Finite Group Databasebeta

← #44561876

Subgroups — #44561876

Order 32128.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548746C211yesmaximal, characteristic, complemented
2C22×#16860797C211yesmaximal, characteristic, complemented
3#30632333C211yesmaximal, characteristic
4#30632079C211yesmaximal, characteristic, complemented
5#30631935C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631909C211yesmaximal, characteristic, complemented
8#30631889C211yesmaximal, characteristic, complemented
9#30631847C211yesmaximal, characteristic, complemented
10#30631843C211yesmaximal, characteristic, complemented
11#30631837C211yesmaximal, characteristic, complemented
12#30631826C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631759C211yesmaximal, characteristic, complemented
15D4×Q8×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18#22548746—24no—
19C2×#16860800C2211yescharacteristic, complemented
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860797—24no—
23C2×#16860797C2211yescharacteristic
24C2×#16860797C2211yescharacteristic
25C2×#16860797C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27C2×#16860797C2211yescharacteristic
28C2×#16860797C2211yescharacteristic
29C2×#16860794C2211yescharacteristic, complemented
30C22×C251⋊C8C2211yescharacteristic
31#22548703C2211yescharacteristic
32#22548700—21no—
33#22548694C2211yescharacteristic
34#22548689C2211yescharacteristic
35#22548688—21no—
36#22548683—21no—
37C4×D1004C2211yescharacteristic
38C22⋊C4×D251—21no—
39#22548648C2211yescharacteristic, complemented
40C4⋊(C251⋊C8)C2211yescharacteristic, complemented
41#22548639C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47C22⋊(C251⋊C8)C2211yescharacteristic
48C22⋊(C251⋊C8)C2211yescharacteristic, complemented
49#22548626C2211yescharacteristic, complemented
50C22×Q8×C251C2211yescharacteristic
51C22×Q8×C251C2211yescharacteristic, complemented
52C4⋊Q8×C251—21no—
53C4⋊Q8×C251C2211yescharacteristic, complemented
54C22⋊Q8×C251—21no—
55C22⋊Q8×C251—21no—
56C22⋊Q8×C251C2211yescharacteristic
57C22⋊Q8×C251C2211yescharacteristic, complemented
58C4×Q8×C251C2211yescharacteristic, complemented
59C4×D4×C251—21no—
60C4×D4×C251C2211yescharacteristic
61C23×D251—21no—
62#16860810—22no—
63#16860810—41no—
64C2×Dic502C2311yescharacteristic
65C2×D1004—22no—
66C2×D1004—21no—
67C2×D1004—41no—
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74#16860800—22no—
75#16860800—22no—
76#16860800—41no—
77#16860797—22no—
78#16860797—22no—
79#16860797—22no—
80#16860797—22no—
81#16860797—41no—
82#16860797D414yescomplemented
83#16860794—22no—
84C2×C251⋊C8—22no—
85C2×C251⋊C8C2311yescharacteristic
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C2×C251⋊C8C2311yescharacteristic
89C4⋊Dic251C2311yescharacteristic
90Dic251⋊C4—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96C2×Q8×C251—21no—
97C2×Q8×C251—21no—
98C2×Q8×C251—22no—
99C2×Q8×C251—22no—
100C2×Q8×C251—41no—
101C2×Q8×C251—41no—
102C2×Q8×C251C2311yescharacteristic
103C2×Q8×C251C2311yescharacteristic
104C2×Q8×C251C2311yescharacteristic
105C2×D4×C251D411yescharacteristic
106C22×C4×C251—21no—
107C22×C4×C251—21no—
108C22×C4×C251C2311yescharacteristic
109C22×C4×C251C2311yescharacteristic
110C4⋊C4×C251—21no—
111C4⋊C4×C251—21no—
112C4⋊C4×C251—21no—
113C4⋊C4×C251—21no—
114C4⋊C4×C251—21no—
115C4⋊C4×C251D411yescharacteristic
116C4⋊C4×C251C2311yescharacteristic
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C22⋊C4×C251D411yescharacteristic
120C22⋊C4×C251D411yescharacteristic
121C42×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—81no—
128Dic502—21no—
129Dic502—21no—
130D1004—21no—
131D1004—21no—
132D1004—21no—
133D1004—42no—
134D1004C2×D412yes—
135C22⋊D251—42no—
136C22⋊D251—81no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C4×D251—42no—
140C4×D251—81no—
141C251⋊C8—21no—
142C251⋊C8—21no—
143C251⋊C8—21no—
144C251⋊C8C2×D412yes—
145C23×C251C2×D411yescharacteristic
146C23×C251C2×D411yescharacteristic
147Q8×C251—21no—
148Q8×C251—21no—
149Q8×C251—21no—
150Q8×C251—21no—
151Q8×C251—21no—
152Q8×C251—42no—
153Q8×C251—42no—
154Q8×C251C2×D412yes—
155D4×C251—22no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—42no—
165C2×C4×C251—42no—
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2×D411yescharacteristic
170C2×C4×C251C2411yescharacteristic
171C22×C251—21no—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—21no—
176C22×C251#13712yescomplemented
177C22×C251C22×D411yescharacteristic
178C2×D251—22no—
179C2×D251—41no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—41no—
184C2×D251—84no—
185C4×C251—21no—
186C4×C251—21no—
187C4×C251—22no—
188C4×C251—21no—
189C4×C251—21no—
190C4×C251—41no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251C22×D411yescharacteristic
194C4×C251H(5,𝔽2)11yescharacteristic
195Dic251—41no—
196Dic251—41no—
197C2×C251—22no—
198C2×C251—41no—
199C2×C251D4211yescharacteristic
200C2×C251C2×#13711yescharacteristic
201C2×C251#57711yescharacteristic
202D251—42no—
203D251—81no—
204C251#326611yescharacteristic, complemented, Sylow
205#3266—2511nomaximal, Sylow
206C2×#137—2511no—
207C22×SD16—2511no—
208D4×Q8D25111yescharacteristic, complemented
209#546—2511no—
210#497—2511no—
211#468—2511no—
212#467—2511no—
213#464—2511no—
214#461—2511no—
215#452—2511no—
216#451—2511no—
217#450—2511no—
218#448—2511no—
219#440—2511no—
220#435—2511no—
221C2×#41—2511no—
222C22×Q8C2×D25111yescharacteristic
223C22×Q8C2×D25111yescharacteristic, complemented
224C22×D4—2511no—
225#137—5024no—
226C2×Q16—2511no—
227C2×Q16—2511no—
228C2×Q16—2511no—
229C2×SD16—5024no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×SD16—2511no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×M16—2511no—
237C22×C8—2511no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D25111yescharacteristic, complemented
240#124—2511no—
241#123—5021no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—2511no—
245C22⋊Q8C2×D25111yescharacteristic
246C22⋊Q8C2×D25111yescharacteristic, complemented
247#121—5021no—
248#121—2511no—
249#120—5021no—
250C4×Q8C2×D25111yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—2511no—
253C4×D4C2×D25111yescharacteristic
254C2×C22⋊C4—5021no—
255#107—2511no—
256C4⋊C8—2511no—
257#103—2511no—
258#103—2511no—
259#103—2511no—
260#103—2511no—
261#102—2511no—
262#102—2511no—
263C22⋊C8—2511no—
264C22⋊C8—2511no—
265#97—2511no—
266C24—5021no—
267#41—10041no—
268#41—5022no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—2511no—
276C2×Q8C22×D25111yescharacteristic
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×D4—10041no—
280C2×D4—5022no—
281C2×D4—5021no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—2511no—
285C2×D4—2511no—
286C2×D4C22⋊D25111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4C22×D25111yescharacteristic
292C22×C4C22×D25111yescharacteristic
293Q16—10041no—
294Q16—5022no—
295Q16—5022no—
296SD16—10041no—
297SD16—5022no—
298SD16—5022no—
299SD16—5022no—
300SD16—5022no—
301SD16—2514no—
302M16—5022no—
303C2×C8—5022no—
304C2×C8—2511no—
305C2×C8—2511no—
306C2×C8—2511no—
307C2×C8—2511no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—5021no—
314C4⋊C4—2511no—
315C4⋊C4C22⋊D25111yescharacteristic
316C4⋊C4C22×D25111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4C22⋊D25111yescharacteristic
325C22⋊C4C22⋊D25111yescharacteristic
326C42—21no—
327C42C22×D25111yescharacteristic
328C23—20081no—
329C23—10042no—
330C23—10042no—
331C23—5021no—
332C23—5021no—
333C23C2×C22⋊D25111yescharacteristic
334C23C2×C22⋊D25111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—5021no—
343Q8—5021no—
344Q8D4×D25112yes—
345D4—22no—
346D4—20081no—
347D4—10042no—
348D4—10042no—
349D4—5021no—
350D4—5021no—
351D4—5021no—
352D4—2512no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—20081no—
364C2×C4—10042no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4C2×C22⋊D25111yescharacteristic
368C2×C4C2×C22⋊D25111yescharacteristic
369C2×C4C2×C22⋊D25111yescharacteristic
370C2×C4C2×C22⋊D25111yescharacteristic
371C2×C4C23×D25111yescharacteristic
372C8—5021no—
373C8—5021no—
374C8—5021no—
375C8—2512no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—20084no—
382C22—10041no—
383C22—10041no—
384C22—10041no—
385C22—10041no—
386C22—10041no—
387C22—5022no—
388C22#2254874612yescomplemented
389C22C22×C22⋊D25111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—10041no—
399C4—10041no—
400C4C2×D4×D25111yescharacteristic
401C4#2254876011yescharacteristic
402C2—22no—
403C2—41no—
404C2—20081no—
405C2—10042no—
406C2D4×C22⋊D25111yescharacteristic
407C2C2×#2254874611yescharacteristic
408C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingD4×Q8×C251C2
socleC22×C251C22×D4
prime radical#30632333C2
hypercenterD4×Q8D251
hypocenterC251#3266
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C22×#16860797C2semidirect
4. #30632079C2semidirect
5. #30631935C2semidirect
6. #30631926C2semidirect
7. #30631909C2semidirect
8. #30631889C2semidirect
9. #30631847C2semidirect
10. #30631843C2semidirect
11. #30631837C2semidirect
12. #30631826C2semidirect
13. #30631785C2semidirect
14. #30631759C2semidirect
15. D4×Q8×C251C2semidirect
19. C2×#16860800C22semidirect
20. C2×#16860800C22semidirect
21. C2×#16860800C22semidirect
29. C2×#16860794C22semidirect
39. #22548648C22semidirect
40. C4⋊(C251⋊C8)C22semidirect
41. #22548639C22semidirect
42. #22548639C22semidirect
43. #22548639C22semidirect
44. #22548638C22semidirect
48. C22⋊(C251⋊C8)C22semidirect
49. #22548626C22semidirect
51. C22×Q8×C251C22semidirect
53. C4⋊Q8×C251C22semidirect
57. C22⋊Q8×C251C22semidirect
58. C4×Q8×C251C22semidirect
82. #16860797D4semidirect
176. C22×C251#137semidirect
204. C251#3266semidirect
208. D4×Q8D251semidirect
223. C22×Q8C2×D251semidirect
239. C4⋊Q8C2×D251semidirect
246. C22⋊Q8C2×D251semidirect
250. C4×Q8C2×D251semidirect
388. C22#22548746semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C22×#168607972
#306323332
#306320792
#306319352
#306319262
#306319092
#306318892
#306318472
#306318432
#306318372
#306318262
#306317852
#306317592
D4×Q8×C2512
#3266251