Finite Group Databasebeta

← #38754252

Subgroups — #38754252

Order 25216.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263353C211yesmaximal, characteristic, complemented
2#27161748C211yesmaximal, characteristic, complemented
3#27161717C211yesmaximal, characteristic, complemented
4#27161465C211yesmaximal, characteristic, complemented
5#27161322C211yesmaximal, characteristic, complemented
6#27161315C211yesmaximal, characteristic, complemented
7#27161303C211yesmaximal, characteristic, complemented
8#27161272C211yesmaximal, characteristic, complemented
9#27161240C211yesmaximal, characteristic, complemented
10#27161224C211yesmaximal, characteristic, complemented
11#27161224C211yesmaximal, characteristic, complemented
12#27161211C211yesmaximal, characteristic, complemented
13#27161180C211yesmaximal, characteristic, complemented
14#27161150C211yesmaximal, characteristic, complemented
15C2×#135×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18#20263353—24no—
19C2×#15691345C2211yescharacteristic
20C2×#15691345C2211yescharacteristic, complemented
21C2×D1576C2211yescharacteristic
22C2×#15691343C2211yescharacteristic
23C2×#15691343C2211yescharacteristic
24C2×#15691341C2211yescharacteristic
25#20263322—21no—
26#20263322C2211yescharacteristic, complemented
27#20263315C2211yescharacteristic
28#20263309—21no—
29#20263302C2211yescharacteristic
30#20263300—21no—
31#20263300C2211yescharacteristic
32#20263300C2211yescharacteristic, complemented
33#20263298—21no—
34#20263297—21no—
35#20263297C2211yescharacteristic
36#20263294—21no—
37Q8×Dic197C2211yescharacteristic, complemented
38D4×Dic197C2211yescharacteristic
39C4×C22⋊D197—21no—
40C4×C22⋊D197C2211yescharacteristic
41C2×#15691335—21no—
42#20263260C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic, complemented
48#20263247C2211yescharacteristic
49#20263247C2211yescharacteristic, complemented
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic
52#20263236C2211yescharacteristic, complemented
53C2×#41×C197C2211yescharacteristic
54C2×#41×C197C2211yescharacteristic, complemented
55#135×C197—24no—
56C2×Q16×C197C2211yescharacteristic, complemented
57C2×SD16×C197C2211yescharacteristic
58C2×SD16×C197C2211yescharacteristic, complemented
59C2×D8×C197C2211yescharacteristic
60C22×C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691359—22no—
63#15691359—41no—
64C2×D788—21no—
65C2×D788C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197D411yescharacteristic
74D4×D197—22no—
75D4×D197—41no—
76C22×Dic197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79#15691345—22no—
80#15691345—41no—
81D1576—22no—
82#15691343—22no—
83#15691343—41no—
84#15691341—22no—
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C4⋊Dic197—21no—
88C4⋊Dic197C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91#15691337D411yescharacteristic
92C22⋊Dic197—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197D411yescharacteristic
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335D411yescharacteristic
101C4×Dic197—21no—
102C4×Dic197C2311yescharacteristic
103#41×C197—22no—
104#41×C197—22no—
105#41×C197—41no—
106#41×C197—41no—
107C2×Q8×C197C2311yescharacteristic
108C2×Q8×C197C2311yescharacteristic
109C2×D4×C197—21no—
110C2×D4×C197—21no—
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—22no—
117SD16×C197—22no—
118SD16×C197D414yescomplemented
119D8×C197—22no—
120C2×C8×C197—22no—
121C2×C8×C197C2311yescharacteristic
122C2×C8×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129D788—21no—
130D788—21no—
131D788—21no—
132D788—21no—
133D788—42no—
134C22⋊D197—41no—
135C22⋊D197—42no—
136C22⋊D197—42no—
137C22⋊D197—81no—
138C22⋊D197—81no—
139C22⋊D197—81no—
140C22⋊D197—81no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—81no—
144C2×Dic197C2×D411yescharacteristic
145C2×Dic197C2×D411yescharacteristic
146C2×Dic197C2×D411yescharacteristic
147C4×D197—21no—
148C4×D197—21no—
149C4×D197—42no—
150#13096802—21no—
151#13096802—21no—
152C23×C197—21no—
153C23×C197—21no—
154C23×C197C2×D411yescharacteristic
155Q8×C197—21no—
156Q8×C197—21no—
157Q8×C197—21no—
158Q8×C197C2×D412yes—
159D4×C197—21no—
160D4×C197—21no—
161D4×C197—21no—
162D4×C197—42no—
163D4×C197—42no—
164D4×C197C2×D412yes—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—42no—
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—21no—
178C22×C197—21no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197C22×D411yescharacteristic
185C2×D197—21no—
186C2×D197—21no—
187C2×D197—21no—
188C2×D197—41no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—84no—
195C4×C197—21no—
196C4×C197—22no—
197C4×C197—41no—
198C4×C197C22×D411yescharacteristic
199C4×C197H(5,𝔽2)11yescharacteristic
200Dic197—21no—
201Dic197—21no—
202Dic197—21no—
203Dic197—41no—
204Dic197—41no—
205C2×C197—42no—
206C2×C197—41no—
207C2×C197—81no—
208C2×C197D4211yescharacteristic
209C2×C197#57511yescharacteristic
210C2×C197#57611yescharacteristic
211D197—41no—
212D197—81no—
213D197—81no—
214C197#326811yescharacteristic, complemented, Sylow
215#3268—1971nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1971no—
217C2×#135D19711yescharacteristic, complemented
218#550—1971no—
219#546—1971no—
220#496—1971no—
221#468—1971no—
222#466—1971no—
223#464—1971no—
224#459—1971no—
225#452—1971no—
226#449—1971no—
227#449—1971no—
228#447—1971no—
229#440—1971no—
230#435—1971no—
231C2×#41—1971no—
232C2×#41C2×D19711yescharacteristic
233C2×#41C2×D19711yescharacteristic, complemented
234C22×D4—1971no—
235AGL(1,ℤ/8)—3944no—
236#135—24no—
237C2×Q16C2×D19711yescharacteristic, complemented
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16C2×D19711yescharacteristic
241C2×SD16C2×D19711yescharacteristic, complemented
242C2×D8—1971no—
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8C2×D19711yescharacteristic
246C2×M16—1971no—
247C22×C8C2×D19711yescharacteristic, complemented
248#127—3941no—
249#127—1971no—
250#124—1971no—
251#123—3941no—
252C22⋊Q8—1971no—
253#121—3941no—
254#121—3941no—
255#121—3941no—
256#121—1971no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260C4×Q8—1971no—
261C4×D4—3941no—
262C4×D4—1971no—
263C4×D4—1971no—
264C2×C22⋊C4—3941no—
265#107—1971no—
266C4⋊C8—1971no—
267#103—1971no—
268#103—1971no—
269#102—1971no—
270#102—1971no—
271#102—1971no—
272#102—1971no—
273C22⋊C8—1971no—
274C22⋊C8—1971no—
275#97—1971no—
276C24—3941no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—7881no—
282#41—3942no—
283C2×Q8C22×D19711yescharacteristic
284C2×Q8C22×D19711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—7881no—
288C2×D4—3941no—
289C2×D4—3942no—
290C2×D4—3941no—
291C2×D4—3941no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4—1971no—
299C2×D4C22×D19711yescharacteristic
300C2×D4C22×D19711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—3941no—
304C22×C4—3941no—
305C22×C4—1971no—
306C22×C4C22×D19711yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—7881no—
310SD16—3942no—
311SD16C22⋊D19714yescomplemented
312D8—22no—
313D8—7881no—
314D8—3942no—
315D8—3942no—
316M16—3942no—
317C2×C8—22no—
318C2×C8—1971no—
319C2×C8—1971no—
320C2×C8C22×D19711yescharacteristic
321C2×C8C22×D19711yescharacteristic
322C4⋊C4—3941no—
323C4⋊C4—3941no—
324C4⋊C4—1971no—
325C4⋊C4—1971no—
326C4⋊C4—1971no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—1971no—
335C22⋊C4—1971no—
336C42—3941no—
337C42—1971no—
338C23—21no—
339C23—21no—
340C23—7882no—
341C23—7882no—
342C23—15761no—
343C23—3941no—
344C23—3941no—
345C23—1971no—
346C23D4×D19711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D19712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—7882no—
357D4—7882no—
358D4—7881no—
359D4—7882no—
360D4—15761no—
361D4—15761no—
362D4—15761no—
363D4—15761no—
364D4—3941no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4C2×C22⋊D19712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—7882no—
376C2×C4—15761no—
377C2×C4—3941no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—1971no—
382C2×C4—1971no—
383C2×C4—1971no—
384C2×C4D4×D19711yescharacteristic
385C2×C4C23×D19711yescharacteristic
386C8—21no—
387C8—3941no—
388C8—3941no—
389C8C2×C22⋊D19712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—7881no—
400C22—7881no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—15764no—
406C22—3941no—
407C22—3941no—
408C22—3941no—
409C22C2×D4×D19711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—7881no—
414C4—7881no—
415C4—3941no—
416C4—3941no—
417C4—3941no—
418C4C22×C22⋊D19711yescharacteristic
419C4#2026337111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—7881no—
424C2—15761no—
425C2—15761no—
426C2D4×C22⋊D19711yescharacteristic
427C2#2716187611yescharacteristic
428C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC22×C197C22×D4
hypercenterC2×#135D197
hypocenterC197#3268
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. #27161748C2semidirect
3. #27161717C2semidirect
4. #27161465C2semidirect
5. #27161322C2semidirect
6. #27161315C2semidirect
7. #27161303C2semidirect
8. #27161272C2semidirect
9. #27161240C2semidirect
10. #27161224C2semidirect
11. #27161224C2semidirect
12. #27161211C2semidirect
13. #27161180C2semidirect
14. #27161150C2semidirect
15. C2×#135×C197C2semidirect
20. C2×#15691345C22semidirect
26. #20263322C22semidirect
32. #20263300C22semidirect
37. Q8×Dic197C22semidirect
42. #20263260C22semidirect
43. #20263255C22semidirect
44. #20263250C22semidirect
45. #20263250C22semidirect
47. #20263248C22semidirect
49. #20263247C22semidirect
50. C22⋊#13096802C22semidirect
52. #20263236C22semidirect
54. C2×#41×C197C22semidirect
56. C2×Q16×C197C22semidirect
58. C2×SD16×C197C22semidirect
60. C22×C8×C197C22semidirect
118. SD16×C197D4semidirect
214. C197#3268semidirect
217. C2×#135D197semidirect
233. C2×#41C2×D197semidirect
237. C2×Q16C2×D197semidirect
241. C2×SD16C2×D197semidirect
247. C22×C8C2×D197semidirect
311. SD16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
#271617482
#271617172
#271614652
#271613222
#271613152
#271613032
#271612722
#271612402
#271612242
#271612112
#271611802
#271611502
C2×#135×C1972
#3268197