Finite Group Databasebeta

← #44487339

Subgroups — #44487339

Order 30592.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#21439941C211yesmaximal, characteristic
2#29504675C211yesmaximal, characteristic
3#29504669C211yesmaximal, characteristic, complemented
4#29504666C211yesmaximal, characteristic, complemented
5#29504666C211yesmaximal, characteristic, complemented
6#29504666C211yesmaximal, characteristic, complemented
7#29504660C211yesmaximal, characteristic, complemented
8#29504659C211yesmaximal, characteristic, complemented
9#29504654C211yesmaximal, characteristic, complemented
10#29504645C211yesmaximal, characteristic, complemented
11#29504630C211yesmaximal, characteristic, complemented
12#29504623C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic239C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic239C211yesmaximal, characteristic, complemented
15#386×C239C211yesmaximal, characteristic
16C23×Dic239C2211yescharacteristic
17#21439941C418yescomplemented
18C2×C4⋊Dic239C2211yescharacteristic
19C2×Dic239⋊C4C2211yescharacteristic
20C2×Dic239⋊C4C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×Dic239⋊C4C2211yescharacteristic
24C2×Dic239⋊C4C2211yescharacteristic
25C2×C22⋊Dic239C2211yescharacteristic
26C2×C22⋊Dic239C2211yescharacteristic
27C2×C22⋊Dic239C2211yescharacteristic
28C2×C22⋊Dic239C2211yescharacteristic
29C2×C22⋊Dic239C2211yescharacteristic
30C2×C22⋊Dic239C2211yescharacteristic
31C2×C22⋊Dic239C2211yescharacteristic
32C2×C4×Dic239C2211yescharacteristic
33C2×C4×Dic239C2211yescharacteristic
34C2×C4×Dic239C2211yescharacteristic
35C2×C4×Dic239C2211yescharacteristic
36C2×C4×Dic239C2211yescharacteristic
37C239⋊#95C2211yescharacteristic
38C239⋊#95C2211yescharacteristic
39C239⋊#95C2211yescharacteristic
40C239⋊#95C2211yescharacteristic
41C239⋊#95C2211yescharacteristic
42C239⋊#95C2211yescharacteristic
43C239⋊#95C2211yescharacteristic
44C239⋊#95C2211yescharacteristic
45C22×D4×C239C2211yescharacteristic
46C23×C4×C239C2211yescharacteristic
47C2×C22⋊C4×C239C2211yescharacteristic
48C2×C22⋊C4×C239C2211yescharacteristic
49C2×C22⋊C4×C239C2211yescharacteristic
50#95×C239C2211yescharacteristic
51#95×C239C2211yescharacteristic
52C22×Dic239—41no—
53C22×Dic239—41no—
54C22×Dic239C2×C414yescomplemented
55C22×Dic239C2311yescharacteristic
56C22×Dic239C2311yescharacteristic
57C22×Dic239C2311yescharacteristic
58C22×Dic239C2311yescharacteristic
59C22×Dic239C2311yescharacteristic
60C22×Dic239C2311yescharacteristic
61C22×Dic239C2311yescharacteristic
62C22×Dic239C2311yescharacteristic
63C4⋊Dic239—41no—
64Dic239⋊C4—22no—
65Dic239⋊C4—41no—
66Dic239⋊C4—41no—
67Dic239⋊C4—41no—
68Dic239⋊C4C2×C414yescomplemented
69Dic239⋊C4C2×C414yescomplemented
70C22⋊Dic239—22no—
71C22⋊Dic239—22no—
72C22⋊Dic239—41no—
73C22⋊Dic239—41no—
74C22⋊Dic239C2×C414yes—
75C22⋊Dic239C2×C414yes—
76C22⋊Dic239C2×C414yescomplemented
77C4×Dic239—22no—
78C4×Dic239—41no—
79C4×Dic239—41no—
80C4×Dic239—41no—
81C4×Dic239—41no—
82C24×C239C2311yescharacteristic
83C24×C239C2311yescharacteristic
84C2×D4×C239—41no—
85C2×D4×C239—41no—
86C2×D4×C239C2×C414yes—
87C22×C4×C239—22no—
88C22×C4×C239—41no—
89C22×C4×C239—41no—
90C22×C4×C239C2311yescharacteristic
91C22×C4×C239C2311yescharacteristic
92C22×C4×C239C2311yescharacteristic
93C22×C4×C239C2311yescharacteristic
94C22×C4×C239C2311yescharacteristic
95C22⋊C4×C239—22no—
96C22⋊C4×C239—41no—
97C22⋊C4×C239—41no—
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—21no—
109C2×Dic239—21no—
110C2×Dic239—21no—
111C2×Dic239—21no—
112C2×Dic239—21no—
113C2×Dic239—21no—
114C2×Dic239—21no—
115C2×Dic239—21no—
116C2×Dic239—21no—
117C2×Dic239—44no—
118C2×Dic239C22×C412yes—
119C2×Dic239C22×C412yes—
120C2×Dic239C22×C412yes—
121C2×Dic239C22×C412yes—
122C2×Dic239#4112yes—
123C23×C239—21no—
124C23×C239—21no—
125C23×C239—21no—
126C23×C239—21no—
127C23×C239—21no—
128C23×C239—21no—
129C23×C239—21no—
130C23×C239—21no—
131C23×C239—21no—
132C23×C239—21no—
133C23×C239—21no—
134C23×C239—21no—
135C23×C239C22×C412yes—
136C23×C239C22×C412yes—
137C23×C239C2411yescharacteristic
138D4×C239—44no—
139C2×C4×C239—21no—
140C2×C4×C239—21no—
141C2×C4×C239—21no—
142C2×C4×C239—21no—
143C2×C4×C239—21no—
144C2×C4×C239—21no—
145C2×C4×C239—21no—
146C2×C4×C239—21no—
147C2×C4×C239—21no—
148C2×C4×C239—21no—
149C2×C4×C239—21no—
150C2×C4×C239—21no—
151C2×C4×C239—21no—
152C2×C4×C239—44no—
153C2×C4×C239C22×C412yes—
154C2×C4×C239#4112yes—
155C22×C239—22no—
156C22×C239—22no—
157C22×C239—22no—
158C22×C239—22no—
159C22×C239—22no—
160C22×C239—22no—
161C22×C239—41no—
162C22×C239—41no—
163C22×C239—41no—
164C22×C239—41no—
165C22×C239—41no—
166C22×C239—41no—
167C22×C239—41no—
168C22×C239—41no—
169C22×C239C23×C411yescharacteristic
170C22×C239C2×#4111yescharacteristic
171C22×C239C2×#4111yescharacteristic
172C22×C239H(5,𝔽2)11yescharacteristic
173C22×C239H(5,𝔽2)11yescharacteristic
174C22×C239H(5,𝔽2)11yescharacteristic
175C22×C2392−1+411yescharacteristic
176C4×C239—22no—
177C4×C239—22no—
178C4×C239—41no—
179C4×C239—41no—
180C4×C239—41no—
181Dic239—22no—
182Dic239—22no—
183Dic239—22no—
184Dic239—22no—
185Dic239—22no—
186Dic239—41no—
187Dic239—41no—
188Dic239—41no—
189C2×C239—42no—
190C2×C239—42no—
191C2×C239C4×#4111yescharacteristic
192C2×C239#51811yescharacteristic
193C2×C239#52011yescharacteristic
194C2×C239#53811yescharacteristic
195C2×C239#53911yescharacteristic
196C2×C239#54011yescharacteristic
197C2×C239#54211yescharacteristic
198C239#233511yescharacteristic, complemented, Sylow
199#2335—2391nomaximal, Sylow
200C2×#123—2391no—
201#390—2391no—
202#389—2391no—
203#388—2391no—
204#388—2391no—
205#388—2391no—
206#388—2391no—
207#387—2391no—
208#387—2391no—
209#386D23911yescharacteristic
210#385—2391no—
211#380—2391no—
212C4×C4⋊C4—2391no—
213C4×C22⋊C4—2391no—
214C4×C22⋊C4—2391no—
215C22×D4C2×D23911yescharacteristic
216C23×C4—2391no—
217C23×C4C2×D23911yescharacteristic
218#123—2398no—
219C2×C4⋊C4—2391no—
220C2×C4⋊C4—2391no—
221C2×C4⋊C4—2391no—
222C2×C4⋊C4—2391no—
223C2×C4⋊C4—2391no—
224C2×C4⋊C4—2391no—
225C2×C4⋊C4—2391no—
226C2×C22⋊C4—2391no—
227C2×C22⋊C4—2391no—
228C2×C22⋊C4—2391no—
229C2×C22⋊C4—2391no—
230C2×C22⋊C4—2391no—
231C2×C22⋊C4—2391no—
232C2×C22⋊C4—2391no—
233C2×C22⋊C4C2×D23911yescharacteristic
234C2×C22⋊C4C2×D23911yescharacteristic
235C2×C22⋊C4C2×D23911yescharacteristic
236C2×C42—2391no—
237C2×C42—2391no—
238C2×C42—2391no—
239C2×C42—2391no—
240C2×C42—2391no—
241#95—2391no—
242#95—2391no—
243#95—2391no—
244#95—2391no—
245#95—2391no—
246#95—2391no—
247#95—2391no—
248#95—2391no—
249#95C2×D23911yescharacteristic
250#95C2×D23911yescharacteristic
251C24C22×D23911yescharacteristic
252C24C22×D23911yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C4×D23914yes—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—9561no—
260C22×C4—9561no—
261C22×C4—2394no—
262C22×C4—2391no—
263C22×C4—2391no—
264C22×C4—2391no—
265C22×C4—2391no—
266C22×C4—2391no—
267C22×C4—2391no—
268C22×C4—2391no—
269C22×C4—2391no—
270C22×C4C22×D23911yescharacteristic
271C22×C4C22×D23911yescharacteristic
272C22×C4C22×D23911yescharacteristic
273C22×C4C22×D23911yescharacteristic
274C22×C4C22×D23911yescharacteristic
275C4⋊C4—9561no—
276C4⋊C4—9561no—
277C4⋊C4—9561no—
278C4⋊C4—9561no—
279C4⋊C4—4782no—
280C4⋊C4—2394no—
281C4⋊C4—2394no—
282C22⋊C4—22no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—9561no—
286C22⋊C4—9561no—
287C22⋊C4—4782no—
288C22⋊C4—4782no—
289C22⋊C4—2394no—
290C22⋊C4—2394no—
291C22⋊C4—2394no—
292C42—9561no—
293C42—9561no—
294C42—9561no—
295C42—9561no—
296C42—4782no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C4×D23912yes—
310C23C2×C4×D23912yes—
311C23C23×D23911yescharacteristic
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—9564no—
328C2×C4—4781no—
329C2×C4—4781no—
330C2×C4—4781no—
331C2×C4—4781no—
332C2×C4—4781no—
333C2×C4—4781no—
334C2×C4—4781no—
335C2×C4—4781no—
336C2×C4—4781no—
337C2×C4—4781no—
338C2×C4—4781no—
339C2×C4—4781no—
340C2×C4—4781no—
341C2×C4—4781no—
342C2×C4—4781no—
343C2×C4—4781no—
344C2×C4—4781no—
345C2×C4—4781no—
346C2×C4—4781no—
347C2×C4—2392no—
348C2×C4—2392no—
349C2×C4—2392no—
350C2×C4—2392no—
351C2×C4—2392no—
352C2×C4C2×C4×D23912yes—
353C2×C4#1683854712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C22×C4×D23911yescharacteristic
369C22#41×D23911yescharacteristic
370C22C2×#1683854711yescharacteristic
371C22#2144000511yescharacteristic
372C22#2144000511yescharacteristic
373C22#2144000511yescharacteristic
374C22#2144000711yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—9561no—
381C4—9561no—
382C4—9561no—
383C4—4782no—
384C4—4782no—
385C4—4782no—
386C4—4782no—
387C4—4782no—
388C2—42no—
389C2—42no—
390C2C4×#1683854711yescharacteristic
391C2#2950524511yescharacteristic
392C2#2950525111yescharacteristic
393C2#2950535711yescharacteristic
394C2#2950536511yescharacteristic
395C2#2950537211yescharacteristic
396C2#2950538911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#386×C239C2
socle, cosocleC23×C239C24
prime radicalC22×D4×C239C22
hypercenter#386D239
hypocenterC239#2335
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #29504669C2semidirect
4. #29504666C2semidirect
5. #29504666C2semidirect
6. #29504666C2semidirect
7. #29504660C2semidirect
8. #29504659C2semidirect
9. #29504654C2semidirect
10. #29504645C2semidirect
11. #29504630C2semidirect
12. #29504623C2semidirect
13. C22⋊C4×Dic239C2semidirect
14. C4×C22⋊Dic239C2semidirect
17. #21439941C4semidirect
54. C22×Dic239C2×C4semidirect
68. Dic239⋊C4C2×C4semidirect
69. Dic239⋊C4C2×C4semidirect
76. C22⋊Dic239C2×C4semidirect
198. C239#2335semidirect

Maximal subgroups

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

subgroupindex
C2×#214399412
#295046752
#295046692
#295046662
#295046602
#295046592
#295046542
#295046452
#295046302
#295046232
C22⋊C4×Dic2392
C4×C22⋊Dic2392
#386×C2392
#2335239