Finite Group Databasebeta

← #44562262

Subgroups — #44562262

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#30632384C211yesmaximal, characteristic, complemented
2#30632347C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632025C211yesmaximal, characteristic, complemented
5#30632007C211yesmaximal, characteristic, complemented
6#30631997C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631824C211yesmaximal, characteristic, complemented
11#30631778C211yesmaximal, characteristic, complemented
12C8×C22⋊D251C211yesmaximal, characteristic, complemented
13#30631684C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#465×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548708—21no—
22#22548706C2211yescharacteristic
23#22548704C2211yescharacteristic
24#22548701—21no—
25#22548696—21no—
26#22548693—21no—
27#22548690C2211yescharacteristic
28#22548688—21no—
29#22548686—21no—
30#22548686C2211yescharacteristic
31#22548680C2211yescharacteristic
32D4×Dic251C2211yescharacteristic
33#22548677—21no—
34C4×C22⋊D251C2211yescharacteristic
35(C22⋊D251)⋊C4—21no—
36#22548674C2211yescharacteristic
37C4⋊C4×D251C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548646C2211yescharacteristic
40Dic251⋊C8C2211yescharacteristic
41#22548640C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C22⋊Q8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63Q8×D251—22no—
64Q8×D251—41no—
65C2×D1004—21no—
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251D411yescharacteristic
70C22×Dic251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74C2×C4×D251C2311yescharacteristic
75#16860798—22no—
76#16860795—41no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C8×D251—41no—
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251D411yescharacteristic
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98C2×Q8×C251C2311yescharacteristic
99C2×D4×C251—21no—
100C2×D4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102SD16×C251—41no—
103C2×C8×C251—41no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C22⋊C4×C251—21no—
111C22×D251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—42no—
115Dic502#4112yes—
116D1004—21no—
117D1004—21no—
118D1004—42no—
119C22⋊D251—22no—
120C22⋊D251—42no—
121C22⋊D251—42no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—81no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—42no—
132C4×D251—42no—
133C4×D251—81no—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251—21no—
137C23×C251C2×D411yescharacteristic
138Q8×C251—21no—
139Q8×C251—21no—
140D4×C251—21no—
141D4×C251—21no—
142D4×C251—81no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251#13512yes—
165C4×C251—21no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251H(5,𝔽2)11yescharacteristic
171Dic251—21no—
172Dic251—22no—
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—41no—
178C2×C251—81no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—22no—
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#465D25111yescharacteristic, complemented
194#463—2511no—
195#449—2511no—
196#448—2511no—
197C4×SD16—2511no—
198C8×D4—2511no—
199#416—2511no—
200C2×#103—2511no—
201C2×#41—2511no—
202C22×Q8—2511no—
203C2×SD16—2511no—
204C2×SD16C2×D25111yescharacteristic
205C2×D8—2511no—
206C22×C8—2511no—
207C22×C8C2×D25111yescharacteristic
208#126—5021no—
209#125—2511no—
210#124—5021no—
211#124—2511no—
212#123—5021no—
213C22⋊Q8—5021no—
214C22⋊Q8C2×D25111yescharacteristic
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#106C2×D25111yescharacteristic
230C4⋊C8—2511no—
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#103C2×D25111yescharacteristic
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102C2×D25111yescharacteristic
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243C4×C8—2511no—
244#41—10041no—
245#41—5022no—
246C2×Q8—10041no—
247C2×Q8—5021no—
248C2×Q8—5022no—
249C2×Q8—2511no—
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—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—2511no—
263C22×C4C22×D25111yescharacteristic
264SD16—41no—
265SD16—5022no—
266D8—10041no—
267C2×C8—41no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8C22×D25111yescharacteristic
272C2×C8C22×D25111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—5021no—
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—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—10042no—
303Q8—5021no—
304Q8—2512no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—10042no—
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—20081no—
320C2×C4—20081no—
321C2×C4—10042no—
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—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—10041no—
344C22—10041no—
345C22—10041no—
346C22—5021no—
347C22—5021no—
348C22—2512no—
349C22C2×D4×D25111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—10041no—
355C4—10041no—
356C4—5021no—
357C4—5021no—
358C4—5022no—
359C4—5021no—
360C4#41×D25111yescharacteristic
361C4#2254876011yescharacteristic
362C2—41no—
363C2—81no—
364C2—20081no—
365C2—5022no—
366C2#3063232611yescharacteristic
367C2#135×D25111yescharacteristic
368C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#465×C251C2
socleC22×C251C22×D4
hypercenter#465D251
hypocenterC251#3301
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #30632384C2semidirect
2. #30632347C2semidirect
3. #30632043C2semidirect
4. #30632025C2semidirect
5. #30632007C2semidirect
6. #30631997C2semidirect
7. #30631922C2semidirect
8. #30631904C2semidirect
9. #30631830C2semidirect
10. #30631824C2semidirect
11. #30631778C2semidirect
12. C8×C22⋊D251C2semidirect
13. #30631684C2semidirect
14. #103×D251C2semidirect
15. #465×C251C2semidirect
184. C251#3301semidirect
193. #465D251semidirect

Maximal subgroups

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

subgroupindex
#306323842
#306323472
#306320432
#306320252
#306320072
#306319972
#306319222
#306319042
#306318302
#306318242
#306317782
C8×C22⋊D2512
#306316842
#103×D2512
#465×C2512
#3301251