Finite Group Databasebeta

← #44561892

Subgroups — #44561892

Order 32128.

500 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×#22548739C211yesmaximal, characteristic, complemented
2C2×#22548734C211yesmaximal, characteristic, complemented
3#30632341C211yesmaximal, characteristic, complemented
4#30632332C211yesmaximal, characteristic, complemented
5#30632081C211yesmaximal, characteristic, complemented
6#30631946C211yesmaximal, characteristic, complemented
7#30631917C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631892C211yesmaximal, characteristic, complemented
10#30631850C211yesmaximal, characteristic, complemented
11#30631850C211yesmaximal, characteristic, complemented
12#30631839C211yesmaximal, characteristic, complemented
13#30631833C211yesmaximal, characteristic, complemented
14#30631760C211yesmaximal, characteristic, complemented
15#442×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860809C2211yescharacteristic
18C2×#16860809C2211yescharacteristic, complemented
19C2×D4×D251C2211yescharacteristic
20#22548739—24no—
21#22548734—24no—
22C2×Dic1004C2211yescharacteristic, complemented
23C2×Dic1004C2211yescharacteristic, complemented
24C2×#16860799C2211yescharacteristic
25C2×#16860799C2211yescharacteristic
26C2×#16860798C2211yescharacteristic
27C2×#16860798C2211yescharacteristic
28C2×#16860798C2211yescharacteristic, complemented
29C2×#16860795C2211yescharacteristic
30C2×#16860793C2211yescharacteristic
31C2×C8×D251C2211yescharacteristic, complemented
32#22548715C2211yescharacteristic, complemented
33#22548703C2211yescharacteristic
34#22548699—21no—
35#22548699—21no—
36#22548699—21no—
37#22548695C2211yescharacteristic
38#22548695C2211yescharacteristic
39#22548695C2211yescharacteristic, complemented
40#22548690C2211yescharacteristic
41#22548689—21no—
42#22548689—21no—
43#22548685—21no—
44C4×D1004C2211yescharacteristic
45C2×C4⋊Dic251—21no—
46C2×(C2×D251)⋊C4—21no—
47C4⋊(C251⋊C8)C2211yescharacteristic, complemented
48#22548638C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548638C2211yescharacteristic, complemented
51#22548635C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53(C2×D251)⋊C8C2211yescharacteristic, complemented
54C2×D8×C251C2211yescharacteristic
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic, complemented
57#106×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic
59#102×C251C2211yescharacteristic, complemented
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860809—22no—
63#16860809—22no—
64#16860809—22no—
65#16860809—41no—
66#16860809—41no—
67#16860809—41no—
68C2×Dic502C2311yescharacteristic
69C2×Dic502C2311yescharacteristic
70C2×Dic502C2311yescharacteristic
71C2×D1004—21no—
72C2×D1004D411yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77C2×C22⋊D251—21no—
78D4×D251—22no—
79D4×D251—41no—
80C22×Dic251—21no—
81C22×Dic251—21no—
82C22×Dic251—21no—
83C2×C4×D251C2311yescharacteristic
84C2×C4×D251C2311yescharacteristic
85Dic1004—22no—
86Dic1004—41no—
87#16860799—22no—
88#16860799—41no—
89#16860798—22no—
90#16860798—22no—
91#16860798—22no—
92#16860795—22no—
93#16860793—22no—
94C2×C251⋊C8C2311yescharacteristic
95C2×C251⋊C8C2311yescharacteristic
96C8×D251—22no—
97C4⋊Dic251—21no—
98C4⋊Dic251—21no—
99C4⋊Dic251—21no—
100C4⋊Dic251—21no—
101C4⋊Dic251—21no—
102C4⋊Dic251D411yescharacteristic
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109(C2×D251)⋊C4D411yescharacteristic
110(C2×D251)⋊C4D411yescharacteristic
111C2×D4×C251C2311yescharacteristic
112C2×D4×C251C2311yescharacteristic
113C22×C4×C251—21no—
114C22×C4×C251—21no—
115D8×C251D414yescomplemented
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C4⋊C4×C251C2311yescharacteristic
119C4⋊C4×C251C2311yescharacteristic
120C22⋊C4×C251—21no—
121C22⋊C4×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—42no—
125C22×D251—42no—
126C22×D251—81no—
127C22×D251C2×D411yescharacteristic
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502—21no—
131Dic502—21no—
132Dic502—21no—
133Dic502—21no—
134Dic502—21no—
135D1004—21no—
136D1004—21no—
137D1004—41no—
138D1004—81no—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—42no—
142C22⋊D251—42no—
143C2×Dic251—21no—
144C2×Dic251—21no—
145C2×Dic251—21no—
146C2×Dic251—42no—
147C2×Dic251—42no—
148C2×Dic251—42no—
149C2×Dic251C2×D411yescharacteristic
150C2×Dic251C2×D411yescharacteristic
151C4×D251—21no—
152C4×D251—21no—
153C4×D251—21no—
154C4×D251—21no—
155C251⋊C8—21no—
156C251⋊C8—21no—
157C23×C251—21no—
158C23×C251—21no—
159D4×C251—21no—
160D4×C251—21no—
161D4×C251C2×D412yes—
162D4×C251C2×D412yes—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—42no—
166C2×C4×C251—81no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2411yescharacteristic
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—22no—
173C22×C251—22no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251C22×D411yescharacteristic
179C2×D251—21no—
180C2×D251—21no—
181C2×D251—21no—
182C2×D251—21no—
183C2×D251—21no—
184C2×D251—21no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—21no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251C22×D411yescharacteristic
194C4×C251H(5,𝔽2)11yescharacteristic
195Dic251—21no—
196Dic251—21no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—42no—
201C2×C251—42no—
202C2×C251D4211yescharacteristic
203C2×C251#57511yescharacteristic
204C2×C251#57711yescharacteristic
205D251—41no—
206D251—41no—
207D251—81no—
208C251#326711yescharacteristic, complemented, Sylow
209#3267—2511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2511no—
211C2×#135—2511no—
212#547—2511no—
213#546—2511no—
214#497—2511no—
215#470—2511no—
216#465—2511no—
217#463—2511no—
218#461—2511no—
219#452—2511no—
220#452—2511no—
221#450—2511no—
222#449—2511no—
223#442D25111yescharacteristic, complemented
224#435—2511no—
225C2×#41—2511no—
226C2×#41—2511no—
227C2×#41—2511no—
228C22×D4—2511no—
229AGL(1,ℤ/8)—5024no—
230#135—5024no—
231C2×Q16—2511no—
232C2×Q16—2511no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16—2511no—
237C2×SD16—2511no—
238C2×D8—2511no—
239C2×D8C2×D25111yescharacteristic
240C2×M16—2511no—
241C22×C8—2511no—
242C4⋊Q8—2511no—
243#124—2511no—
244#123—5021no—
245#123—5021no—
246#123—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8—2511no—
249C22⋊Q8—2511no—
250#121—5021no—
251#121—5021no—
252#121—2511no—
253#120—5021no—
254C4×D4—2511no—
255C4×D4C2×D25111yescharacteristic
256C4×D4C2×D25111yescharacteristic, complemented
257C2×C4⋊C4—5021no—
258C2×C22⋊C4—5021no—
259#106C2×D25111yescharacteristic, complemented
260C4⋊C8—2511no—
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102C2×D25111yescharacteristic
266#102C2×D25111yescharacteristic, complemented
267C22⋊C8—2511no—
268C22⋊C8—2511no—
269#97C2×D25111yescharacteristic, complemented
270C24—5021no—
271#41—10041no—
272#41—10041no—
273#41—10041no—
274#41—5022no—
275#41—5022no—
276#41—5022no—
277C2×Q8—2511no—
278C2×Q8—2511no—
279C2×Q8—2511no—
280C2×D4—10041no—
281C2×D4—5021no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—5021no—
285C2×D4—5022no—
286C2×D4—5021no—
287C2×D4—2511no—
288C2×D4—2511no—
289C2×D4C22×D25111yescharacteristic
290C2×D4C22×D25111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4—2511no—
298Q16—10041no—
299Q16—5022no—
300SD16—10041no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—5022no—
305D8—5022no—
306D8D100414yescomplemented
307M16—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—5021no—
317C4⋊C4—5021no—
318C4⋊C4—2511no—
319C4⋊C4C22×D25111yescharacteristic
320C4⋊C4C22×D25111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4—2511no—
330C22⋊C4—2511no—
331C42C22×D25111yescharacteristic
332C23—21no—
333C23—21no—
334C23—20081no—
335C23—10042no—
336C23—10042no—
337C23—5021no—
338C23—2511no—
339C23—2511no—
340Q8—5021no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8—5021no—
345Q8—5021no—
346D4—21no—
347D4—21no—
348D4—20081no—
349D4—10041no—
350D4—10042no—
351D4—10042no—
352D4—10042no—
353D4—10042no—
354D4—5021no—
355D4—5021no—
356D4C2×D100412yes—
357D4C2×D100412yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—10042no—
363C2×C4—10042no—
364C2×C4—10042no—
365C2×C4—5021no—
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×C4D4×D25111yescharacteristic
375C2×C4D4×D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—5021no—
379C8—5021no—
380C8C2×D100412yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—20084no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—5021no—
392C22—5021no—
393C22—5021no—
394C22—5021no—
395C22—5021no—
396C22—5021no—
397C22C2×D4×D25111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—10041no—
403C4—10041no—
404C4—10041no—
405C4—5021no—
406C4—5021no—
407C4C22×D100411yescharacteristic
408C4#2254876111yescharacteristic
409C2—42no—
410C2—42no—
411C2—20081no—
412C2—10041no—
413C2—10041no—
414C2D4×D100411yescharacteristic
415C2#3063248311yescharacteristic
416C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#442×C251C2
socleC22×C251C22×D4
hypercenter#442D251
hypocenterC251#3267
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×#22548734C2semidirect
3. #30632341C2semidirect
4. #30632332C2semidirect
5. #30632081C2semidirect
6. #30631946C2semidirect
7. #30631917C2semidirect
8. #30631902C2semidirect
9. #30631892C2semidirect
10. #30631850C2semidirect
11. #30631850C2semidirect
12. #30631839C2semidirect
13. #30631833C2semidirect
14. #30631760C2semidirect
15. #442×C251C2semidirect
18. C2×#16860809C22semidirect
22. C2×Dic1004C22semidirect
23. C2×Dic1004C22semidirect
28. C2×#16860798C22semidirect
31. C2×C8×D251C22semidirect
32. #22548715C22semidirect
39. #22548695C22semidirect
47. C4⋊(C251⋊C8)C22semidirect
48. #22548638C22semidirect
49. #22548638C22semidirect
50. #22548638C22semidirect
53. (C2×D251)⋊C8C22semidirect
56. C4×D4×C251C22semidirect
57. #106×C251C22semidirect
59. #102×C251C22semidirect
60. #97×C251C22semidirect
115. D8×C251D4semidirect
208. C251#3267semidirect
223. #442D251semidirect
256. C4×D4C2×D251semidirect
259. #106C2×D251semidirect
266. #102C2×D251semidirect
269. #97C2×D251semidirect
306. D8D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×#225487342
#306323412
#306323322
#306320812
#306319462
#306319172
#306319022
#306318922
#306318502
#306318392
#306318332
#306317602
#442×C2512
#3267251