Finite Group Databasebeta

← #44562352

Subgroups — #44562352

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#30632348C211yesmaximal, characteristic, complemented
2#30632338C211yesmaximal, characteristic, complemented
3#30632048C211yesmaximal, characteristic, complemented
4#30632021C211yesmaximal, characteristic, complemented
5#30632000C211yesmaximal, characteristic, complemented
6#30631980C211yesmaximal, characteristic, complemented
7#30631936C211yesmaximal, characteristic, complemented
8#30631923C211yesmaximal, characteristic, complemented
9#30631905C211yesmaximal, characteristic, complemented
10#30631835C211yesmaximal, characteristic, complemented
11#30631826C211yesmaximal, characteristic, complemented
12#30631790C211yesmaximal, characteristic, complemented
13#30631758C211yesmaximal, characteristic, complemented
14#30631687C211yesmaximal, characteristic, complemented
15C2×#103×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860810—21no—
18C2×#16860797C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21C2×#16860793C2211yescharacteristic
22#22548714C2211yescharacteristic
23#22548704—21no—
24#22548704C2211yescharacteristic
25#22548700—21no—
26#22548698—21no—
27#22548693—21no—
28#22548691—21no—
29#22548691C2211yescharacteristic
30#22548690C2211yescharacteristic
31#22548689—21no—
32#22548689C2211yescharacteristic
33#22548689C2211yescharacteristic
34Q8×Dic251C2211yescharacteristic
35#22548677C2211yescharacteristic
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38C4⋊C4×D251C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548648C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51#22548625C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C2×C4⋊C4×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#103×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—41no—
61#16860810—44no—
62C2×Dic502—21no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004C2311yescharacteristic
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#16860797—22no—
75#16860797—41no—
76D2008—41no—
77#16860793—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87C22⋊Dic251D411yescharacteristic
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98C2×Q8×C251—21no—
99C2×Q8×C251—22no—
100C2×Q8×C251—41no—
101C2×Q8×C251C2311yescharacteristic
102C2×Q8×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251—21no—
105C22×C4×C251C2311yescharacteristic
106C2×C8×C251—41no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251—21no—
110C4⋊C4×C251C2311yescharacteristic
111C4⋊C4×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251C2×D411yescharacteristic
115Dic502—42no—
116D1004—21no—
117D1004—21no—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121D1004—42no—
122C22⋊D251—22no—
123C22⋊D251—42no—
124C22⋊D251—42no—
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×C251C2×D411yescharacteristic
138Q8×C251—21no—
139Q8×C251—21no—
140Q8×C251—21no—
141Q8×C251—42no—
142Q8×C251#4112yescomplemented
143C2×C4×C251—21no—
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×C251AGL(1,ℤ/8)12yes—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C4×C251—21no—
169C4×C251—22no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251—41no—
173C4×C251C2×#4111yescharacteristic
174C4×C251H(5,𝔽2)11yescharacteristic
175Dic251—21no—
176Dic251—21no—
177Dic251—21no—
178Dic251—41no—
179Dic251—41no—
180C2×C251—22no—
181C2×C251#54611yescharacteristic
182C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C251#57811yescharacteristic
184D251—41no—
185D251—81no—
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#417—2511no—
203C2×#103D25111yescharacteristic, complemented
204C2×#41—5021no—
205C2×#41—2511no—
206C22×Q8C2×D25111yescharacteristic
207C2×SD16—2511no—
208C2×SD16—2511no—
209C2×D8—2511no—
210C2×M16—2511no—
211C22×C8C2×D25111yescharacteristic
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#103C2×D25111yescharacteristic
237#103C2×D25111yescharacteristic
238#103C2×D25111yescharacteristic
239#103C2×D25111yescharacteristic
240#102—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244C22⋊C8—2511no—
245C22⋊C8—2511no—
246#97—2511no—
247#41—10044no—
248#41—10041no—
249#41—5022no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—5021no—
254C2×Q8C22×D25111yescharacteristic
255C2×Q8C22×D25111yescharacteristic
256C2×D4—5021no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—2511no—
261C2×D4—2511no—
262C2×D4—2511no—
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—10041no—
274C2×C8—41no—
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—5021no—
301C23—5021no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—10042no—
309Q8#1686081012yescomplemented
310D4—10042no—
311D4—10042no—
312D4—10042no—
313D4—10042no—
314D4—5022no—
315D4—5021no—
316D4—5021no—
317D4—5021no—
318D4—5021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—10042no—
328C2×C4—10042no—
329C2×C4—10042no—
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—10041no—
346C22—10041no—
347C22—10041no—
348C22—10041no—
349C22—10041no—
350C22—10041no—
351C22—5021no—
352C22—5021no—
353C22—5021no—
354C22#2254874212yes—
355C22C2×D4×D25111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—10041no—
362C4—10041no—
363C4—5021no—
364C4—5021no—
365C4—5021no—
366C4C2×#1686081011yescharacteristic
367C4#2254876011yescharacteristic
368C2—22no—
369C2—20081no—
370C2—20081no—
371C2—10041no—
372C2#3063232711yescharacteristic
373C2C2×#2254874211yescharacteristic
374C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#103×C251C2
socleC22×C251C22×D4
hypercenterC2×#103D251
hypocenterC251#3308
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #30632348C2semidirect
2. #30632338C2semidirect
3. #30632048C2semidirect
4. #30632021C2semidirect
5. #30632000C2semidirect
6. #30631980C2semidirect
7. #30631936C2semidirect
8. #30631923C2semidirect
9. #30631905C2semidirect
10. #30631835C2semidirect
11. #30631826C2semidirect
12. #30631790C2semidirect
13. #30631758C2semidirect
14. #30631687C2semidirect
15. C2×#103×C251C2semidirect
142. Q8×C251#41semidirect
187. C251#3308semidirect
203. C2×#103D251semidirect
309. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306323482
#306323382
#306320482
#306320212
#306320002
#306319802
#306319362
#306319232
#306319052
#306318352
#306318262
#306317902
#306317582
#306316872
C2×#103×C2512
#3308251