Finite Group Databasebeta

← #38751950

Subgroups — #38751950

Order 25216.

758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C211yesmaximal, characteristic, complemented
2C2×#20263355C211yesmaximal, characteristic, complemented
3C2×#20263350C211yesmaximal, characteristic, complemented
4#27161589C211yesmaximal, characteristic, complemented
5#27161244C211yesmaximal, characteristic, complemented
6#27161243C211yesmaximal, characteristic, complemented
7#27161242C211yesmaximal, characteristic, complemented
8#27161241C211yesmaximal, characteristic, complemented
9#27161223C211yesmaximal, characteristic, complemented
10#27161222C211yesmaximal, characteristic, complemented
11#27161221C211yesmaximal, characteristic, complemented
12#27161086C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic, complemented
14#27161041C211yesmaximal, characteristic, complemented
15#449×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691358—21no—
18C2×#15691358C2211yescharacteristic
19C2×#15691357—21no—
20C2×#15691357C2211yescharacteristic
21C2×#15691357C2211yescharacteristic
22#41×D197—24no—
23#41×D197—24no—
24C2×Q8×D197C2211yescharacteristic
25C2×D4×D197—21no—
26C2×D4×D197C2211yescharacteristic
27C22×C4×D197—21no—
28C22×C4×D197C2211yescharacteristic
29#20263355—24no—
30#20263350—24no—
31C2×Dic788C2211yescharacteristic, complemented
32C2×#15691348C2211yescharacteristic, complemented
33C2×#15691347C2211yescharacteristic, complemented
34C2×#15691346C2211yescharacteristic, complemented
35C2×#15691346C2211yescharacteristic, complemented
36C2×#15691343C2211yescharacteristic, complemented
37C2×#15691341C2211yescharacteristic
38C2×#15691341C2211yescharacteristic
39#20263308—21no—
40#20263306C2211yescharacteristic, complemented
41#20263305C2211yescharacteristic, complemented
42#20263297C2211yescharacteristic, complemented
43#20263294—21no—
44D4×Dic197—21no—
45#20263286—21no—
46#20263284C2211yescharacteristic
47#20263251C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263247C2211yescharacteristic, complemented
52#20263246C2211yescharacteristic, complemented
53C22⋊#13096802C2211yescharacteristic, complemented
54#20263238C2211yescharacteristic, complemented
55#20263238C2211yescharacteristic, complemented
56C2×#41×C197C2211yescharacteristic
57C2×SD16×C197C2211yescharacteristic, complemented
58C2×D8×C197C2211yescharacteristic, complemented
59#121×C197C2211yescharacteristic, complemented
60#103×C197C2211yescharacteristic, complemented
61#102×C197C2211yescharacteristic, complemented
62C22⋊C8×C197C2211yescharacteristic, complemented
63C23×D197—21no—
64C23×D197D411yescharacteristic
65#15691359—22no—
66#15691359—22no—
67#15691358—28no—
68#15691358—22no—
69#15691358—22no—
70#15691357—28no—
71#15691357—22no—
72#15691357—22no—
73#15691357—22no—
74#15691357—22no—
75C2×Dic394—21no—
76C2×Dic394C2311yescharacteristic
77C2×Dic394C2311yescharacteristic
78Q8×D197—22no—
79Q8×D197D414yescomplemented
80C2×D788—21no—
81C2×D788C2311yescharacteristic
82C2×C22⋊D197—21no—
83C2×C22⋊D197—21no—
84C2×C22⋊D197—21no—
85C2×C22⋊D197—21no—
86C2×C22⋊D197—21no—
87D4×D197—28no—
88D4×D197—22no—
89D4×D197D414yescomplemented
90C22×Dic197—21no—
91C22×Dic197—21no—
92C22×Dic197D411yescharacteristic
93C2×C4×D197—28no—
94C2×C4×D197—21no—
95C2×C4×D197—21no—
96C2×C4×D197—22no—
97C2×C4×D197—22no—
98C2×C4×D197—21no—
99C2×C4×D197D411yescharacteristic
100C2×C4×D197D411yescharacteristic
101C2×C4×D197C2311yescharacteristic
102C2×C4×D197C2311yescharacteristic
103Dic788—22no—
104#15691348—22no—
105#15691347—22no—
106#15691346—22no—
107#15691346—22no—
108#15691343—22no—
109#15691341—22no—
110#15691341—22no—
111C2×#13096802C2311yescharacteristic
112C2×#13096802C2311yescharacteristic
113C4⋊Dic197—21no—
114C4⋊Dic197C2311yescharacteristic
115#15691337—21no—
116C22⋊Dic197—21no—
117C22⋊Dic197—21no—
118#15691335—21no—
119#15691335—21no—
120C4×Dic197—21no—
121#41×C197—22no—
122#41×C197—22no—
123C2×Q8×C197C2311yescharacteristic
124C2×D4×C197—21no—
125C2×D4×C197—21no—
126C2×D4×C197C2311yescharacteristic
127C2×D4×C197C2311yescharacteristic
128C22×C4×C197—21no—
129C22×C4×C197C2311yescharacteristic
130SD16×C197—22no—
131D8×C197—22no—
132C2×C8×C197C2311yescharacteristic
133C2×C8×C197C2311yescharacteristic
134C4⋊C4×C197C2311yescharacteristic
135C22⋊C4×C197—21no—
136C22×D197—21no—
137C22×D197—22no—
138C22×D197—24no—
139C22×D197—41no—
140C22×D197—41no—
141C22×D197—42no—
142C22×D197—42no—
143C22×D197C2×D411yescharacteristic
144C22×D197C2×D411yescharacteristic
145Dic394—21no—
146Dic394—24no—
147Dic394—21no—
148Dic394—21no—
149Dic394C2×D412yes—
150D788—24no—
151D788—21no—
152D788C2×D412yes—
153C22⋊D197—24no—
154C22⋊D197—24no—
155C22⋊D197—24no—
156C22⋊D197—42no—
157C22⋊D197—42no—
158C2×Dic197—24no—
159C2×Dic197—21no—
160C2×Dic197—22no—
161C2×Dic197—21no—
162C2×Dic197—21no—
163C2×Dic197—42no—
164C2×Dic197C2×D411yescharacteristic
165C2×Dic197C2×D411yescharacteristic
166C4×D197—24no—
167C4×D197—24no—
168C4×D197—22no—
169C4×D197—24no—
170C4×D197—21no—
171C4×D197—21no—
172C4×D197—22no—
173C4×D197C2×D412yes—
174C4×D197C2×D412yes—
175#13096802—21no—
176#13096802—21no—
177C23×C197—21no—
178C23×C197—21no—
179C23×C197C2×D411yescharacteristic
180Q8×C197—21no—
181Q8×C197C2×D412yes—
182D4×C197—24no—
183D4×C197—21no—
184D4×C197—21no—
185D4×C197—21no—
186D4×C197—42no—
187D4×C197C2×D412yes—
188C2×C4×C197—24no—
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197—21no—
193C2×C4×C197C2×D411yescharacteristic
194C2×C4×C197C2411yescharacteristic
195C8×C197—21no—
196C8×C197—21no—
197C22×C197—21no—
198C22×C197—22no—
199C22×C197—21no—
200C22×C197—21no—
201C22×C197—41no—
202C22×C197—41no—
203C22×C197—41no—
204C22×C197—41no—
205C22×C197—41no—
206C22×C197C22×D411yescharacteristic
207C2×D197—21no—
208C2×D197—21no—
209C2×D197—21no—
210C2×D197—22no—
211C2×D197—21no—
212C2×D197—21no—
213C2×D197—44no—
214C2×D197—48no—
215C2×D197—41no—
216C2×D197—41no—
217C2×D197#12012yes—
218C4×C197—21no—
219C4×C197—22no—
220C4×C197—41no—
221C4×C197C22×D411yescharacteristic
222C4×C197C22×D411yescharacteristic
223Dic197—21no—
224Dic197—22no—
225Dic197—41no—
226Dic197—41no—
227Dic197#12012yes—
228C2×C197—41no—
229C2×C197—42no—
230C2×C197—81no—
231C2×C197C2×#12011yescharacteristic
232C2×C197#57511yescharacteristic
233C2×C197#57511yescharacteristic
234D197—22no—
235D197—41no—
236D197—42no—
237C197#300511yescharacteristic, complemented, Sylow
238#3005—1971nomaximal, Sylow
239C22×#41—1971no—
240C2×#137—1971no—
241C2×AGL(1,ℤ/8)—1971no—
242#525—1971no—
243#452—1971no—
244#452—1971no—
245#452—1971no—
246#452—1971no—
247#449—1971no—
248#449—1971no—
249#449—1971no—
250#449D19711yescharacteristic, complemented
251#419—1971no—
252#418—1971no—
253#407—1971no—
254C2×#41—3941no—
255C2×#41—3944no—
256C2×#41—3944no—
257C2×#41—3941no—
258C2×#41—1971no—
259C2×#41—1971no—
260C2×#41—1971no—
261C2×#41—1971no—
262C2×#41C2×D19711yescharacteristic
263C22×Q8—1971no—
264C22×D4—3941no—
265C22×D4—1971no—
266C23×C4—3941no—
267C23×C4—1971no—
268#137—3944no—
269AGL(1,ℤ/8)—3944no—
270C2×Q16—1971no—
271C2×Q16—1971no—
272C2×SD16—1971no—
273C2×SD16—1971no—
274C2×SD16—1971no—
275C2×SD16C2×D19711yescharacteristic, complemented
276C2×D8—1971no—
277C2×D8C2×D19711yescharacteristic, complemented
278C2×M16—1971no—
279C2×M16—1971no—
280#123—3941no—
281C22⋊Q8—1971no—
282C22⋊Q8—1971no—
283#121—1971no—
284#121C2×D19711yescharacteristic, complemented
285#120—3941no—
286C4×D4—3941no—
287C4×D4—3941no—
288#117—1971no—
289#103—1971no—
290#103—1971no—
291#103—1971no—
292#103C2×D19711yescharacteristic, complemented
293#102—1971no—
294#102—1971no—
295#102—1971no—
296#102C2×D19711yescharacteristic, complemented
297C22⋊C8—1971no—
298C22⋊C8—1971no—
299C22⋊C8—1971no—
300C22⋊C8C2×D19711yescharacteristic, complemented
301C24—3941no—
302C24—1971no—
303#41—22no—
304#41—22no—
305#41—3948no—
306#41—3942no—
307#41—3942no—
308#41—3942no—
309#41—3942no—
310#41—3948no—
311#41—3942no—
312#41—3942no—
313#41—3942no—
314#41—3942no—
315C2×Q8—3941no—
316C2×Q8—3942no—
317C2×Q8—1974no—
318C2×Q8—1971no—
319C2×Q8—1971no—
320C2×Q8C22×D19711yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—3941no—
324C2×D4—3948no—
325C2×D4—3941no—
326C2×D4—3941no—
327C2×D4—3941no—
328C2×D4—3941no—
329C2×D4—3942no—
330C2×D4—3941no—
331C2×D4—1974no—
332C2×D4—1971no—
333C2×D4C22×D19711yescharacteristic
334C2×D4C22×D19711yescharacteristic
335C22×C4—21no—
336C22×C4—3948no—
337C22×C4—3941no—
338C22×C4—3941no—
339C22×C4—3942no—
340C22×C4—3942no—
341C22×C4—3941no—
342C22×C4—3941no—
343C22×C4—3941no—
344C22×C4—1971no—
345C22×C4—1971no—
346C22×C4—1971no—
347C22×C4—1971no—
348C22×C4—1971no—
349C22×C4C22×D19711yescharacteristic
350Q16—3942no—
351Q16—3942no—
352SD16—22no—
353SD16—3942no—
354SD16—3942no—
355SD16—3942no—
356D8—22no—
357D8—3942no—
358M16—3942no—
359M16—3942no—
360C2×C8—1971no—
361C2×C8—1971no—
362C2×C8C22×D19711yescharacteristic
363C2×C8C22×D19711yescharacteristic
364C4⋊C4—3941no—
365C4⋊C4—3941no—
366C4⋊C4—1971no—
367C4⋊C4C22×D19711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3941no—
370C22⋊C4—3941no—
371C22⋊C4—3941no—
372C22⋊C4—3941no—
373C42—3941no—
374C23—21no—
375C23—21no—
376C23—7881no—
377C23—7881no—
378C23—7882no—
379C23—7882no—
380C23—3942no—
381C23—3944no—
382C23—3941no—
383C23—1971no—
384C23—1971no—
385C23D4×D19711yescharacteristic
386Q8—21no—
387Q8—3944no—
388Q8—3941no—
389Q8—3941no—
390Q8—3941no—
391Q8—1972no—
392Q8D4×D19712yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—7882no—
399D4—7882no—
400D4—3944no—
401D4—3944no—
402D4—3944no—
403D4—3944no—
404D4—3941no—
405D4—1972no—
406D4D4×D19712yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—7882no—
413C2×C4—3944no—
414C2×C4—3944no—
415C2×C4—3942no—
416C2×C4—3944no—
417C2×C4—3942no—
418C2×C4—3941no—
419C2×C4—3941no—
420C2×C4—3944no—
421C2×C4—3941no—
422C2×C4—3942no—
423C2×C4—3941no—
424C2×C4—3941no—
425C2×C4—1971no—
426C2×C4—1971no—
427C2×C4—1972no—
428C2×C4—1972no—
429C2×C4D4×D19711yescharacteristic
430C2×C4C23×D19711yescharacteristic
431C8—21no—
432C8—21no—
433C8—3941no—
434C8—3941no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—7884no—
445C22—7888no—
446C22—7881no—
447C22—7881no—
448C22—3941no—
449C22—3941no—
450C22—3941no—
451C22—3942no—
452C22—3941no—
453C22—3941no—
454C22—1972no—
455C22C2×D4×D19711yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—7881no—
460C4—7881no—
461C4—3941no—
462C4—3942no—
463C4—1972no—
464C4C2×D4×D19711yescharacteristic
465C4C2×D4×D19711yescharacteristic
466C2—41no—
467C2—42no—
468C2—81no—
469C2—7882no—
470C2—7881no—
471C2—3942no—
472C2#120×D19711yescharacteristic
473C2#2716187511yescharacteristic
474C2#2716187611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#449×C197C2
socleC22×C197C22×D4
hypercenter#449D197
hypocenterC197#3005
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D197C2semidirect
2. C2×#20263355C2semidirect
3. C2×#20263350C2semidirect
4. #27161589C2semidirect
5. #27161244C2semidirect
6. #27161243C2semidirect
7. #27161242C2semidirect
8. #27161241C2semidirect
9. #27161223C2semidirect
10. #27161222C2semidirect
11. #27161221C2semidirect
12. #27161086C2semidirect
13. #27161082C2semidirect
14. #27161041C2semidirect
15. #449×C197C2semidirect
31. C2×Dic788C22semidirect
32. C2×#15691348C22semidirect
33. C2×#15691347C22semidirect
34. C2×#15691346C22semidirect
35. C2×#15691346C22semidirect
36. C2×#15691343C22semidirect
40. #20263306C22semidirect
41. #20263305C22semidirect
42. #20263297C22semidirect
47. #20263251C22semidirect
48. #20263250C22semidirect
49. #20263249C22semidirect
50. #20263248C22semidirect
51. #20263247C22semidirect
52. #20263246C22semidirect
53. C22⋊#13096802C22semidirect
54. #20263238C22semidirect
55. #20263238C22semidirect
57. C2×SD16×C197C22semidirect
58. C2×D8×C197C22semidirect
59. #121×C197C22semidirect
60. #103×C197C22semidirect
61. #102×C197C22semidirect
62. C22⋊C8×C197C22semidirect
79. Q8×D197D4semidirect
89. D4×D197D4semidirect
237. C197#3005semidirect
250. #449D197semidirect
275. C2×SD16C2×D197semidirect
277. C2×D8C2×D197semidirect
284. #121C2×D197semidirect
292. #103C2×D197semidirect
296. #102C2×D197semidirect
300. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×#202633552
C2×#202633502
#271615892
#271612442
#271612432
#271612422
#271612412
#271612232
#271612222
#271612212
#271610862
#271610822
#271610412
#449×C1972
#3005197