Finite Group Databasebeta

← #44559854

Subgroups — #44559854

Order 32128.

500 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×#22548739C211yesmaximal, characteristic
3#124×D251C211yesmaximal, characteristic, complemented
4C4×#16860809C211yesmaximal, characteristic, complemented
5#30631912C211yesmaximal, characteristic, complemented
6#30631910C211yesmaximal, characteristic, complemented
7#30631908C211yesmaximal, characteristic, complemented
8#30631906C211yesmaximal, characteristic, complemented
9#30631904C211yesmaximal, characteristic, complemented
10#30631902C211yesmaximal, characteristic, complemented
11#30631900C211yesmaximal, characteristic, complemented
12#30631709C211yesmaximal, characteristic, complemented
13#30631694C211yesmaximal, characteristic, complemented
14#30631690C211yesmaximal, characteristic, complemented
15#463×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548744—24no—
20#22548739—24no—
21C2×Dic1004C2211yescharacteristic, complemented
22C2×#16860800C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic, complemented
26C2×#16860795C2211yescharacteristic
27C2×#16860793C2211yescharacteristic
28C2×#16860793C2211yescharacteristic
29#22548704C2211yescharacteristic
30#22548703C2211yescharacteristic
31#22548702C2211yescharacteristic
32#22548701—21no—
33#22548701—21no—
34C4×Dic502C2211yescharacteristic, complemented
35#22548680C2211yescharacteristic, complemented
36D4×Dic251C2211yescharacteristic, complemented
37C4×C22⋊D251—21no—
38(C22⋊D251)⋊C4—21no—
39#22548673C2211yescharacteristic
40#22548671—21no—
41C22⋊C4×D251—21no—
42C22⋊C4×D251—21no—
43C2×C4×Dic251—21no—
44C42×D251C2211yescharacteristic
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46Dic251⋊C8C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548639C2211yescharacteristic, complemented
50#22548638C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic
52#22548636C2211yescharacteristic
53#22548635C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic, complemented
55C2×D8×C251C2211yescharacteristic
56#124×C251C2211yescharacteristic
57C4×D4×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic
61C23×D251—21no—
62#16860809—22no—
63#16860809D414yes—
64C2×Dic502—21no—
65C2×Dic502C2311yescharacteristic
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—22no—
69C2×D1004C2311yescharacteristic
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72D4×D251—22no—
73D4×D251—22no—
74C22×Dic251—21no—
75C22×Dic251—21no—
76C2×C4×D251—21no—
77C2×C4×D251D411yescharacteristic
78C2×C4×D251D411yescharacteristic
79C2×C4×D251C2311yescharacteristic
80Dic1004—22no—
81#16860800—22no—
82#16860799—22no—
83#16860798—22no—
84#16860798—22no—
85#16860795—22no—
86#16860793—22no—
87#16860793—22no—
88C2×C251⋊C8C2311yescharacteristic
89C2×C251⋊C8C2311yescharacteristic
90C4⋊Dic251C2311yescharacteristic
91Dic251⋊C4—21no—
92Dic251⋊C4—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95C22⋊Dic251—21no—
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—
101C4×Dic251—21no—
102C4×Dic251—21no—
103C4×Dic251—21no—
104C4×Dic251D411yescharacteristic
105C4×Dic251D411yescharacteristic
106C4×Dic251C2311yescharacteristic
107C2×Q8×C251C2311yescharacteristic
108C2×D4×C251C2311yescharacteristic
109C2×D4×C251C2311yescharacteristic
110C22×C4×C251—21no—
111SD16×C251—22no—
112D8×C251—22no—
113C2×C8×C251C2311yescharacteristic
114C2×C8×C251C2311yescharacteristic
115C4⋊C4×C251C2311yescharacteristic
116C22⋊C4×C251—21no—
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C42×C251C2311yescharacteristic
120C22×D251—21no—
121C22×D251—42no—
122C22×D251—42no—
123C22×D251—42no—
124C22×D251C2×D411yescharacteristic
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502—42no—
129Dic502C2×D412yes—
130D1004—21no—
131D1004—21no—
132C22⋊D251—24no—
133C22⋊D251—42no—
134C2×Dic251—21no—
135C2×Dic251—24no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—42no—
139C2×Dic251C2×D411yescharacteristic
140C2×Dic251C2×D411yescharacteristic
141C2×Dic251C2×D411yescharacteristic
142C4×D251—41no—
143C4×D251—41no—
144C4×D251—42no—
145C4×D251C2×D412yes—
146C4×D251#4112yes—
147C251⋊C8—21no—
148C251⋊C8—21no—
149C23×C251—21no—
150C23×C251—21no—
151Q8×C251—21no—
152Q8×C251—21no—
153D4×C251—21no—
154D4×C251—21no—
155D4×C251—21no—
156D4×C251C2×D412yes—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—42no—
160C2×C4×C251C2×D411yescharacteristic
161C2×C4×C251C2×D411yescharacteristic
162C2×C4×C251C2411yescharacteristic
163C8×C251—21no—
164C8×C251—21no—
165C22×C251—22no—
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—21no—
180C4×C251—21no—
181C4×C251—41no—
182C4×C251—41no—
183C4×C251C22×D411yescharacteristic
184C4×C251C2×#4111yescharacteristic
185Dic251—22no—
186Dic251—21no—
187Dic251—21no—
188Dic251—41no—
189Dic251—41no—
190Dic251#12112yes—
191C2×C251—42no—
192C2×C251—81no—
193C2×C251C2×#12111yescharacteristic
194C2×C251#57511yescharacteristic
195C2×C251#57511yescharacteristic
196D251—22no—
197D251—81no—
198C251#304011yescharacteristic, complemented, Sylow
199#3040—2511nomaximal, Sylow
200C2×#137—2511no—
201C2×AGL(1,ℤ/8)—2511no—
202C2×#124—2511no—
203C4×#41—2511no—
204#464—2511no—
205#464—2511no—
206#464—2511no—
207#464—2511no—
208#463—2511no—
209#463—2511no—
210#463—2511no—
211#463D25111yescharacteristic, complemented
212#423—2511no—
213#419—2511no—
214#418—2511no—
215C2×#41—2511no—
216C22×Q8—2511no—
217C22×D4—2511no—
218#137—5024no—
219AGL(1,ℤ/8)—5024no—
220C2×Q16—2511no—
221C2×Q16—2511no—
222C2×SD16—2511no—
223C2×SD16—2511no—
224C2×SD16—2511no—
225C2×SD16C2×D25111yescharacteristic, complemented
226C2×D8—2511no—
227C2×D8C2×D25111yescharacteristic
228C2×M16—2511no—
229C2×M16—2511no—
230#124—5021no—
231#124—5021no—
232#124—2511no—
233#124—2511no—
234#124—2511no—
235#124C2×D25111yescharacteristic
236C4×Q8—2511no—
237C4×Q8—2511no—
238C4×D4—5021no—
239C4×D4—5021no—
240C4×D4—2511no—
241C4×D4C2×D25111yescharacteristic, complemented
242#117—5021no—
243#117—2511no—
244C2×C22⋊C4—5021no—
245C2×C22⋊C4—5021no—
246C2×C42—5021no—
247C2×C42—2511no—
248C4⋊C8—2511no—
249C4⋊C8—2511no—
250C4⋊C8—2511no—
251C4⋊C8C2×D25111yescharacteristic, complemented
252#103—2511no—
253#103—2511no—
254#103—2511no—
255#103C2×D25111yescharacteristic, complemented
256#102—2511no—
257#102—2511no—
258#102—2511no—
259#102C2×D25111yescharacteristic
260C24—5021no—
261#41—5022no—
262#41—2514no—
263C2×Q8—5021no—
264C2×Q8—5022no—
265C2×Q8—5022no—
266C2×Q8—2511no—
267C2×Q8—2511no—
268C2×Q8C22×D25111yescharacteristic
269C2×D4—5021no—
270C2×D4—5022no—
271C2×D4—5022no—
272C2×D4—5021no—
273C2×D4—2511no—
274C2×D4C22×D25111yescharacteristic
275C2×D4C22×D25111yescharacteristic
276C22×C4—21no—
277C22×C4—5021no—
278C22×C4—5021no—
279C22×C4—5021no—
280C22×C4—2511no—
281C22×C4—2511no—
282C22×C4—2511no—
283Q16—5022no—
284Q16—5022no—
285SD16—22no—
286SD16—5022no—
287SD16—5022no—
288SD16—5022no—
289D8—22no—
290D8—5022no—
291M16—5022no—
292M16—5022no—
293C2×C8—2511no—
294C2×C8—2511no—
295C2×C8C22×D25111yescharacteristic
296C2×C8C22×D25111yescharacteristic
297C4⋊C4—5021no—
298C4⋊C4—5021no—
299C4⋊C4—2511no—
300C4⋊C4C22×D25111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—5021no—
305C22⋊C4—5021no—
306C22⋊C4—5021no—
307C22⋊C4—5021no—
308C22⋊C4—5021no—
309C22⋊C4—5021no—
310C22⋊C4—5021no—
311C22⋊C4—5021no—
312C42—5021no—
313C42—5021no—
314C42—5021no—
315C42—2511no—
316C42—2511no—
317C42—2511no—
318C42C22×D25111yescharacteristic
319C23—21no—
320C23—21no—
321C23—10042no—
322C23—10042no—
323C23—10042no—
324C23—5021no—
325C23—2511no—
326Q8—21no—
327Q8—21no—
328Q8—10042no—
329Q8—5021no—
330Q8—5021no—
331Q8—5021no—
332Q8—2512no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—10042no—
337D4—5024no—
338D4—5021no—
339D4—5021no—
340D4D4×D25112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—10041no—
345C2×C4—10041no—
346C2×C4—10042no—
347C2×C4—10042no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5024no—
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—22no—
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—41no—
381C4—41no—
382C4—10041no—
383C4—10041no—
384C4—5021no—
385C4—5021no—
386C4—5022no—
387C4—2512no—
388C4C2×D4×D25111yescharacteristic
389C4#41×D25111yescharacteristic
390C2—42no—
391C2—81no—
392C2—20081no—
393C2—5022no—
394C2#121×D25111yescharacteristic
395C2#3063248311yescharacteristic
396C2#3063248411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#463×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548739C2
hypercenter#463D251
hypocenterC251#3040
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22548744C2semidirect
3. #124×D251C2semidirect
4. C4×#16860809C2semidirect
5. #30631912C2semidirect
6. #30631910C2semidirect
7. #30631908C2semidirect
8. #30631906C2semidirect
9. #30631904C2semidirect
10. #30631902C2semidirect
11. #30631900C2semidirect
12. #30631709C2semidirect
13. #30631694C2semidirect
14. #30631690C2semidirect
15. #463×C251C2semidirect
21. C2×Dic1004C22semidirect
22. C2×#16860800C22semidirect
25. C2×#16860798C22semidirect
34. C4×Dic502C22semidirect
35. #22548680C22semidirect
36. D4×Dic251C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. Dic251⋊C8C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548639C22semidirect
50. #22548638C22semidirect
54. C2×SD16×C251C22semidirect
57. C4×D4×C251C22semidirect
58. C4⋊C8×C251C22semidirect
59. #103×C251C22semidirect
198. C251#3040semidirect
211. #463D251semidirect
225. C2×SD16C2×D251semidirect
241. C4×D4C2×D251semidirect
251. C4⋊C8C2×D251semidirect
255. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487442
C2×#225487392
#124×D2512
C4×#168608092
#306319122
#306319102
#306319082
#306319062
#306319042
#306319022
#306319002
#306317092
#306316942
#306316902
#463×C2512
#3040251