Finite Group Databasebeta

← #44562076

Subgroups — #44562076

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×#22548734C211yesmaximal, characteristic, complemented
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632343C211yesmaximal, characteristic, complemented
4#30632331C211yesmaximal, characteristic, complemented
5#30632070C211yesmaximal, characteristic, complemented
6#30631915C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631908C211yesmaximal, characteristic, complemented
9#30631900C211yesmaximal, characteristic, complemented
10#30631850C211yesmaximal, characteristic, complemented
11#30631845C211yesmaximal, characteristic, complemented
12#30631829C211yesmaximal, characteristic, complemented
13#30631778C211yesmaximal, characteristic, complemented
14(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860809C2211yescharacteristic, complemented
18C22×Dic502C2211yescharacteristic
19#22548734—24no—
20C2×Dic1004C2211yescharacteristic, complemented
21C2×Dic1004C2211yescharacteristic, complemented
22C2×#16860798—24no—
23C2×#16860798C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic
26C2×#16860798C2211yescharacteristic
27C2×#16860798C2211yescharacteristic, complemented
28C22×C251⋊C8C2211yescharacteristic
29C2×C8×D251C2211yescharacteristic, complemented
30#22548702C2211yescharacteristic
31#22548702C2211yescharacteristic, complemented
32#22548701—21no—
33#22548696—21no—
34#22548695C2211yescharacteristic, complemented
35#22548694C2211yescharacteristic
36#22548692—21no—
37#22548687—21no—
38#22548687—21no—
39#22548686C2211yescharacteristic
40#22548684—21no—
41#22548680C2211yescharacteristic, complemented
42D4×Dic251—21no—
43D4×Dic251C2211yescharacteristic
44C4×C22⋊D251C2211yescharacteristic
45C2×C22⋊Dic251—21no—
46#22548646C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548640C2211yescharacteristic, complemented
50#22548638C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53C8×Dic251C2211yescharacteristic, complemented
54C22×D4×C251C2211yescharacteristic
55C2×D8×C251C2211yescharacteristic
56C2×D8×C251C2211yescharacteristic
57#121×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#102×C251C2211yescharacteristic
60C22⋊C8×C251C2211yescharacteristic
61#16860809—22no—
62#16860809—22no—
63#16860809—41no—
64#16860809—41no—
65C2×Dic502—22no—
66C2×Dic502—21no—
67C2×Dic502—41no—
68C2×Dic502C2311yescharacteristic
69C2×Dic502C2311yescharacteristic
70C2×Dic502C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251D411yescharacteristic
75C22×Dic251—21no—
76C22×Dic251—21no—
77C22×Dic251—21no—
78C22×Dic251—21no—
79C2×C4×D251C2311yescharacteristic
80Dic1004—22no—
81Dic1004—41no—
82#16860798—22no—
83#16860798—22no—
84#16860798—22no—
85#16860798—22no—
86#16860798D414yescomplemented
87C2×C251⋊C8—22no—
88C2×C251⋊C8C2311yescharacteristic
89C2×C251⋊C8C2311yescharacteristic
90C8×D251—22no—
91C4⋊Dic251—21no—
92C4⋊Dic251C2311yescharacteristic
93Dic251⋊C4—21no—
94Dic251⋊C4—21no—
95Dic251⋊C4D411yescharacteristic
96C22⋊Dic251—21no—
97C22⋊Dic251—21no—
98C22⋊Dic251—21no—
99C22⋊Dic251—21no—
100C22⋊Dic251—21no—
101C22⋊Dic251—21no—
102C22⋊Dic251—21no—
103C22⋊Dic251D411yescharacteristic
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4D411yescharacteristic
106C4×Dic251—21no—
107C4×Dic251C2311yescharacteristic
108C24×C251—21no—
109C2×D4×C251—22no—
110C2×D4×C251—21no—
111C2×D4×C251—21no—
112C2×D4×C251—41no—
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115C2×D4×C251C2311yescharacteristic
116C22×C4×C251C2311yescharacteristic
117D8×C251—22no—
118D8×C251—41no—
119C2×C8×C251C2311yescharacteristic
120C2×C8×C251C2311yescharacteristic
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22×D251C2×D411yescharacteristic
124Dic502—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502—21no—
129Dic502—42no—
130Dic502C2×D412yes—
131C22⋊D251—22no—
132C22⋊D251—42no—
133C22⋊D251—42no—
134C22⋊D251—81no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—42no—
140C2×Dic251—42no—
141C2×Dic251—42no—
142C2×Dic251—81no—
143C2×Dic251C2×D411yescharacteristic
144C2×Dic251C2×D411yescharacteristic
145C2×Dic251C2×D411yescharacteristic
146C4×D251—21no—
147C4×D251—21no—
148C251⋊C8—21no—
149C251⋊C8C2×D412yes—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—42no—
154C23×C251—42no—
155C23×C251—81no—
156C23×C251C2×D411yescharacteristic
157D4×C251—21no—
158D4×C251—21no—
159D4×C251—21no—
160D4×C251—21no—
161D4×C251—21no—
162D4×C251—42no—
163D4×C251—42no—
164D4×C251C2×D412yes—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2411yescharacteristic
170C8×C251—21no—
171C8×C251—21no—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—22no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—84no—
184C22×C251#13512yescomplemented
185C22×C251C22×D411yescharacteristic
186C2×D251—21no—
187C2×D251—21no—
188C2×D251—21no—
189C4×C251—21no—
190C4×C251—41no—
191C4×C251C22×D411yescharacteristic
192C4×C251H(5,𝔽2)11yescharacteristic
193Dic251—21no—
194Dic251—22no—
195Dic251—21no—
196Dic251—21no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—22no—
201C2×C251—42no—
202C2×C251—81no—
203C2×C251—81no—
204C2×C251D4211yescharacteristic
205C2×C251C2×#13511yescharacteristic
206C2×C251#57711yescharacteristic
207D251—41no—
208C251#328211yescharacteristic, complemented, Sylow
209#3282—2511nomaximal, Sylow
210C2×#135—2511no—
211C22×SD16—2511no—
212#548—2511no—
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#447D25111yescharacteristic, complemented
223C4×SD16—2511no—
224C8×D4—2511no—
225C2×#41—2511no—
226C2×#41—2511no—
227C22×Q8—2511no—
228C22×D4C2×D25111yescharacteristic
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×D8C2×D25111yescharacteristic
239C2×D8C2×D25111yescharacteristic
240C22×C8—2511no—
241C22×C8—2511no—
242#124—5021no—
243#124—2511no—
244#124—2511no—
245#123—5021no—
246C22⋊Q8—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8—2511no—
249#121—5021no—
250#121—5021no—
251#121—2511no—
252#121C2×D25111yescharacteristic
253#120—5021no—
254C4×Q8—2511no—
255C4×D4—5021no—
256C4×D4—2511no—
257C4×D4—2511no—
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8—2511no—
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102C2×D25111yescharacteristic
266#102C2×D25111yescharacteristic
267C22⋊C8—2511no—
268C22⋊C8C2×D25111yescharacteristic
269C4×C8—2511no—
270C24—21no—
271#41—10041no—
272#41—10041no—
273#41—5022no—
274#41—5022no—
275C2×Q8—10041no—
276C2×Q8—5022no—
277C2×Q8—5021no—
278C2×Q8—2511no—
279C2×Q8—2511no—
280C2×Q8—2511no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—5021no—
288C2×D4—2511no—
289C2×D4C22×D25111yescharacteristic
290C2×D4C22×D25111yescharacteristic
291C2×D4C22×D25111yescharacteristic
292C22×C4—5021no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298Q16—10041no—
299Q16—5022no—
300SD16—5022no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—2514no—
305D8—22no—
306D8—41no—
307C2×C8—5022no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—5021no—
314C4⋊C4—5021no—
315C4⋊C4—5021no—
316C4⋊C4—2511no—
317C4⋊C4—2511no—
318C4⋊C4C22×D25111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—2511no—
329C22⋊C4—2511no—
330C42—5021no—
331C42—2511no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—2511no—
339C23D4×D25111yescharacteristic
340Q8—10042no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8—5021no—
345Q8—5021no—
346Q8—2512no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—20081no—
355D4—10042no—
356D4—10042no—
357D4—5022no—
358D4D4×D25112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—20081no—
363C2×C4—10042no—
364C2×C4—10042no—
365C2×C4—10042no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—2511no—
373C2×C4—2511no—
374C2×C4—2511no—
375C2×C4D4×D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—21no—
379C8—5021no—
380C8—2512no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—5021no—
394C22—5021no—
395C22—5021no—
396C22#2254873412yescomplemented
397C22C2×D4×D25111yescharacteristic
398C4—21no—
399C4—41no—
400C4—10041no—
401C4—10041no—
402C4—10041no—
403C4—5021no—
404C4—5021no—
405C4—5021no—
406C4—5022no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876011yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—10041no—
414C2#3063231911yescharacteristic
415C2C2×#2254873411yescharacteristic
416C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3282
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#22548734C2semidirect
2. C22×#16860798C2semidirect
3. #30632343C2semidirect
4. #30632331C2semidirect
5. #30632070C2semidirect
6. #30631915C2semidirect
7. #30631914C2semidirect
8. #30631908C2semidirect
9. #30631900C2semidirect
10. #30631850C2semidirect
11. #30631845C2semidirect
12. #30631829C2semidirect
13. #30631778C2semidirect
14. (C22⋊D251)⋊C8C2semidirect
15. #447×C251C2semidirect
17. C2×#16860809C22semidirect
20. C2×Dic1004C22semidirect
21. C2×Dic1004C22semidirect
27. C2×#16860798C22semidirect
29. C2×C8×D251C22semidirect
31. #22548702C22semidirect
34. #22548695C22semidirect
41. #22548680C22semidirect
46. #22548646C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548640C22semidirect
50. #22548638C22semidirect
51. #22548636C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. C8×Dic251C22semidirect
86. #16860798D4semidirect
184. C22×C251#135semidirect
208. C251#3282semidirect
222. #447D251semidirect
396. C22#22548734semidirect

Maximal subgroups

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

subgroupindex
C2×#225487342
C22×#168607982
#306323432
#306323312
#306320702
#306319152
#306319142
#306319082
#306319002
#306318502
#306318452
#306318292
#306317782
(C22⋊D251)⋊C82
#447×C2512
#3282251