Finite Group Databasebeta

← #44560953

Subgroups — #44560953

Order 32128.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632232C211yesmaximal, characteristic, complemented
2#30632223C211yesmaximal, characteristic, complemented
3#30632095C211yesmaximal, characteristic, complemented
4#30632073C211yesmaximal, characteristic, complemented
5#30632050C211yesmaximal, characteristic, complemented
6#30632048C211yesmaximal, characteristic, complemented
7#30631940C211yesmaximal, characteristic, complemented
8#30631938C211yesmaximal, characteristic, complemented
9#30631935C211yesmaximal, characteristic, complemented
10SD16×Dic251C211yesmaximal, characteristic, complemented
11#30631772C211yesmaximal, characteristic, complemented
12#30631769C211yesmaximal, characteristic, complemented
13#30631767C211yesmaximal, characteristic, complemented
14M16×Dic251C211yesmaximal, characteristic, complemented
15#468×C251C211yesmaximal, characteristic, complemented
16C2×#16860809—21no—
17C2×#16860797C2211yescharacteristic
18C2×D2008C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×#16860794C2211yescharacteristic
21#22548714C2211yescharacteristic
22#22548711C2211yescharacteristic
23#22548710—21no—
24#22548706C2211yescharacteristic
25#22548704C2211yescharacteristic
26#22548701—21no—
27#22548699—21no—
28#22548689C2211yescharacteristic
29#22548689C2211yescharacteristic
30#22548687—21no—
31Q8×Dic251C2211yescharacteristic
32D4×Dic251C2211yescharacteristic
33#22548677C2211yescharacteristic
34#22548677C2211yescharacteristic
35(C22⋊D251)⋊C4—21no—
36(C22⋊D251)⋊C4—21no—
37#22548673—21no—
38C2×C4×Dic251C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548647C2211yescharacteristic
41#22548646C2211yescharacteristic
42#22548639C2211yescharacteristic
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548625C2211yescharacteristic
49#22548625C2211yescharacteristic
50C8×Dic251C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C2×M16×C251C2211yescharacteristic
54C22⋊Q8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#107×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860809—44no—
60C2×Dic502—21no—
61C2×D1004C2311yescharacteristic
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C22×Dic251—21no—
68C22×Dic251D411yescharacteristic
69C22×Dic251D411yescharacteristic
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72#16860797—22no—
73D2008—22no—
74#16860795—22no—
75#16860794—22no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C4⋊Dic251—21no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84C22⋊Dic251—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91C4×Dic251—21no—
92C4×Dic251D411yescharacteristic
93C4×Dic251D411yescharacteristic
94C4×Dic251C2311yescharacteristic
95C4×Dic251C2311yescharacteristic
96C2×Q8×C251C2311yescharacteristic
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100SD16×C251—22no—
101M16×C251—22no—
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C4⋊C4×C251—21no—
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22⋊C4×C251—21no—
109C22×D251—21no—
110C22×D251—21no—
111Dic502—42no—
112D1004—21no—
113D1004—21no—
114D1004—21no—
115D1004—21no—
116C22⋊D251—42no—
117C22⋊D251—42no—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C2×Dic251—22no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—41no—
124C2×Dic251—42no—
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—42no—
130C4×D251—42no—
131C251⋊C8—21no—
132C251⋊C8#4112yescomplemented
133C23×C251—21no—
134C23×C251C2×D411yescharacteristic
135Q8×C251—21no—
136Q8×C251—21no—
137D4×C251—21no—
138D4×C251—21no—
139D4×C251—42no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251#4112yescomplemented
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251C2×#4111yescharacteristic
167C4×C251C2×#4111yescharacteristic
168Dic251—21no—
169Dic251—21no—
170Dic251—21no—
171Dic251—41no—
172Dic251—41no—
173Dic251AGL(1,ℤ/8)12yes—
174C2×C251—41no—
175C2×C251—81no—
176C2×C251#53211yescharacteristic
177C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C251#57511yescharacteristic
179D251—81no—
180D251—81no—
181C251#317111yescharacteristic, complemented, Sylow
182#3171—2511nomaximal, Sylow
183#532—2511no—
184#529—2511no—
185#501—2511no—
186#496—2511no—
187#490—2511no—
188#489—2511no—
189#469—2511no—
190#469—2511no—
191#468—2511no—
192#468D25111yescharacteristic, complemented
193C4×SD16—2511no—
194C4×SD16—2511no—
195C4×D8—2511no—
196C4×D8—2511no—
197C4×M16—2511no—
198C2×#41—5021no—
199C2×SD16—2511no—
200C2×SD16C2×D25111yescharacteristic
201C2×D8—2511no—
202C2×D8—2511no—
203C2×M16—2511no—
204C2×M16C2×D25111yescharacteristic
205C4⋊Q8—2511no—
206#127—2511no—
207#126—5021no—
208#125—2511no—
209#124—5021no—
210#124—2511no—
211#123—5021no—
212C22⋊Q8C2×D25111yescharacteristic
213#121—5021no—
214#121—2511no—
215#121—2511no—
216#121C2×D25111yescharacteristic
217C4×Q8—2511no—
218C4×D4—5021no—
219C4×D4—5021no—
220C4×D4—2511no—
221C4×D4—2511no—
222C4×D4—2511no—
223#117—5021no—
224C2×C42—2511no—
225#107—2511no—
226#107C2×D25111yescharacteristic
227#106—2511no—
228#106—2511no—
229#103—2511no—
230#103C2×D25111yescharacteristic
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236#102C2×D25111yescharacteristic
237#97—2511no—
238#97—2511no—
239C4×C8—2511no—
240C4×C8—2511no—
241#41—10044no—
242C2×Q8—5021no—
243C2×Q8C22×D25111yescharacteristic
244C2×D4—21no—
245C2×D4—5021no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—5021no—
249C2×D4—2511no—
250C2×D4—2511no—
251C2×D4C22×D25111yescharacteristic
252C22×C4—5021no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—2511no—
256C22×C4—2511no—
257C22×C4C22×D25111yescharacteristic
258SD16—22no—
259SD16—5022no—
260D8—5022no—
261D8—5022no—
262M16—22no—
263M16—5022no—
264C2×C8—2511no—
265C2×C8—2511no—
266C2×C8C22×D25111yescharacteristic
267C2×C8C22×D25111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—2511no—
274C4⋊C4—2511no—
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C42—5021no—
287C42—2511no—
288C42—2511no—
289C42—2511no—
290C42—2511no—
291C23—21no—
292C23—5021no—
293C23—5021no—
294C23D4×D25111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—10042no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—10042no—
302D4—10042no—
303D4—10042no—
304D4—10042no—
305D4—5021no—
306D4—5021no—
307D4—5021no—
308D4—5021no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—10042no—
315C2×C4—10042no—
316C2×C4—10042no—
317C2×C4—10041no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—5022no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4—2511no—
325C2×C4D4×D25111yescharacteristic
326C2×C4C23×D25111yescharacteristic
327C8—21no—
328C8—5021no—
329C8—2512no—
330C8#1686080912yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—10041no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—10041no—
349C4—10041no—
350C4—5021no—
351C4—5021no—
352C4—5021no—
353C4—2512no—
354C4#41×D25111yescharacteristic
355C4C2×#1686080911yescharacteristic
356C2—41no—
357C2—81no—
358C2—20081no—
359C2—20081no—
360C2#3063223211yescharacteristic
361C2AGL(1,ℤ/8)×D25111yescharacteristic
362C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#468×C251C2
socleC22×C251C22×D4
hypercenter#468D251
hypocenterC251#3171
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #30632232C2semidirect
2. #30632223C2semidirect
3. #30632095C2semidirect
4. #30632073C2semidirect
5. #30632050C2semidirect
6. #30632048C2semidirect
7. #30631940C2semidirect
8. #30631938C2semidirect
9. #30631935C2semidirect
10. SD16×Dic251C2semidirect
11. #30631772C2semidirect
12. #30631769C2semidirect
13. #30631767C2semidirect
14. M16×Dic251C2semidirect
15. #468×C251C2semidirect
132. C251⋊C8#41semidirect
148. C8×C251#41semidirect
181. C251#3171semidirect
192. #468D251semidirect
330. C8#16860809semidirect

Maximal subgroups

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

subgroupindex
#306322322
#306322232
#306320952
#306320732
#306320502
#306320482
#306319402
#306319382
#306319352
SD16×Dic2512
#306317722
#306317692
#306317672
M16×Dic2512
#468×C2512
#3171251