Finite Group Databasebeta

← #44559881

Subgroups — #44559881

Order 32128.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548744C211yesmaximal, characteristic, complemented
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632250C211yesmaximal, characteristic
4#30632163C211yesmaximal, characteristic, complemented
5#30631910C211yesmaximal, characteristic, complemented
6#30631909C211yesmaximal, characteristic, complemented
7#30631908C211yesmaximal, characteristic, complemented
8#30631907C211yesmaximal, characteristic, complemented
9#30631886C211yesmaximal, characteristic, complemented
10#30631885C211yesmaximal, characteristic, complemented
11#30631884C211yesmaximal, characteristic, complemented
12#30631712C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic, complemented
14#30631680C211yesmaximal, characteristic, complemented
15#460×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548744—24no—
20C2×Dic1004C2211yescharacteristic, complemented
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860797C2211yescharacteristic
26SD16×D251—24no—
27C2×#16860793C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic
29#22548711C2211yescharacteristic
30#22548703C2211yescharacteristic
31#22548702C2211yescharacteristic
32#22548687—21no—
33#22548687—21no—
34#22548683—21no—
35#22548683—21no—
36Q8×Dic251C2211yescharacteristic, complemented
37C4×Dic502—21no—
38C4×Dic502C2211yescharacteristic, complemented
39#22548680—21no—
40#22548680C2211yescharacteristic, complemented
41#22548673—21no—
42#22548673C2211yescharacteristic
43#22548672—21no—
44#22548672C2211yescharacteristic
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46Dic251⋊C8C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548636C2211yescharacteristic
52#22548636C2211yescharacteristic, complemented
53#22548635C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C2×SD16×C251C2211yescharacteristic, complemented
56#127×C251C2211yescharacteristic
57C4×Q8×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860809—22no—
63#16860809—22no—
64C2×Dic502—21no—
65C2×Dic502C2311yescharacteristic
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251D414yescomplemented
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74D4×D251—22no—
75D4×D251—22no—
76C22×Dic251—21no—
77C2×C4×D251—21no—
78C2×C4×D251C2311yescharacteristic
79Dic1004—22no—
80#16860800—22no—
81#16860799—22no—
82#16860798—22no—
83#16860798—22no—
84#16860797—22no—
85#16860793—22no—
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C8×D251—22no—
89C4⋊Dic251—21no—
90C4⋊Dic251C2311yescharacteristic
91Dic251⋊C4—21no—
92Dic251⋊C4—21no—
93Dic251⋊C4D411yescharacteristic
94Dic251⋊C4D411yescharacteristic
95C22⋊Dic251—21no—
96C22⋊Dic251—21no—
97(C2×D251)⋊C4—21no—
98(C2×D251)⋊C4—21no—
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4D411yescharacteristic
101(C2×D251)⋊C4D411yescharacteristic
102C4×Dic251—21no—
103C4×Dic251—21no—
104C4×Dic251—21no—
105C4×Dic251—21no—
106C4×Dic251C2311yescharacteristic
107C2×Q8×C251C2311yescharacteristic
108C2×D4×C251—21no—
109C2×D4×C251—21no—
110C2×D4×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112SD16×C251—22no—
113SD16×C251—22no—
114C2×C8×C251C2311yescharacteristic
115C2×C8×C251C2311yescharacteristic
116C4⋊C4×C251—21no—
117C4⋊C4×C251C2311yescharacteristic
118C42×C251—21no—
119C42×C251C2311yescharacteristic
120C22×D251—21no—
121C22×D251—42no—
122C22×D251—81no—
123C22×D251—81no—
124C22×D251C2×D411yescharacteristic
125Dic502—21no—
126Dic502—24no—
127Dic502—21no—
128Dic502—21no—
129Dic502C2×D412yes—
130D1004—21no—
131D1004—21no—
132C22⋊D251—42no—
133C22⋊D251—42no—
134C22⋊D251—81no—
135C22⋊D251—81no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—42no—
140C2×Dic251C2×D411yescharacteristic
141C2×Dic251C2×D411yescharacteristic
142C2×Dic251C2×D411yescharacteristic
143C4×D251—24no—
144C4×D251C2×D412yes—
145C4×D251#4112yes—
146C251⋊C8—21no—
147C251⋊C8—21no—
148C23×C251—21no—
149C23×C251—21no—
150Q8×C251—21no—
151Q8×C251C2×D412yes—
152D4×C251—21no—
153D4×C251—21no—
154D4×C251—21no—
155D4×C251—21no—
156D4×C251—81no—
157D4×C251—81no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251C2×D411yescharacteristic
161C2×C4×C251C2×D411yescharacteristic
162C2×C4×C251C2411yescharacteristic
163C8×C251—21no—
164C8×C251—21no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251—41no—
171C22×C251C22×D411yescharacteristic
172C2×D251—21no—
173C2×D251—21no—
174C2×D251—41no—
175C2×D251—41no—
176C2×D251—41no—
177C2×D251—84no—
178C2×D251#12112yes—
179C4×C251—22no—
180C4×C251—21no—
181C4×C251—21no—
182C4×C251—41no—
183C4×C251C22×D411yescharacteristic
184C4×C251C2×#4111yescharacteristic
185Dic251—22no—
186Dic251—21no—
187Dic251—21no—
188Dic251—41no—
189Dic251—41no—
190Dic251#12112yes—
191C2×C251—81no—
192C2×C251—81no—
193C2×C251C2×#12111yescharacteristic
194C2×C251#57711yescharacteristic
195C2×C251#57711yescharacteristic
196D251—22no—
197D251—81no—
198C251#304311yescharacteristic, complemented, Sylow
199#3043—2511nomaximal, Sylow
200C2×#137—2511no—
201C22×SD16—2511no—
202#535—2511no—
203#519—2511no—
204#464—2511no—
205#464—2511no—
206#464—2511no—
207#464—2511no—
208#460—2511no—
209#460—2511no—
210#460—2511no—
211#460D25111yescharacteristic, complemented
212#424—2511no—
213#418—2511no—
214#416—2511no—
215C2×#41—2511no—
216C22×Q8—2511no—
217C22×D4—2511no—
218#137—5024no—
219C2×Q16—2511no—
220C2×Q16—2511no—
221C2×SD16—5024no—
222C2×SD16—2511no—
223C2×SD16—2511no—
224C2×SD16—2511no—
225C2×SD16—2511no—
226C2×SD16C2×D25111yescharacteristic
227C2×SD16C2×D25111yescharacteristic, complemented
228C2×M16—2511no—
229C22×C8—2511no—
230#127—2511no—
231#127C2×D25111yescharacteristic
232#124—2511no—
233#124—2511no—
234#121—5021no—
235#121—5021no—
236#120—5021no—
237#120—5021no—
238C4×Q8—5021no—
239C4×Q8—5021no—
240C4×Q8—2511no—
241C4×Q8—2511no—
242C4×Q8—2511no—
243C4×Q8C2×D25111yescharacteristic, complemented
244#117—5021no—
245#117—5021no—
246#117—2511no—
247#117—2511no—
248C4⋊C8—2511no—
249C4⋊C8—2511no—
250C4⋊C8—2511no—
251C4⋊C8C2×D25111yescharacteristic, complemented
252#103—2511no—
253#103—2511no—
254#102—2511no—
255#102—2511no—
256#102—2511no—
257#102—2511no—
258#102C2×D25111yescharacteristic
259#102C2×D25111yescharacteristic, complemented
260C24—5021no—
261#41—5022no—
262#41—5022no—
263C2×Q8—5021no—
264C2×Q8—5022no—
265C2×Q8—2514no—
266C2×Q8—2511no—
267C2×Q8—2511no—
268C2×Q8C22×D25111yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—5021no—
272C2×D4—5022no—
273C2×D4—5022no—
274C2×D4—5021no—
275C2×D4—5021no—
276C2×D4—5021no—
277C2×D4—2511no—
278C2×D4C22×D25111yescharacteristic
279C2×D4C22×D25111yescharacteristic
280C22×C4—5021no—
281C22×C4—5021no—
282C22×C4—2511no—
283Q16—5022no—
284Q16—5022no—
285SD16—22no—
286SD16—22no—
287SD16—5022no—
288SD16—5022no—
289SD16—5022no—
290SD16—5022no—
291M16—5022no—
292C2×C8—5022no—
293C2×C8—2511no—
294C2×C8—2511no—
295C2×C8C22×D25111yescharacteristic
296C2×C8C22×D25111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—5021no—
299C4⋊C4—5021no—
300C4⋊C4—5021no—
301C4⋊C4—2511no—
302C4⋊C4—2511no—
303C4⋊C4—2511no—
304C4⋊C4C22×D25111yescharacteristic
305C22⋊C4—5021no—
306C22⋊C4—5021no—
307C22⋊C4—5021no—
308C22⋊C4—5021no—
309C22⋊C4—5021no—
310C22⋊C4—2511no—
311C22⋊C4—2511no—
312C42—21no—
313C42—5021no—
314C42—5021no—
315C42—5021no—
316C42—5021no—
317C42—2511no—
318C42C22×D25111yescharacteristic
319C23—21no—
320C23—21no—
321C23—20081no—
322C23—20081no—
323C23—10042no—
324C23—5021no—
325C23—2511no—
326Q8—21no—
327Q8—5024no—
328Q8—5021no—
329Q8—5021no—
330Q8—5021no—
331Q8—2512no—
332Q8D4×D25112yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—20081no—
340D4—20081no—
341D4—10042no—
342D4—10042no—
343D4—5021no—
344D4—5021no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—10042no—
348C2×C4—5024no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—2511no—
353C2×C4—2511no—
354C2×C4—2511no—
355C2×C4—2512no—
356C2×C4—2512no—
357C2×C4D4×D25111yescharacteristic
358C2×C4D4×D25111yescharacteristic
359C2×C4C23×D25111yescharacteristic
360C8—21no—
361C8—21no—
362C8—5021no—
363C8—5021no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—20084no—
371C22—10041no—
372C22—10041no—
373C22—10041no—
374C22—5021no—
375C22—5021no—
376C22—2512no—
377C22C2×D4×D25111yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—10041no—
383C4—10041no—
384C4—5021no—
385C4—5021no—
386C4—5022no—
387C4—2512no—
388C4C2×D4×D25111yescharacteristic
389C4#41×D25111yescharacteristic
390C2—81no—
391C2—81no—
392C2—20081no—
393C2—5022no—
394C2#121×D25111yescharacteristic
395C2#3063249311yescharacteristic
396C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#460×C251C2
socleC22×C251C22×D4
prime radical#30632250C2
hypercenter#460D251
hypocenterC251#3043
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#22548744C2semidirect
2. C2×SD16×D251C2semidirect
4. #30632163C2semidirect
5. #30631910C2semidirect
6. #30631909C2semidirect
7. #30631908C2semidirect
8. #30631907C2semidirect
9. #30631886C2semidirect
10. #30631885C2semidirect
11. #30631884C2semidirect
12. #30631712C2semidirect
13. #30631690C2semidirect
14. #30631680C2semidirect
15. #460×C251C2semidirect
20. C2×Dic1004C22semidirect
21. C2×#16860800C22semidirect
24. C2×#16860798C22semidirect
36. Q8×Dic251C22semidirect
38. C4×Dic502C22semidirect
40. #22548680C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. Dic251⋊C8C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548638C22semidirect
52. #22548636C22semidirect
55. C2×SD16×C251C22semidirect
57. C4×Q8×C251C22semidirect
58. C4⋊C8×C251C22semidirect
60. #102×C251C22semidirect
68. Q8×D251D4semidirect
198. C251#3043semidirect
211. #460D251semidirect
227. C2×SD16C2×D251semidirect
243. C4×Q8C2×D251semidirect
251. C4⋊C8C2×D251semidirect
259. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487442
C2×SD16×D2512
#306322502
#306321632
#306319102
#306319092
#306319082
#306319072
#306318862
#306318852
#306318842
#306317122
#306316902
#306316802
#460×C2512
#3043251