Finite Group Databasebeta

← #38745852

Subgroups — #38745852

Order 25216.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263296C211yesmaximal, characteristic
2#27160981C211yesmaximal, characteristic, complemented
3#27160980C211yesmaximal, characteristic, complemented
4#27160979C211yesmaximal, characteristic, complemented
5#27160972C211yesmaximal, characteristic, complemented
6#27160971C211yesmaximal, characteristic, complemented
7#27160970C211yesmaximal, characteristic, complemented
8#27160969C211yesmaximal, characteristic, complemented
9#27160968C211yesmaximal, characteristic, complemented
10#27160957C211yesmaximal, characteristic, complemented
11#27160957C211yesmaximal, characteristic, complemented
12#27160955C211yesmaximal, characteristic, complemented
13#27160934C211yesmaximal, characteristic, complemented
14#27160934C211yesmaximal, characteristic, complemented
15#388×C197C211yesmaximal, characteristic, complemented
16C24×D197C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×C22⋊D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic, complemented
21#20263296C418yescomplemented
22C2×C4⋊Dic197C2211yescharacteristic, complemented
23C2×#15691337C2211yescharacteristic, complemented
24C2×C22⋊Dic197C2211yescharacteristic
25C2×C22⋊Dic197C2211yescharacteristic
26C2×#15691335C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic, complemented
37C2×#15691335C2211yescharacteristic, complemented
38C2×#15691335C2211yescharacteristic, complemented
39C2×C4×Dic197C2211yescharacteristic, complemented
40C2×C4×Dic197C2211yescharacteristic, complemented
41#20263233C2211yescharacteristic, complemented
42#20263233C2211yescharacteristic, complemented
43#20263233C2211yescharacteristic, complemented
44#20263233C2211yescharacteristic, complemented
45C2×C4⋊C4×C197C2211yescharacteristic, complemented
46C2×C22⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic, complemented
50#95×C197C2211yescharacteristic, complemented
51#95×C197C2211yescharacteristic, complemented
52C23×D197—41no—
53C23×D197—41no—
54C23×D197—41no—
55C23×D197—41no—
56C23×D197—41no—
57C23×D197—41no—
58C23×D197C2×C414yescomplemented
59C23×D197C2311yescharacteristic
60C23×D197C2311yescharacteristic
61C23×D197C2311yescharacteristic
62C2×D788—41no—
63C2×D788—41no—
64C2×D788—41no—
65C2×D788—41no—
66C2×D788C2×C414yes—
67C2×D788C2×C414yes—
68C2×C22⋊D197—41no—
69C2×C22⋊D197—41no—
70C2×C22⋊D197C2×C414yes—
71C22×Dic197C2311yescharacteristic
72C22×Dic197C2311yescharacteristic
73C22×Dic197C2311yescharacteristic
74C22×Dic197C2311yescharacteristic
75C22×Dic197C2311yescharacteristic
76C2×C4×D197—41no—
77C2×C4×D197—41no—
78C2×C4×D197—41no—
79C4⋊Dic197—41no—
80#15691337—41no—
81C22⋊Dic197—41no—
82C22⋊Dic197—41no—
83#15691335—41no—
84#15691335—41no—
85#15691335—41no—
86#15691335—41no—
87#15691335—41no—
88#15691335—41no—
89#15691335—41no—
90#15691335—41no—
91#15691335—41no—
92#15691335—41no—
93#15691335—41no—
94#15691335C2×C414yescomplemented
95#15691335C2×C414yescomplemented
96C4×Dic197—41no—
97C4×Dic197—41no—
98C24×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102C22×C4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104C22×C4×C197C2311yescharacteristic
105C4⋊C4×C197—41no—
106C22⋊C4×C197—41no—
107C22⋊C4×C197—41no—
108C22⋊C4×C197C2×C414yescomplemented
109C42×C197—41no—
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×D197—21no—
127C22×D197—21no—
128C22×D197—44no—
129C22×D197—44no—
130C22×D197—44no—
131C22×D197—44no—
132C22×D197—44no—
133C22×D197—44no—
134C22×D197—44no—
135C22×D197C22×C412yes—
136C22×D197C22×C412yes—
137C22×D197C22×C412yes—
138D788—44no—
139D788—44no—
140C22⋊D197—44no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—21no—
144C2×Dic197—21no—
145C2×Dic197—21no—
146C2×Dic197—21no—
147C2×Dic197—21no—
148C2×Dic197—21no—
149C2×Dic197—21no—
150C2×Dic197—21no—
151C2×Dic197—21no—
152C2×Dic197—21no—
153C2×Dic197—21no—
154C2×Dic197—21no—
155C2×Dic197C22×C412yes—
156C4×D197—44no—
157C23×C197—21no—
158C23×C197—21no—
159C23×C197—21no—
160C23×C197—21no—
161C23×C197—21no—
162C23×C197—21no—
163C23×C197C22×C412yes—
164C23×C197C2411yescharacteristic
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×C197—21no—
173C2×C4×C197—21no—
174C2×C4×C197—21no—
175C2×C4×C197—21no—
176C2×C4×C197—21no—
177C2×C4×C197—21no—
178C2×C4×C197—21no—
179C2×C4×C197—21no—
180C2×C4×C197—21no—
181C2×C4×C197C22×C412yes—
182C2×C4×C197C22×C412yes—
183C22×C197—22no—
184C22×C197—22no—
185C22×C197—22no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197C23×C411yescharacteristic
191C22×C197H(5,𝔽2)11yescharacteristic
192C22×C197H(5,𝔽2)11yescharacteristic
193C22×C197H(5,𝔽2)11yescharacteristic
194C22×C197H(5,𝔽2)11yescharacteristic
195C22×C197H(5,𝔽2)11yescharacteristic
196C22×C197H(5,𝔽2)11yescharacteristic
197C2×D197—22no—
198C2×D197—22no—
199C2×D197—22no—
200C2×D197—22no—
201C2×D197—22no—
202C2×D197—22no—
203C2×D197—22no—
204C2×D197—22no—
205C2×D197—22no—
206C2×D197—416no—
207C2×D197—41no—
208C2×D197—41no—
209C2×D197—41no—
210C2×D197—41no—
211C2×D197—41no—
212C2×D197—41no—
213C2×D197—41no—
214C2×D197—41no—
215C2×D197—41no—
216C2×D197—41no—
217C2×D197—41no—
218C2×D197—41no—
219C4×C197—22no—
220C4×C197—22no—
221C4×C197—41no—
222C4×C197—41no—
223C4×C197—41no—
224C4×C197—41no—
225Dic197—22no—
226Dic197—41no—
227Dic197—41no—
228Dic197—41no—
229Dic197—41no—
230C2×C197—42no—
231C2×C197#51811yescharacteristic
232C2×C197#51811yescharacteristic
233C2×C197#51811yescharacteristic
234C2×C197#56011yescharacteristic
235C2×C197#56011yescharacteristic
236C2×C197#56011yescharacteristic
237C2×C197H(3,𝔽4)11yescharacteristic
238D197—42no—
239D197—42no—
240D197—42no—
241C197#237511yescharacteristic, complemented, Sylow
242#2375—1971nomaximal, Sylow
243C2×#120—1971no—
244#390—1971no—
245#390—1971no—
246#390—1971no—
247#388—1971no—
248#388—1971no—
249#388—1971no—
250#388—1971no—
251#388—1971no—
252#388D19711yescharacteristic, complemented
253#386—1971no—
254#386—1971no—
255#386—1971no—
256#379—1971no—
257#379—1971no—
258C25—1971no—
259C22×D4—1971no—
260C22×D4—1971no—
261C22×D4—1971no—
262C23×C4—1971no—
263#120—1978no—
264C2×C4⋊C4—1971no—
265C2×C4⋊C4—1971no—
266C2×C4⋊C4C2×D19711yescharacteristic, complemented
267C2×C22⋊C4—1971no—
268C2×C22⋊C4—1971no—
269C2×C22⋊C4—1971no—
270C2×C22⋊C4—1971no—
271C2×C22⋊C4—1971no—
272C2×C22⋊C4—1971no—
273C2×C22⋊C4—1971no—
274C2×C22⋊C4—1971no—
275C2×C22⋊C4—1971no—
276C2×C22⋊C4—1971no—
277C2×C22⋊C4—1971no—
278C2×C22⋊C4—1971no—
279C2×C22⋊C4—1971no—
280C2×C22⋊C4—1971no—
281C2×C22⋊C4—1971no—
282C2×C22⋊C4C2×D19711yescharacteristic
283C2×C22⋊C4C2×D19711yescharacteristic
284C2×C22⋊C4C2×D19711yescharacteristic
285C2×C42—1971no—
286C2×C42—1971no—
287C2×C42C2×D19711yescharacteristic, complemented
288#95—1971no—
289#95—1971no—
290#95—1971no—
291#95—1971no—
292#95C2×D19711yescharacteristic, complemented
293#95C2×D19711yescharacteristic, complemented
294C24—7881no—
295C24—7881no—
296C24—7881no—
297C24—7881no—
298C24—7881no—
299C24—7881no—
300C24—1974no—
301C24—1971no—
302C24—1971no—
303C24—1971no—
304C24C22×D19711yescharacteristic
305C2×D4—7881no—
306C2×D4—7881no—
307C2×D4—7881no—
308C2×D4—7881no—
309C2×D4—7881no—
310C2×D4—7881no—
311C2×D4—1974no—
312C2×D4—1974no—
313C2×D4—1974no—
314C22×C4—7881no—
315C22×C4—7881no—
316C22×C4—7881no—
317C22×C4—1971no—
318C22×C4—1971no—
319C22×C4—1971no—
320C22×C4—1971no—
321C22×C4—1971no—
322C22×C4C22×D19711yescharacteristic
323C22×C4C22×D19711yescharacteristic
324C22×C4C22×D19711yescharacteristic
325C22×C4C22×D19711yescharacteristic
326C22×C4C22×D19711yescharacteristic
327C22×C4C22×D19711yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—7881no—
330C4⋊C4—7881no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—7881no—
334C22⋊C4—7881no—
335C22⋊C4—7881no—
336C22⋊C4—7881no—
337C22⋊C4—7881no—
338C22⋊C4—7881no—
339C22⋊C4—7881no—
340C22⋊C4—7881no—
341C22⋊C4—7881no—
342C22⋊C4—7881no—
343C22⋊C4—7881no—
344C22⋊C4—7881no—
345C22⋊C4—7881no—
346C22⋊C4—1974no—
347C22⋊C4—1974no—
348C22⋊C4C4×D19714yescomplemented
349C42—41no—
350C42—7881no—
351C42—7881no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—7884no—
359C23—7884no—
360C23—7884no—
361C23—7884no—
362C23—7884no—
363C23—7884no—
364C23—7884no—
365C23—3941no—
366C23—3941no—
367C23—3941no—
368C23—3941no—
369C23—3941no—
370C23—3941no—
371C23—3941no—
372C23—3941no—
373C23—3941no—
374C23—3941no—
375C23—3941no—
376C23—3941no—
377C23—3941no—
378C23—3941no—
379C23—3941no—
380C23—3941no—
381C23—3941no—
382C23—3941no—
383C23—1972no—
384C23—1972no—
385C23—1972no—
386C23C2×C4×D19712yes—
387C23C23×D19711yescharacteristic
388D4—7884no—
389D4—7884no—
390D4—7884no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—7884no—
408C2×C4—3941no—
409C2×C4—3941no—
410C2×C4—3941no—
411C2×C4—3941no—
412C2×C4—3941no—
413C2×C4—3941no—
414C2×C4—3941no—
415C2×C4—3941no—
416C2×C4—3941no—
417C2×C4—3941no—
418C2×C4—3941no—
419C2×C4—3941no—
420C2×C4—3941no—
421C2×C4—3941no—
422C2×C4—1972no—
423C2×C4C2×C4×D19712yes—
424C2×C4C2×C4×D19712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—78816no—
433C22—7881no—
434C22—7881no—
435C22—7881no—
436C22—7881no—
437C22—7881no—
438C22—7881no—
439C22—7881no—
440C22—7881no—
441C22—7881no—
442C22—7881no—
443C22—7881no—
444C22—7881no—
445C22—3942no—
446C22—3942no—
447C22—3942no—
448C22—3942no—
449C22—3942no—
450C22—3942no—
451C22—3942no—
452C22—3942no—
453C22—3942no—
454C22C22×C4×D19711yescharacteristic
455C22#2026337111yescharacteristic
456C22#2026337111yescharacteristic
457C22#2026337211yescharacteristic
458C22#2026337211yescharacteristic
459C22#2026337211yescharacteristic
460C22#2026337211yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—7881no—
468C4—7881no—
469C4—7881no—
470C4—7881no—
471C4—3942no—
472C2—42no—
473C2—7882no—
474C2—7882no—
475C2—7882no—
476C2#2716155111yescharacteristic
477C2#2716155111yescharacteristic
478C2#2716155311yescharacteristic
479C2#2716181711yescharacteristic
480C2#2716181911yescharacteristic
481C2#2716182011yescharacteristic
482C2#2716182111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#388×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263296C2
hypercenter#388D197
hypocenterC197#2375
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #27160981C2semidirect
3. #27160980C2semidirect
4. #27160979C2semidirect
5. #27160972C2semidirect
6. #27160971C2semidirect
7. #27160970C2semidirect
8. #27160969C2semidirect
9. #27160968C2semidirect
10. #27160957C2semidirect
11. #27160957C2semidirect
12. #27160955C2semidirect
13. #27160934C2semidirect
14. #27160934C2semidirect
15. #388×C197C2semidirect
20. C22×C4×D197C22semidirect
21. #20263296C4semidirect
22. C2×C4⋊Dic197C22semidirect
23. C2×#15691337C22semidirect
36. C2×#15691335C22semidirect
37. C2×#15691335C22semidirect
38. C2×#15691335C22semidirect
39. C2×C4×Dic197C22semidirect
40. C2×C4×Dic197C22semidirect
41. #20263233C22semidirect
42. #20263233C22semidirect
43. #20263233C22semidirect
44. #20263233C22semidirect
45. C2×C4⋊C4×C197C22semidirect
49. C2×C42×C197C22semidirect
50. #95×C197C22semidirect
51. #95×C197C22semidirect
58. C23×D197C2×C4semidirect
94. #15691335C2×C4semidirect
95. #15691335C2×C4semidirect
108. C22⋊C4×C197C2×C4semidirect
241. C197#2375semidirect
252. #388D197semidirect
266. C2×C4⋊C4C2×D197semidirect
287. C2×C42C2×D197semidirect
292. #95C2×D197semidirect
293. #95C2×D197semidirect
348. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632962
#271609812
#271609802
#271609792
#271609722
#271609712
#271609702
#271609692
#271609682
#271609572
#271609552
#271609342
#388×C1972
#2375197