Finite Group Databasebeta

← #44561895

Subgroups — #44561895

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×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548737C211yesmaximal, characteristic, complemented
3#30632342C211yesmaximal, characteristic, complemented
4#30632333C211yesmaximal, characteristic, complemented
5#30632081C211yesmaximal, characteristic, complemented
6#30631945C211yesmaximal, characteristic, complemented
7#30631916C211yesmaximal, characteristic, complemented
8#30631903C211yesmaximal, characteristic, complemented
9#30631893C211yesmaximal, characteristic, complemented
10#30631851C211yesmaximal, characteristic, complemented
11#30631851C211yesmaximal, characteristic, complemented
12#30631840C211yesmaximal, characteristic, complemented
13#30631834C211yesmaximal, characteristic, complemented
14#30631800C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic
18C2×#16860810C2211yescharacteristic, complemented
19C22×D1004C2211yescharacteristic
20#22548743—24no—
21#22548737—24no—
22C2×Dic1004C2211yescharacteristic, complemented
23C2×Dic1004C2211yescharacteristic, complemented
24C2×#16860799C2211yescharacteristic
25C2×#16860799C2211yescharacteristic
26C2×#16860799C2211yescharacteristic
27C2×#16860799C2211yescharacteristic
28C2×#16860799C2211yescharacteristic, complemented
29C2×D2008C2211yescharacteristic
30C2×D2008C2211yescharacteristic
31#22548715C2211yescharacteristic, complemented
32#22548703C2211yescharacteristic
33#22548700—21no—
34#22548697—21no—
35#22548697—21no—
36#22548694C2211yescharacteristic
37#22548694C2211yescharacteristic
38#22548694C2211yescharacteristic, complemented
39#22548689C2211yescharacteristic
40#22548688—21no—
41#22548686—21no—
42#22548683—21no—
43C4×D1004C2211yescharacteristic
44C4×D1004C2211yescharacteristic, complemented
45C4⋊C4×D251—21no—
46C22⋊C4×D251—21no—
47#22548647C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548640C2211yescharacteristic, complemented
50#22548640C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic
52#22548637C2211yescharacteristic
53#22548637C2211yescharacteristic, complemented
54C2×M16×C251C2211yescharacteristic
55C22×C8×C251C2211yescharacteristic, complemented
56C4×D4×C251C2211yescharacteristic
57C4⋊C8×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C22⋊C8×C251C2211yescharacteristic, complemented
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860810—22no—
63#16860810—22no—
64#16860810—22no—
65#16860810—41no—
66#16860810—41no—
67#16860810—41no—
68C2×Dic502C2311yescharacteristic
69C2×Dic502C2311yescharacteristic
70C2×Dic502C2311yescharacteristic
71C2×D1004—22no—
72C2×D1004—21no—
73C2×D1004—41no—
74C2×D1004C2311yescharacteristic
75C2×D1004C2311yescharacteristic
76C2×D1004C2311yescharacteristic
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79C2×C22⋊D251—21no—
80C2×C22⋊D251—21no—
81C2×C4×D251—21no—
82C2×C4×D251—21no—
83C2×C4×D251—21no—
84C2×C4×D251—21no—
85C2×C4×D251—21no—
86Dic1004—22no—
87Dic1004—41no—
88#16860799—22no—
89#16860799—22no—
90#16860799—22no—
91#16860799—22no—
92#16860799—41no—
93D2008—22no—
94D2008D414yescomplemented
95C4⋊Dic251—21no—
96C4⋊Dic251—21no—
97C4⋊Dic251C2311yescharacteristic
98C4⋊Dic251C2311yescharacteristic
99Dic251⋊C4—21no—
100Dic251⋊C4—21no—
101Dic251⋊C4—21no—
102C22⋊Dic251—21no—
103C22⋊Dic251—21no—
104C22⋊Dic251—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109(C2×D251)⋊C4—21no—
110C2×D4×C251D411yescharacteristic
111C22×C4×C251C2311yescharacteristic
112C22×C4×C251C2311yescharacteristic
113M16×C251—22no—
114C2×C8×C251—22no—
115C2×C8×C251C2311yescharacteristic
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C2×C8×C251C2311yescharacteristic
119C4⋊C4×C251D411yescharacteristic
120C22⋊C4×C251D411yescharacteristic
121C22⋊C4×C251D411yescharacteristic
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—81no—
129Dic502—21no—
130Dic502—21no—
131Dic502—21no—
132Dic502—21no—
133Dic502—21no—
134Dic502—21no—
135D1004—21no—
136D1004—21no—
137D1004—21no—
138D1004—21no—
139D1004—42no—
140D1004C2×D412yes—
141D1004C2×D412yes—
142C22⋊D251—42no—
143C22⋊D251—42no—
144C22⋊D251—42no—
145C22⋊D251—81no—
146C2×Dic251—21no—
147C2×Dic251—21no—
148C2×Dic251—21no—
149C2×Dic251—21no—
150C2×Dic251—21no—
151C4×D251—42no—
152C4×D251—42no—
153C4×D251—42no—
154C4×D251—42no—
155C4×D251—81no—
156C23×C251C2×D411yescharacteristic
157C23×C251C2×D411yescharacteristic
158D4×C251—41no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251C2×D411yescharacteristic
164C2×C4×C251C2×D411yescharacteristic
165C2×C4×C251C2×D411yescharacteristic
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2411yescharacteristic
168C8×C251—21no—
169C8×C251—21no—
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—21no—
176C22×C251—21no—
177C22×C251—21no—
178C22×C251C22×D411yescharacteristic
179C2×D251—22no—
180C2×D251—22no—
181C2×D251—41no—
182C2×D251—41no—
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—21no—
192C4×C251—21no—
193C4×C251C22×D411yescharacteristic
194C4×C251H(5,𝔽2)11yescharacteristic
195Dic251—41no—
196Dic251—41no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—41no—
201C2×C251—41no—
202C2×C251D4211yescharacteristic
203C2×C251#57511yescharacteristic
204C2×C251#57711yescharacteristic
205D251—42no—
206D251—42no—
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#442—2511no—
224#435D25111yescharacteristic, complemented
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×D8—2511no—
240C2×M16C2×D25111yescharacteristic
241C22×C8C2×D25111yescharacteristic, complemented
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×D4—2511no—
256C4×D4C2×D25111yescharacteristic
257C2×C4⋊C4—5021no—
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8C2×D25111yescharacteristic, complemented
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8C2×D25111yescharacteristic
268C22⋊C8C2×D25111yescharacteristic, complemented
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—5022no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—5021no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—2511no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4D100411yescharacteristic
291C22×C4—5021no—
292C22×C4—5021no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4C22×D25111yescharacteristic
297C22×C4C22×D25111yescharacteristic
298Q16—10041no—
299Q16—5022no—
300SD16—10041no—
301SD16—5022no—
302SD16—5022no—
303SD16—5022no—
304SD16—5022no—
305D8—5022no—
306D8—2514no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D25111yescharacteristic
310C2×C8C22×D25111yescharacteristic
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⋊C4—2511no—
320C4⋊C4D100411yescharacteristic
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4D100411yescharacteristic
330C22⋊C4D100411yescharacteristic
331C42C22×D25111yescharacteristic
332C23—20081no—
333C23—10042no—
334C23—10042no—
335C23—5021no—
336C23—5021no—
337C23—5021no—
338C23C2×D100411yescharacteristic
339C23C2×D100411yescharacteristic
340Q8—5021no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8—5021no—
345Q8—5021no—
346D4—41no—
347D4—20081no—
348D4—10042no—
349D4—10042no—
350D4—10042no—
351D4—10042no—
352D4—5021no—
353D4—5021no—
354D4—5021no—
355D4—5021no—
356D4—2512no—
357D4—2512no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—20081no—
363C2×C4—10042no—
364C2×C4—10042no—
365C2×C4—10042no—
366C2×C4—10042no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4C2×D100411yescharacteristic
373C2×C4C2×D100411yescharacteristic
374C2×C4C2×D100411yescharacteristic
375C2×C4C2×D100411yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D25112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—20084no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—10041no—
392C22—10041no—
393C22—10041no—
394C22—10041no—
395C22—5022no—
396C22—5022no—
397C22C22×D100411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—10041no—
403C4—10041no—
404C4—10041no—
405C4—10041no—
406C4—10041no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876111yescharacteristic
409C2—41no—
410C2—41no—
411C2—20081no—
412C2—10042no—
413C2—10042no—
414C2D4×D100411yescharacteristic
415C2#3063248711yescharacteristic
416C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#435×C251C2
socleC22×C251C22×D4
hypercenter#435D251
hypocenterC251#3267
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×#22548737C2semidirect
3. #30632342C2semidirect
4. #30632333C2semidirect
5. #30632081C2semidirect
6. #30631945C2semidirect
7. #30631916C2semidirect
8. #30631903C2semidirect
9. #30631893C2semidirect
10. #30631851C2semidirect
11. #30631851C2semidirect
12. #30631840C2semidirect
13. #30631834C2semidirect
14. #30631800C2semidirect
15. #435×C251C2semidirect
18. C2×#16860810C22semidirect
22. C2×Dic1004C22semidirect
23. C2×Dic1004C22semidirect
28. C2×#16860799C22semidirect
31. #22548715C22semidirect
38. #22548694C22semidirect
44. C4×D1004C22semidirect
47. #22548647C22semidirect
48. #22548640C22semidirect
49. #22548640C22semidirect
50. #22548640C22semidirect
53. #22548637C22semidirect
55. C22×C8×C251C22semidirect
57. C4⋊C8×C251C22semidirect
59. C22⋊C8×C251C22semidirect
60. #97×C251C22semidirect
94. D2008D4semidirect
208. C251#3267semidirect
224. #435D251semidirect
241. C22×C8C2×D251semidirect
260. C4⋊C8C2×D251semidirect
268. C22⋊C8C2×D251semidirect
269. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487372
#306323422
#306323332
#306320812
#306319452
#306319162
#306319032
#306318932
#306318512
#306318402
#306318342
#306318002
#435×C2512
#3267251