Finite Group Databasebeta

← #38745411

Subgroups — #38745411

Order 25216.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263301C211yesmaximal, characteristic
2C2×C4×D788C211yesmaximal, characteristic, complemented
3C2×#20263288C211yesmaximal, characteristic, complemented
4#27160970C211yesmaximal, characteristic, complemented
5#27160968C211yesmaximal, characteristic, complemented
6#27160965C211yesmaximal, characteristic, complemented
7#27160957C211yesmaximal, characteristic, complemented
8#27160957C211yesmaximal, characteristic, complemented
9#27160957C211yesmaximal, characteristic, complemented
10#27160957C211yesmaximal, characteristic, complemented
11#27160953C211yesmaximal, characteristic, complemented
12#27160950C211yesmaximal, characteristic, complemented
13#27160938C211yesmaximal, characteristic, complemented
14#95×D197C211yesmaximal, characteristic, complemented
15#387×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2211yescharacteristic
24#20263301C418yescomplemented
25C4×D788—24no—
26#20263288—24no—
27C2×C4⋊Dic197C2211yescharacteristic
28C2×#15691337C2211yescharacteristic
29C2×#15691337C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×#15691335C2211yescharacteristic
38C2×#15691335C2211yescharacteristic
39C2×#15691335C2211yescharacteristic
40C2×C4×Dic197C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46#20263233C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C4⋊C4×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53#95×C197C2211yescharacteristic
54C23×D197C2311yescharacteristic
55C23×D197C2311yescharacteristic
56C23×D197C2311yescharacteristic
57C2×D788—22no—
58C2×D788—22no—
59C2×D788—22no—
60C2×D788—22no—
61C2×D788—41no—
62C2×D788—41no—
63C2×D788C2×C414yescomplemented
64C2×D788C2×C414yescomplemented
65C2×D788C2×C414yescomplemented
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C22×Dic197C2311yescharacteristic
69C22×Dic197C2311yescharacteristic
70C22×Dic197C2311yescharacteristic
71C2×C4×D197—22no—
72C2×C4×D197—22no—
73C2×C4×D197—22no—
74C2×C4×D197—41no—
75C2×C4×D197—41no—
76C2×C4×D197—41no—
77C2×C4×D197—41no—
78C2×C4×D197—41no—
79C2×C4×D197—41no—
80C2×C4×D197—41no—
81C2×C4×D197—41no—
82C2×C4×D197—41no—
83C2×C4×D197—41no—
84C2×C4×D197C2×C414yescomplemented
85C2×C4×D197D414yescomplemented
86C4⋊Dic197—22no—
87#15691337—41no—
88#15691337—41no—
89#15691335—22no—
90#15691335—22no—
91#15691335—22no—
92#15691335—22no—
93#15691335—41no—
94#15691335—41no—
95#15691335—41no—
96#15691335—41no—
97#15691335C2×C414yescomplemented
98#15691335C2×C414yescomplemented
99C4×Dic197—22no—
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102C22×C4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104C22×C4×C197C2311yescharacteristic
105C22×C4×C197C2311yescharacteristic
106C22×C4×C197C2311yescharacteristic
107C4⋊C4×C197—22no—
108C4⋊C4×C197C2×C414yescomplemented
109C42×C197—22no—
110C22×D197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—21no—
116C22×D197—21no—
117C22×D197—21no—
118C22×D197—21no—
119C22×D197—21no—
120C22×D197—21no—
121C22×D197—21no—
122C22×D197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197C22×C412yes—
127C22×D197C22×C412yes—
128C22×D197C22×C412yes—
129C22×D197C2×D412yes—
130C22×D197#4112yes—
131D788—28no—
132D788—28no—
133D788—44no—
134C2×Dic197—21no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C2×Dic197—21no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—21no—
144C2×Dic197—21no—
145C2×Dic197—21no—
146C2×Dic197—21no—
147C2×Dic197C22×C412yes—
148C2×Dic197C2×D412yes—
149C4×D197—44no—
150C4×D197—44no—
151C4×D197—44no—
152C4×D197—44no—
153C4×D197—44no—
154C23×C197C2411yescharacteristic
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197C22×C412yes—
173C2×C4×C197C22×C412yes—
174C2×C4×C197C22×C412yes—
175C2×C4×C197C2×D412yes—
176C22×C197C23×C411yescharacteristic
177C22×C197C22×D411yescharacteristic
178C22×C197C2×#4111yescharacteristic
179C22×C197H(5,𝔽2)11yescharacteristic
180C22×C197H(5,𝔽2)11yescharacteristic
181C22×C197H(5,𝔽2)11yescharacteristic
182C22×C1972−1+411yescharacteristic
183C2×D197—22no—
184C2×D197—22no—
185C2×D197—22no—
186C2×D197—22no—
187C2×D197—22no—
188C2×D197—22no—
189C2×D197—22no—
190C2×D197—22no—
191C2×D197—22no—
192C2×D197—22no—
193C2×D197—22no—
194C2×D197—22no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—41no—
203C2×D197C4×D414yes—
204C4×C197—22no—
205C4×C197—22no—
206C4×C197—22no—
207C4×C197—22no—
208C4×C197—41no—
209C4×C197—41no—
210C4×C197—41no—
211Dic197—22no—
212Dic197—22no—
213Dic197—41no—
214Dic197—41no—
215Dic197—41no—
216C2×C197C2×C4×D411yescharacteristic
217C2×C197#51811yescharacteristic
218C2×C197#52011yescharacteristic
219C2×C197#53411yescharacteristic
220C2×C197#53711yescharacteristic
221C2×C197#53811yescharacteristic
222C2×C197#53911yescharacteristic
223D197—24no—
224D197—42no—
225D197—42no—
226C197#232511yescharacteristic, complemented, Sylow
227#2325—1971nomaximal, Sylow
228C2×#121—1971no—
229C2×C4×D4—1971no—
230C2×C4×D4—1971no—
231#388—1971no—
232#388—1971no—
233#387—1971no—
234#387D19711yescharacteristic, complemented
235#386—1971no—
236#386—1971no—
237#386—1971no—
238#386—1971no—
239#385—1971no—
240#385—1971no—
241#380—1971no—
242C2×#95—1971no—
243C22×D4—1971no—
244C22×D4—1971no—
245C22×D4—1971no—
246C23×C4—1971no—
247C23×C4—1971no—
248C23×C4—1971no—
249C23×C4—1971no—
250C23×C4—1971no—
251#121—1978no—
252C4×D4—3944no—
253C4×D4—3944no—
254C2×C4⋊C4—1971no—
255C2×C4⋊C4—1971no—
256C2×C4⋊C4—1971no—
257C2×C4⋊C4C2×D19711yescharacteristic
258C2×C4⋊C4C2×D19711yescharacteristic
259C2×C22⋊C4—1971no—
260C2×C22⋊C4—1971no—
261C2×C22⋊C4—1971no—
262C2×C22⋊C4—1971no—
263C2×C22⋊C4—1971no—
264C2×C22⋊C4—1971no—
265C2×C22⋊C4—1971no—
266C2×C22⋊C4—1971no—
267C2×C22⋊C4—1971no—
268C2×C22⋊C4—1971no—
269C2×C42—1971no—
270C2×C42C2×D19711yescharacteristic
271#95—1971no—
272#95—1971no—
273#95—1971no—
274#95—1971no—
275#95—1971no—
276#95—1971no—
277#95C2×D19711yescharacteristic
278#95C2×D19711yescharacteristic
279#95C2×D19711yescharacteristic
280#95C2×D19711yescharacteristic
281C24—1971no—
282C24—1971no—
283C24—1971no—
284C2×D4—7881no—
285C2×D4—7881no—
286C2×D4—3942no—
287C2×D4—3942no—
288C2×D4—3942no—
289C2×D4—3942no—
290C2×D4—1974no—
291C2×D4—1974no—
292C2×D4—1974no—
293C22×C4—7881no—
294C22×C4—7881no—
295C22×C4—7881no—
296C22×C4—7881no—
297C22×C4—7881no—
298C22×C4—7881no—
299C22×C4—7881no—
300C22×C4—7881no—
301C22×C4—7881no—
302C22×C4—7881no—
303C22×C4—3942no—
304C22×C4—3942no—
305C22×C4—3942no—
306C22×C4—1974no—
307C22×C4—1974no—
308C22×C4—1971no—
309C22×C4—1971no—
310C22×C4—1971no—
311C22×C4—1971no—
312C22×C4—1971no—
313C22×C4C22×D19711yescharacteristic
314C22×C4C22×D19711yescharacteristic
315C22×C4C22×D19711yescharacteristic
316C22×C4C22×D19711yescharacteristic
317C22×C4C22×D19711yescharacteristic
318C22×C4C22×D19711yescharacteristic
319C22×C4C22×D19711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—7881no—
322C4⋊C4—7881no—
323C4⋊C4—3942no—
324C4⋊C4C4×D19714yescomplemented
325C22⋊C4—7881no—
326C22⋊C4—7881no—
327C22⋊C4—7881no—
328C22⋊C4—7881no—
329C22⋊C4—3942no—
330C22⋊C4—3942no—
331C22⋊C4—3942no—
332C22⋊C4—3942no—
333C22⋊C4—1974no—
334C22⋊C4—1974no—
335C42—22no—
336C42—3942no—
337C23—3941no—
338C23—3941no—
339C23—3941no—
340C23—3941no—
341C23—3941no—
342C23—3941no—
343C23—3941no—
344C23—3941no—
345C23—3941no—
346C23—3941no—
347C23—3941no—
348C23—3941no—
349C23—3941no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23—1972no—
354C23—1972no—
355C23—1972no—
356C23—1972no—
357C23—1972no—
358C23C23×D19711yescharacteristic
359D4—7884no—
360D4—3948no—
361D4—3948no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—7884no—
380C2×C4—7884no—
381C2×C4—7884no—
382C2×C4—7884no—
383C2×C4—7884no—
384C2×C4—3941no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4—3941no—
389C2×C4—3941no—
390C2×C4—3941no—
391C2×C4—3941no—
392C2×C4—3941no—
393C2×C4—3941no—
394C2×C4—3941no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—1972no—
398C2×C4—1972no—
399C2×C4C2×C4×D19712yes—
400C2×C4C2×C4×D19712yes—
401C2×C4C2×C4×D19712yes—
402C2×C4D4×D19712yes—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—7881no—
411C22—3942no—
412C22—3942no—
413C22—3942no—
414C22—3942no—
415C22—3942no—
416C22—3942no—
417C22—3942no—
418C22—3942no—
419C22—3942no—
420C22—3942no—
421C22—3942no—
422C22—3942no—
423C22—1974no—
424C22C22×C4×D19711yescharacteristic
425C22C2×D4×D19711yescharacteristic
426C22#41×D19711yescharacteristic
427C22#2026337211yescharacteristic
428C22#2026337211yescharacteristic
429C22#2026337211yescharacteristic
430C22#2026337411yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—7881no—
439C4—7881no—
440C4—7881no—
441C4—3942no—
442C4—3942no—
443C2—7882no—
444C2—7882no—
445C2—3944no—
446C2C4×D4×D19711yescharacteristic
447C2#2716155311yescharacteristic
448C2#2716156111yescharacteristic
449C2#2716164011yescharacteristic
450C2#2716165911yescharacteristic
451C2#2716166811yescharacteristic
452C2#2716167711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263301C2
hypercenter#387D197
hypocenterC197#2325
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D788C2semidirect
3. C2×#20263288C2semidirect
4. #27160970C2semidirect
5. #27160968C2semidirect
6. #27160965C2semidirect
7. #27160957C2semidirect
8. #27160957C2semidirect
9. #27160957C2semidirect
10. #27160957C2semidirect
11. #27160953C2semidirect
12. #27160950C2semidirect
13. #27160938C2semidirect
14. #95×D197C2semidirect
15. #387×C197C2semidirect
24. #20263301C4semidirect
63. C2×D788C2×C4semidirect
64. C2×D788C2×C4semidirect
65. C2×D788C2×C4semidirect
84. C2×C4×D197C2×C4semidirect
85. C2×C4×D197D4semidirect
97. #15691335C2×C4semidirect
98. #15691335C2×C4semidirect
108. C4⋊C4×C197C2×C4semidirect
226. C197#2325semidirect
234. #387D197semidirect
324. C4⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633012
C2×C4×D7882
C2×#202632882
#271609702
#271609682
#271609652
#271609572
#271609532
#271609502
#271609382
#95×D1972
#387×C1972
#2325197