Finite Group Databasebeta

← #38754317

Subgroups — #38754317

Order 25216.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2#27161723C211yesmaximal, characteristic, complemented
3#27161717C211yesmaximal, characteristic, complemented
4#27161460C211yesmaximal, characteristic, complemented
5#27161325C211yesmaximal, characteristic, complemented
6#27161322C211yesmaximal, characteristic, complemented
7#27161296C211yesmaximal, characteristic, complemented
8#27161292C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161238C211yesmaximal, characteristic, complemented
11#27161213C211yesmaximal, characteristic, complemented
12#27161211C211yesmaximal, characteristic, complemented
13D8×Dic197C211yesmaximal, characteristic, complemented
14#27161155C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic, complemented
18#20263350—24no—
19C2×#15691348C2211yescharacteristic, complemented
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691343C2211yescharacteristic
23C2×#15691343C2211yescharacteristic
24C2×#15691341C2211yescharacteristic, complemented
25#20263315C2211yescharacteristic, complemented
26#20263313C2211yescharacteristic, complemented
27#20263309—21no—
28#20263307—21no—
29#20263305C2211yescharacteristic, complemented
30#20263302C2211yescharacteristic, complemented
31#20263300C2211yescharacteristic
32#20263298—21no—
33#20263298—21no—
34#20263297C2211yescharacteristic
35#20263295—21no—
36#20263294—21no—
37D4×Dic197C2211yescharacteristic, complemented
38D4×Dic197C2211yescharacteristic, complemented
39C4×C22⋊D197C2211yescharacteristic
40C2×C22⋊Dic197—21no—
41C2×#15691335—21no—
42#20263260C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic, complemented
48#20263247C2211yescharacteristic, complemented
49#20263247C2211yescharacteristic, complemented
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C2×#41×C197C2211yescharacteristic
54C22×D4×C197C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C197—24no—
56C2×SD16×C197C2211yescharacteristic, complemented
57C2×SD16×C197C2211yescharacteristic, complemented
58C2×D8×C197C2211yescharacteristic
59C2×D8×C197C2211yescharacteristic
60C2×M16×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691357—22no—
63#15691357—41no—
64C2×Dic394C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197D411yescharacteristic
71D4×D197—22no—
72D4×D197—41no—
73C22×Dic197—21no—
74C22×Dic197—21no—
75C22×Dic197—21no—
76C2×C4×D197C2311yescharacteristic
77#15691348—41no—
78#15691347—22no—
79#15691346—22no—
80#15691343—22no—
81#15691343—41no—
82#15691341—22no—
83C2×#13096802C2311yescharacteristic
84C2×#13096802C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197—21no—
95C22⋊Dic197—21no—
96C22⋊Dic197D411yescharacteristic
97#15691335—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335—21no—
101#15691335D411yescharacteristic
102C4×Dic197C2311yescharacteristic
103C24×C197—21no—
104#41×C197—22no—
105#41×C197—41no—
106C2×Q8×C197C2311yescharacteristic
107C2×D4×C197—22no—
108C2×D4×C197—21no—
109C2×D4×C197—21no—
110C2×D4×C197—41no—
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116SD16×C197—22no—
117SD16×C197—22no—
118D8×C197—22no—
119D8×C197D414yescomplemented
120M16×C197—22no—
121C2×C8×C197C2311yescharacteristic
122C2×C8×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—42no—
125C22×D197—42no—
126C22×D197—81no—
127C22×D197C2×D411yescharacteristic
128Dic394—21no—
129Dic394—21no—
130D788—21no—
131D788—21no—
132C22⋊D197—41no—
133C22⋊D197—42no—
134C22⋊D197—42no—
135C22⋊D197—81no—
136C22⋊D197—81no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C2×Dic197—42no—
141C2×Dic197—81no—
142C2×Dic197—81no—
143C2×Dic197C2×D411yescharacteristic
144C2×Dic197C2×D411yescharacteristic
145C2×Dic197C2×D411yescharacteristic
146C4×D197—21no—
147C4×D197—21no—
148#13096802—21no—
149#13096802—21no—
150C23×C197—21no—
151C23×C197—21no—
152C23×C197—21no—
153C23×C197—42no—
154C23×C197—42no—
155C23×C197—81no—
156C23×C197C2×D411yescharacteristic
157Q8×C197—21no—
158Q8×C197—21no—
159D4×C197—21no—
160D4×C197—21no—
161D4×C197—21no—
162D4×C197—21no—
163D4×C197—42no—
164D4×C197—42no—
165D4×C197C2×D412yes—
166D4×C197C2×D412yes—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—42no—
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2411yescharacteristic
173C8×C197—21no—
174C8×C197C2×D412yes—
175C22×C197—21no—
176C22×C197—22no—
177C22×C197—22no—
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—84no—
188C22×C197C22×D411yescharacteristic
189C2×D197—21no—
190C2×D197—21no—
191C2×D197—21no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—84no—
196C4×C197—21no—
197C4×C197—41no—
198C4×C197C22×D411yescharacteristic
199C4×C197H(5,𝔽2)11yescharacteristic
200Dic197—21no—
201Dic197—21no—
202Dic197—21no—
203Dic197—41no—
204Dic197—41no—
205Dic197—41no—
206C2×C197—42no—
207C2×C197—41no—
208C2×C197—42no—
209C2×C197—81no—
210C2×C197D4211yescharacteristic
211C2×C197#57511yescharacteristic
212C2×C197#57611yescharacteristic
213D197—41no—
214D197—81no—
215C197#327311yescharacteristic, complemented, Sylow
216#3273—1971nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1971no—
218C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
219#546—1971no—
220#546—1971no—
221#495—1971no—
222#468—1971no—
223#468—1971no—
224#463—1971no—
225#463—1971no—
226#452—1971no—
227#452—1971no—
228#447—1971no—
229#447—1971no—
230C4×D8—1971no—
231#436—1971no—
232C2×#41—1971no—
233C2×#41C2×D19711yescharacteristic
234C22×D4—1971no—
235C22×D4C2×D19711yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—3944no—
238C2×Q16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16C2×D19711yescharacteristic, complemented
242C2×SD16C2×D19711yescharacteristic, complemented
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8C2×D19711yescharacteristic
246C2×D8C2×D19711yescharacteristic
247C2×M16—1971no—
248C2×M16C2×D19711yescharacteristic, complemented
249#124—1971no—
250#124—1971no—
251#123—3941no—
252#123—3941no—
253C22⋊Q8—1971no—
254C22⋊Q8—1971no—
255#121—3941no—
256#121—3941no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260#120—3941no—
261C4×D4—1971no—
262C4×D4—1971no—
263C4×D4—1971no—
264C2×C22⋊C4—3941no—
265C2×C22⋊C4—3941no—
266#107—1971no—
267C4⋊C8—1971no—
268#103—1971no—
269#103—1971no—
270#102—1971no—
271#102—1971no—
272#102—1971no—
273#102—1971no—
274C22⋊C8—1971no—
275C22⋊C8—1971no—
276C4×C8—1971no—
277C24—21no—
278C24—3941no—
279#41—22no—
280#41—41no—
281#41—7881no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8C22×D19711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—7881no—
290C2×D4—3941no—
291C2×D4—3942no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—1971no—
296C2×D4—1971no—
297C2×D4C22×D19711yescharacteristic
298C2×D4C22×D19711yescharacteristic
299C2×D4C22×D19711yescharacteristic
300C22×C4—21no—
301C22×C4—3941no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—1971no—
305C22×C4C22×D19711yescharacteristic
306Q16—7881no—
307SD16—22no—
308SD16—22no—
309SD16—3942no—
310SD16—3942no—
311D8—22no—
312D8—7881no—
313D8—3942no—
314D8C22⋊D19714yescomplemented
315M16—22no—
316M16—3942no—
317C2×C8—1971no—
318C2×C8—1971no—
319C2×C8C22×D19711yescharacteristic
320C2×C8C22×D19711yescharacteristic
321C4⋊C4—3941no—
322C4⋊C4—3941no—
323C4⋊C4—1971no—
324C4⋊C4—1971no—
325C4⋊C4—1971no—
326C22⋊C4—3941no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—1971no—
337C22⋊C4—1971no—
338C42—1971no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—7882no—
346C23—7882no—
347C23—15761no—
348C23—3941no—
349C23—1971no—
350C23D4×D19711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3941no—
354Q8—3941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—7882no—
362D4—7882no—
363D4—7881no—
364D4—15761no—
365D4—15761no—
366D4—3941no—
367D4—3941no—
368D4C2×C22⋊D19712yes—
369D4C2×C22⋊D19712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—7882no—
375C2×C4—15761no—
376C2×C4—15761no—
377C2×C4—3941no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4—1971no—
383C2×C4—1971no—
384C2×C4—1971no—
385C2×C4D4×D19711yescharacteristic
386C2×C4C23×D19711yescharacteristic
387C8—21no—
388C8—3941no—
389C8—3941no—
390C8C2×C22⋊D19712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—15764no—
408C22—3941no—
409C22—3941no—
410C22—3941no—
411C22C2×D4×D19711yescharacteristic
412C4—21no—
413C4—41no—
414C4—7881no—
415C4—7881no—
416C4—7881no—
417C4—3941no—
418C4—3941no—
419C4—3941no—
420C4C22×C22⋊D19711yescharacteristic
421C4#2026337111yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—7881no—
427C2—15761no—
428C2D4×C22⋊D19711yescharacteristic
429C2#2716187511yescharacteristic
430C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3273
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161723C2semidirect
3. #27161717C2semidirect
4. #27161460C2semidirect
5. #27161325C2semidirect
6. #27161322C2semidirect
7. #27161296C2semidirect
8. #27161292C2semidirect
9. #27161241C2semidirect
10. #27161238C2semidirect
11. #27161213C2semidirect
12. #27161211C2semidirect
13. D8×Dic197C2semidirect
14. #27161155C2semidirect
15. C2×AGL(1,ℤ/8)×C197C2semidirect
17. C2×D4×D197C22semidirect
19. C2×#15691348C22semidirect
20. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
24. C2×#15691341C22semidirect
25. #20263315C22semidirect
26. #20263313C22semidirect
29. #20263305C22semidirect
30. #20263302C22semidirect
37. D4×Dic197C22semidirect
38. D4×Dic197C22semidirect
42. #20263260C22semidirect
43. #20263255C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263248C22semidirect
47. #20263247C22semidirect
48. #20263247C22semidirect
49. #20263247C22semidirect
50. C22⋊#13096802C22semidirect
51. #20263238C22semidirect
52. C8×Dic197C22semidirect
54. C22×D4×C197C22semidirect
56. C2×SD16×C197C22semidirect
57. C2×SD16×C197C22semidirect
60. C2×M16×C197C22semidirect
119. D8×C197D4semidirect
215. C197#3273semidirect
218. C2×AGL(1,ℤ/8)D197semidirect
235. C22×D4C2×D197semidirect
241. C2×SD16C2×D197semidirect
242. C2×SD16C2×D197semidirect
248. C2×M16C2×D197semidirect
314. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617232
#271617172
#271614602
#271613252
#271613222
#271612962
#271612922
#271612412
#271612382
#271612132
#271612112
D8×Dic1972
#271611552
C2×AGL(1,ℤ/8)×C1972
#3273197