Finite Group Databasebeta

← #44561873

Subgroups — #44561873

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×#22548745C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632350C211yesmaximal, characteristic, complemented
4#30632325C211yesmaximal, characteristic
5#30632080C211yesmaximal, characteristic, complemented
6#30631935C211yesmaximal, characteristic, complemented
7#30631927C211yesmaximal, characteristic, complemented
8#30631912C211yesmaximal, characteristic, complemented
9#30631890C211yesmaximal, characteristic, complemented
10#30631841C211yesmaximal, characteristic, complemented
11#30631839C211yesmaximal, characteristic, complemented
12#30631828C211yesmaximal, characteristic, complemented
13#30631784C211yesmaximal, characteristic, complemented
14#30631761C211yesmaximal, characteristic, complemented
15#452×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic, complemented
17C2×Q8×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548745—24no—
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860799C2211yescharacteristic
23C2×#16860799C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27SD16×D251—24no—
28C2×#16860794C2211yescharacteristic, complemented
29C2×C8×D251C2211yescharacteristic
30#22548714—21no—
31#22548713C2211yescharacteristic, complemented
32#22548704C2211yescharacteristic
33#22548698—21no—
34#22548695C2211yescharacteristic
35#22548693—21no—
36#22548692—21no—
37#22548691C2211yescharacteristic
38#22548689C2211yescharacteristic
39#22548687—21no—
40#22548683—21no—
41#22548680C2211yescharacteristic, complemented
42D4×Dic251C2211yescharacteristic
43C4×C22⋊D251C2211yescharacteristic
44(C22⋊D251)⋊C4—21no—
45C2×(C2×D251)⋊C4—21no—
46#22548648C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548640C2211yescharacteristic, complemented
50#22548639C2211yescharacteristic, complemented
51#22548635C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53#22548625C2211yescharacteristic, complemented
54C2×#41×C251C2211yescharacteristic
55C2×Q16×C251C2211yescharacteristic, complemented
56C2×SD16×C251C2211yescharacteristic
57C22⋊Q8×C251C2211yescharacteristic, complemented
58#103×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic
60C22⋊C8×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C2×Dic502—21no—
63C2×Dic502—22no—
64C2×Dic502—21no—
65C2×Dic502—41no—
66C2×Dic502C2311yescharacteristic
67C2×Dic502C2311yescharacteristic
68Q8×D251—22no—
69Q8×D251—41no—
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251D411yescharacteristic
74D4×D251—22no—
75D4×D251—41no—
76C22×Dic251—21no—
77C22×Dic251—21no—
78C2×C4×D251—21no—
79C2×C4×D251C2311yescharacteristic
80#16860800—22no—
81#16860800—22no—
82#16860799—22no—
83#16860799—41no—
84#16860798—22no—
85#16860798D414yescomplemented
86#16860797—22no—
87#16860794—22no—
88C2×C251⋊C8C2311yescharacteristic
89C2×C251⋊C8C2311yescharacteristic
90C8×D251—22no—
91C4⋊Dic251—21no—
92C4⋊Dic251C2311yescharacteristic
93Dic251⋊C4—21no—
94Dic251⋊C4—21no—
95Dic251⋊C4—21no—
96Dic251⋊C4—21no—
97Dic251⋊C4D411yescharacteristic
98C22⋊Dic251—21no—
99C22⋊Dic251D411yescharacteristic
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4D411yescharacteristic
106C4×Dic251—21no—
107C4×Dic251C2311yescharacteristic
108#41×C251—22no—
109#41×C251—41no—
110C2×Q8×C251C2311yescharacteristic
111C2×Q8×C251C2311yescharacteristic
112C2×D4×C251—21no—
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116Q16×C251—41no—
117SD16×C251—22no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251—21no—
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22×D251—21no—
124C22×D251—42no—
125C22×D251—42no—
126C22×D251—81no—
127C22×D251C2×D411yescharacteristic
128Dic502—21no—
129Dic502—21no—
130Dic502—21no—
131Dic502—42no—
132Dic502—42no—
133Dic502C2×D412yes—
134D1004—21no—
135D1004—21no—
136C22⋊D251—22no—
137C22⋊D251—42no—
138C22⋊D251—81no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251—42no—
143C2×Dic251—81no—
144C2×Dic251C2×D411yescharacteristic
145C2×Dic251C2×D411yescharacteristic
146C2×Dic251C2×D411yescharacteristic
147C4×D251—21no—
148C4×D251—21no—
149C4×D251—42no—
150C251⋊C8—21no—
151C251⋊C8C2×D412yes—
152C23×C251—21no—
153C23×C251C2×D411yescharacteristic
154Q8×C251—21no—
155Q8×C251—21no—
156Q8×C251—21no—
157Q8×C251—21no—
158D4×C251—21no—
159D4×C251—42no—
160D4×C251C2×D412yes—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—42no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2411yescharacteristic
169C8×C251—21no—
170C8×C251—21no—
171C22×C251—21no—
172C22×C251—22no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251C22×D411yescharacteristic
178C2×D251—21no—
179C2×D251—21no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—84no—
184C2×D251#13712yescomplemented
185C4×C251—21no—
186C4×C251—41no—
187C4×C251—41no—
188C4×C251—41no—
189C4×C251C22×D411yescharacteristic
190C4×C251H(5,𝔽2)11yescharacteristic
191Dic251—21no—
192Dic251—22no—
193Dic251—21no—
194Dic251—21no—
195Dic251—41no—
196Dic251—41no—
197C2×C251—42no—
198C2×C251—41no—
199C2×C251D4211yescharacteristic
200C2×C251C2×#13711yescharacteristic
201C2×C251#57711yescharacteristic
202D251—22no—
203D251—81no—
204C251#326611yescharacteristic, complemented, Sylow
205#3266—2511nomaximal, Sylow
206C2×#137—2511no—
207C22×SD16—2511no—
208D4×Q8—2511no—
209#546—2511no—
210#497—2511no—
211#468—2511no—
212#467—2511no—
213#464—2511no—
214#461—2511no—
215#452D25111yescharacteristic, complemented
216#451—2511no—
217#450—2511no—
218#448—2511no—
219#440—2511no—
220#435—2511no—
221C2×#41C2×D25111yescharacteristic
222C22×Q8—2511no—
223C22×Q8—2511no—
224C22×D4—2511no—
225#137—5024no—
226C2×Q16—2511no—
227C2×Q16—2511no—
228C2×Q16C2×D25111yescharacteristic, complemented
229C2×SD16—5024no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×SD16—2511no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16C2×D25111yescharacteristic
236C2×M16—2511no—
237C22×C8—2511no—
238C4⋊Q8—5021no—
239C4⋊Q8—2511no—
240#124—2511no—
241#123—5021no—
242C22⋊Q8—5021no—
243C22⋊Q8—5021no—
244C22⋊Q8—2511no—
245C22⋊Q8—2511no—
246C22⋊Q8C2×D25111yescharacteristic, complemented
247#121—5021no—
248#121—2511no—
249#120—5021no—
250C4×Q8—2511no—
251C4×D4—5021no—
252C4×D4—2511no—
253C4×D4—2511no—
254C2×C22⋊C4—5021no—
255#107—2511no—
256C4⋊C8—2511no—
257#103—2511no—
258#103—2511no—
259#103—2511no—
260#103C2×D25111yescharacteristic, complemented
261#102—2511no—
262#102C2×D25111yescharacteristic
263C22⋊C8—2511no—
264C22⋊C8C2×D25111yescharacteristic, complemented
265#97—2511no—
266C24—5021no—
267#41—22no—
268#41—41no—
269C2×Q8—10041no—
270C2×Q8—10041no—
271C2×Q8—5022no—
272C2×Q8—5021no—
273C2×Q8—5022no—
274C2×Q8—5021no—
275C2×Q8—2511no—
276C2×Q8—2511no—
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×D4—21no—
280C2×D4—10041no—
281C2×D4—5021no—
282C2×D4—5022no—
283C2×D4—5021no—
284C2×D4—2511no—
285C2×D4—2511no—
286C2×D4C22×D25111yescharacteristic
287C22×C4—21no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4—2511no—
292C22×C4C22×D25111yescharacteristic
293Q16—41no—
294Q16—5022no—
295Q16—5022no—
296SD16—22no—
297SD16—10041no—
298SD16—5022no—
299SD16—5022no—
300SD16—5022no—
301SD16—2514no—
302M16—5022no—
303C2×C8—5022no—
304C2×C8—2511no—
305C2×C8—2511no—
306C2×C8C22×D25111yescharacteristic
307C2×C8C22×D25111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—5021no—
310C4⋊C4—5021no—
311C4⋊C4—5021no—
312C4⋊C4—5021no—
313C4⋊C4—5021no—
314C4⋊C4—2511no—
315C4⋊C4—2511no—
316C4⋊C4C22×D25111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—5021no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—2511no—
325C22⋊C4—2511no—
326C42—5021no—
327C42—2511no—
328C23—21no—
329C23—20081no—
330C23—10042no—
331C23—10042no—
332C23—5021no—
333C23—2511no—
334C23D4×D25111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—10042no—
340Q8—10042no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8—2512no—
345D4—21no—
346D4—42no—
347D4—20081no—
348D4—10042no—
349D4—5022no—
350D4—5021no—
351D4—5021no—
352D4D4×D25112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—20081no—
360C2×C4—10042no—
361C2×C4—10042no—
362C2×C4—5021no—
363C2×C4—5021no—
364C2×C4—5021no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4—2511no—
368C2×C4—2511no—
369C2×C4—2511no—
370C2×C4D4×D25111yescharacteristic
371C2×C4C23×D25111yescharacteristic
372C8—21no—
373C8—21no—
374C8—5021no—
375C8—2512no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—20084no—
383C22—10041no—
384C22—10041no—
385C22—10041no—
386C22—5021no—
387C22—5021no—
388C22—2512no—
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—10041no—
395C4—10041no—
396C4—5021no—
397C4—5021no—
398C4—5022no—
399C4—5021no—
400C4C2×D4×D25111yescharacteristic
401C4#2254876011yescharacteristic
402C2—41no—
403C2—42no—
404C2—20081no—
405C2—5022no—
406C2#3063231911yescharacteristic
407C2#137×D25111yescharacteristic
408C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#452×C251C2
socleC22×C251C22×D4
prime radical#30632325C2
hypercenter#452D251
hypocenterC251#3266
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22548745C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632350C2semidirect
5. #30632080C2semidirect
6. #30631935C2semidirect
7. #30631927C2semidirect
8. #30631912C2semidirect
9. #30631890C2semidirect
10. #30631841C2semidirect
11. #30631839C2semidirect
12. #30631828C2semidirect
13. #30631784C2semidirect
14. #30631761C2semidirect
15. #452×C251C2semidirect
16. C22×Dic502C22semidirect
20. C2×#16860800C22semidirect
21. C2×#16860800C22semidirect
28. C2×#16860794C22semidirect
31. #22548713C22semidirect
41. #22548680C22semidirect
46. #22548648C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548640C22semidirect
50. #22548639C22semidirect
53. #22548625C22semidirect
55. C2×Q16×C251C22semidirect
57. C22⋊Q8×C251C22semidirect
58. #103×C251C22semidirect
60. C22⋊C8×C251C22semidirect
85. #16860798D4semidirect
184. C2×D251#137semidirect
204. C251#3266semidirect
215. #452D251semidirect
228. C2×Q16C2×D251semidirect
246. C22⋊Q8C2×D251semidirect
260. #103C2×D251semidirect
264. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C2×SD16×D2512
#306323502
#306323252
#306320802
#306319352
#306319272
#306319122
#306318902
#306318412
#306318392
#306318282
#306317842
#306317612
#452×C2512
#3266251