Finite Group Databasebeta

← #44562357

Subgroups — #44562357

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#30632338C211yesmaximal, characteristic, complemented
3#30632045C211yesmaximal, characteristic, complemented
4#30632022C211yesmaximal, characteristic, complemented
5#30631999C211yesmaximal, characteristic, complemented
6#30631980C211yesmaximal, characteristic, complemented
7#30631933C211yesmaximal, characteristic, complemented
8#30631922C211yesmaximal, characteristic, complemented
9#30631901C211yesmaximal, characteristic, complemented
10#30631832C211yesmaximal, characteristic, complemented
11#30631827C211yesmaximal, characteristic, complemented
12#30631790C211yesmaximal, characteristic, complemented
13#30631756C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#417×C251C211yesmaximal, characteristic, complemented
16C2×#16860810—21no—
17C2×Q8×D251C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22#22548714C2211yescharacteristic
23#22548702—21no—
24#22548702C2211yescharacteristic
25#22548700—21no—
26#22548698—21no—
27#22548694—21no—
28#22548694C2211yescharacteristic
29#22548693—21no—
30#22548690C2211yescharacteristic
31#22548689C2211yescharacteristic
32#22548686—21no—
33#22548686C2211yescharacteristic
34Q8×Dic251C2211yescharacteristic
35#22548677C2211yescharacteristic
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38C4⋊C4×D251C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548648C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51#22548625C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C2×M16×C251C2211yescharacteristic
54C2×C4⋊C4×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860810—44no—
60C2×Dic502—21no—
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
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74#16860799—41no—
75#16860797—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⋊Dic251—21no—
91C22⋊Dic251D411yescharacteristic
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98#41×C251—22no—
99#41×C251—41no—
100C2×Q8×C251C2311yescharacteristic
101C2×D4×C251—21no—
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251—21no—
105C22×C4×C251C2311yescharacteristic
106M16×C251—41no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251—21no—
110C4⋊C4×C251C2311yescharacteristic
111C4⋊C4×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251C2×D411yescharacteristic
114Dic502—21no—
115Dic502—21no—
116Dic502—42no—
117Dic502—42no—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121C22⋊D251—22no—
122C22⋊D251—42no—
123C22⋊D251—42no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—42no—
133C4×D251—42no—
134C4×D251—42no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251—21no—
138C23×C251C2×D411yescharacteristic
139Q8×C251—21no—
140Q8×C251#4112yescomplemented
141D4×C251—21no—
142D4×C251—21no—
143D4×C251—42no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—42no—
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×C251C22×D411yescharacteristic
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251AGL(1,ℤ/8)12yes—
168C4×C251—21no—
169C4×C251—22no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251C2×#4111yescharacteristic
173C4×C251H(5,𝔽2)11yescharacteristic
174Dic251—21no—
175Dic251—21no—
176Dic251—21no—
177Dic251—41no—
178Dic251—41no—
179Dic251—41no—
180C2×C251—41no—
181C2×C251—81no—
182C2×C251#54611yescharacteristic
183C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C251#57811yescharacteristic
185D251—22no—
186D251—81no—
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#449—2511no—
199#448—2511no—
200#440—2511no—
201#435—2511no—
202#417D25111yescharacteristic, complemented
203C2×#103—2511no—
204C2×#41—5021no—
205C2×#41C2×D25111yescharacteristic
206C22×Q8—2511no—
207C2×SD16—2511no—
208C2×SD16—2511no—
209C2×D8—2511no—
210C2×M16C2×D25111yescharacteristic
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#121—2511no—
224C4×Q8—2511no—
225C4×D4—5021no—
226C4×D4—2511no—
227C4×D4—2511no—
228C2×C4⋊C4—2511no—
229C2×C4⋊C4C2×D25111yescharacteristic
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#103C2×D25111yescharacteristic
239#103C2×D25111yescharacteristic
240#102—2511no—
241#102—2511no—
242#102C2×D25111yescharacteristic
243#102C2×D25111yescharacteristic
244C22⋊C8—2511no—
245C22⋊C8—2511no—
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—5021no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—2511no—
261C2×D4—2511no—
262C2×D4C22×D25111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—5021no—
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4—2511no—
269C22×C4C22×D25111yescharacteristic
270SD16—10041no—
271SD16—5022no—
272D8—10041no—
273M16—41no—
274C2×C8—10041no—
275C2×C8—2511no—
276C2×C8—2511no—
277C2×C8C22×D25111yescharacteristic
278C2×C8C22×D25111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—5021no—
281C4⋊C4—5021no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—2511no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4C22×D25111yescharacteristic
288C4⋊C4C22×D25111yescharacteristic
289C22⋊C4—5021no—
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—5021no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—10042no—
306Q8—10042no—
307Q8—5021no—
308Q8—5021no—
309Q8#1686081012yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—10042no—
314D4—10042no—
315D4—10042no—
316D4—5022no—
317D4—5021no—
318D4—5021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—10042no—
327C2×C4—10042no—
328C2×C4—10042no—
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—10041no—
350C22—10041no—
351C22—10041no—
352C22—5021no—
353C22—5021no—
354C22—2512no—
355C22C2×D4×D25111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—10041no—
361C4—10041no—
362C4—10041no—
363C4—5021no—
364C4—5021no—
365C4—5021no—
366C4C2×#1686081011yescharacteristic
367C4#2254876011yescharacteristic
368C2—41no—
369C2—81no—
370C2—20081no—
371C2—5022no—
372C2#3063232711yescharacteristic
373C2AGL(1,ℤ/8)×D25111yescharacteristic
374C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
hypercenter#417D251
hypocenterC251#3308
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #30632344C2semidirect
2. #30632338C2semidirect
3. #30632045C2semidirect
4. #30632022C2semidirect
5. #30631999C2semidirect
6. #30631980C2semidirect
7. #30631933C2semidirect
8. #30631922C2semidirect
9. #30631901C2semidirect
10. #30631832C2semidirect
11. #30631827C2semidirect
12. #30631790C2semidirect
13. #30631756C2semidirect
14. #103×D251C2semidirect
15. #417×C251C2semidirect
140. Q8×C251#41semidirect
187. C251#3308semidirect
202. #417D251semidirect
309. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306323442
#306323382
#306320452
#306320222
#306319992
#306319802
#306319332
#306319222
#306319012
#306318322
#306318272
#306317902
#306317562
#103×D2512
#417×C2512
#3308251