Finite Group Databasebeta

← #44487102

Subgroups — #44487102

Order 30592.

516 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#21439953C211yesmaximal, characteristic
2#29504666C211yesmaximal, characteristic
3#29504660C211yesmaximal, characteristic
4#29504656C211yesmaximal, characteristic, complemented
5#29504655C211yesmaximal, characteristic, complemented
6#29504653C211yesmaximal, characteristic, complemented
7#29504639C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic239C211yesmaximal, characteristic, complemented
9#29504623C211yesmaximal, characteristic, complemented
10#29504623C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic239C211yesmaximal, characteristic
12C22⋊C4×Dic239C211yesmaximal, characteristic
13C22⋊C4×Dic239C211yesmaximal, characteristic
14C42×Dic239C211yesmaximal, characteristic, complemented
15#388×C239C211yesmaximal, characteristic
16C23×Dic239C2211yescharacteristic
17#21439953C418yescomplemented
18C2×C4⋊Dic239C2211yescharacteristic
19C2×Dic239⋊C4C2211yescharacteristic
20C2×Dic239⋊C4C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×C22⋊Dic239C2211yescharacteristic
24C2×C22⋊Dic239C2211yescharacteristic
25C2×C22⋊Dic239C2211yescharacteristic
26C2×C4×Dic239C2211yescharacteristic
27C2×C4×Dic239C2211yescharacteristic
28C2×C4×Dic239C2211yescharacteristic
29C2×C4×Dic239C2211yescharacteristic
30C2×C4×Dic239C2211yescharacteristic
31C2×C4×Dic239C2211yescharacteristic
32C2×C4×Dic239C2211yescharacteristic
33C2×C4×Dic239C2211yescharacteristic
34C2×C4×Dic239C2211yescharacteristic
35C239⋊#95C2211yescharacteristic
36C239⋊#95C2211yescharacteristic
37C239⋊#95C2211yescharacteristic
38C239⋊#95C2211yescharacteristic
39C239⋊#95C2211yescharacteristic
40C239⋊#95C2211yescharacteristic
41C239⋊#95C2211yescharacteristic
42C239⋊#95C2211yescharacteristic
43C239⋊#95C2211yescharacteristic
44C239⋊#95C2211yescharacteristic
45C2×C4⋊C4×C239C2211yescharacteristic
46C2×C22⋊C4×C239C2211yescharacteristic
47C2×C22⋊C4×C239C2211yescharacteristic
48C2×C22⋊C4×C239C2211yescharacteristic
49C2×C42×C239C2211yescharacteristic
50#95×C239C2211yescharacteristic
51#95×C239C2211yescharacteristic
52C22×Dic239—22no—
53C22×Dic239—22no—
54C22×Dic239—22no—
55C22×Dic239C2311yescharacteristic
56C22×Dic239C2311yescharacteristic
57C22×Dic239C2311yescharacteristic
58C22×Dic239C2311yescharacteristic
59C22×Dic239C2311yescharacteristic
60C22×Dic239C2311yescharacteristic
61C22×Dic239C2311yescharacteristic
62C22×Dic239C2311yescharacteristic
63C4⋊Dic239C2×C414yescomplemented
64Dic239⋊C4—22no—
65Dic239⋊C4—22no—
66Dic239⋊C4C2×C414yescomplemented
67Dic239⋊C4C2×C414yescomplemented
68C22⋊Dic239—22no—
69C22⋊Dic239C2×C414yes—
70C22⋊Dic239C2×C414yes—
71C4×Dic239—22no—
72C4×Dic239—22no—
73C4×Dic239—22no—
74C4×Dic239—22no—
75C4×Dic239—22no—
76C4×Dic239—22no—
77C4×Dic239—22no—
78C4×Dic239—22no—
79C4×Dic239C2×C414yescomplemented
80C24×C239C2311yescharacteristic
81C22×C4×C239C2311yescharacteristic
82C22×C4×C239C2311yescharacteristic
83C22×C4×C239C2311yescharacteristic
84C22×C4×C239C2311yescharacteristic
85C22×C4×C239C2311yescharacteristic
86C22×C4×C239C2311yescharacteristic
87C4⋊C4×C239—22no—
88C22⋊C4×C239—22no—
89C22⋊C4×C239—22no—
90C22⋊C4×C239C2×C414yes—
91C42×C239—22no—
92C2×Dic239—21no—
93C2×Dic239—21no—
94C2×Dic239—21no—
95C2×Dic239—21no—
96C2×Dic239—21no—
97C2×Dic239—21no—
98C2×Dic239—21no—
99C2×Dic239—21no—
100C2×Dic239—21no—
101C2×Dic239—21no—
102C2×Dic239—21no—
103C2×Dic239—21no—
104C2×Dic239—21no—
105C2×Dic239—21no—
106C2×Dic239—21no—
107C2×Dic239—21no—
108C2×Dic239—44no—
109C2×Dic239C22×C412yes—
110C2×Dic239C22×C412yes—
111C2×Dic239C22×C412yes—
112C2×Dic239C22×C412yes—
113C2×Dic239#4112yes—
114C2×Dic239#4112yes—
115C2×Dic239#4112yes—
116C2×Dic239#4112yes—
117C23×C239—21no—
118C23×C239—21no—
119C23×C239—21no—
120C23×C239—21no—
121C23×C239—21no—
122C23×C239—21no—
123C23×C239C22×C412yes—
124C23×C239C2411yescharacteristic
125C2×C4×C239—21no—
126C2×C4×C239—21no—
127C2×C4×C239—21no—
128C2×C4×C239—21no—
129C2×C4×C239—21no—
130C2×C4×C239—21no—
131C2×C4×C239—21no—
132C2×C4×C239—21no—
133C2×C4×C239—21no—
134C2×C4×C239—21no—
135C2×C4×C239—21no—
136C2×C4×C239—21no—
137C2×C4×C239—21no—
138C2×C4×C239—21no—
139C2×C4×C239C22×C412yes—
140C2×C4×C239C22×C412yes—
141C2×C4×C239#4112yes—
142C2×C4×C239#4112yes—
143C22×C239—22no—
144C22×C239—22no—
145C22×C239—22no—
146C22×C239—41no—
147C22×C239—41no—
148C22×C239—41no—
149C22×C239—41no—
150C22×C239C23×C411yescharacteristic
151C22×C239C2×#4111yescharacteristic
152C22×C239C2×#4111yescharacteristic
153C22×C239C2×#4111yescharacteristic
154C22×C239C2×#4111yescharacteristic
155C22×C239C2×#4111yescharacteristic
156C22×C239C2×#4111yescharacteristic
157C4×C239—22no—
158C4×C239—22no—
159C4×C239—22no—
160C4×C239—22no—
161C4×C239—41no—
162C4×C239—41no—
163Dic239—22no—
164Dic239—22no—
165Dic239—22no—
166Dic239—22no—
167Dic239—22no—
168Dic239—41no—
169Dic239—41no—
170Dic239#12614yescomplemented
171C2×C239—42no—
172C2×C239C4×#4111yescharacteristic
173C2×C239C4×#4111yescharacteristic
174C2×C239C4×#4111yescharacteristic
175C2×C239C2×#12611yescharacteristic
176C2×C239#52911yescharacteristic
177C2×C239#52911yescharacteristic
178C2×C239#52911yescharacteristic
179C239C4×#12611yescharacteristic, complemented, Sylow
180C4×#126—2391nomaximal, Sylow
181C2×#126—2391no—
182#388—2391no—
183#388—2391no—
184#388D23911yescharacteristic
185#387—2391no—
186#387—2391no—
187#387—2391no—
188#383—2391no—
189C4×C4⋊C4—2391no—
190C4×C4⋊C4—2391no—
191C4×C4⋊C4—2391no—
192C4×C22⋊C4—2391no—
193C4×C22⋊C4—2391no—
194C4×C22⋊C4—2391no—
195C43—2391no—
196C23×C4—2391no—
197#126—2398no—
198C2×C4⋊C4—2391no—
199C2×C4⋊C4—2391no—
200C2×C4⋊C4—2391no—
201C2×C4⋊C4—2391no—
202C2×C4⋊C4—2391no—
203C2×C4⋊C4C2×D23911yescharacteristic
204C2×C22⋊C4—2391no—
205C2×C22⋊C4—2391no—
206C2×C22⋊C4—2391no—
207C2×C22⋊C4C2×D23911yescharacteristic
208C2×C22⋊C4C2×D23911yescharacteristic
209C2×C22⋊C4C2×D23911yescharacteristic
210C2×C42—2391no—
211C2×C42—2391no—
212C2×C42—2391no—
213C2×C42—2391no—
214C2×C42—2391no—
215C2×C42—2391no—
216C2×C42—2391no—
217C2×C42—2391no—
218C2×C42—2391no—
219C2×C42C2×D23911yescharacteristic
220#95—2391no—
221#95—2391no—
222#95—2391no—
223#95—2391no—
224#95—2391no—
225#95—2391no—
226#95—2391no—
227#95—2391no—
228#95—2391no—
229#95—2391no—
230#95C2×D23911yescharacteristic
231#95C2×D23911yescharacteristic
232C24C22×D23911yescharacteristic
233C22×C4—4782no—
234C22×C4—4782no—
235C22×C4—4782no—
236C22×C4—2391no—
237C22×C4—2391no—
238C22×C4—2391no—
239C22×C4—2391no—
240C22×C4—2391no—
241C22×C4—2391no—
242C22×C4—2391no—
243C22×C4—2391no—
244C22×C4C22×D23911yescharacteristic
245C22×C4C22×D23911yescharacteristic
246C22×C4C22×D23911yescharacteristic
247C22×C4C22×D23911yescharacteristic
248C22×C4C22×D23911yescharacteristic
249C22×C4C22×D23911yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—4782no—
252C4⋊C4—4782no—
253C4⋊C4—2394no—
254C4⋊C4—2394no—
255C4⋊C4—2394no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—4782no—
259C22⋊C4—2394no—
260C22⋊C4—2394no—
261C22⋊C4C4×D23914yes—
262C42—22no—
263C42—4782no—
264C42—4782no—
265C42—4782no—
266C42—4782no—
267C42—4782no—
268C42—4782no—
269C42—4782no—
270C42—4782no—
271C42—2394no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D23912yes—
279C23C23×D23911yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—9564no—
295C2×C4—4781no—
296C2×C4—4781no—
297C2×C4—4781no—
298C2×C4—4781no—
299C2×C4—4781no—
300C2×C4—4781no—
301C2×C4—4781no—
302C2×C4—4781no—
303C2×C4—4781no—
304C2×C4—4781no—
305C2×C4—4781no—
306C2×C4—4781no—
307C2×C4—4781no—
308C2×C4—4781no—
309C2×C4—4781no—
310C2×C4—4781no—
311C2×C4—2392no—
312C2×C4—2392no—
313C2×C4—2392no—
314C2×C4—2392no—
315C2×C4—2392no—
316C2×C4—2392no—
317C2×C4—2392no—
318C2×C4—2392no—
319C2×C4C2×C4×D23912yes—
320C2×C4C2×C4×D23912yes—
321C2×C4#1683854712yes—
322C2×C4#1683854712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D23911yescharacteristic
331C22#41×D23911yescharacteristic
332C22#41×D23911yescharacteristic
333C22#41×D23911yescharacteristic
334C22#41×D23911yescharacteristic
335C22C2×#1683854711yescharacteristic
336C22C2×#1683854711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—9561no—
344C4—9561no—
345C4—4782no—
346C4—4782no—
347C4—4782no—
348C4—4782no—
349C4—4782no—
350C4—2394no—
351C2—42no—
352C2C4×#1683854711yescharacteristic
353C2C4×#1683854711yescharacteristic
354C2#2950524011yescharacteristic
355C2#126×D23911yescharacteristic
356C2#2950529911yescharacteristic
357C2#2950530111yescharacteristic
358C2#2950530311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#388×C239C2
socle, cosocleC23×C239C24
prime radicalC24×C239C23
hypercenter#388D239
hypocenterC239C4×#126
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
4. #29504656C2semidirect
5. #29504655C2semidirect
6. #29504653C2semidirect
7. #29504639C2semidirect
8. C4⋊C4×Dic239C2semidirect
9. #29504623C2semidirect
10. #29504623C2semidirect
14. C42×Dic239C2semidirect
17. #21439953C4semidirect
63. C4⋊Dic239C2×C4semidirect
66. Dic239⋊C4C2×C4semidirect
67. Dic239⋊C4C2×C4semidirect
79. C4×Dic239C2×C4semidirect
170. Dic239#126semidirect
179. C239C4×#126semidirect

Maximal subgroups

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

subgroupindex
C2×#214399532
#295046662
#295046602
#295046562
#295046552
#295046532
#295046392
C4⋊C4×Dic2392
#295046232
C22⋊C4×Dic2392
C42×Dic2392
#388×C2392
C4×#126239