Subgroups — #13144408

Order 1728.

660 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88544	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720576	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720528	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720526	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720516	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592713	—	3	1	no	maximal
9	#10588697	—	3	3	no	maximal
10	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊D₉)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₃⋊D₉	—	2	1	no	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	#88544	C₄	1	4	yes	complemented
18	#88536	—	2	1	no	—
19	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
22	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
23	C₂×#3753	—	3	1	no	—
24	C₄×C₄⋊(C₃⋊D₃)	—	3	1	no	—
25	#63659	—	3	1	no	—
26	#63636	—	3	1	no	—
27	#63635	—	3	1	no	—
28	#63630	—	3	1	no	—
29	C₄×D₄×C₉	D₃	1	3	yes	complemented
30	C₂×#3673	—	3	3	no	—
31	C₄×D₃₆	—	3	3	no	—
32	#63392	—	3	3	no	—
33	#63369	—	3	3	no	—
34	#63367	—	3	3	no	—
35	#63362	—	3	3	no	—
36	C₂³×C₃×C₉	—	2	1	no	—
37	C₂²×C₃⋊D₉	—	2	1	no	—
38	C₃×D₄×C₉	—	2	1	no	—
39	C₃×D₄×C₉	C₂×C₄	1	2	yes	complemented
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	4	2	no	—
42	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊Dic₉	—	2	1	no	—
46	C₄⋊(C₃⋊D₉)	—	2	1	no	—
47	C₄⋊(C₃⋊D₉)	C₂×C₄	1	2	yes	complemented
48	C₄×C₃⋊D₉	—	4	2	no	—
49	C₃⋊(C₉⋊C₈)	—	2	1	no	—
50	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	complemented
51	#5014	—	9	3	no	—
52	#5014	—	9	1	no	—
53	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃²×C₄	—	2	1	no	—
55	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
56	C₂×C₄×C₃⋊D₃	—	6	1	no	—
57	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₂²⋊C₄	—	2	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	#3753	—	3	4	no	—
61	#3752	—	6	1	no	—
62	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
63	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
64	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
65	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
66	C₂²×C₄×C₉	—	2	3	no	—
67	C₂×D₃₆	—	3	3	no	—
68	C₂×C₄×D₉	—	6	3	no	—
69	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
70	C₉×C₂²⋊C₄	—	2	3	no	—
71	C₄²×C₉	C₂×D₃	1	3	yes	—
72	#3673	—	3	12	no	—
73	(C₂×D₉)⋊C₄	—	6	3	no	—
74	C₄⋊Dic₉	—	3	3	no	—
75	C₂×C₉⋊C₈	—	3	3	no	—
76	C₂×C₉⋊C₈	—	3	3	no	—
77	C₂²×C₃×C₉	—	2	2	no	—
78	C₂²×C₃×C₉	—	4	1	no	—
79	C₂²×C₃×C₉	—	4	1	no	—
80	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃⋊D₉	—	2	2	no	—
82	C₂×C₃⋊D₉	—	4	1	no	—
83	C₂×C₃⋊D₉	—	4	1	no	—
84	C₃×C₄×C₉	—	2	1	no	—
85	C₃×C₄×C₉	—	2	1	no	—
86	C₃×C₄×C₉	—	4	1	no	—
87	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄×C₉	#41	1	1	yes	characteristic
89	C₃⋊Dic₉	—	4	1	no	—
90	C₃×C₄×D₄	D₉	1	3	yes	complemented
91	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
92	C₂×#213	—	9	3	no	—
93	C₂×#213	—	9	1	no	—
94	C₄×D₁₂	—	9	3	no	—
95	C₄×D₁₂	—	9	1	no	—
96	#832	—	9	3	no	—
97	#832	—	9	1	no	—
98	#809	—	9	3	no	—
99	#809	—	9	1	no	—
100	#808	—	9	3	no	—
101	#808	—	9	1	no	—
102	#803	—	9	3	no	—
103	#803	—	9	1	no	—
104	C₂³×C₃²	—	2	1	no	—
105	C₂²×C₃⋊D₃	—	6	1	no	—
106	C₃²×D₄	—	2	1	no	—
107	C₃²×D₄	C₄×D₃	1	2	yes	—
108	C₂×C₃²×C₄	—	2	1	no	—
109	C₂×C₃²×C₄	—	4	2	no	—
110	C₂×C₃²×C₄	#67	1	1	yes	characteristic
111	C₂×C₃²×C₄	#67	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	—	6	1	no	—
114	C₄⋊(C₃⋊D₃)	—	3	2	no	—
115	C₄⋊(C₃⋊D₃)	—	6	1	no	—
116	C₄×C₃⋊D₃	—	12	2	no	—
117	C₂³×C₉	—	2	3	no	—
118	C₂²×D₉	—	6	3	no	—
119	C₃⋊(C₃⋊C₈)	—	3	2	no	—
120	C₃⋊(C₃⋊C₈)	—	6	1	no	—
121	D₄×C₉	—	2	3	no	—
122	D₄×C₉	C₄×D₃	1	6	yes	complemented
123	C₂×C₄×C₉	—	2	3	no	—
124	C₂×C₄×C₉	—	4	6	no	—
125	C₂×C₄×C₉	#67	1	3	yes	—
126	C₂×C₄×C₉	#67	1	3	yes	—
127	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
128	C₂×Dic₉	—	6	3	no	—
129	D₃₆	—	3	6	no	—
130	D₃₆	—	6	3	no	—
131	C₄×D₉	—	12	6	no	—
132	C₉⋊C₈	—	3	6	no	—
133	C₉⋊C₈	—	6	3	no	—
134	#442	—	27	1	no	Sylow
135	C₂×C₃×C₉	—	4	2	no	—
136	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₃⋊D₉	—	4	2	no	—
140	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
141	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
142	C₂²×C₃×C₄	—	2	3	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂×D₁₂	—	9	3	no	—
145	C₂×D₁₂	—	9	1	no	—
146	C₂×C₄×D₃	—	18	3	no	—
147	C₂×C₄×D₃	—	18	1	no	—
148	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
149	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
150	C₃×C₂²⋊C₄	—	2	3	no	—
151	C₃×C₂²⋊C₄	—	2	1	no	—
152	C₃×C₄²	C₂×D₉	1	3	yes	—
153	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
154	#213	—	9	12	no	—
155	#213	—	9	4	no	—
156	(C₂×D₃)⋊C₄	—	18	3	no	—
157	(C₂×D₃)⋊C₄	—	18	1	no	—
158	C₄⋊Dic₃	—	9	3	no	—
159	C₄⋊Dic₃	—	9	1	no	—
160	C₂×C₃⋊C₈	—	9	3	no	—
161	C₂×C₃⋊C₈	—	9	1	no	—
162	C₂×C₃⋊C₈	—	9	3	no	—
163	C₂×C₃⋊C₈	—	9	1	no	—
164	C₂²×C₃²	—	2	2	no	—
165	C₂²×C₃²	—	4	1	no	—
166	C₂²×C₃²	—	4	1	no	—
167	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
168	C₂×C₃⋊D₃	—	6	2	no	—
169	C₂×C₃⋊D₃	—	12	1	no	—
170	C₂×C₃⋊D₃	—	12	1	no	—
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	—	4	1	no	—
174	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₃²×C₄	#235	1	1	yes	characteristic
176	C₃⋊Dic₃	—	12	1	no	—
177	C₂²×C₉	—	2	6	no	—
178	C₂²×C₉	—	4	3	no	—
179	C₂²×C₉	—	4	3	no	—
180	C₂²×C₉	C₂×#67	1	3	yes	—
181	C₂×D₉	—	6	6	no	—
182	C₂×D₉	—	12	3	no	—
183	C₂×D₉	—	12	3	no	—
184	C₄×C₉	—	2	3	no	—
185	C₄×C₉	—	2	3	no	—
186	C₄×C₉	—	4	3	no	—
187	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
188	C₄×C₉	#235	1	3	yes	—
189	Dic₉	—	12	3	no	—
190	C₂×D₈	—	27	1	no	—
191	C₄×D₄	—	27	1	no	—
192	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic, complemented
193	#106	—	27	1	no	—
194	#102	—	27	1	no	—
195	#102	—	27	1	no	—
196	#97	—	27	1	no	—
197	C₃×C₉	#442	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₃	—	2	3	no	—
199	C₂³×C₃	—	2	1	no	—
200	C₂²×D₃	—	18	3	no	—
201	C₂²×D₃	—	18	1	no	—
202	C₃×D₄	—	2	3	no	—
203	C₃×D₄	—	2	1	no	—
204	C₃×D₄	C₄×D₉	1	6	yes	complemented
205	C₃×D₄	C₄×C₃⋊D₃	1	2	yes	—
206	C₂×C₃×C₄	—	2	3	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	6	no	—
210	C₂×C₃×C₄	#611	1	3	yes	—
211	C₂×C₃×C₄	#611	1	3	yes	—
212	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
213	C₂×C₃×C₄	#638	1	1	yes	characteristic
214	C₂×C₃×C₄	#638	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
216	C₂×Dic₃	—	18	3	no	—
217	C₂×Dic₃	—	18	1	no	—
218	D₁₂	—	9	6	no	—
219	D₁₂	—	9	2	no	—
220	D₁₂	—	18	3	no	—
221	D₁₂	—	18	1	no	—
222	C₄×D₃	—	36	2	no	—
223	C₄×D₃	—	36	6	no	—
224	C₃⋊C₈	—	9	6	no	—
225	C₃⋊C₈	—	9	2	no	—
226	C₃⋊C₈	—	18	3	no	—
227	C₃⋊C₈	—	18	1	no	—
228	C₂×C₃²	—	4	2	no	—
229	C₂×C₃²	C₄×#67	1	1	yes	characteristic
230	C₂×C₃²	#932	1	1	yes	characteristic
231	C₂×C₃²	#949	1	1	yes	characteristic
232	C₃⋊D₃	—	12	2	no	—
233	C₂×C₉	—	4	6	no	—
234	C₂×C₉	C₄×#67	1	3	yes	—
235	C₂×C₉	#932	1	3	yes	—
236	C₂×C₉	#949	1	3	yes	—
237	D₉	—	12	6	no	—
238	C₂×D₄	—	27	1	no	—
239	C₂×D₄	C₂×C₃⋊D₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	54	1	no	—
242	D₈	—	27	4	no	—
243	C₂×C₈	—	27	1	no	—
244	C₂×C₈	—	27	1	no	—
245	C₄⋊C₄	—	27	1	no	—
246	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	54	1	no	—
249	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
250	C₂²×C₃	—	2	6	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	4	3	no	—
253	C₂²×C₃	—	4	3	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	C₂×#611	1	3	yes	—
257	C₂²×C₃	C₂×#638	1	1	yes	characteristic
258	C₂×D₃	—	18	6	no	—
259	C₂×D₃	—	18	2	no	—
260	C₂×D₃	—	36	3	no	—
261	C₂×D₃	—	36	3	no	—
262	C₂×D₃	—	36	1	no	—
263	C₂×D₃	—	36	1	no	—
264	C₃×C₄	—	2	3	no	—
265	C₃×C₄	—	2	3	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	4	3	no	—
269	C₃×C₄	—	4	1	no	—
270	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
271	C₃×C₄	#3697	1	3	yes	—
272	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
273	C₃×C₄	#3828	1	1	yes	characteristic
274	Dic₃	—	36	3	no	—
275	Dic₃	—	36	1	no	—
276	C₃²	#5014	1	1	yes	characteristic
277	C₉	#5014	1	3	yes	complemented
278	C₂³	—	2	1	no	—
279	C₂³	—	54	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	27	2	no	—
282	D₄	—	54	1	no	—
283	D₄	C₄×C₃⋊D₉	1	2	yes	complemented
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	54	1	no	—
287	C₂×C₄	—	108	2	no	—
288	C₂×C₄	#6233	1	1	yes	characteristic
289	C₂×C₄	#6233	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
291	C₈	—	27	2	no	—
292	C₈	—	54	1	no	—
293	C₂×C₃	—	4	6	no	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	C₄×#611	1	3	yes	—
296	C₂×C₃	#63495	1	3	yes	—
297	C₂×C₃	#63512	1	3	yes	—
298	C₂×C₃	C₄×#638	1	1	yes	characteristic
299	C₂×C₃	#64141	1	1	yes	characteristic
300	C₂×C₃	#64158	1	1	yes	characteristic
301	D₃	—	36	6	no	—
302	D₃	—	36	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	54	2	no	—
307	C₂²	—	108	1	no	—
308	C₂²	—	108	1	no	—
309	C₂²	C₂×#6233	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	108	1	no	—
314	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
315	C₄	#88736	1	1	yes	characteristic
316	C₃	#10588697	1	3	yes	complemented
317	C₃	#10592713	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	108	2	no	—
320	C₂	C₄×#6233	1	1	yes	characteristic
321	C₂	#11721833	1	1	yes	characteristic
322	C₂	#11721867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×#88544	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	#442
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×D₄×C₉	C₂	semidirect
3. C₄×C₄⋊(C₃⋊D₉)	C₂	semidirect
4. #11720576	C₂	semidirect
5. #11720528	C₂	semidirect
6. #11720526	C₂	semidirect
7. #11720516	C₂	semidirect
17. #88544	C₄	semidirect
29. C₄×D₄×C₉	D₃	semidirect
39. C₃×D₄×C₉	C₂×C₄	semidirect
47. C₄⋊(C₃⋊D₉)	C₂×C₄	semidirect
50. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
90. C₃×C₄×D₄	D₉	semidirect
122. D₄×C₉	C₄×D₃	semidirect
192. C₄×D₄	C₃⋊D₉	semidirect
197. C₃×C₉	#442	semidirect
204. C₃×D₄	C₄×D₉	semidirect
277. C₉	#5014	semidirect
283. D₄	C₄×C₃⋊D₉	semidirect
316. C₃	#10588697	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₂×#88544	2
C₄×C₄⋊(C₃⋊D₉)	2
#11720576	2
#11720528	2
#11720526	2
#11720516	2
#10592713	3
#10588697	3