Finite Group Databasebeta

← #44562267

Subgroups — #44562267

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632383C211yesmaximal, characteristic, complemented
2#30632041C211yesmaximal, characteristic, complemented
3#30632025C211yesmaximal, characteristic, complemented
4#30632003C211yesmaximal, characteristic, complemented
5#30631994C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631899C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631826C211yesmaximal, characteristic, complemented
11C4×#16860797C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14C2×#22548639C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic
21#22548710—21no—
22#22548707C2211yescharacteristic
23#22548697—21no—
24#22548689C2211yescharacteristic
25#22548689C2211yescharacteristic
26#22548687—21no—
27C4×D1004C2211yescharacteristic
28(C22⋊D251)⋊C4—21no—
29#22548674C2211yescharacteristic
30C2×C4⋊Dic251C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548639C2211yescharacteristic
36#22548639C2211yescharacteristic
37#22548639C2211yescharacteristic
38#22548639C2211yescharacteristic
39#22548639C2211yescharacteristic
40#22548639C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548636C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48C2×#41×C251C2211yescharacteristic
49C22×Q8×C251C2211yescharacteristic
50#124×C251—21no—
51#124×C251C2211yescharacteristic
52C22⋊Q8×C251—21no—
53C22⋊Q8×C251C2211yescharacteristic
54#121×C251—21no—
55#121×C251C2211yescharacteristic
56C4×Q8×C251C2211yescharacteristic
57C4×D4×C251—21no—
58C4×D4×C251C2211yescharacteristic
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C22×Dic251—21no—
63C2×C4×D251—21no—
64#16860797—22no—
65#16860797—41no—
66#16860795—41no—
67C2×C251⋊C8—41no—
68C2×C251⋊C8—41no—
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79C22⋊Dic251—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83C4×Dic251—21no—
84#41×C251—22no—
85#41×C251—41no—
86C2×Q8×C251—21no—
87C2×Q8×C251—22no—
88C2×Q8×C251—41no—
89C2×Q8×C251C2311yescharacteristic
90C2×Q8×C251C2311yescharacteristic
91C2×D4×C251—21no—
92C2×D4×C251—21no—
93C2×D4×C251D411yescharacteristic
94C2×D4×C251C2311yescharacteristic
95C22×C4×C251—21no—
96C22×C4×C251—21no—
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C4⋊C4×C251—21no—
100C4⋊C4×C251—21no—
101C4⋊C4×C251D411yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112D1004—21no—
113D1004—21no—
114C22⋊D251—42no—
115C22⋊D251—81no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—81no—
120C4×D251—81no—
121C251⋊C8—21no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C251⋊C8—21no—
125C23×C251—21no—
126C23×C251C2×D411yescharacteristic
127C23×C251C2×D411yescharacteristic
128Q8×C251—21no—
129Q8×C251—21no—
130Q8×C251—21no—
131Q8×C251—42no—
132Q8×C251#4112yes—
133D4×C251—22no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—42no—
137D4×C251—42no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—42no—
146C2×C4×C251—42no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2411yescharacteristic
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251#13512yes—
161C22×C251C22×D411yescharacteristic
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—22no—
168C4×C251—21no—
169C4×C251—21no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251C2×#4111yescharacteristic
173C4×C251H(5,𝔽2)11yescharacteristic
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—22no—
178C2×C251—41no—
179C2×C251—81no—
180C2×C251#54611yescharacteristic
181C2×C251C2×#13511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—81no—
184C251#330111yescharacteristic, complemented, Sylow
185#3301—2511nomaximal, Sylow
186#553—2511no—
187#548D25111yescharacteristic, complemented
188#488—2511no—
189#485—2511no—
190#481—2511no—
191#479—2511no—
192#466—2511no—
193#465—2511no—
194#463—2511no—
195#449—2511no—
196#448—2511no—
197C4×SD16—2511no—
198C8×D4—2511no—
199#416—2511no—
200C2×#103—2511no—
201C2×#41C2×D25111yescharacteristic
202C22×Q8C2×D25111yescharacteristic
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×D8—2511no—
206C22×C8—2511no—
207C22×C8—2511no—
208#126—5021no—
209#125—2511no—
210#124—21no—
211#124C2×D25111yescharacteristic
212#123—5021no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D25111yescharacteristic
215#121—21no—
216#121—5021no—
217#121—2511no—
218#121—2511no—
219#121C2×D25111yescharacteristic
220C4×Q8C2×D25111yescharacteristic
221C4×D4—21no—
222C4×D4—5021no—
223C4×D4—2511no—
224C4×D4C2×D25111yescharacteristic
225#117—2511no—
226C2×C4⋊C4—2511no—
227#107—2511no—
228#106—2511no—
229#106—2511no—
230C4⋊C8—2511no—
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#103—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243C4×C8—2511no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D25111yescharacteristic
250C2×Q8C22×D25111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4C22⋊D25111yescharacteristic
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4C22×D25111yescharacteristic
263C22×C4C22×D25111yescharacteristic
264SD16—10041no—
265SD16—5022no—
266D8—10041no—
267C2×C8—10041no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8—2511no—
272C2×C8—2511no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4C22⋊D25111yescharacteristic
282C4⋊C4C22×D25111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4C22⋊D25111yescharacteristic
292C22⋊C4C22⋊D25111yescharacteristic
293C42—21no—
294C42—5021no—
295C42C22×D25111yescharacteristic
296C23—21no—
297C23—5021no—
298C23C2×C22⋊D25111yescharacteristic
299C23C2×C22⋊D25111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686080912yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—20081no—
311D4—10042no—
312D4—5021no—
313D4—5021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—20081no—
324C2×C4—20081no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4C2×C22⋊D25111yescharacteristic
329C2×C4C2×C22⋊D25111yescharacteristic
330C2×C4C2×C22⋊D25111yescharacteristic
331C2×C4C2×C22⋊D25111yescharacteristic
332C2×C4C23×D25111yescharacteristic
333C8—5021no—
334C8—5021no—
335C8—5021no—
336C8—5021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—10041no—
346C22—10041no—
347C22—10041no—
348C22#2254873512yes—
349C22C22×C22⋊D25111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—10041no—
360C4C2×#1686080911yescharacteristic
361C4#2254876011yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—20081no—
366C2#3063233111yescharacteristic
367C2C2×#2254873511yescharacteristic
368C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#548×C251C2
socleC22×C251C22×D4
hypercenter#548D251
hypocenterC251#3301
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #30632383C2semidirect
2. #30632041C2semidirect
3. #30632025C2semidirect
4. #30632003C2semidirect
5. #30631994C2semidirect
6. #30631921C2semidirect
7. #30631918C2semidirect
8. #30631899C2semidirect
9. #30631832C2semidirect
10. #30631826C2semidirect
11. C4×#16860797C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631683C2semidirect
14. C2×#22548639C2semidirect
15. #548×C251C2semidirect
184. C251#3301semidirect
187. #548D251semidirect

Maximal subgroups

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

subgroupindex
#306323832
#306320412
#306320252
#306320032
#306319942
#306319212
#306319182
#306318992
#306318322
#306318262
C4×#168607972
D4×C251⋊C82
#306316832
C2×#225486392
#548×C2512
#3301251