Finite Group Databasebeta

← #44561860

Subgroups — #44561860

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×#22548742C211yesmaximal, characteristic, complemented
2#30632348C211yesmaximal, characteristic, complemented
3D4×C22⋊D251C211yesmaximal, characteristic, complemented
4#30632073C211yesmaximal, characteristic, complemented
5#30631930C211yesmaximal, characteristic, complemented
6#30631923C211yesmaximal, characteristic, complemented
7#30631904C211yesmaximal, characteristic, complemented
8#30631885C211yesmaximal, characteristic, complemented
9#30631836C211yesmaximal, characteristic, complemented
10#30631831C211yesmaximal, characteristic, complemented
11#30631826C211yesmaximal, characteristic, complemented
12#30631821C211yesmaximal, characteristic, complemented
13#30631788C211yesmaximal, characteristic, complemented
14#30631758C211yesmaximal, characteristic, complemented
15C22×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic, complemented
17C22×C22⋊D251—21no—
18C2×D4×D251C2211yescharacteristic
19#22548742—24no—
20C2×#16860797C2211yescharacteristic, complemented
21C2×#16860797C2211yescharacteristic, complemented
22C2×D2008C2211yescharacteristic, complemented
23C2×#16860795C2211yescharacteristic
24C2×#16860795C2211yescharacteristic, complemented
25C2×#16860793C2211yescharacteristic, complemented
26#22548711C2211yescharacteristic
27#22548704—21no—
28#22548704C2211yescharacteristic, complemented
29#22548691—21no—
30#22548691C2211yescharacteristic, complemented
31#22548689—21no—
32#22548689C2211yescharacteristic
33#22548689C2211yescharacteristic, complemented
34#22548687—21no—
35#22548686C2211yescharacteristic
36#22548684—21no—
37#22548683—21no—
38Q8×Dic251C2211yescharacteristic, complemented
39D4×Dic251C2211yescharacteristic
40C4×C22⋊D251—21no—
41C4×C22⋊D251C2211yescharacteristic
42#22548649C2211yescharacteristic, complemented
43Dic251⋊C8C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic, complemented
47#22548637C2211yescharacteristic, complemented
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic, complemented
50C22⋊(C251⋊C8)C2211yescharacteristic, complemented
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#22548625C2211yescharacteristic, complemented
53C22×Q8×C251C2211yescharacteristic, complemented
54C22×D4×C251C2211yescharacteristic
55C2×SD16×C251—24no—
56C2×SD16×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic
58C2×SD16×C251C2211yescharacteristic, complemented
59C2×SD16×C251C2211yescharacteristic, complemented
60C22×C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860811—22no—
63#16860811—41no—
64C2×D1004—21no—
65C2×D1004C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—44no—
74C2×C22⋊D251D411yescharacteristic
75D4×D251—22no—
76D4×D251—41no—
77C22×Dic251—21no—
78C2×C4×D251—21no—
79C2×C4×D251C2311yescharacteristic
80#16860797—22no—
81#16860797—41no—
82D2008—22no—
83#16860795—22no—
84#16860795—41no—
85#16860793—22no—
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C4⋊Dic251—21no—
89C4⋊Dic251C2311yescharacteristic
90C4⋊Dic251C2311yescharacteristic
91Dic251⋊C4—21no—
92Dic251⋊C4D411yescharacteristic
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95C22⋊Dic251D411yescharacteristic
96(C2×D251)⋊C4—21no—
97(C2×D251)⋊C4—21no—
98(C2×D251)⋊C4—21no—
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4D411yescharacteristic
102C4×Dic251—21no—
103C4×Dic251C2311yescharacteristic
104C24×C251—21no—
105C2×Q8×C251—21no—
106C2×Q8×C251—22no—
107C2×Q8×C251—41no—
108C2×Q8×C251C2311yescharacteristic
109C2×Q8×C251C2311yescharacteristic
110C2×D4×C251—22no—
111C2×D4×C251—21no—
112C2×D4×C251—41no—
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115C22×C4×C251—21no—
116C22×C4×C251C2311yescharacteristic
117SD16×C251—22no—
118SD16×C251—22no—
119SD16×C251—22no—
120SD16×C251D414yescomplemented
121C2×C8×C251—22no—
122C2×C8×C251C2311yescharacteristic
123C2×C8×C251C2311yescharacteristic
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—42no—
127C22×D251—42no—
128C22×D251—81no—
129C22×D251C2×D411yescharacteristic
130D1004—21no—
131D1004—21no—
132D1004—21no—
133D1004—21no—
134D1004—42no—
135C22⋊D251—22no—
136C22⋊D251—42no—
137C22⋊D251—42no—
138C22⋊D251—42no—
139C22⋊D251—42no—
140C22⋊D251—42no—
141C22⋊D251—81no—
142C2×Dic251—21no—
143C2×Dic251—21no—
144C2×Dic251—42no—
145C2×Dic251C2×D411yescharacteristic
146C2×Dic251C2×D411yescharacteristic
147C2×Dic251C2×D411yescharacteristic
148C4×D251—21no—
149C4×D251—21no—
150C4×D251—42no—
151C251⋊C8—21no—
152C251⋊C8—21no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251—42no—
156C23×C251—42no—
157C23×C251—81no—
158C23×C251C2×D411yescharacteristic
159Q8×C251—21no—
160Q8×C251—21no—
161Q8×C251—21no—
162Q8×C251—42no—
163Q8×C251C2×D412yescomplemented
164D4×C251—21no—
165D4×C251—21no—
166D4×C251—21no—
167D4×C251—42no—
168D4×C251C2×D412yes—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—42no—
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2411yescharacteristic
176C8×C251—21no—
177C8×C251C2×D412yescomplemented
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—22no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—84no—
187C22×C251AGL(1,ℤ/8)12yescomplemented
188C22×C251C22×D411yescharacteristic
189C2×D251—21no—
190C2×D251—21no—
191C2×D251—21no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—41no—
198C2×D251—84no—
199C4×C251—21no—
200C4×C251—22no—
201C4×C251—41no—
202C4×C251C22×D411yescharacteristic
203C4×C251H(5,𝔽2)11yescharacteristic
204Dic251—21no—
205Dic251—21no—
206Dic251—21no—
207Dic251—41no—
208Dic251—41no—
209C2×C251—22no—
210C2×C251—42no—
211C2×C251—81no—
212C2×C251D4211yescharacteristic
213C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C251#57711yescharacteristic
215D251—41no—
216D251—81no—
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16D25111yescharacteristic, complemented
222#548—2511no—
223D42—2511no—
224#496—2511no—
225#468—2511no—
226#466—2511no—
227#463—2511no—
228#460—2511no—
229#450—2511no—
230#449—2511no—
231#448—2511no—
232#447—2511no—
233#440—2511no—
234#435—2511no—
235C2×#41—2511no—
236C22×Q8C2×D25111yescharacteristic, complemented
237C22×D4—5021no—
238C22×D4—2511no—
239C22×D4C2×D25111yescharacteristic
240AGL(1,ℤ/8)—5024no—
241C2×SD16—24no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16C2×D25111yescharacteristic
245C2×SD16C2×D25111yescharacteristic
246C2×SD16C2×D25111yescharacteristic, complemented
247C2×SD16C2×D25111yescharacteristic, complemented
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16—2511no—
252C22×C8C2×D25111yescharacteristic, complemented
253#127—2511no—
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8—2511no—
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#102—2511no—
276#102—2511no—
277C22⋊C8—2511no—
278C22⋊C8—2511no—
279#97—2511no—
280C24—21no—
281C24—5021no—
282#41—10041no—
283#41—5022no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D25111yescharacteristic
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—21no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4—2511no—
311C22×C4C22×D25111yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—22no—
315SD16—10041no—
316SD16—5022no—
317SD16C22⋊D25114yescomplemented
318D8—10041no—
319D8—5022no—
320D8—5022no—
321M16—5022no—
322C2×C8—22no—
323C2×C8—2511no—
324C2×C8—2511no—
325C2×C8C22×D25111yescharacteristic
326C2×C8C22×D25111yescharacteristic
327C4⋊C4—5021no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4—2511no—
331C4⋊C4—2511no—
332C22⋊C4—5021no—
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—21no—
358Q8—42no—
359Q8C2×C22⋊D25112yescomplemented
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—
376D4C2×C22⋊D25112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
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—5021no—
395C8—5021no—
396C8C2×C22⋊D25112yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—84no—
406C22—20084no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—5021no—
414C22—5021no—
415C22—5021no—
416C22#2254874212yescomplemented
417C22C2×D4×D25111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—10041no—
422C4—10041no—
423C4—5021no—
424C4—5021no—
425C4—5021no—
426C4C22×C22⋊D25111yescharacteristic
427C4#2254876011yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—20081no—
432C2—20081no—
433C2—10041no—
434C2D4×C22⋊D25111yescharacteristic
435C2C2×#2254874211yescharacteristic
436C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC22×C251C22×D4
hypercenterC22×SD16D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. #30632348C2semidirect
3. D4×C22⋊D251C2semidirect
4. #30632073C2semidirect
5. #30631930C2semidirect
6. #30631923C2semidirect
7. #30631904C2semidirect
8. #30631885C2semidirect
9. #30631836C2semidirect
10. #30631831C2semidirect
11. #30631826C2semidirect
12. #30631821C2semidirect
13. #30631788C2semidirect
14. #30631758C2semidirect
15. C22×SD16×C251C2semidirect
16. C2×#16860811C22semidirect
20. C2×#16860797C22semidirect
21. C2×#16860797C22semidirect
22. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
25. C2×#16860793C22semidirect
28. #22548704C22semidirect
30. #22548691C22semidirect
33. #22548689C22semidirect
38. Q8×Dic251C22semidirect
42. #22548649C22semidirect
43. Dic251⋊C8C22semidirect
44. #22548639C22semidirect
45. #22548639C22semidirect
46. #22548637C22semidirect
47. #22548637C22semidirect
49. #22548636C22semidirect
50. C22⋊(C251⋊C8)C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. #22548625C22semidirect
53. C22×Q8×C251C22semidirect
58. C2×SD16×C251C22semidirect
59. C2×SD16×C251C22semidirect
60. C22×C8×C251C22semidirect
120. SD16×C251D4semidirect
163. Q8×C251C2×D4semidirect
177. C8×C251C2×D4semidirect
187. C22×C251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
221. C22×SD16D251semidirect
236. C22×Q8C2×D251semidirect
246. C2×SD16C2×D251semidirect
247. C2×SD16C2×D251semidirect
252. C22×C8C2×D251semidirect
317. SD16C22⋊D251semidirect
359. Q8C2×C22⋊D251semidirect
396. C8C2×C22⋊D251semidirect
416. C22#22548742semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
#306323482
D4×C22⋊D2512
#306320732
#306319302
#306319232
#306319042
#306318852
#306318362
#306318312
#306318262
#306318212
#306317882
#306317582
C22×SD16×C2512
#3265251