Finite Group Databasebeta

← #44562073

Subgroups — #44562073

Order 32128.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548735C211yesmaximal, characteristic, complemented
2C22×#16860797C211yesmaximal, characteristic, complemented
3#30632333C211yesmaximal, characteristic, complemented
4#30632069C211yesmaximal, characteristic, complemented
5#30631918C211yesmaximal, characteristic, complemented
6#30631914C211yesmaximal, characteristic, complemented
7#30631909C211yesmaximal, characteristic, complemented
8#30631899C211yesmaximal, characteristic, complemented
9#30631847C211yesmaximal, characteristic, complemented
10#30631843C211yesmaximal, characteristic, complemented
11#30631832C211yesmaximal, characteristic, complemented
12#30631820C211yesmaximal, characteristic, complemented
13C4×#16860797C211yesmaximal, characteristic, complemented
14D4×C251⋊C8C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18#22548735—24no—
19C2×#16860800C2211yescharacteristic, complemented
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860798C2211yescharacteristic, complemented
22C2×#16860797—24no—
23C2×#16860797C2211yescharacteristic
24C2×#16860797C2211yescharacteristic
25C2×#16860797C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic
29C22×C251⋊C8C2211yescharacteristic
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#22548646C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548636C2211yescharacteristic, complemented
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47C22⋊(C251⋊C8)C2211yescharacteristic
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49C4×C251⋊C8C2211yescharacteristic, complemented
50C2×#41×C251C2211yescharacteristic, complemented
51C22×Q8×C251C2211yescharacteristic
52#124×C251—21no—
53#124×C251C2211yescharacteristic, complemented
54C22⋊Q8×C251—21no—
55C22⋊Q8×C251C2211yescharacteristic, complemented
56#121×C251—21no—
57#121×C251C2211yescharacteristic
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#16860800—41no—
76#16860798—22no—
77#16860797—22no—
78#16860797—22no—
79#16860797—22no—
80#16860797D414yescomplemented
81#16860795—22no—
82#16860795—41no—
83C2×C251⋊C8—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—41no—
98C2×Q8×C251—21no—
99C2×Q8×C251—22no—
100C2×Q8×C251—41no—
101C2×Q8×C251C2311yescharacteristic
102C2×Q8×C251C2311yescharacteristic
103C2×D4×C251—21no—
104C2×D4×C251—21no—
105C2×D4×C251D411yescharacteristic
106C2×D4×C251C2311yescharacteristic
107C22×C4×C251—21no—
108C22×C4×C251—21no—
109C22×C4×C251C2311yescharacteristic
110C22×C4×C251C2311yescharacteristic
111C4⋊C4×C251—21no—
112C4⋊C4×C251—21no—
113C4⋊C4×C251D411yescharacteristic
114C4⋊C4×C251C2311yescharacteristic
115C22⋊C4×C251—21no—
116C22⋊C4×C251—21no—
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×C251C2×D411yescharacteristic
147C23×C251C2×D411yescharacteristic
148Q8×C251—21no—
149Q8×C251—21no—
150Q8×C251—21no—
151Q8×C251—42no—
152Q8×C251C2×D412yes—
153D4×C251—22no—
154D4×C251—21no—
155D4×C251—21no—
156D4×C251—42no—
157D4×C251—42no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—42no—
166C2×C4×C251—42no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2×D411yescharacteristic
170C2×C4×C251C2×D411yescharacteristic
171C2×C4×C251C2411yescharacteristic
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—21no—
176C22×C251—21no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251#13512yescomplemented
181C22×C251C22×D411yescharacteristic
182C2×D251—22no—
183C2×D251—41no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—21no—
191C4×C251—22no—
192C4×C251—21no—
193C4×C251—21no—
194C4×C251—41no—
195C4×C251—41no—
196C4×C251C22×D411yescharacteristic
197C4×C251H(5,𝔽2)11yescharacteristic
198Dic251—41no—
199Dic251—41no—
200C2×C251—22no—
201C2×C251—41no—
202C2×C251—81no—
203C2×C251D4211yescharacteristic
204C2×C251C2×#13511yescharacteristic
205C2×C251#57711yescharacteristic
206D251—42no—
207D251—81no—
208C251#328211yescharacteristic, complemented, Sylow
209#3282—2511nomaximal, Sylow
210C2×#135—2511no—
211C22×SD16—2511no—
212#548D25111yescharacteristic, complemented
213#546—2511no—
214#495—2511no—
215#465—2511no—
216#465—2511no—
217#464—2511no—
218#463—2511no—
219#452—2511no—
220#451—2511no—
221#449—2511no—
222#447—2511no—
223C4×SD16—2511no—
224C8×D4—2511no—
225C2×#41—2511no—
226C2×#41C2×D25111yescharacteristic, complemented
227C22×Q8C2×D25111yescharacteristic
228C22×D4—2511no—
229#135—5024no—
230C2×Q16—2511no—
231C2×Q16—2511no—
232C2×SD16—5024no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16—2511no—
237C2×SD16—2511no—
238C2×D8—2511no—
239C2×D8—2511no—
240C22×C8—2511no—
241C22×C8—2511no—
242#124—21no—
243#124—2511no—
244#124C2×D25111yescharacteristic, complemented
245#123—5021no—
246C22⋊Q8—21no—
247C22⋊Q8—2511no—
248C22⋊Q8C2×D25111yescharacteristic, complemented
249#121—21no—
250#121—5021no—
251#121—2511no—
252#121C2×D25111yescharacteristic
253#120—5021no—
254C4×Q8C2×D25111yescharacteristic, complemented
255C4×D4—21no—
256C4×D4—2511no—
257C4×D4C2×D25111yescharacteristic
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8—2511no—
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8—2511no—
268C22⋊C8—2511no—
269C4×C8—2511no—
270C24—5021no—
271#41—22no—
272#41—41no—
273#41—10041no—
274#41—5022no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—2511no—
279C2×Q8C22×D25111yescharacteristic
280C2×Q8C22×D25111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—10041no—
284C2×D4—5022no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—5021no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4C22⋊D25111yescharacteristic
291C2×D4C22×D25111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4C22×D25111yescharacteristic
297C22×C4C22×D25111yescharacteristic
298Q16—10041no—
299Q16—5022no—
300SD16—5022no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—2514no—
305D8—10041no—
306D8—5022no—
307C2×C8—5022no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8—2511no—
312C2×C8—2511no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—5021no—
316C4⋊C4—2511no—
317C4⋊C4C22⋊D25111yescharacteristic
318C4⋊C4C22×D25111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4C22⋊D25111yescharacteristic
329C22⋊C4C22⋊D25111yescharacteristic
330C42—21no—
331C42C22×D25111yescharacteristic
332C23—21no—
333C23—20081no—
334C23—10042no—
335C23—10042no—
336C23—5021no—
337C23—5021no—
338C23C2×C22⋊D25111yescharacteristic
339C23C2×C22⋊D25111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—5021no—
345Q8—5021no—
346Q8D4×D25112yes—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—20081no—
353D4—10042no—
354D4—10042no—
355D4—5021no—
356D4—5021no—
357D4—5021no—
358D4—2512no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—20081no—
369C2×C4—10042no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4C2×C22⋊D25111yescharacteristic
373C2×C4C2×C22⋊D25111yescharacteristic
374C2×C4C2×C22⋊D25111yescharacteristic
375C2×C4C2×C22⋊D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—5021no—
378C8—5021no—
379C8—5021no—
380C8—2512no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—20084no—
390C22—10041no—
391C22—10041no—
392C22—10041no—
393C22—10041no—
394C22—10041no—
395C22—5022no—
396C22#2254873512yescomplemented
397C22C22×C22⋊D25111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—10041no—
406C4—10041no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876011yescharacteristic
409C2—22no—
410C2—41no—
411C2—81no—
412C2—20081no—
413C2—10042no—
414C2D4×C22⋊D25111yescharacteristic
415C2C2×#2254873511yescharacteristic
416C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#548×C251C2
socleC22×C251C22×D4
hypercenter#548D251
hypocenterC251#3282
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22548735C2semidirect
2. C22×#16860797C2semidirect
3. #30632333C2semidirect
4. #30632069C2semidirect
5. #30631918C2semidirect
6. #30631914C2semidirect
7. #30631909C2semidirect
8. #30631899C2semidirect
9. #30631847C2semidirect
10. #30631843C2semidirect
11. #30631832C2semidirect
12. #30631820C2semidirect
13. C4×#16860797C2semidirect
14. D4×C251⋊C8C2semidirect
15. #548×C251C2semidirect
19. C2×#16860800C22semidirect
20. C2×#16860800C22semidirect
21. C2×#16860798C22semidirect
30. C22×C251⋊C8C22semidirect
39. #22548646C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
43. #22548638C22semidirect
44. #22548636C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. C4×C251⋊C8C22semidirect
50. C2×#41×C251C22semidirect
53. #124×C251C22semidirect
55. C22⋊Q8×C251C22semidirect
58. C4×Q8×C251C22semidirect
80. #16860797D4semidirect
180. C22×C251#135semidirect
208. C251#3282semidirect
212. #548D251semidirect
226. C2×#41C2×D251semidirect
244. #124C2×D251semidirect
248. C22⋊Q8C2×D251semidirect
254. C4×Q8C2×D251semidirect
396. C22#22548735semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
C22×#168607972
#306323332
#306320692
#306319182
#306319142
#306319092
#306318992
#306318472
#306318432
#306318322
#306318202
C4×#168607972
D4×C251⋊C82
#548×C2512
#3282251