Finite Group Databasebeta

← #44561858

Subgroups — #44561858

Order 32128.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548741C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632347C211yesmaximal, characteristic, complemented
4D4×C22⋊D251C211yesmaximal, characteristic, complemented
5#30632073C211yesmaximal, characteristic, complemented
6#30631935C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631886C211yesmaximal, characteristic, complemented
10#30631834C211yesmaximal, characteristic, complemented
11#30631824C211yesmaximal, characteristic, complemented
12#30631822C211yesmaximal, characteristic, complemented
13#30631784C211yesmaximal, characteristic, complemented
14#30631761C211yesmaximal, characteristic, complemented
15#450×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic, complemented
18C22×C22⋊D251—21no—
19C2×D4×D251C2211yescharacteristic
20#22548741—24no—
21C2×#16860799C2211yescharacteristic, complemented
22C2×#16860798C2211yescharacteristic
23C2×#16860798C2211yescharacteristic, complemented
24C2×#16860797C2211yescharacteristic, complemented
25SD16×D251—24no—
26C2×D2008C2211yescharacteristic, complemented
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860794C2211yescharacteristic, complemented
30C2×C8×D251C2211yescharacteristic, complemented
31#22548711C2211yescharacteristic
32#22548704C2211yescharacteristic, complemented
33#22548701—21no—
34#22548693—21no—
35#22548690C2211yescharacteristic, complemented
36#22548689C2211yescharacteristic
37#22548688—21no—
38#22548687—21no—
39#22548686C2211yescharacteristic
40#22548684—21no—
41#22548683—21no—
42#22548680C2211yescharacteristic, complemented
43D4×Dic251C2211yescharacteristic
44#22548677—21no—
45C4×C22⋊D251C2211yescharacteristic
46#22548648C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548639C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic, complemented
51#22548635C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548625C2211yescharacteristic, complemented
54C22×D4×C251C2211yescharacteristic
55C2×SD16×C251C2211yescharacteristic
56C2×SD16×C251C2211yescharacteristic, complemented
57C22⋊Q8×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic
59#102×C251C2211yescharacteristic, complemented
60C22⋊C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860810—22no—
63#16860810—41no—
64C2×Dic502—21no—
65C2×Dic502C2311yescharacteristic
66Q8×D251—22no—
67Q8×D251—41no—
68C2×D1004—21no—
69C2×D1004C2311yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77C2×C22⋊D251—44no—
78C2×C22⋊D251D411yescharacteristic
79D4×D251—22no—
80D4×D251—41no—
81C22×Dic251—21no—
82C2×C4×D251—21no—
83C2×C4×D251—21no—
84C2×C4×D251C2311yescharacteristic
85#16860799—22no—
86#16860798—22no—
87#16860798D414yescomplemented
88#16860797—22no—
89D2008—41no—
90#16860795—22no—
91#16860795—22no—
92#16860794—22no—
93C2×C251⋊C8C2311yescharacteristic
94C2×C251⋊C8C2311yescharacteristic
95C8×D251—22no—
96C4⋊Dic251C2311yescharacteristic
97Dic251⋊C4—21no—
98Dic251⋊C4D411yescharacteristic
99C22⋊Dic251—21no—
100C22⋊Dic251—21no—
101C22⋊Dic251D411yescharacteristic
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4D411yescharacteristic
107C4×Dic251—21no—
108C4×Dic251C2311yescharacteristic
109C24×C251—21no—
110C2×Q8×C251C2311yescharacteristic
111C2×D4×C251—22no—
112C2×D4×C251—21no—
113C2×D4×C251—41no—
114C2×D4×C251C2311yescharacteristic
115C2×D4×C251C2311yescharacteristic
116C22×C4×C251C2311yescharacteristic
117SD16×C251—22no—
118SD16×C251—41no—
119C2×C8×C251C2311yescharacteristic
120C2×C8×C251C2311yescharacteristic
121C4⋊C4×C251—21no—
122C4⋊C4×C251C2311yescharacteristic
123C22⋊C4×C251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—81no—
129C22×D251C2×D411yescharacteristic
130Dic502—21no—
131Dic502—42no—
132Dic502C2×D412yescomplemented
133D1004—21no—
134D1004—21no—
135D1004—21no—
136D1004—21no—
137D1004—42no—
138C22⋊D251—22no—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—42no—
142C22⋊D251—42no—
143C22⋊D251—42no—
144C22⋊D251—81no—
145C2×Dic251—21no—
146C2×Dic251—21no—
147C2×Dic251—42no—
148C2×Dic251C2×D411yescharacteristic
149C2×Dic251C2×D411yescharacteristic
150C2×Dic251C2×D411yescharacteristic
151C4×D251—21no—
152C4×D251—21no—
153C4×D251—42no—
154C4×D251—42no—
155C251⋊C8—21no—
156C251⋊C8C2×D412yescomplemented
157C23×C251—21no—
158C23×C251—21no—
159C23×C251—42no—
160C23×C251—42no—
161C23×C251—81no—
162C23×C251C2×D411yescharacteristic
163Q8×C251—21no—
164Q8×C251—21no—
165D4×C251—21no—
166D4×C251—21no—
167D4×C251—21no—
168D4×C251—42no—
169D4×C251C2×D412yes—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251—21no—
178C22×C251—21no—
179C22×C251—22no—
180C22×C251—21no—
181C22×C251—21no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—84no—
188C22×C251C22×D411yescharacteristic
189C2×D251—21no—
190C2×D251—21no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—84no—
198C2×D251AGL(1,ℤ/8)12yescomplemented
199C4×C251—21no—
200C4×C251—41no—
201C4×C251—41no—
202C4×C251C22×D411yescharacteristic
203C4×C251H(5,𝔽2)11yescharacteristic
204Dic251—21no—
205Dic251—22no—
206Dic251—21no—
207Dic251—21no—
208Dic251—41no—
209C2×C251—42no—
210C2×C251—41no—
211C2×C251—81no—
212C2×C251D4211yescharacteristic
213C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C251#57711yescharacteristic
215D251—22no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42—2511no—
224#496—2511no—
225#468—2511no—
226#466—2511no—
227#463—2511no—
228#460—2511no—
229#450D25111yescharacteristic, complemented
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435—2511no—
235C2×#41—2511no—
236C22×Q8—2511no—
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4C2×D25111yescharacteristic
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16C2×D25111yescharacteristic
247C2×SD16C2×D25111yescharacteristic, complemented
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16—2511no—
252C22×C8—2511no—
253#127—2511no—
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8C2×D25111yescharacteristic, complemented
258#121—5021no—
259#121—5021no—
260#121—2511no—
261#121—2511no—
262#121—2511no—
263#120—5021no—
264#120—5021no—
265C4×Q8—2511no—
266C4×D4—5021no—
267C4×D4—2511no—
268C4×D4—2511no—
269#107—2511no—
270C4⋊C8—2511no—
271#103—2511no—
272#103—2511no—
273#102—2511no—
274#102—2511no—
275#102C2×D25111yescharacteristic
276#102C2×D25111yescharacteristic, complemented
277C22⋊C8—2511no—
278C22⋊C8C2×D25111yescharacteristic, complemented
279#97—2511no—
280C24—21no—
281C24—5021no—
282#41—10041no—
283#41—5022no—
284C2×Q8—10041no—
285C2×Q8—5021no—
286C2×Q8—5022no—
287C2×Q8—2511no—
288C2×Q8C22×D25111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—10044no—
293C2×D4—10041no—
294C2×D4—5021no—
295C2×D4—5021no—
296C2×D4—5021no—
297C2×D4—5021no—
298C2×D4—5022no—
299C2×D4—5021no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4—2511no—
304C2×D4—2511no—
305C2×D4C22×D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—5021no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4C22×D25111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—5022no—
315SD16—5022no—
316SD16—5022no—
317SD16—2514no—
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—5022no—
322C2×C8—5022no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4—2511no—
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—5021no—
334C22⋊C4—5021no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4—2511no—
340C22⋊C4—2511no—
341C42—5021no—
342C42—2511no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—20081no—
349C23—10042no—
350C23—10042no—
351C23—5021no—
352C23—5021no—
353C23—2511no—
354C23D4×D25111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—10042no—
358Q8—5021no—
359Q8—2512no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—20081no—
365D4—10042no—
366D4—10042no—
367D4—10042no—
368D4—10042no—
369D4—10042no—
370D4—10042no—
371D4—5022no—
372D4—5021no—
373D4—5021no—
374D4—5021no—
375D4—5021no—
376D4D4×D25112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—10042no—
382C2×C4—10042no—
383C2×C4—10042no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—2511no—
389C2×C4—2511no—
390C2×C4—2511no—
391C2×C4D4×D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—21no—
394C8—21no—
395C8—5021no—
396C8—2512no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—20084no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—10041no—
414C22—5021no—
415C22—5021no—
416C22—2512no—
417C22C2×D4×D25111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—10041no—
422C4—5021no—
423C4—5021no—
424C4—5022no—
425C4—5021no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876011yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—20081no—
432C2—20081no—
433C2—5022no—
434C2#3063231911yescharacteristic
435C2AGL(1,ℤ/8)×D25111yescharacteristic
436C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
hypercenter#450D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632347C2semidirect
4. D4×C22⋊D251C2semidirect
5. #30632073C2semidirect
6. #30631935C2semidirect
7. #30631922C2semidirect
8. #30631904C2semidirect
9. #30631886C2semidirect
10. #30631834C2semidirect
11. #30631824C2semidirect
12. #30631822C2semidirect
13. #30631784C2semidirect
14. #30631761C2semidirect
15. #450×C251C2semidirect
16. C2×#16860810C22semidirect
17. C2×Q8×D251C22semidirect
21. C2×#16860799C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860797C22semidirect
26. C2×D2008C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860794C22semidirect
30. C2×C8×D251C22semidirect
32. #22548704C22semidirect
35. #22548690C22semidirect
42. #22548680C22semidirect
46. #22548648C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548639C22semidirect
50. #22548637C22semidirect
51. #22548635C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548625C22semidirect
56. C2×SD16×C251C22semidirect
57. C22⋊Q8×C251C22semidirect
59. #102×C251C22semidirect
60. C22⋊C8×C251C22semidirect
87. #16860798D4semidirect
132. Dic502C2×D4semidirect
156. C251⋊C8C2×D4semidirect
198. C2×D251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
229. #450D251semidirect
247. C2×SD16C2×D251semidirect
257. C22⋊Q8C2×D251semidirect
276. #102C2×D251semidirect
278. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C2×SD16×D2512
#306323472
D4×C22⋊D2512
#306320732
#306319352
#306319222
#306319042
#306318862
#306318342
#306318242
#306318222
#306317842
#306317612
#450×C2512
#3265251