Finite Group Databasebeta

← #44561856

Subgroups — #44561856

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
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632346C211yesmaximal, characteristic, complemented
4#30632074C211yesmaximal, characteristic, complemented
5#30631931C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631884C211yesmaximal, characteristic, complemented
9#30631838C211yesmaximal, characteristic, complemented
10#30631830C211yesmaximal, characteristic, complemented
11#30631825C211yesmaximal, characteristic, complemented
12#30631821C211yesmaximal, characteristic, complemented
13#30631787C211yesmaximal, characteristic, complemented
14#30631759C211yesmaximal, characteristic, complemented
15D42×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C22×Dic502C2211yescharacteristic, complemented
18#22548741—24no—
19C2×#16860798—24no—
20C2×#16860798C2211yescharacteristic
21C2×#16860798C2211yescharacteristic
22C2×#16860798C2211yescharacteristic, complemented
23C2×#16860798C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860798C2211yescharacteristic, complemented
26C2×#16860795C2211yescharacteristic
27C2×#16860795C2211yescharacteristic, complemented
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860794C2211yescharacteristic, complemented
30C22×C251⋊C8C2211yescharacteristic, complemented
31#22548703C2211yescharacteristic, complemented
32#22548701—21no—
33#22548694C2211yescharacteristic, complemented
34#22548693—21no—
35#22548689C2211yescharacteristic, complemented
36#22548687—21no—
37C4×Dic502C2211yescharacteristic, complemented
38(C22⋊D251)⋊C4—21no—
39#22548648C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548638C2211yescharacteristic, complemented
42#22548638C2211yescharacteristic, complemented
43#22548636C2211yescharacteristic
44#22548636C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic, complemented
46#22548635C2211yescharacteristic, complemented
47C22⋊(C251⋊C8)C2211yescharacteristic, complemented
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49#22548626C2211yescharacteristic, complemented
50C22×D4×C251—21no—
51C22×D4×C251C2211yescharacteristic
52C22×D4×C251C2211yescharacteristic
53#127×C251C2211yescharacteristic
54#121×C251—21no—
55#121×C251C2211yescharacteristic
56#121×C251C2211yescharacteristic
57#120×C251—21no—
58#120×C251—21no—
59C4×D4×C251C2211yescharacteristic
60C4×D4×C251C2211yescharacteristic
61#16860810—22no—
62#16860810—41no—
63C2×Dic502—22no—
64C2×Dic502—21no—
65C2×Dic502—41no—
66C2×Dic502C2311yescharacteristic
67C2×Dic502C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73#16860798—22no—
74#16860798—22no—
75#16860798—22no—
76#16860798—22no—
77#16860798—41no—
78#16860798D414yescomplemented
79#16860795—22no—
80#16860795—22no—
81#16860795—41no—
82#16860794—22no—
83C2×C251⋊C8—22no—
84C2×C251⋊C8C2311yescharacteristic
85C2×C251⋊C8C2311yescharacteristic
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C4⋊Dic251C2311yescharacteristic
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95C4×Dic251—21no—
96C24×C251—21no—
97C24×C251—21no—
98C2×D4×C251—22no—
99C2×D4×C251—22no—
100C2×D4×C251—21no—
101C2×D4×C251—21no—
102C2×D4×C251—21no—
103C2×D4×C251—21no—
104C2×D4×C251—21no—
105C2×D4×C251—21no—
106C2×D4×C251—41no—
107C2×D4×C251—41no—
108C2×D4×C251—44no—
109C2×D4×C251D411yescharacteristic
110C2×D4×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112C2×D4×C251C2311yescharacteristic
113C22×C4×C251—21no—
114C22×C4×C251C2311yescharacteristic
115C22×C4×C251C2311yescharacteristic
116C4⋊C4×C251D411yescharacteristic
117C4⋊C4×C251C2311yescharacteristic
118C22⋊C4×C251—21no—
119C22⋊C4×C251—21no—
120C22⋊C4×C251—21no—
121C22⋊C4×C251D411yescharacteristic
122C22⋊C4×C251D411yescharacteristic
123C42×C251C2311yescharacteristic
124C22×D251—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502—42no—
129Dic502C2×D412yescomplemented
130D1004—21no—
131D1004—21no—
132C22⋊D251—42no—
133C22⋊D251—42no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—42no—
138C4×D251—42no—
139C251⋊C8—21no—
140C251⋊C8—21no—
141C251⋊C8—21no—
142C251⋊C8C2×D412yescomplemented
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251—42no—
147C23×C251—42no—
148C23×C251—42no—
149C23×C251—42no—
150C23×C251—81no—
151C23×C251—81no—
152C23×C251C2×D411yescharacteristic
153C23×C251C2×D411yescharacteristic
154D4×C251—22no—
155D4×C251—21no—
156D4×C251—21no—
157D4×C251—21no—
158D4×C251—21no—
159D4×C251—21no—
160D4×C251—42no—
161D4×C251—42no—
162D4×C251—42no—
163D4×C251—42no—
164D4×C251—42no—
165D4×C251—81no—
166D4×C251C2×D412yes—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—42no—
173C2×C4×C251C2×D411yescharacteristic
174C2×C4×C251C2×D411yescharacteristic
175C2×C4×C251C2×D411yescharacteristic
176C2×C4×C251C2×D411yescharacteristic
177C2×C4×C251C2411yescharacteristic
178C22×C251—21no—
179C22×C251—21no—
180C22×C251—21no—
181C22×C251—22no—
182C22×C251—21no—
183C22×C251—21no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251—41no—
189C22×C251—41no—
190C22×C251—41no—
191C22×C251—41no—
192C22×C251—84no—
193C22×C251—84no—
194C22×C251AGL(1,ℤ/8)12yescomplemented
195C22×C251C22×D411yescharacteristic
196C2×D251—41no—
197C2×D251—41no—
198C2×D251—41no—
199C4×C251—21no—
200C4×C251—21no—
201C4×C251—21no—
202C4×C251—21no—
203C4×C251—41no—
204C4×C251C22×D411yescharacteristic
205C4×C251H(5,𝔽2)11yescharacteristic
206Dic251—22no—
207Dic251—41no—
208Dic251—41no—
209C2×C251—22no—
210C2×C251—41no—
211C2×C251—42no—
212C2×C251—81no—
213C2×C251—81no—
214C2×C251D4211yescharacteristic
215C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C251#57711yescharacteristic
217D251—81no—
218C251#326511yescharacteristic, complemented, Sylow
219#3265—2511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2511no—
221C22×SD16—2511no—
222#548—2511no—
223D42D25111yescharacteristic, complemented
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×Q8—2511no—
237C22×D4—21no—
238C22×D4C2×D25111yescharacteristic
239C22×D4C2×D25111yescharacteristic
240AGL(1,ℤ/8)—5024no—
241C2×SD16—5024no—
242C2×SD16—2511no—
243C2×SD16—2511no—
244C2×SD16—2511no—
245C2×SD16—2511no—
246C2×SD16—2511no—
247C2×SD16—2511no—
248C2×D8—2511no—
249C2×D8—2511no—
250C2×D8—2511no—
251C2×M16—2511no—
252C22×C8—2511no—
253#127C2×D25111yescharacteristic
254#124—5021no—
255#124—2511no—
256C22⋊Q8—5021no—
257C22⋊Q8—2511no—
258#121—21no—
259#121—5021no—
260#121—2511no—
261#121C2×D25111yescharacteristic
262#121C2×D25111yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—2511no—
266C4×D4—5021no—
267C4×D4C2×D25111yescharacteristic
268C4×D4C2×D25111yescharacteristic
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—21no—
282#41—10041no—
283#41—5022no—
284C2×Q8—10041no—
285C2×Q8—5022no—
286C2×Q8—5021no—
287C2×Q8—2511no—
288C2×Q8—2511no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—5021no—
301C2×D4—5021no—
302C2×D4—2511no—
303C2×D4C22⋊D25111yescharacteristic
304C2×D4C22×D25111yescharacteristic
305C2×D4C22×D25111yescharacteristic
306C2×D4C22×D25111yescharacteristic
307C22×C4—21no—
308C22×C4—5021no—
309C22×C4—5021no—
310C22×C4C22×D25111yescharacteristic
311C22×C4C22×D25111yescharacteristic
312SD16—10041no—
313SD16—5022no—
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×C8—2511no—
326C2×C8—2511no—
327C4⋊C4—5021no—
328C4⋊C4—5021no—
329C4⋊C4—2511no—
330C4⋊C4C22⋊D25111yescharacteristic
331C4⋊C4C22×D25111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—5021no—
336C22⋊C4—5021no—
337C22⋊C4—5021no—
338C22⋊C4—5021no—
339C22⋊C4C22⋊D25111yescharacteristic
340C22⋊C4C22⋊D25111yescharacteristic
341C42—5021no—
342C42C22×D25111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—5021no—
353C23C2×C22⋊D25111yescharacteristic
354C23C2×C22⋊D25111yescharacteristic
355Q8—10042no—
356Q8—5021no—
357Q8—5021no—
358Q8—5021no—
359Q8—2512no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—10042no—
373D4—10042no—
374D4—5021no—
375D4—5021no—
376D4D4×D25112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—10042no—
384C2×C4—10042no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4C2×C22⋊D25111yescharacteristic
389C2×C4C2×C22⋊D25111yescharacteristic
390C2×C4C2×C22⋊D25111yescharacteristic
391C2×C4C2×C22⋊D25111yescharacteristic
392C2×C4C23×D25111yescharacteristic
393C8—5021no—
394C8—5021no—
395C8—5021no—
396C8—2512no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—10041no—
414C22—10041no—
415C22—10041no—
416C22#2254874112yescomplemented
417C22C22×C22⋊D25111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—10041no—
424C4—10041no—
425C4—5022no—
426C4C2×D4×D25111yescharacteristic
427C4#2254876011yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—20081no—
434C2D4×C22⋊D25111yescharacteristic
435C2C2×#2254874111yescharacteristic
436C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingD42×C251C2
socleC22×C251C22×D4
hypercenterD42D251
hypocenterC251#3265
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C22×#16860798C2semidirect
3. #30632346C2semidirect
4. #30632074C2semidirect
5. #30631931C2semidirect
6. #30631921C2semidirect
7. #30631902C2semidirect
8. #30631884C2semidirect
9. #30631838C2semidirect
10. #30631830C2semidirect
11. #30631825C2semidirect
12. #30631821C2semidirect
13. #30631787C2semidirect
14. #30631759C2semidirect
15. D42×C251C2semidirect
16. C2×#16860810C22semidirect
17. C22×Dic502C22semidirect
22. C2×#16860798C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860798C22semidirect
25. C2×#16860798C22semidirect
27. C2×#16860795C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860794C22semidirect
30. C22×C251⋊C8C22semidirect
31. #22548703C22semidirect
33. #22548694C22semidirect
35. #22548689C22semidirect
37. C4×Dic502C22semidirect
39. #22548648C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548638C22semidirect
42. #22548638C22semidirect
44. #22548636C22semidirect
45. #22548636C22semidirect
46. #22548635C22semidirect
47. C22⋊(C251⋊C8)C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. #22548626C22semidirect
78. #16860798D4semidirect
129. Dic502C2×D4semidirect
142. C251⋊C8C2×D4semidirect
194. C22×C251AGL(1,ℤ/8)semidirect
218. C251#3265semidirect
223. D42D251semidirect
416. C22#22548741semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C22×#168607982
#306323462
#306320742
#306319312
#306319212
#306319022
#306318842
#306318382
#306318302
#306318252
#306318212
#306317872
#306317592
D42×C2512
#3265251