Subgroups — #13139092

Order 1728.

336 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#88420	C₂	1	1	yes	maximal, characteristic
3	D₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720555	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63381	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720286	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720285	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590973	—	3	1	no	maximal
9	#10588799	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	#88532	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₉×#208	C₂²	1	1	yes	characteristic
15	C₃×#3667	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₉×ΓL(1,𝔽₂₅)	—	2	1	no	—
18	D₃×C₈×C₉	—	2	1	no	—
19	#88420	—	2	1	no	—
20	#88420	C₄	1	2	yes	—
21	#88420	C₄	1	2	yes	—
22	D₃×C₉⋊C₈	—	2	1	no	—
23	C₃×D₃×M₁₆	D₃	1	1	yes	characteristic
24	C₂×#3711	—	3	1	no	—
25	D₃×#208	—	3	1	no	—
26	#63649	—	3	1	no	—
27	C₃×#822	—	3	1	no	—
28	#63578	—	3	1	no	—
29	#63577	—	3	1	no	—
30	C₂×C₉×M₁₆	—	3	1	no	—
31	C₂×#3667	—	3	1	no	—
32	C₂×#3667	—	3	1	no	—
33	#63381	—	3	1	no	—
34	#63381	—	3	1	no	—
35	#63381	—	3	1	no	—
36	#63381	—	6	1	no	—
37	#63381	D₃	1	1	yes	characteristic, complemented
38	C₂²×D₃×C₉	Q₈	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₉×Dic₃	Q₈	1	1	yes	characteristic
41	C₄×D₃×C₉	C₂×C₄	1	1	yes	characteristic
42	C₄×D₃×C₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₃×C₉	D₄	1	1	yes	characteristic
44	C₄×D₃×C₉	D₄	1	1	yes	characteristic
45	C₃×C₈×C₉	—	2	1	no	—
46	C₃⋊(C₉⋊C₈)	—	2	1	no	—
47	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
48	C₉×C₃⋊C₈	—	2	1	no	—
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
51	#5124	—	9	1	no	—
52	#4893	—	9	1	no	—
53	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
54	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
55	#3748	—	3	1	no	—
56	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
57	C₃×#208	—	3	1	no	—
58	C₃×#208	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₃⋊C₈	—	3	1	no	—
60	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
61	C₃×D₃×C₈	—	2	1	no	—
62	#3711	—	3	2	no	—
63	#3711	—	3	2	no	—
64	#3711	—	6	1	no	—
65	D₃×C₃⋊C₈	—	6	1	no	—
66	C₂²×C₄×C₉	—	3	1	no	—
67	C₉×M₁₆	—	2	1	no	—
68	C₉×M₁₆	—	3	1	no	—
69	C₉×M₁₆	—	6	1	no	—
70	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
71	C₂×C₈×C₉	—	6	1	no	—
72	#3667	—	3	2	no	—
73	#3667	—	3	2	no	—
74	#3667	—	3	1	no	—
75	#3667	—	6	1	no	—
76	#3667	—	6	1	no	—
77	#3667	C₂×D₃	1	1	yes	characteristic
78	C₂×C₉⋊C₈	—	3	1	no	—
79	C₂×C₉⋊C₈	—	6	1	no	—
80	C₂×C₉⋊C₈	—	6	1	no	—
81	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
83	C₂×D₃×C₉	—	4	1	no	—
84	C₂×D₃×C₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂×D₃×C₉	C₄⋊C₄	1	1	yes	characteristic
86	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₉×Dic₃	C₄⋊C₄	1	1	yes	characteristic
89	C₉×Dic₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₃×M₁₆	—	3	1	no	—
91	C₂×#208	—	9	1	no	—
92	C₂×#208	—	9	1	no	—
93	D₃×M₁₆	—	9	1	no	—
94	D₃×M₁₆	D₉	1	1	yes	characteristic
95	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
96	C₃×#108	—	9	1	no	—
97	#822	—	3	1	no	—
98	#822	—	9	1	no	—
99	#822	—	9	1	no	—
100	#822	—	9	1	no	—
101	#822	—	9	1	no	—
102	#822	—	9	1	no	—
103	#822	—	18	1	no	—
104	#819	—	9	1	no	—
105	C₂²×C₃×D₃	Dic₆	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	Dic₆	1	1	yes	characteristic
108	C₃×C₄×D₃	C₄×D₃	1	1	yes	characteristic
109	C₃×C₄×D₃	C₄×D₃	1	1	yes	characteristic
110	C₃×C₄×D₃	#67	1	1	yes	characteristic
111	C₃×C₄×D₃	#67	1	1	yes	characteristic
112	C₂³×C₉	—	3	1	no	—
113	C₃²×C₈	—	2	1	no	—
114	C₃⋊(C₃⋊C₈)	—	3	2	no	—
115	C₃⋊(C₃⋊C₈)	—	6	1	no	—
116	C₃×C₃⋊C₈	—	2	1	no	—
117	C₃×C₃⋊C₈	—	3	2	no	—
118	C₃×C₃⋊C₈	—	6	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	3	1	no	—
121	C₂×C₄×C₉	—	3	1	no	—
122	C₂×C₄×C₉	—	3	1	no	—
123	C₂×C₄×C₉	—	3	1	no	—
124	C₂×C₄×C₉	—	3	1	no	—
125	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
126	C₈×C₉	—	2	1	no	—
127	C₈×C₉	—	4	1	no	—
128	C₈×C₉	—	6	1	no	—
129	C₉⋊C₈	—	2	1	no	—
130	C₉⋊C₈	—	3	2	no	—
131	C₉⋊C₈	—	6	2	no	—
132	C₉⋊C₈	—	6	1	no	—
133	C₉⋊C₈	—	12	1	no	—
134	C₉⋊C₈	C₄×D₃	1	2	yes	—
135	#430	—	27	1	no	Sylow
136	C₂×C₃×C₉	—	2	1	no	—
137	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
138	D₃×C₉	—	2	1	no	—
139	D₃×C₉	—	2	1	no	—
140	C₂²×C₃×C₄	—	3	1	no	—
141	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
142	C₃×M₁₆	—	2	1	no	—
143	C₃×M₁₆	—	3	1	no	—
144	C₃×M₁₆	—	6	1	no	—
145	C₃×M₁₆	—	9	1	no	—
146	C₃×M₁₆	C₂×D₉	1	1	yes	characteristic
147	C₃×M₁₆	D₃²	1	1	yes	characteristic
148	C₂×C₃×C₈	—	6	1	no	—
149	C₂×C₃×C₈	—	9	1	no	—
150	#208	—	3	1	no	—
151	#208	—	9	2	no	—
152	#208	—	9	2	no	—
153	#208	—	9	1	no	—
154	#208	—	9	1	no	—
155	#208	—	18	1	no	—
156	#208	—	18	1	no	—
157	#208	C₂×D₉	1	1	yes	characteristic
158	C₂×C₃⋊C₈	—	3	1	no	—
159	C₂×C₃⋊C₈	—	9	1	no	—
160	C₂×C₃⋊C₈	—	9	1	no	—
161	C₂×C₃⋊C₈	—	18	1	no	—
162	C₂×C₃⋊C₈	—	18	1	no	—
163	ΓL(1,𝔽₂₅)	—	2	1	no	—
164	ΓL(1,𝔽₂₅)	—	9	2	no	—
165	ΓL(1,𝔽₂₅)	—	9	2	no	—
166	ΓL(1,𝔽₂₅)	—	18	1	no	—
167	D₃×C₈	—	2	1	no	—
168	D₃×C₈	—	18	1	no	—
169	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
170	C₂×C₃×D₃	—	4	1	no	—
171	C₂×C₃×D₃	Dic₃⋊C₄	1	1	yes	characteristic
172	C₂×C₃×D₃	Dic₃⋊C₄	1	1	yes	characteristic
173	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₃²×C₄	C₂×#67	1	1	yes	characteristic
175	C₃×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
176	C₃×Dic₃	Dic₃⋊C₄	1	1	yes	characteristic
177	C₂²×C₉	—	2	1	no	—
178	C₂²×C₉	—	3	1	no	—
179	C₂²×C₉	—	3	1	no	—
180	C₂²×C₉	—	12	1	no	—
181	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
182	C₄×C₉	—	2	1	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	3	1	no	—
185	C₄×C₉	—	3	1	no	—
186	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
187	C₄×C₉	D₃×D₄	1	1	yes	characteristic
188	C₂×M₁₆	—	3	1	no	—
189	C₂×M₁₆	—	27	1	no	—
190	C₂×M₁₆	—	27	1	no	—
191	#108	—	9	1	no	—
192	#108	—	27	1	no	—
193	#108	—	27	1	no	—
194	#108	—	27	1	no	—
195	C₃×C₉	#430	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	3	1	no	—
197	C₂²×D₃	Dic₁₈	1	1	yes	characteristic
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	3	1	no	—
201	C₂×C₃×C₄	—	3	1	no	—
202	C₂×C₃×C₄	—	3	1	no	—
203	C₂×C₃×C₄	—	3	1	no	—
204	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
206	C₂×Dic₃	Dic₁₈	1	1	yes	characteristic
207	C₄×D₃	C₄×D₉	1	1	yes	characteristic
208	C₄×D₃	C₄×D₉	1	1	yes	characteristic
209	C₄×D₃	#611	1	1	yes	characteristic
210	C₄×D₃	#611	1	1	yes	characteristic
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	4	1	no	—
214	C₃×C₈	—	6	1	no	—
215	C₃×C₈	—	9	2	no	—
216	C₃×C₈	—	18	1	no	—
217	C₃⋊C₈	—	2	1	no	—
218	C₃⋊C₈	—	3	2	no	—
219	C₃⋊C₈	—	6	1	no	—
220	C₃⋊C₈	—	9	2	no	—
221	C₃⋊C₈	—	9	2	no	—
222	C₃⋊C₈	—	18	2	no	—
223	C₃⋊C₈	—	18	1	no	—
224	C₃⋊C₈	—	18	1	no	—
225	C₃⋊C₈	—	36	1	no	—
226	C₂×C₃²	—	2	1	no	—
227	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
228	C₃×D₃	—	2	1	no	—
229	C₃×D₃	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	2	1	no	—
232	C₂×C₉	—	4	1	no	—
233	C₂×C₉	—	6	1	no	—
234	C₂×C₉	—	6	1	no	—
235	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	—	3	1	no	—
237	M₁₆	—	3	1	no	—
238	M₁₆	—	6	1	no	—
239	M₁₆	—	9	1	no	—
240	M₁₆	—	27	2	no	—
241	M₁₆	—	27	2	no	—
242	M₁₆	—	27	1	no	—
243	M₁₆	—	54	1	no	—
244	M₁₆	D₃×D₉	1	1	yes	characteristic
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	9	1	no	—
247	C₂×C₈	—	27	1	no	—
248	C₂×C₈	—	54	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	3	1	no	—
251	C₂²×C₃	—	3	1	no	—
252	C₂²×C₃	—	12	1	no	—
253	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
254	C₂²×C₃	D₃×Dic₆	1	1	yes	characteristic
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	Dic₉⋊C₄	1	1	yes	characteristic
257	C₂×D₃	Dic₉⋊C₄	1	1	yes	characteristic
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	3	1	no	—
261	C₃×C₄	—	3	1	no	—
262	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
263	C₃×C₄	C₂×#611	1	1	yes	characteristic
264	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
265	C₃×C₄	D₃×#67	1	1	yes	characteristic
266	Dic₃	Dic₉⋊C₄	1	1	yes	characteristic
267	Dic₃	Dic₉⋊C₄	1	1	yes	characteristic
268	C₃²	#5124	1	1	yes	characteristic
269	C₉	—	2	1	no	—
270	C₉	#4893	1	1	yes	characteristic, complemented
271	C₂³	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	6	1	no	—
280	C₈	—	9	2	no	—
281	C₈	—	18	1	no	—
282	C₈	—	27	2	no	—
283	C₈	—	54	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	4	1	no	—
288	C₂×C₃	—	6	1	no	—
289	C₂×C₃	—	6	1	no	—
290	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
291	C₂×C₃	D₃×Dic₃⋊C₄	1	1	yes	characteristic
292	D₃	—	2	1	no	—
293	D₃	—	2	1	no	—
294	C₂²	—	3	1	no	—
295	C₂²	—	3	1	no	—
296	C₂²	—	12	1	no	—
297	C₂²	D₃×Dic₁₈	1	1	yes	characteristic
298	C₄	—	3	1	no	—
299	C₄	—	3	1	no	—
300	C₄	C₄×D₃×D₉	1	1	yes	characteristic
301	C₄	D₃×#611	1	1	yes	characteristic
302	C₃	—	2	1	no	—
303	C₃	#10588799	1	1	yes	characteristic, complemented
304	C₃	#10590973	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	6	1	no	—
307	C₂	—	6	1	no	—
308	C₂	D₃×Dic₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×#611
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₉×M₁₆	C₂²
socle	C₂×C₃²	C₂×Dic₃⋊C₄
prime radical	C₂²×C₃×D₃	Dic₆
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	M₁₆	D₃×D₉
hypocenter	C₃×C₉	#430
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
4. #11720555	C₂	semidirect
5. C₃×#63381	C₂	semidirect
6. #11720286	C₂	semidirect
7. #11720285	C₂	semidirect
37. #63381	D₃	semidirect
195. C₃×C₉	#430	semidirect
270. C₉	#4893	semidirect
303. C₃	#10588799	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₉×M₁₆	2
C₂×#88420	2
D₃×#3667	2
#11720555	2
C₃×#63381	2
#11720286	2
#11720285	2
#10590973	3
#10588799	3