidirect
182. C₃³	C₄×M₁₆	semidirect
204. C₃×C₈	D₃×Dic₃	semidirect
205. C₃×C₈	C₂×C₃⋊Dic₃	semidirect
210. C₃⋊C₈	C₂×C₃⋊Dic₃	semidirect
251. C₃²	#4707	semidirect
252. C₃²	Dic₃×M₁₆	semidirect
264. C₈	D₃×C₃⋊Dic₃	semidirect
282. C₃	#10590667	semidirect
283. C₃	M₁₆×C₃⋊Dic₃	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
C₄×D₃×C₃⋊Dic₃	2
C₂×#88786	2
C₈×C₃⋊(C₃⋊Dic₃)	2
C₃×#63642	2
#11722020	2
#11722018	2
M₁₆×C₃⋊Dic₃	3
#10590667	3