Finite Group Databasebeta

← #44562349

Subgroups — #44562349

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632344C211yesmaximal, characteristic, complemented
2#30632337C211yesmaximal, characteristic, complemented
3#30632045C211yesmaximal, characteristic, complemented
4#30632023C211yesmaximal, characteristic, complemented
5#30631999C211yesmaximal, characteristic, complemented
6#30631980C211yesmaximal, characteristic, complemented
7#30631931C211yesmaximal, characteristic, complemented
8#30631920C211yesmaximal, characteristic, complemented
9#30631900C211yesmaximal, characteristic, complemented
10#30631824C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12#30631761C211yesmaximal, characteristic, complemented
13#30631688C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#449×C251C211yesmaximal, characteristic, complemented
16C2×#16860809—21no—
17C2×Q8×D251C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860794C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548714C2211yescharacteristic
22#22548702—21no—
23#22548702C2211yescharacteristic
24#22548697—21no—
25#22548696—21no—
26#22548694—21no—
27#22548694C2211yescharacteristic
28#22548692—21no—
29#22548686—21no—
30#22548686C2211yescharacteristic
31#22548686C2211yescharacteristic
32Q8×Dic251C2211yescharacteristic
33D4×Dic251C2211yescharacteristic
34C4×C22⋊D251—21no—
35C4×C22⋊D251C2211yescharacteristic
36C2×C4⋊Dic251C2211yescharacteristic
37C4⋊C4×D251C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548648C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51#22548625C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860809—44no—
60C2×Dic502—21no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63Q8×D251—22no—
64Q8×D251—41no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D411yescharacteristic
68C22×Dic251—21no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73#16860798—41no—
74#16860794—41no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—41no—
78C4⋊Dic251—21no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
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⋊Dic251—21no—
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251D411yescharacteristic
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251—21no—
102C2×D4×C251C2311yescharacteristic
103C2×D4×C251C2311yescharacteristic
104C22×C4×C251—21no—
105C22×C4×C251C2311yescharacteristic
106SD16×C251—22no—
107D8×C251—41no—
108C2×C8×C251C2311yescharacteristic
109C2×C8×C251C2311yescharacteristic
110C4⋊C4×C251C2311yescharacteristic
111C22⋊C4×C251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116Dic502—42no—
117C22⋊D251—22no—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—42no—
125C2×Dic251—42no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—42no—
132C4×D251—42no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251C2×D411yescharacteristic
138Q8×C251—21no—
139Q8×C251#4112yescomplemented
140D4×C251—21no—
141D4×C251—21no—
142D4×C251—21no—
143D4×C251—21no—
144D4×C251—42no—
145D4×C251—42no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—42no—
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251C22×D411yescharacteristic
165C2×D251—21no—
166C2×D251—21no—
167C2×D251AGL(1,ℤ/8)12yes—
168C4×C251—21no—
169C4×C251—22no—
170C4×C251—41no—
171C4×C251C2×#4111yescharacteristic
172C4×C251H(5,𝔽2)11yescharacteristic
173Dic251—21no—
174Dic251—21no—
175Dic251—21no—
176Dic251—41no—
177Dic251—41no—
178Dic251—41no—
179Dic251—41no—
180C2×C251—41no—
181C2×C251—81no—
182C2×C251—81no—
183C2×C251#54611yescharacteristic
184C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C251#57811yescharacteristic
186D251—22no—
187C251#330811yescharacteristic, complemented, Sylow
188#3308—2511nomaximal, Sylow
189#548—2511no—
190#547—2511no—
191#489—2511no—
192#484—2511no—
193#480—2511no—
194#477—2511no—
195#468—2511no—
196#466—2511no—
197#463—2511no—
198#449D25111yescharacteristic, complemented
199#448—2511no—
200#440—2511no—
201#435—2511no—
202#417—2511no—
203C2×#103—2511no—
204C2×#41—5021no—
205C2×#41C2×D25111yescharacteristic
206C22×Q8—2511no—
207C2×SD16—2511no—
208C2×SD16C2×D25111yescharacteristic
209C2×D8C2×D25111yescharacteristic
210C2×M16—2511no—
211C22×C8—2511no—
212C4⋊Q8—2511no—
213#124—5021no—
214#124—2511no—
215#123—5021no—
216#123—5021no—
217C22⋊Q8—5021no—
218C22⋊Q8—5021no—
219C22⋊Q8—2511no—
220#121—5021no—
221#121—2511no—
222#121—2511no—
223#121C2×D25111yescharacteristic
224C4×Q8—2511no—
225C4×D4—5021no—
226C4×D4—2511no—
227C4×D4—2511no—
228C2×C4⋊C4—2511no—
229C2×C4⋊C4—2511no—
230#107—2511no—
231#107—2511no—
232#107—2511no—
233C4⋊C8—2511no—
234#103—2511no—
235#103—2511no—
236#103—2511no—
237#103—2511no—
238#103—2511no—
239#103C2×D25111yescharacteristic
240#102—2511no—
241#102—2511no—
242#102—2511no—
243#102C2×D25111yescharacteristic
244C22⋊C8—2511no—
245C22⋊C8C2×D25111yescharacteristic
246#97—2511no—
247#41—22no—
248#41—41no—
249#41—10044no—
250C2×Q8—10041no—
251C2×Q8—5021no—
252C2×Q8—5021no—
253C2×Q8—5022no—
254C2×Q8—2511no—
255C2×Q8C22×D25111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—2511no—
261C2×D4C22×D25111yescharacteristic
262C2×D4C22×D25111yescharacteristic
263C22×C4—21no—
264C22×C4—5021no—
265C22×C4—5021no—
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4—2511no—
269C22×C4C22×D25111yescharacteristic
270SD16—22no—
271SD16—10041no—
272D8—41no—
273M16—10041no—
274C2×C8—10041no—
275C2×C8—2511no—
276C2×C8—2511no—
277C2×C8C22×D25111yescharacteristic
278C2×C8C22×D25111yescharacteristic
279C4⋊C4—5021no—
280C4⋊C4—5021no—
281C4⋊C4—5021no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—2511no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4—2511no—
288C4⋊C4C22×D25111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4—2511no—
297C22⋊C4—2511no—
298C42—5021no—
299C42—2511no—
300C23—21no—
301C23—21no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—10042no—
306Q8—10042no—
307Q8—5021no—
308Q8—5021no—
309Q8#1686080912yescomplemented
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—10042no—
317D4—10042no—
318D4—5022no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—10042no—
325C2×C4—10042no—
326C2×C4—10042no—
327C2×C4—10042no—
328C2×C4—5021no—
329C2×C4—5021no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—2511no—
335C2×C4—2511no—
336C2×C4—2511no—
337C2×C4D4×D25111yescharacteristic
338C2×C4C23×D25111yescharacteristic
339C8—21no—
340C8—21no—
341C8—5021no—
342C8—5021no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—5021no—
353C22—5021no—
354C22—2512no—
355C22C2×D4×D25111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—10041no—
360C4—10041no—
361C4—10041no—
362C4—10041no—
363C4—5021no—
364C4—5021no—
365C4—5021no—
366C4C2×#1686080911yescharacteristic
367C4#2254876011yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—5022no—
372C2#3063232411yescharacteristic
373C2AGL(1,ℤ/8)×D25111yescharacteristic
374C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#449×C251C2
socleC22×C251C22×D4
hypercenter#449D251
hypocenterC251#3308
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #30632344C2semidirect
2. #30632337C2semidirect
3. #30632045C2semidirect
4. #30632023C2semidirect
5. #30631999C2semidirect
6. #30631980C2semidirect
7. #30631931C2semidirect
8. #30631920C2semidirect
9. #30631900C2semidirect
10. #30631824C2semidirect
11. #30631788C2semidirect
12. #30631761C2semidirect
13. #30631688C2semidirect
14. #103×D251C2semidirect
15. #449×C251C2semidirect
139. Q8×C251#41semidirect
187. C251#3308semidirect
198. #449D251semidirect
309. Q8#16860809semidirect

Maximal subgroups

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

subgroupindex
#306323442
#306323372
#306320452
#306320232
#306319992
#306319802
#306319312
#306319202
#306319002
#306318242
#306317882
#306317612
#306316882
#103×D2512
#449×C2512
#3308251