Finite Group Databasebeta

← #44561906

Subgroups — #44561906

Order 32128.

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×#22548740C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3#30632333C211yesmaximal, characteristic, complemented
4#30632074C211yesmaximal, characteristic, complemented
5#30631931C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631909C211yesmaximal, characteristic, complemented
8#30631879C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631832C211yesmaximal, characteristic, complemented
11#30631830C211yesmaximal, characteristic, complemented
12#30631820C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631759C211yesmaximal, characteristic, complemented
15#550×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18#22548740—24no—
19#22548735—24no—
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860798C2211yescharacteristic, complemented
22C2×#16860798C2211yescharacteristic, complemented
23C2×#16860797C2211yescharacteristic
24C2×#16860797C2211yescharacteristic
25C2×#16860795C2211yescharacteristic
26C2×#16860795C2211yescharacteristic
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic
29C2×#16860794C2211yescharacteristic
30C22×C251⋊C8C2211yescharacteristic, complemented
31#22548703C2211yescharacteristic
32#22548700—21no—
33#22548694C2211yescharacteristic
34#22548689C2211yescharacteristic
35#22548688—21no—
36#22548683—21no—
37C4×D1004C2211yescharacteristic
38C22⋊C4×D251—21no—
39#22548648C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548638C2211yescharacteristic, complemented
43#22548636C2211yescharacteristic, complemented
44#22548636C2211yescharacteristic, complemented
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47C22⋊(C251⋊C8)C2211yescharacteristic
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49#22548626C2211yescharacteristic, complemented
50C2×#41×C251C2211yescharacteristic
51C2×#41×C251C2211yescharacteristic, complemented
52#127×C251—21no—
53#127×C251C2211yescharacteristic, complemented
54#121×C251—21no—
55#121×C251—21no—
56#121×C251C2211yescharacteristic
57#121×C251C2211yescharacteristic, complemented
58C4×Q8×C251C2211yescharacteristic, complemented
59C4×D4×C251—21no—
60C4×D4×C251C2211yescharacteristic
61C23×D251—21no—
62#16860810—22no—
63#16860810—41no—
64C2×Dic502C2311yescharacteristic
65C2×D1004—22no—
66C2×D1004—21no—
67C2×D1004—41no—
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74#16860800—22no—
75#16860798—22no—
76#16860798—41no—
77#16860797—22no—
78#16860797D414yescomplemented
79#16860795—22no—
80#16860795—22no—
81#16860795—22no—
82#16860795—41no—
83#16860794—22no—
84C2×C251⋊C8—22no—
85C2×C251⋊C8C2311yescharacteristic
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C2×C251⋊C8C2311yescharacteristic
89C4⋊Dic251C2311yescharacteristic
90Dic251⋊C4—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96#41×C251—22no—
97#41×C251—22no—
98#41×C251—41no—
99#41×C251—41no—
100C2×Q8×C251C2311yescharacteristic
101C2×D4×C251—21no—
102C2×D4×C251—21no—
103C2×D4×C251—21no—
104C2×D4×C251—21no—
105C2×D4×C251—21no—
106C2×D4×C251—21no—
107C2×D4×C251D411yescharacteristic
108C2×D4×C251C2311yescharacteristic
109C2×D4×C251C2311yescharacteristic
110C22×C4×C251—21no—
111C22×C4×C251—21no—
112C22×C4×C251C2311yescharacteristic
113C22×C4×C251C2311yescharacteristic
114C4⋊C4×C251—21no—
115C4⋊C4×C251D411yescharacteristic
116C4⋊C4×C251C2311yescharacteristic
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C22⋊C4×C251D411yescharacteristic
120C22⋊C4×C251D411yescharacteristic
121C42×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—81no—
128Dic502—21no—
129Dic502—21no—
130D1004—21no—
131D1004—21no—
132D1004—21no—
133D1004—42no—
134D1004C2×D412yes—
135C22⋊D251—42no—
136C22⋊D251—81no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C4×D251—42no—
140C4×D251—81no—
141C251⋊C8—21no—
142C251⋊C8—21no—
143C251⋊C8—21no—
144C251⋊C8C2×D412yes—
145C23×C251—21no—
146C23×C251—21no—
147C23×C251C2×D411yescharacteristic
148C23×C251C2×D411yescharacteristic
149Q8×C251—21no—
150Q8×C251C2×D412yes—
151D4×C251—21no—
152D4×C251—21no—
153D4×C251—21no—
154D4×C251—21no—
155D4×C251—41no—
156D4×C251—42no—
157D4×C251—42no—
158D4×C251—42no—
159D4×C251—81no—
160D4×C251—81no—
161D4×C251—81no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—42no—
169C2×C4×C251—42no—
170C2×C4×C251C2×D411yescharacteristic
171C2×C4×C251C2×D411yescharacteristic
172C2×C4×C251C2×D411yescharacteristic
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2411yescharacteristic
175C22×C251—21no—
176C22×C251—21no—
177C22×C251—21no—
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—21no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251C22×D411yescharacteristic
188C2×D251—22no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—84no—
195C4×C251—21no—
196C4×C251—21no—
197C4×C251—22no—
198C4×C251—21no—
199C4×C251—21no—
200C4×C251—41no—
201C4×C251C22×D411yescharacteristic
202C4×C251H(5,𝔽2)11yescharacteristic
203Dic251—41no—
204Dic251—41no—
205C2×C251—41no—
206C2×C251—41no—
207C2×C251—81no—
208C2×C251—81no—
209C2×C251D4211yescharacteristic
210C2×C251#57511yescharacteristic
211C2×C251#57611yescharacteristic
212D251—42no—
213D251—81no—
214C251#326811yescharacteristic, complemented, Sylow
215#3268—2511nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2511no—
217C2×#135—2511no—
218#550D25111yescharacteristic, complemented
219#546—2511no—
220#496—2511no—
221#468—2511no—
222#466—2511no—
223#464—2511no—
224#459—2511no—
225#452—2511no—
226#449—2511no—
227#449—2511no—
228#447—2511no—
229#440—2511no—
230#435—2511no—
231C2×#41—2511no—
232C2×#41C2×D25111yescharacteristic
233C2×#41C2×D25111yescharacteristic, complemented
234C22×D4—2511no—
235AGL(1,ℤ/8)—5024no—
236#135—5024no—
237C2×Q16—2511no—
238C2×SD16—2511no—
239C2×SD16—2511no—
240C2×SD16—2511no—
241C2×SD16—2511no—
242C2×D8—2511no—
243C2×D8—2511no—
244C2×D8—2511no—
245C2×D8—2511no—
246C2×M16—2511no—
247C22×C8—2511no—
248#127—21no—
249#127C2×D25111yescharacteristic, complemented
250#124—2511no—
251#123—5021no—
252C22⋊Q8—2511no—
253#121—21no—
254#121—21no—
255#121—5021no—
256#121—2511no—
257#121C2×D25111yescharacteristic
258#121C2×D25111yescharacteristic, complemented
259#120—5021no—
260C4×Q8C2×D25111yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—2511no—
263C4×D4C2×D25111yescharacteristic
264C2×C22⋊C4—5021no—
265#107—2511no—
266C4⋊C8—2511no—
267#103—2511no—
268#103—2511no—
269#102—2511no—
270#102—2511no—
271#102—2511no—
272#102—2511no—
273C22⋊C8—2511no—
274C22⋊C8—2511no—
275#97—2511no—
276C24—5021no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—10041no—
282#41—5022no—
283C2×Q8—2511no—
284C2×Q8C22×D25111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—10041no—
292C2×D4—5022no—
293C2×D4—5021no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—2511no—
297C2×D4—2511no—
298C2×D4C22⋊D25111yescharacteristic
299C2×D4C22×D25111yescharacteristic
300C2×D4C22×D25111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—5021no—
304C22×C4—5021no—
305C22×C4C22×D25111yescharacteristic
306C22×C4C22×D25111yescharacteristic
307Q16—5022no—
308SD16—10041no—
309SD16—5022no—
310SD16—5022no—
311SD16—2514no—
312D8—10041no—
313D8—5022no—
314D8—5022no—
315D8—5022no—
316M16—5022no—
317C2×C8—5022no—
318C2×C8—2511no—
319C2×C8—2511no—
320C2×C8—2511no—
321C2×C8—2511no—
322C4⋊C4—21no—
323C4⋊C4—5021no—
324C4⋊C4—2511no—
325C4⋊C4C22⋊D25111yescharacteristic
326C4⋊C4C22×D25111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—5021no—
330C22⋊C4—5021no—
331C22⋊C4—5021no—
332C22⋊C4—5021no—
333C22⋊C4—5021no—
334C22⋊C4C22⋊D25111yescharacteristic
335C22⋊C4C22⋊D25111yescharacteristic
336C42—21no—
337C42C22×D25111yescharacteristic
338C23—21no—
339C23—21no—
340C23—20081no—
341C23—10042no—
342C23—10042no—
343C23—5021no—
344C23—5021no—
345C23C2×C22⋊D25111yescharacteristic
346C23C2×C22⋊D25111yescharacteristic
347Q8—21no—
348Q8—5021no—
349Q8—5021no—
350Q8D4×D25112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—20081no—
363D4—10042no—
364D4—10042no—
365D4—5021no—
366D4—5021no—
367D4—5021no—
368D4—2512no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—20081no—
378C2×C4—10042no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4C2×C22⋊D25111yescharacteristic
382C2×C4C2×C22⋊D25111yescharacteristic
383C2×C4C2×C22⋊D25111yescharacteristic
384C2×C4C2×C22⋊D25111yescharacteristic
385C2×C4C23×D25111yescharacteristic
386C8—5021no—
387C8—5021no—
388C8—5021no—
389C8—2512no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—20084no—
403C22—10041no—
404C22—10041no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—5022no—
409C22C22×C22⋊D25111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—10041no—
417C4—10041no—
418C4C2×D4×D25111yescharacteristic
419C4#2254876011yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—20081no—
425C2—10042no—
426C2D4×C22⋊D25111yescharacteristic
427C2#3063248511yescharacteristic
428C2#3063249011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#550×C251C2
socleC22×C251C22×D4
hypercenter#550D251
hypocenterC251#3268
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C2×#22548735C2semidirect
3. #30632333C2semidirect
4. #30632074C2semidirect
5. #30631931C2semidirect
6. #30631921C2semidirect
7. #30631909C2semidirect
8. #30631879C2semidirect
9. #30631849C2semidirect
10. #30631832C2semidirect
11. #30631830C2semidirect
12. #30631820C2semidirect
13. #30631785C2semidirect
14. #30631759C2semidirect
15. #550×C251C2semidirect
20. C2×#16860800C22semidirect
21. C2×#16860798C22semidirect
22. C2×#16860798C22semidirect
30. C22×C251⋊C8C22semidirect
39. #22548648C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548639C22semidirect
42. #22548638C22semidirect
43. #22548636C22semidirect
44. #22548636C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. #22548626C22semidirect
51. C2×#41×C251C22semidirect
53. #127×C251C22semidirect
57. #121×C251C22semidirect
58. C4×Q8×C251C22semidirect
78. #16860797D4semidirect
214. C251#3268semidirect
218. #550D251semidirect
233. C2×#41C2×D251semidirect
249. #127C2×D251semidirect
258. #121C2×D251semidirect
260. C4×Q8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C2×#225487352
#306323332
#306320742
#306319312
#306319212
#306319092
#306318792
#306318492
#306318322
#306318302
#306318202
#306317852
#306317592
#550×C2512
#3268251