Subgroups — #13141050

Order 1728.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721865	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#950	C₂	1	1	yes	maximal, characteristic
3	#11720878	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63396	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720301	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720280	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#10591675	—	3	1	no	maximal
9	#10588876	—	3	1	no	maximal
10	C₃×C₉×#41	C₂²	1	1	yes	characteristic
11	C₉×#235	C₂²	1	1	yes	characteristic
12	#88532	—	2	1	no	—
13	#88532	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₉×#216	C₄	1	1	yes	characteristic, complemented
17	C₉×#215	C₄	1	1	yes	characteristic, complemented
18	C₉×#214	C₄	1	1	yes	characteristic, complemented
19	C₉×#213	C₄	1	1	yes	characteristic, complemented
20	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×#3667	C₂²	1	1	yes	characteristic
23	#88420	—	2	1	no	—
24	D₃×C₉⋊C₈	—	2	1	no	—
25	#64157	—	3	1	no	—
26	C₃×#950	D₃	1	1	yes	characteristic
27	#63815	—	3	1	no	—
28	C₃×#837	—	3	1	no	—
29	#63577	—	3	1	no	—
30	#63569	—	3	1	no	—
31	Dic₃×C₃⋊C₈	—	3	1	no	—
32	C₉×#135	—	3	1	no	—
33	#63510	—	3	1	no	—
34	#63510	—	3	1	no	—
35	#63510	—	6	1	no	—
36	#63396	—	3	1	no	—
37	#63396	D₃	1	1	yes	characteristic, complemented
38	C₈×Dic₉	—	3	1	no	—
39	#63372	—	3	1	no	—
40	C₃×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
41	C₃×D₄×C₉	—	2	1	no	—
42	C₃×D₄×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×#67	—	2	1	no	—
46	C₂×C₃×Dic₉	—	2	1	no	—
47	C₉×D₁₂	C₂×C₄	1	1	yes	characteristic
48	C₄×D₃×C₉	—	2	1	no	—
49	C₉×Dic₆	C₂×C₄	1	1	yes	characteristic
50	C₃⋊(C₉⋊C₈)	—	2	1	no	—
51	C₃⋊(C₉⋊C₈)	D₄	1	1	yes	characteristic
52	C₃⋊(C₉⋊C₈)	D₄	1	1	yes	characteristic
53	C₉×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₉×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×C₉⋊C₈	—	2	1	no	—
56	#5196	—	9	1	no	—
57	#4824	—	9	1	no	—
58	C₃²×#41	C₂×D₃	1	1	yes	characteristic
59	C₃×#235	C₂×D₃	1	1	yes	characteristic
60	#3748	—	3	1	no	—
61	#3748	—	6	1	no	—
62	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
63	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
64	C₃×#216	Dic₃	1	1	yes	characteristic
65	C₃×#215	Dic₃	1	1	yes	characteristic
66	C₃×#214	Dic₃	1	1	yes	characteristic
67	C₃×#213	Dic₃	1	1	yes	characteristic
68	C₃×C₄×Dic₃	—	3	1	no	—
69	C₃×#208	—	3	1	no	—
70	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
71	#3711	—	6	1	no	—
72	D₃×C₃⋊C₈	—	6	1	no	—
73	C₉×#41	—	2	1	no	—
74	C₉×#41	—	3	1	no	—
75	C₉×#41	C₂×D₃	1	1	yes	characteristic
76	C₉×Q₁₆	—	3	1	no	—
77	C₉×SD₁₆	—	3	1	no	—
78	C₉×SD₁₆	—	3	1	no	—
79	C₉×D₈	—	3	1	no	—
80	C₂×C₈×C₉	—	3	1	no	—
81	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
82	#3667	—	3	1	no	—
83	#3667	—	6	1	no	—
84	#3667	—	6	1	no	—
85	#3667	—	6	1	no	—
86	#3667	—	6	2	no	—
87	#3667	C₂×D₃	1	1	yes	characteristic
88	C₂×C₉⋊C₈	—	3	1	no	—
89	C₂×C₉⋊C₈	—	6	1	no	—
90	C₂×C₉⋊C₈	—	6	1	no	—
91	C₂×C₉⋊C₈	—	6	1	no	—
92	C₂×C₉⋊C₈	—	6	2	no	—
93	C₂²×C₃×C₉	—	2	1	no	—
94	C₂²×C₃×C₉	#41	1	1	yes	characteristic
95	C₂×D₃×C₉	—	2	1	no	—
96	C₃×C₄×C₉	—	2	1	no	—
97	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
98	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₉×Dic₃	—	2	1	no	—
100	C₃×Dic₉	—	2	2	no	—
101	C₃×#135	—	3	1	no	—
102	#950	D₉	1	1	yes	characteristic
103	#948	—	9	1	no	—
104	#948	—	9	1	no	—
105	#948	—	9	1	no	—
106	#948	—	18	1	no	—
107	#907	—	9	1	no	—
108	C₃×#104	—	9	1	no	—
109	#837	—	3	1	no	—
110	#837	—	9	1	no	—
111	#822	—	9	1	no	—
112	#819	—	9	1	no	—
113	C₈×Dic₃	—	9	1	no	—
114	#805	—	9	1	no	—
115	C₄×C₃⋊C₈	—	9	1	no	—
116	C₃²×Q₈	C₂×Dic₃	1	1	yes	characteristic
117	C₃²×D₄	—	2	1	no	—
118	C₃²×D₄	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₃²×C₄	—	2	1	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₃×#67	—	2	1	no	—
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₃×D₁₂	C₂×Dic₃	1	1	yes	characteristic
124	C₃×C₄×D₃	—	2	1	no	—
125	C₃×Dic₆	C₂×Dic₃	1	1	yes	characteristic
126	C₃⋊(C₃⋊C₈)	—	3	1	no	—
127	C₃⋊(C₃⋊C₈)	—	3	1	no	—
128	C₃⋊(C₃⋊C₈)	—	6	1	no	—
129	C₃×C₃⋊C₈	—	6	1	no	—
130	C₃×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
131	C₃×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
132	Q₈×C₉	—	2	1	no	—
133	Q₈×C₉	—	3	1	no	—
134	Q₈×C₉	C₄×D₃	1	1	yes	characteristic
135	D₄×C₉	—	2	2	no	—
136	D₄×C₉	—	2	1	no	—
137	D₄×C₉	—	2	1	no	—
138	D₄×C₉	—	3	1	no	—
139	D₄×C₉	—	6	1	no	—
140	D₄×C₉	C₄×D₃	1	1	yes	characteristic
141	C₂×C₄×C₉	—	2	2	no	—
142	C₂×C₄×C₉	—	2	1	no	—
143	C₂×C₄×C₉	—	2	1	no	—
144	C₂×C₄×C₉	—	6	1	no	—
145	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
146	C₂×Dic₉	—	2	1	no	—
147	C₈×C₉	—	3	1	no	—
148	C₈×C₉	—	3	1	no	—
149	C₉⋊C₈	—	2	1	no	—
150	C₉⋊C₈	—	3	1	no	—
151	C₉⋊C₈	—	3	1	no	—
152	C₉⋊C₈	—	6	1	no	—
153	C₉⋊C₈	—	6	1	no	—
154	C₉⋊C₈	—	6	2	no	—
155	C₉⋊C₈	—	6	1	no	—
156	#443	—	27	1	no	Sylow
157	C₂×C₃×C₉	—	2	1	no	—
158	C₂×C₃×C₉	—	4	1	no	—
159	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
160	D₃×C₉	—	4	1	no	—
161	C₃×#41	—	2	1	no	—
162	C₃×#41	—	3	1	no	—
163	C₃×#41	C₂×D₉	1	1	yes	characteristic
164	C₃×#41	D₃²	1	1	yes	characteristic
165	#235	C₂×D₉	1	1	yes	characteristic
166	C₃×Q₁₆	—	3	1	no	—
167	C₃×SD₁₆	—	3	1	no	—
168	C₃×SD₁₆	—	3	1	no	—
169	C₃×D₈	—	3	1	no	—
170	C₃×M₁₆	—	9	1	no	—
171	C₂×C₃×C₈	—	3	1	no	—
172	C₃×C₄²	—	9	1	no	—
173	#216	Dic₉	1	1	yes	characteristic, complemented
174	#215	Dic₉	1	1	yes	characteristic, complemented
175	#214	Dic₉	1	1	yes	characteristic, complemented
176	#213	Dic₉	1	1	yes	characteristic, complemented
177	C₄×Dic₃	—	3	1	no	—
178	#208	—	3	1	no	—
179	#208	—	9	1	no	—
180	#208	—	9	1	no	—
181	#208	—	18	1	no	—
182	#208	—	18	1	no	—
183	#208	—	18	1	no	—
184	#208	—	18	1	no	—
185	#208	—	18	2	no	—
186	C₂×C₃⋊C₈	—	9	1	no	—
187	C₂×C₃⋊C₈	—	9	1	no	—
188	C₂×C₃⋊C₈	—	18	1	no	—
189	C₂×C₃⋊C₈	—	18	1	no	—
190	C₂×C₃⋊C₈	—	18	1	no	—
191	C₂×C₃⋊C₈	—	18	1	no	—
192	C₂×C₃⋊C₈	—	18	2	no	—
193	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
194	ΓL(1,𝔽₂₅)	—	18	1	no	—
195	D₃×C₈	—	18	1	no	—
196	C₂²×C₃²	—	2	1	no	—
197	C₂²×C₃²	#237	1	1	yes	characteristic
198	C₂×C₃×D₃	—	2	1	no	—
199	C₃²×C₄	—	2	1	no	—
200	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
201	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
202	C₃×Dic₃	—	2	1	no	—
203	C₃×Dic₃	—	6	2	no	—
204	C₂²×C₉	—	2	2	no	—
205	C₂²×C₉	—	2	1	no	—
206	C₂²×C₉	—	2	1	no	—
207	C₂²×C₉	—	6	1	no	—
208	C₂²×C₉	#237	1	1	yes	characteristic
209	C₄×C₉	—	2	2	no	—
210	C₄×C₉	—	2	1	no	—
211	C₄×C₉	—	2	1	no	—
212	C₄×C₉	—	2	1	no	—
213	C₄×C₉	—	6	1	no	—
214	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
215	C₄×C₉	D₃×D₄	1	1	yes	characteristic
216	Dic₉	—	2	2	no	—
217	#135	—	3	1	no	—
218	#131	—	27	1	no	—
219	#131	—	27	1	no	—
220	#108	—	27	1	no	—
221	#104	—	9	1	no	—
222	#104	—	27	1	no	—
223	C₄×C₈	—	27	1	no	—
224	C₃×C₉	#443	1	1	yes	characteristic, complemented, Sylow
225	C₃×Q₈	—	2	1	no	—
226	C₃×Q₈	—	3	1	no	—
227	C₃×Q₈	C₂×Dic₉	1	1	yes	characteristic
228	C₃×Q₈	D₃×Dic₃	1	1	yes	characteristic
229	C₃×D₄	—	2	2	no	—
230	C₃×D₄	—	2	1	no	—
231	C₃×D₄	—	2	1	no	—
232	C₃×D₄	—	2	1	no	—
233	C₃×D₄	—	3	1	no	—
234	C₃×D₄	—	6	1	no	—
235	C₃×D₄	C₂×Dic₉	1	1	yes	characteristic
236	C₃×D₄	D₃×Dic₃	1	1	yes	characteristic
237	C₂×C₃×C₄	—	2	2	no	—
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	6	1	no	—
242	C₂×C₃×C₄	—	18	1	no	—
243	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
245	#67	—	2	1	no	—
246	C₂×Dic₃	—	6	1	no	—
247	D₁₂	C₂×Dic₉	1	1	yes	characteristic
248	C₄×D₃	—	2	1	no	—
249	Dic₆	C₂×Dic₉	1	1	yes	characteristic
250	C₃×C₈	—	3	1	no	—
251	C₃×C₈	—	3	1	no	—
252	C₃×C₈	—	18	1	no	—
253	C₃⋊C₈	—	6	1	no	—
254	C₃⋊C₈	—	9	1	no	—
255	C₃⋊C₈	—	9	1	no	—
256	C₃⋊C₈	—	9	1	no	—
257	C₃⋊C₈	—	9	1	no	—
258	C₃⋊C₈	—	18	1	no	—
259	C₃⋊C₈	—	18	1	no	—
260	C₃⋊C₈	—	18	2	no	—
261	C₃⋊C₈	—	18	1	no	—
262	C₃⋊C₈	—	18	1	no	—
263	C₃⋊C₈	C₂×Dic₉	1	1	yes	characteristic
264	C₃⋊C₈	C₂×Dic₉	1	1	yes	characteristic
265	C₂×C₃²	—	2	1	no	—
266	C₂×C₃²	—	4	1	no	—
267	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
268	C₃×D₃	—	4	1	no	—
269	C₂×C₉	—	2	1	no	—
270	C₂×C₉	—	2	1	no	—
271	C₂×C₉	—	4	1	no	—
272	C₂×C₉	—	4	2	no	—
273	C₂×C₉	—	4	1	no	—
274	C₂×C₉	—	12	1	no	—
275	C₂×C₉	#67⋊C₄	1	1	yes	characteristic
276	#41	—	3	1	no	—
277	#41	D₃×D₉	1	1	yes	characteristic
278	Q₁₆	—	3	1	no	—
279	SD₁₆	—	3	1	no	—
280	SD₁₆	—	3	1	no	—
281	D₈	—	3	1	no	—
282	M₁₆	—	9	1	no	—
283	M₁₆	—	27	1	no	—
284	M₁₆	—	54	1	no	—
285	M₁₆	—	54	1	no	—
286	C₂×C₈	—	3	1	no	—
287	C₂×C₈	—	27	1	no	—
288	C₂×C₈	—	54	1	no	—
289	C₂×C₈	—	54	1	no	—
290	C₄²	—	9	1	no	—
291	C₂²×C₃	—	2	2	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	2	1	no	—
295	C₂²×C₃	—	6	1	no	—
296	C₂²×C₃	#3699	1	1	yes	characteristic
297	C₂²×C₃	#3805	1	1	yes	characteristic
298	C₂×D₃	—	2	1	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	6	1	no	—
305	C₃×C₄	—	18	2	no	—
306	C₃×C₄	D₄×D₉	1	1	yes	characteristic
307	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
308	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
309	C₃×C₄	#3811	1	1	yes	characteristic
310	Dic₃	—	2	1	no	—
311	Dic₃	—	6	2	no	—
312	C₃²	#5196	1	1	yes	characteristic
313	C₉	—	2	1	no	—
314	C₉	#4824	1	1	yes	characteristic, complemented
315	Q₈	—	3	1	no	—
316	Q₈	D₃×Dic₉	1	1	yes	characteristic
317	D₄	—	2	1	no	—
318	D₄	—	3	1	no	—
319	D₄	—	6	1	no	—
320	D₄	D₃×Dic₉	1	1	yes	characteristic
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	6	1	no	—
323	C₂×C₄	—	18	1	no	—
324	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
325	C₈	—	3	1	no	—
326	C₈	—	3	1	no	—
327	C₈	—	18	1	no	—
328	C₈	—	27	1	no	—
329	C₈	—	27	1	no	—
330	C₈	—	54	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	4	1	no	—
335	C₂×C₃	—	4	2	no	—
336	C₂×C₃	—	4	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	12	1	no	—
339	C₂×C₃	D₄×Dic₉	1	1	yes	characteristic
340	C₂×C₃	#67⋊Dic₃	1	1	yes	characteristic
341	D₃	—	4	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	6	1	no	—
344	C₂²	#88663	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	6	1	no	—
347	C₄	—	18	2	no	—
348	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
349	C₄	#88668	1	1	yes	characteristic
350	C₃	—	2	1	no	—
351	C₃	#10588876	1	1	yes	characteristic, complemented
352	C₃	#10591675	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	4	1	no	—
355	C₂	—	12	1	no	—
356	C₂	#67⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88668
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₉×#41	C₂²
socle	C₂×C₃²	D₄×Dic₃
prime radical	C₃×#950	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#41	D₃×D₉
hypocenter	C₃×C₉	#443
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #11721865	C₂	semidirect
3. #11720878	C₂	semidirect
4. C₃×#63396	C₂	semidirect
5. #11720301	C₂	semidirect
6. #11720280	C₂	semidirect
7. C₃⋊C₈×Dic₉	C₂	semidirect
16. C₉×#216	C₄	semidirect
17. C₉×#215	C₄	semidirect
18. C₉×#214	C₄	semidirect
19. C₉×#213	C₄	semidirect
37. #63396	D₃	semidirect
173. #216	Dic₉	semidirect
174. #215	Dic₉	semidirect
175. #214	Dic₉	semidirect
176. #213	Dic₉	semidirect
224. C₃×C₉	#443	semidirect
314. C₉	#4824	semidirect
351. C₃	#10588876	semidirect

Maximal subgroups

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

subgroup	index
#11721865	2
C₉×#950	2
#11720878	2
C₃×#63396	2
#11720301	2
#11720280	2
C₃⋊C₈×Dic₉	2
#10591675	3
#10588876	3