Finite Group Databasebeta

← #44562271

Subgroups — #44562271

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#30632389C211yesmaximal, characteristic, complemented
2#30632348C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632026C211yesmaximal, characteristic, complemented
5#30632006C211yesmaximal, characteristic, complemented
6#30631992C211yesmaximal, characteristic, complemented
7#30631921C211yesmaximal, characteristic, complemented
8#30631913C211yesmaximal, characteristic, complemented
9#30631904C211yesmaximal, characteristic, complemented
10#30631831C211yesmaximal, characteristic, complemented
11SD16×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14C2×#22548639C211yesmaximal, characteristic, complemented
15#448×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548708—21no—
21#22548706C2211yescharacteristic
22#22548704—21no—
23#22548704C2211yescharacteristic
24#22548699—21no—
25#22548691—21no—
26#22548691C2211yescharacteristic
27#22548689—21no—
28#22548689C2211yescharacteristic
29#22548687—21no—
30#22548686C2211yescharacteristic
31Q8×Dic251C2211yescharacteristic
32D4×Dic251—21no—
33D4×Dic251C2211yescharacteristic
34C4×C22⋊D251—21no—
35C4×C22⋊D251C2211yescharacteristic
36#22548673C2211yescharacteristic
37C2×C4⋊Dic251C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548647C2211yescharacteristic
40#22548647C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548639C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—41no—
61C2×D1004—21no—
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251D411yescharacteristic
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251C2311yescharacteristic
70#16860795—41no—
71C2×C251⋊C8—41no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C8×D251—41no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251—21no—
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251D411yescharacteristic
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C2×Q8×C251—21no—
97C2×Q8×C251—22no—
98C2×Q8×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×Q8×C251C2311yescharacteristic
101C2×D4×C251—21no—
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251C2311yescharacteristic
105SD16×C251—22no—
106SD16×C251—41no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251—21no—
112C22×D251C2×D411yescharacteristic
113D1004—21no—
114D1004—21no—
115D1004—42no—
116C22⋊D251—22no—
117C22⋊D251—42no—
118C22⋊D251—81no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—81no—
123C2×Dic251—81no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—21no—
128C4×D251—21no—
129C4×D251—42no—
130C251⋊C8—21no—
131C251⋊C8—21no—
132C23×C251—21no—
133C23×C251C2×D411yescharacteristic
134Q8×C251—21no—
135Q8×C251—21no—
136Q8×C251—21no—
137Q8×C251—42no—
138Q8×C251#4112yes—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—42no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—42no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251#13512yes—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C4×C251—21no—
166C4×C251—22no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251H(5,𝔽2)11yescharacteristic
171Dic251—21no—
172Dic251—21no—
173Dic251—21no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—22no—
178C2×C251—81no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—41no—
183D251—81no—
184C251#330111yescharacteristic, complemented, Sylow
185#3301—2511nomaximal, Sylow
186#553—2511no—
187#548—2511no—
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#448D25111yescharacteristic, complemented
197C4×SD16—2511no—
198C8×D4—2511no—
199#416—2511no—
200C2×#103—2511no—
201C2×#41—2511no—
202C22×Q8C2×D25111yescharacteristic
203C2×SD16C2×D25111yescharacteristic
204C2×SD16C2×D25111yescharacteristic
205C2×D8—2511no—
206C22×C8—2511no—
207C22×C8—2511no—
208#126—5021no—
209#125—2511no—
210#124—5021no—
211#124—2511no—
212#123—5021no—
213C22⋊Q8—5021no—
214C22⋊Q8—2511no—
215#121—5021no—
216#121—5021no—
217#121—2511no—
218#121—2511no—
219#121C2×D25111yescharacteristic
220C4×Q8—2511no—
221C4×D4—5021no—
222C4×D4—5021no—
223C4×D4—2511no—
224C4×D4—2511no—
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#103C2×D25111yescharacteristic
236#103C2×D25111yescharacteristic
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241C22⋊C8—2511no—
242C22⋊C8C2×D25111yescharacteristic
243C4×C8—2511no—
244#41—10041no—
245#41—5022no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D25111yescharacteristic
250C2×Q8C22×D25111yescharacteristic
251C2×D4—21no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—2511no—
263C22×C4C22×D25111yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—10041no—
267C2×C8—10041no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8C22×D25111yescharacteristic
272C2×C8C22×D25111yescharacteristic
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4—2511no—
282C4⋊C4C22×D25111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—2511no—
292C22⋊C4—2511no—
293C42—5021no—
294C42—5021no—
295C42—2511no—
296C23—21no—
297C23—5021no—
298C23—2511no—
299C23D4×D25111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686080912yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—20081no—
309D4—10042no—
310D4—10042no—
311D4—5022no—
312D4—5021no—
313D4—5021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—20081no—
321C2×C4—20081no—
322C2×C4—10042no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—2511no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4D4×D25111yescharacteristic
332C2×C4C23×D25111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5021no—
336C8—5021no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—10041no—
343C22—10041no—
344C22—10041no—
345C22—5021no—
346C22—5021no—
347C22—5021no—
348C22#2254873612yes—
349C22C2×D4×D25111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—10041no—
355C4—10041no—
356C4—10041no—
357C4—5021no—
358C4—5021no—
359C4—5021no—
360C4C2×#1686080911yescharacteristic
361C4#2254876011yescharacteristic
362C2—22no—
363C2—81no—
364C2—20081no—
365C2—10041no—
366C2#3063232411yescharacteristic
367C2C2×#2254873611yescharacteristic
368C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#448×C251C2
socleC22×C251C22×D4
hypercenter#448D251
hypocenterC251#3301
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #30632389C2semidirect
2. #30632348C2semidirect
3. #30632043C2semidirect
4. #30632026C2semidirect
5. #30632006C2semidirect
6. #30631992C2semidirect
7. #30631921C2semidirect
8. #30631913C2semidirect
9. #30631904C2semidirect
10. #30631831C2semidirect
11. SD16×Dic251C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631682C2semidirect
14. C2×#22548639C2semidirect
15. #448×C251C2semidirect
184. C251#3301semidirect
196. #448D251semidirect

Maximal subgroups

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

subgroupindex
#306323892
#306323482
#306320432
#306320262
#306320062
#306319922
#306319212
#306319132
#306319042
#306318312
SD16×Dic2512
(C22⋊D251)⋊C82
#306316822
C2×#225486392
#448×C2512
#3301251