Subgroups — #13158951

Order 1728.

1188 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#88788	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11722050	C₂	1	2	yes	maximal, complemented
6	#10592536	—	3	1	no	maximal
7	#10590340	—	3	4	no	maximal
8	C₂×C₄×C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
9	C₂×C₄×C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
13	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
14	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
17	#88788	—	2	4	no	—
18	C₂×#3743	—	3	2	no	—
19	C₄²×C₃⋊D₃	—	3	1	no	—
20	C₄²×C₃⋊D₃	—	3	4	no	—
21	C₄²×C₃⋊D₃	—	6	4	no	—
22	C₂×#3712	—	3	8	no	—
23	C₃²×C₄⋊C₈	—	3	1	no	—
24	#63641	—	3	2	no	—
25	C₄⋊(C₃⋊(C₃⋊C₈))	D₃	1	1	yes	characteristic, complemented
26	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
27	C₃×C₄⋊(C₃⋊C₈)	D₃	1	4	yes	complemented
28	#63572	—	3	8	no	—
29	C₂²×C₃⋊(C₃⋊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×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
36	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
37	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
38	C₄×C₃⋊(C₃⋊D₃)	D₄	1	2	yes	—
39	C₄×C₃⋊(C₃⋊D₃)	Q₈	1	2	yes	—
40	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
41	C₃²×C₃⋊C₈	—	2	2	no	—
42	#4837	—	9	4	no	—
43	#4837	—	9	1	no	—
44	C₂×C₄×C₃⋊D₃	—	3	1	no	—
45	C₂×C₄×C₃⋊D₃	—	3	4	no	—
46	C₂×C₄×C₃⋊D₃	—	3	1	no	—
47	C₂×C₄×C₃⋊D₃	—	3	1	no	—
48	C₂×C₄×C₃⋊D₃	—	3	4	no	—
49	C₂×C₄×C₃⋊D₃	—	3	4	no	—
50	C₂×C₄×C₃⋊D₃	—	6	4	no	—
51	C₂×C₄×C₃⋊D₃	—	6	4	no	—
52	C₂×C₄×C₃⋊D₃	—	6	4	no	—
53	C₂×C₃²×C₈	—	3	2	no	—
54	C₃²×C₄²	—	2	4	no	—
55	C₃²×C₄²	C₂×D₃	1	4	yes	—
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₄×C₃⋊Dic₃	—	3	1	no	—
58	C₄×C₃⋊Dic₃	—	3	1	no	—
59	C₄×C₃⋊Dic₃	—	3	4	no	—
60	C₄×C₃⋊Dic₃	—	3	4	no	—
61	C₄×C₃⋊Dic₃	—	3	1	no	—
62	C₄×C₃⋊Dic₃	—	3	4	no	—
63	C₄×C₃⋊Dic₃	—	6	4	no	—
64	C₄×C₃⋊Dic₃	—	6	4	no	—
65	C₄×C₃⋊Dic₃	—	6	4	no	—
66	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
67	#3743	—	6	4	no	—
68	C₂×C₃×C₃⋊C₈	—	3	8	no	—
69	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
70	#3712	—	6	16	no	—
71	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
73	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
74	C₂×C₃⋊(C₃⋊D₃)	C₄⋊C₄	1	2	yes	—
75	C₃³×C₄	—	2	1	no	—
76	C₃³×C₄	M₁₆	1	2	yes	complemented
77	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
80	C₃⋊(C₃⋊Dic₃)	C₄⋊C₄	1	2	yes	—
81	C₃⋊(C₃⋊Dic₃)	M₁₆	1	2	yes	complemented
82	C₂×ΓL(1,𝔽₂₅)	—	9	8	no	—
83	C₂×ΓL(1,𝔽₂₅)	—	9	2	no	—
84	C₄²×D₃	—	9	4	no	—
85	C₄²×D₃	—	9	1	no	—
86	C₄²×D₃	—	18	4	no	—
87	C₃×C₄⋊C₈	—	3	4	no	—
88	C₃×C₄⋊C₈	—	9	1	no	—
89	Dic₃⋊C₈	—	9	8	no	—
90	Dic₃⋊C₈	—	9	2	no	—
91	C₄⋊(C₃⋊C₈)	—	3	4	no	—
92	C₄⋊(C₃⋊C₈)	C₃⋊D₃	1	1	yes	characteristic, complemented
93	C₂²×C₃⋊D₃	—	3	1	no	—
94	C₂²×C₃⋊D₃	—	3	4	no	—
95	C₂²×C₃⋊D₃	—	6	4	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
100	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
101	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃⋊Dic₃	—	3	1	no	—
106	C₂×C₃⋊Dic₃	—	3	1	no	—
107	C₂×C₃⋊Dic₃	—	3	1	no	—
108	C₂×C₃⋊Dic₃	—	3	4	no	—
109	C₂×C₃⋊Dic₃	—	3	4	no	—
110	C₂×C₃⋊Dic₃	—	3	4	no	—
111	C₂×C₃⋊Dic₃	—	6	4	no	—
112	C₂×C₃⋊Dic₃	—	6	4	no	—
113	C₂×C₃⋊Dic₃	—	6	4	no	—
114	C₄×C₃⋊D₃	—	3	2	no	—
115	C₄×C₃⋊D₃	—	3	2	no	—
116	C₄×C₃⋊D₃	—	3	8	no	—
117	C₄×C₃⋊D₃	—	3	8	no	—
118	C₄×C₃⋊D₃	—	6	8	no	—
119	C₄×C₃⋊D₃	—	6	8	no	—
120	C₄×C₃⋊D₃	—	6	2	no	—
121	C₄×C₃⋊D₃	—	6	8	no	—
122	C₄×C₃⋊D₃	—	6	16	no	—
123	C₄×C₃⋊D₃	—	6	16	no	—
124	C₄×C₃⋊D₃	—	6	8	no	—
125	C₄×C₃⋊D₃	—	6	2	no	—
126	C₃²×C₈	—	6	2	no	—
127	C₃⋊(C₃⋊C₈)	—	2	2	no	—
128	C₃×C₃⋊C₈	—	2	8	no	—
129	C₃×C₃⋊C₈	—	6	8	no	—
130	#423	—	27	1	no	Sylow
131	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
134	C₃⋊(C₃⋊D₃)	—	2	2	no	—
135	C₂×C₄×D₃	—	9	4	no	—
136	C₂×C₄×D₃	—	9	1	no	—
137	C₂×C₄×D₃	—	9	4	no	—
138	C₂×C₄×D₃	—	9	4	no	—
139	C₂×C₄×D₃	—	9	1	no	—
140	C₂×C₄×D₃	—	9	1	no	—
141	C₂×C₄×D₃	—	18	4	no	—
142	C₂×C₄×D₃	—	18	4	no	—
143	C₂×C₄×D₃	—	18	4	no	—
144	C₂×C₃×C₈	—	3	8	no	—
145	C₂×C₃×C₈	—	9	2	no	—
146	C₃×C₄²	—	2	4	no	—
147	C₃×C₄²	D₃²	1	4	yes	—
148	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
149	C₄×Dic₃	—	9	4	no	—
150	C₄×Dic₃	—	9	4	no	—
151	C₄×Dic₃	—	9	1	no	—
152	C₄×Dic₃	—	9	1	no	—
153	C₄×Dic₃	—	9	4	no	—
154	C₄×Dic₃	—	9	1	no	—
155	C₄×Dic₃	—	18	4	no	—
156	C₄×Dic₃	—	18	4	no	—
157	C₄×Dic₃	—	18	4	no	—
158	C₂×C₃⋊C₈	—	3	8	no	—
159	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
160	ΓL(1,𝔽₂₅)	—	18	4	no	—
161	ΓL(1,𝔽₂₅)	—	18	16	no	—
162	C₂²×C₃²	—	2	4	no	—
163	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
164	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₃⋊D₃	—	3	2	no	—
166	C₂×C₃⋊D₃	—	3	8	no	—
167	C₂×C₃⋊D₃	—	6	8	no	—
168	C₂×C₃⋊D₃	—	6	8	no	—
169	C₂×C₃⋊D₃	—	6	8	no	—
170	C₂×C₃⋊D₃	—	6	4	no	—
171	C₂×C₃⋊D₃	—	6	1	no	—
172	C₂×C₃⋊D₃	—	6	4	no	—
173	C₂×C₃⋊D₃	—	6	1	no	—
174	C₃²×C₄	—	2	8	no	—
175	C₃²×C₄	—	2	4	no	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	—	2	8	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	2	4	no	—
180	C₃²×C₄	ΓL(1,𝔽₂₅)	1	8	yes	complemented
181	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	complemented
182	C₃²×C₄	D₃×D₄	1	4	yes	—
183	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
184	C₃²×C₄	D₃×Q₈	1	4	yes	—
185	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
186	C₃⋊Dic₃	—	3	2	no	—
187	C₃⋊Dic₃	—	3	2	no	—
188	C₃⋊Dic₃	—	3	8	no	—
189	C₃⋊Dic₃	—	3	8	no	—
190	C₃⋊Dic₃	—	6	8	no	—
191	C₃⋊Dic₃	—	6	8	no	—
192	C₃⋊Dic₃	—	6	8	no	—
193	C₃⋊Dic₃	—	6	4	no	—
194	C₃⋊Dic₃	—	6	1	no	—
195	C₂×M₁₆	—	27	2	no	—
196	C₂×C₄²	—	27	1	no	—
197	C₄⋊C₈	—	3	1	no	—
198	C₄⋊C₈	—	9	1	no	—
199	C₄⋊C₈	—	27	2	no	—
200	C₃³	#423	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₃	—	9	4	no	—
202	C₂²×D₃	—	9	1	no	—
203	C₂²×D₃	—	18	4	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	#624	1	4	yes	—
208	C₂×C₃×C₄	#624	1	4	yes	—
209	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
212	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
213	C₂×Dic₃	—	9	4	no	—
214	C₂×Dic₃	—	9	4	no	—
215	C₂×Dic₃	—	9	1	no	—
216	C₂×Dic₃	—	9	1	no	—
217	C₂×Dic₃	—	9	4	no	—
218	C₂×Dic₃	—	9	1	no	—
219	C₂×Dic₃	—	18	4	no	—
220	C₂×Dic₃	—	18	4	no	—
221	C₂×Dic₃	—	18	4	no	—
222	C₄×D₃	—	9	8	no	—
223	C₄×D₃	—	9	8	no	—
224	C₄×D₃	—	9	2	no	—
225	C₄×D₃	—	9	2	no	—
226	C₄×D₃	—	18	8	no	—
227	C₄×D₃	—	18	8	no	—
228	C₄×D₃	—	18	8	no	—
229	C₄×D₃	—	18	2	no	—
230	C₄×D₃	—	18	16	no	—
231	C₄×D₃	—	18	16	no	—
232	C₄×D₃	—	18	8	no	—
233	C₄×D₃	—	18	2	no	—
234	C₃×C₈	—	6	8	no	—
235	C₃×C₈	—	18	2	no	—
236	C₃⋊C₈	—	2	2	no	—
237	C₃⋊C₈	—	6	8	no	—
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	2	4	no	—
240	C₂×C₃²	—	2	4	no	—
241	C₂×C₃²	D₃×C₄⋊C₄	1	4	yes	—
242	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
244	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
245	C₂×C₃²	D₃×M₁₆	1	4	yes	—
246	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
247	C₃⋊D₃	—	6	16	no	—
248	C₃⋊D₃	—	6	8	no	—
249	C₃⋊D₃	—	6	2	no	—
250	C₂²×C₄	—	27	1	no	—
251	C₂²×C₄	—	27	1	no	—
252	C₂²×C₄	—	27	1	no	—
253	M₁₆	—	54	4	no	—
254	C₂×C₈	—	3	2	no	—
255	C₂×C₈	—	9	2	no	—
256	C₄²	—	27	1	no	—
257	C₄²	—	27	1	no	—
258	C₄²	—	27	1	no	—
259	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	C₂×#624	1	4	yes	—
262	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
263	C₂×D₃	—	9	2	no	—
264	C₂×D₃	—	9	8	no	—
265	C₂×D₃	—	18	8	no	—
266	C₂×D₃	—	18	8	no	—
267	C₂×D₃	—	18	8	no	—
268	C₂×D₃	—	18	4	no	—
269	C₂×D₃	—	18	4	no	—
270	C₂×D₃	—	18	1	no	—
271	C₂×D₃	—	18	1	no	—
272	C₃×C₄	—	2	8	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	8	no	—
278	C₃×C₄	#3712	1	8	yes	complemented
279	C₃×C₄	#3743	1	2	yes	complemented
280	C₃×C₄	#3797	1	4	yes	—
281	C₃×C₄	#3802	1	4	yes	—
282	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
283	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
284	Dic₃	—	9	8	no	—
285	Dic₃	—	9	2	no	—
286	Dic₃	—	9	2	no	—
287	Dic₃	—	9	8	no	—
288	Dic₃	—	18	8	no	—
289	Dic₃	—	18	8	no	—
290	Dic₃	—	18	8	no	—
291	Dic₃	—	18	4	no	—
292	Dic₃	—	18	1	no	—
293	C₃²	—	2	4	no	—
294	C₃²	#4837	1	4	yes	complemented
295	C₃²	#4837	1	1	yes	characteristic, complemented
296	C₂³	—	27	1	no	—
297	C₂×C₄	—	27	1	no	—
298	C₂×C₄	—	27	1	no	—
299	C₂×C₄	—	27	2	no	—
300	C₂×C₄	—	27	2	no	—
301	C₂×C₄	—	27	1	no	—
302	C₂×C₄	—	54	2	no	—
303	C₂×C₄	—	54	2	no	—
304	C₂×C₄	#6289	1	1	yes	characteristic
305	C₂×C₄	#6289	1	1	yes	characteristic
306	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
307	C₈	—	6	2	no	—
308	C₈	—	18	2	no	—
309	C₂×C₃	—	2	4	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	2	4	no	—
312	C₂×C₃	#63818	1	4	yes	—
313	C₂×C₃	C₂×#3712	1	4	yes	—
314	C₂×C₃	#63900	1	4	yes	—
315	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
316	C₂×C₃	C₂×#3743	1	1	yes	characteristic
317	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
318	D₃	—	18	16	no	—
319	D₃	—	18	2	no	—
320	D₃	—	18	8	no	—
321	C₂²	—	27	2	no	—
322	C₂²	—	54	1	no	—
323	C₂²	—	54	1	no	—
324	C₂²	C₂×#6289	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	27	2	no	—
327	C₄	—	27	2	no	—
328	C₄	—	54	1	no	—
329	C₄	#88788	1	2	yes	complemented
330	C₄	#89019	1	1	yes	characteristic
331	C₄	#89026	1	1	yes	characteristic
332	C₃	—	2	4	no	—
333	C₃	#10590340	1	4	yes	complemented
334	C₃	#10592536	1	1	yes	characteristic, complemented
335	C₂	—	54	2	no	—
336	C₂	#11723358	1	1	yes	characteristic
337	C₂	C₂×#88788	1	1	yes	characteristic
338	C₂	#11723503	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₃³×C₄²	C₂²
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄²	D₃×C₃⋊D₃
hypocenter	C₃³	#423
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
4. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
5. #11722050	C₂	semidirect
25. C₄⋊(C₃⋊(C₃⋊C₈))	D₃	semidirect
27. C₃×C₄⋊(C₃⋊C₈)	D₃	semidirect
76. C₃³×C₄	M₁₆	semidirect
81. C₃⋊(C₃⋊Dic₃)	M₁₆	semidirect
92. C₄⋊(C₃⋊C₈)	C₃⋊D₃	semidirect
180. C₃²×C₄	ΓL(1,𝔽₂₅)	semidirect
181. C₃²×C₄	ΓL(1,𝔽₂₅)	semidirect
200. C₃³	#423	semidirect
278. C₃×C₄	#3712	semidirect
279. C₃×C₄	#3743	semidirect
294. C₃²	#4837	semidirect
295. C₃²	#4837	semidirect
329. C₄	#88788	semidirect
333. C₃	#10590340	semidirect
334. C₃	#10592536	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₃⋊(C₃⋊D₃)	2
C₂×#88788	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃²×C₄⋊(C₃⋊C₈)	2
#11722050	2
#10592536	3
#10590340	3