Subgroups — #13135879

Order 1728.

710 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	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#88551	C₂	1	1	yes	maximal, characteristic
3	C₂×#88551	C₂	1	1	yes	maximal, characteristic
4	#10589393	—	3	1	no	maximal
5	#10588204	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	1	no	—
8	C₂²×C₃⋊Dic₉	—	2	1	no	—
9	C₂²×C₃⋊Dic₉	C₄	1	2	yes	complemented
10	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
11	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	complemented
12	#88551	—	2	1	no	—
13	#88551	—	2	1	no	—
14	#88551	—	2	1	no	—
15	#88551	C₄	1	2	yes	complemented
16	C₂×C₃²×C₂²⋊C₄	D₃	1	1	yes	characteristic
17	C₂×#3757	—	3	1	no	—
18	C₂×#3757	—	3	1	no	—
19	C₂×C₉×C₂²⋊C₄	D₃	1	3	yes	—
20	C₂×#3676	—	3	3	no	—
21	C₂×#3676	—	3	3	no	—
22	C₂³×C₃×C₉	—	4	2	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂³×C₃×C₉	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	4	2	no	—
32	C₂×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₃⋊Dic₉	—	2	2	no	—
34	C₂×C₃⋊Dic₉	—	4	2	no	—
35	C₂×C₃⋊Dic₉	—	4	2	no	—
36	#4527	—	9	3	no	—
37	#4527	—	9	1	no	—
38	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	—	2	1	no	—
40	C₂²×C₃⋊Dic₃	—	3	2	no	—
41	C₂²×C₃⋊Dic₃	—	6	1	no	—
42	C₂⁴×C₉	C₂×D₃	1	3	yes	—
43	C₃²×C₂²⋊C₄	—	2	1	no	—
44	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
45	#3757	—	3	2	no	—
46	#3757	—	6	1	no	—
47	#3757	—	6	1	no	—
48	#3757	—	6	1	no	—
49	C₂²×C₄×C₉	—	2	3	no	—
50	C₂²×Dic₉	—	3	6	no	—
51	C₂²×Dic₉	—	6	3	no	—
52	C₉×C₂²⋊C₄	—	2	3	no	—
53	C₉×C₂²⋊C₄	Dic₃	1	6	yes	complemented
54	#3676	—	3	6	no	—
55	#3676	—	6	3	no	—
56	#3676	—	6	3	no	—
57	#3676	—	6	3	no	—
58	C₂²×C₃×C₉	—	2	2	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	—	4	1	no	—
62	C₂²×C₃×C₉	—	4	1	no	—
63	C₂²×C₃×C₉	—	4	1	no	—
64	C₂²×C₃×C₉	—	4	1	no	—
65	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
72	C₃×C₄×C₉	—	4	2	no	—
73	C₃⋊Dic₉	—	4	2	no	—
74	C₃⋊Dic₉	—	4	2	no	—
75	C₂×C₃×C₂²⋊C₄	D₉	1	3	yes	—
76	C₂×C₃×C₂²⋊C₄	C₃⋊D₃	1	1	yes	characteristic
77	C₂×#217	—	9	3	no	—
78	C₂×#217	—	9	1	no	—
79	C₂×#217	—	9	3	no	—
80	C₂×#217	—	9	1	no	—
81	C₂³×C₃²	—	4	2	no	—
82	C₂³×C₃²	Dic₆	1	1	yes	characteristic
83	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
84	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
85	C₂³×C₃²	D₁₂	1	1	yes	characteristic
86	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
87	C₂³×C₃²	#67	1	1	yes	characteristic
88	C₂³×C₃²	#67	1	1	yes	characteristic
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	—	4	2	no	—
91	C₂×C₃⋊Dic₃	—	6	2	no	—
92	C₂×C₃⋊Dic₃	—	6	2	no	—
93	C₂×C₃⋊Dic₃	—	12	2	no	—
94	C₂×C₃⋊Dic₃	—	12	2	no	—
95	C₂³×C₉	—	4	6	no	—
96	C₂³×C₉	Dic₆	1	3	yes	—
97	C₂³×C₉	C₄×D₃	1	3	yes	—
98	C₂³×C₉	C₄×D₃	1	3	yes	—
99	C₂³×C₉	D₁₂	1	3	yes	—
100	C₂³×C₉	C₂×Dic₃	1	3	yes	—
101	C₂³×C₉	#67	1	3	yes	—
102	C₂³×C₉	#67	1	3	yes	—
103	C₂×C₄×C₉	—	2	6	no	—
104	C₂×C₄×C₉	—	4	6	no	—
105	C₂×Dic₉	—	6	6	no	—
106	C₂×Dic₉	—	6	6	no	—
107	C₂×Dic₉	—	12	6	no	—
108	C₂×Dic₉	—	12	6	no	—
109	#342	—	27	1	no	Sylow
110	C₂×C₃×C₉	—	2	1	no	—
111	C₂×C₃×C₉	—	2	1	no	—
112	C₂×C₃×C₉	—	2	1	no	—
113	C₂×C₃×C₉	—	2	1	no	—
114	C₂×C₃×C₉	—	2	1	no	—
115	C₂×C₃×C₉	—	2	1	no	—
116	C₂×C₃×C₉	#95	1	1	yes	characteristic
117	C₂×C₃×C₉	#99	1	2	yes	—
118	C₂⁴×C₃	C₂×D₉	1	3	yes	—
119	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₂²×C₃×C₄	—	2	3	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×Dic₃	—	9	6	no	—
123	C₂²×Dic₃	—	9	2	no	—
124	C₂²×Dic₃	—	18	3	no	—
125	C₂²×Dic₃	—	18	1	no	—
126	C₃×C₂²⋊C₄	—	2	3	no	—
127	C₃×C₂²⋊C₄	—	2	1	no	—
128	C₃×C₂²⋊C₄	Dic₉	1	6	yes	complemented
129	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
130	#217	—	9	6	no	—
131	#217	—	9	2	no	—
132	#217	—	18	3	no	—
133	#217	—	18	3	no	—
134	#217	—	18	3	no	—
135	#217	—	18	1	no	—
136	#217	—	18	1	no	—
137	#217	—	18	1	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	—	2	2	no	—
141	C₂²×C₃²	—	4	1	no	—
142	C₂²×C₃²	—	4	1	no	—
143	C₂²×C₃²	—	4	1	no	—
144	C₂²×C₃²	—	4	1	no	—
145	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
146	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
147	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
149	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
150	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
151	C₂²×C₃²	#217	1	1	yes	characteristic
152	C₃²×C₄	—	4	2	no	—
153	C₃⋊Dic₃	—	12	2	no	—
154	C₃⋊Dic₃	—	12	2	no	—
155	C₂²×C₉	—	2	6	no	—
156	C₂²×C₉	—	2	6	no	—
157	C₂²×C₉	—	2	6	no	—
158	C₂²×C₉	—	4	3	no	—
159	C₂²×C₉	—	4	3	no	—
160	C₂²×C₉	—	4	3	no	—
161	C₂²×C₉	—	4	3	no	—
162	C₂²×C₉	C₄×Dic₃	1	3	yes	—
163	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
164	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
165	C₂²×C₉	C₄⋊Dic₃	1	3	yes	—
166	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
167	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
168	C₂²×C₉	#217	1	3	yes	—
169	C₄×C₉	—	4	6	no	—
170	Dic₉	—	12	6	no	—
171	Dic₉	—	12	6	no	—
172	C₂×C₂²⋊C₄	—	27	1	no	—
173	C₂×C₂²⋊C₄	—	27	1	no	—
174	C₂×C₂²⋊C₄	C₃⋊D₉	1	1	yes	characteristic
175	C₃×C₉	#342	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₃	—	4	2	no	—
177	C₂³×C₃	—	4	6	no	—
178	C₂³×C₃	Dic₁₈	1	3	yes	—
179	C₂³×C₃	C₄×D₉	1	3	yes	—
180	C₂³×C₃	C₄×D₉	1	3	yes	—
181	C₂³×C₃	D₃₆	1	3	yes	—
182	C₂³×C₃	C₂×Dic₉	1	3	yes	—
183	C₂³×C₃	#611	1	3	yes	—
184	C₂³×C₃	#611	1	3	yes	—
185	C₂³×C₃	#634	1	1	yes	characteristic
186	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
187	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
188	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
189	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
190	C₂³×C₃	#638	1	1	yes	characteristic
191	C₂³×C₃	#638	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	6	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	—	4	6	no	—
196	C₂×Dic₃	—	18	6	no	—
197	C₂×Dic₃	—	18	6	no	—
198	C₂×Dic₃	—	18	2	no	—
199	C₂×Dic₃	—	18	2	no	—
200	C₂×Dic₃	—	36	2	no	—
201	C₂×Dic₃	—	36	6	no	—
202	C₂×Dic₃	—	36	2	no	—
203	C₂×Dic₃	—	36	6	no	—
204	C₂×C₃²	—	2	1	no	—
205	C₂×C₃²	—	2	1	no	—
206	C₂×C₃²	—	2	1	no	—
207	C₂×C₃²	—	2	1	no	—
208	C₂×C₃²	—	2	1	no	—
209	C₂×C₃²	—	2	1	no	—
210	C₂×C₃²	#806	1	2	yes	—
211	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
212	C₂×C₉	—	2	3	no	—
213	C₂×C₉	—	2	3	no	—
214	C₂×C₉	—	2	3	no	—
215	C₂×C₉	—	2	3	no	—
216	C₂×C₉	—	2	3	no	—
217	C₂×C₉	—	2	3	no	—
218	C₂×C₉	#806	1	6	yes	—
219	C₂×C₉	C₃⋊#95	1	3	yes	—
220	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	27	2	no	—
223	C₂²×C₄	—	54	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	27	2	no	—
226	C₂²⋊C₄	—	54	1	no	—
227	C₂²⋊C₄	—	54	1	no	—
228	C₂²⋊C₄	—	54	1	no	—
229	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	complemented
230	C₂²×C₃	—	2	6	no	—
231	C₂²×C₃	—	2	6	no	—
232	C₂²×C₃	—	2	6	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	4	3	no	—
237	C₂²×C₃	—	4	3	no	—
238	C₂²×C₃	—	4	3	no	—
239	C₂²×C₃	—	4	3	no	—
240	C₂²×C₃	—	4	1	no	—
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	—	4	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	C₄×Dic₉	1	3	yes	—
245	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
246	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
247	C₂²×C₃	C₄⋊Dic₉	1	3	yes	—
248	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
249	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
250	C₂²×C₃	#3676	1	3	yes	—
251	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
252	C₂²×C₃	#3750	1	1	yes	characteristic
253	C₂²×C₃	#3750	1	1	yes	characteristic
254	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
255	C₂²×C₃	#3752	1	1	yes	characteristic
256	C₂²×C₃	#3752	1	1	yes	characteristic
257	C₂²×C₃	#3757	1	1	yes	characteristic
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	6	no	—
260	Dic₃	—	36	2	no	—
261	Dic₃	—	36	6	no	—
262	Dic₃	—	36	6	no	—
263	Dic₃	—	36	2	no	—
264	C₃²	#4527	1	1	yes	characteristic
265	C₉	#4527	1	3	yes	complemented
266	C₂³	—	4	2	no	—
267	C₂³	#6226	1	1	yes	characteristic
268	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
269	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
270	C₂³	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
271	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
272	C₂³	#6233	1	1	yes	characteristic
273	C₂³	#6233	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	54	2	no	—
277	C₂×C₄	—	54	2	no	—
278	C₂×C₄	—	108	2	no	—
279	C₂×C₄	—	108	2	no	—
280	C₂×C₃	—	2	3	no	—
281	C₂×C₃	—	2	3	no	—
282	C₂×C₃	—	2	3	no	—
283	C₂×C₃	—	2	3	no	—
284	C₂×C₃	—	2	3	no	—
285	C₂×C₃	—	2	3	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	#63365	1	6	yes	—
293	C₂×C₃	C₉⋊#95	1	3	yes	—
294	C₂×C₃	#63633	1	2	yes	—
295	C₂×C₃	#63658	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₄×C₃⋊Dic₉	1	1	yes	characteristic
304	C₂²	#88534	1	1	yes	characteristic
305	C₂²	#88534	1	1	yes	characteristic
306	C₂²	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
307	C₂²	#88536	1	1	yes	characteristic
308	C₂²	#88536	1	1	yes	characteristic
309	C₂²	#88551	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	108	2	no	—
312	C₄	—	108	2	no	—
313	C₃	#10588204	1	3	yes	complemented
314	C₃	#10589393	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	#11720523	1	2	yes	—
322	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88536
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂⁴×C₃	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×C₂²⋊C₄	C₂
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂⁴×C₃×C₉	C₂²
hypercenter	C₂×C₂²⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	#342
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂²×C₃⋊Dic₉	C₄	semidirect
11. C₃×C₉×C₂²⋊C₄	C₄	semidirect
15. #88551	C₄	semidirect
53. C₉×C₂²⋊C₄	Dic₃	semidirect
128. C₃×C₂²⋊C₄	Dic₉	semidirect
175. C₃×C₉	#342	semidirect
229. C₂²⋊C₄	C₃⋊Dic₉	semidirect
265. C₉	#4527	semidirect
313. C₃	#10588204	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₂²⋊C₄	2
C₂×#88551	2
#10589393	3
#10588204	3