Finite Group Databasebeta

← #44560395

Subgroups — #44560395

Order 32128.

428 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632235C211yesmaximal, characteristic
2C4×Q8×D251C211yesmaximal, characteristic, complemented
3#30632004C212yesmaximal, complemented
4#30631973C211yesmaximal, characteristic, complemented
5#30631972C211yesmaximal, characteristic, complemented
6#30631884C211yesmaximal, characteristic, complemented
7#30631824C212yesmaximal, complemented
8#30631778C212yesmaximal, complemented
9SD16×Dic251C212yesmaximal, complemented
10#30631739C211yesmaximal, characteristic, complemented
11#460×C251C211yesmaximal, characteristic, complemented
12C2×#16860809—21no—
13C2×Q8×D251C2211yescharacteristic
14C2×#16860798C2212yes—
15#22548715C2211yescharacteristic
16#22548702—21no—
17#22548686—21no—
18#22548686C2212yes—
19Q8×Dic251C2211yescharacteristic
20C4×Dic502—21no—
21C4×Dic502C2211yescharacteristic
22#22548680—21no—
23#22548680C2211yescharacteristic
24D4×Dic251—21no—
25D4×Dic251C2212yes—
26C4⋊C4×D251—21no—
27C4⋊C4×D251C2211yescharacteristic
28C42×D251—21no—
29C42×D251C2211yescharacteristic
30#22548647C2212yes—
31#22548646C2212yes—
32C4⋊(C251⋊C8)C2211yescharacteristic
33#22548640C2212yes—
34#22548639C2212yes—
35#22548636C2212yes—
36(C2×D251)⋊C8C2212yes—
37C8×Dic251C2212yes—
38C2×SD16×C251C2212yes—
39#127×C251C2211yescharacteristic
40C4×Q8×C251C2211yescharacteristic
41C4⋊C8×C251C2211yescharacteristic
42#102×C251C2212yes—
43#16860809—44no—
44C2×Dic502—21no—
45C2×Dic502—21no—
46C2×Dic502C2311yescharacteristic
47Q8×D251—22no—
48Q8×D251—22no—
49C2×C22⋊D251—22no—
50C22×Dic251—22no—
51C2×C4×D251—21no—
52C2×C4×D251—21no—
53C2×C4×D251D411yescharacteristic
54C2×C4×D251D411yescharacteristic
55C2×C4×D251C2311yescharacteristic
56#16860798—24no—
57C2×C251⋊C8C2312yes—
58C4⋊Dic251—21no—
59C4⋊Dic251—21no—
60C4⋊Dic251C2311yescharacteristic
61C4⋊Dic251C2312yes—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65C22⋊Dic251—22no—
66C22⋊Dic251—22no—
67C4×Dic251—21no—
68C4×Dic251—21no—
69C4×Dic251—21no—
70C4×Dic251D411yescharacteristic
71C4×Dic251D411yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251—21no—
76C2×D4×C251C2312yes—
77SD16×C251—24no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251—21no—
80C4⋊C4×C251C2311yescharacteristic
81C42×C251—21no—
82C42×C251C2311yescharacteristic
83C22×D251C2×D411yescharacteristic
84Dic502—24no—
85Dic502—21no—
86Dic502—42no—
87Dic502#4112yescomplemented
88C22⋊D251—44no—
89C2×Dic251—21no—
90C2×Dic251—21no—
91C2×Dic251—22no—
92C2×Dic251—44no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—24no—
97C4×D251—21no—
98C4×D251—21no—
99C4×D251—41no—
100C4×D251—42no—
101C4×D251SD1614yescomplemented
102C251⋊C8—22no—
103C23×C251—22no—
104Q8×C251—21no—
105Q8×C251#4112yescomplemented
106D4×C251—22no—
107D4×C251—22no—
108D4×C251—42no—
109D4×C251—42no—
110C2×C4×C251—21no—
111C2×C4×C251—21no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C8×C251—22no—
116C22×C251—42no—
117C22×C251—42no—
118C22×C251—42no—
119C22×C251C22×D411yescharacteristic
120C2×D251—21no—
121C2×D251—21no—
122C2×D251C2×SD1612yes—
123C4×C251—22no—
124C4×C251—21no—
125C4×C251—41no—
126C4×C251C2×SD1612yes—
127C4×C251C2×#4111yescharacteristic
128C4×C251C2×#4111yescharacteristic
129Dic251—22no—
130Dic251—21no—
131Dic251—21no—
132Dic251—42no—
133Dic251—41no—
134Dic251C2×SD1612yes—
135C2×C251—82no—
136C2×C251#52511yescharacteristic
137C2×C251C22×SD1611yescharacteristic
138C2×C251#57511yescharacteristic
139D251—22no—
140C251#310311yescharacteristic, complemented, Sylow
141#3103—2511nomaximal, Sylow
142#532—2511no—
143C2×C4×Q8—2511no—
144#481—2512no—
145#475—2511no—
146#475—2511no—
147#460—2511no—
148#460D25111yescharacteristic, complemented
149#448—2512no—
150C4×SD16—2512no—
151C4×SD16—2512no—
152#431—2511no—
153C2×#41—5021no—
154C22×Q8—2511no—
155C2×SD16—2512no—
156C2×SD16C2×D25112yes—
157C4⋊Q8—2511no—
158#127C2×D25111yescharacteristic
159#124—5021no—
160#121—5021no—
161#121—2512no—
162C4×Q8—5021no—
163C4×Q8—5021no—
164C4×Q8—2511no—
165C4×Q8—2511no—
166C4×Q8—2511no—
167C4×Q8C2×D25111yescharacteristic
168C4×D4—5021no—
169C4×D4—2512no—
170C2×C4⋊C4—5021no—
171C2×C4⋊C4—2511no—
172C2×C42—5021no—
173C2×C42—2511no—
174#106—2512no—
175#106—2512no—
176C4⋊C8—2511no—
177C4⋊C8C2×D25111yescharacteristic
178#103—2512no—
179#103—2512no—
180#102—2512no—
181#102C2×D25112yes—
182C22⋊C8—2512no—
183C4×C8—2512no—
184#41—10044no—
185C2×Q8—5021no—
186C2×Q8—5022no—
187C2×Q8—5022no—
188C2×Q8—5021no—
189C2×Q8—2511no—
190C2×Q8C22×D25111yescharacteristic
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—5022no—
194C2×D4C22×D25112yes—
195C22×C4—5021no—
196C22×C4—5021no—
197C22×C4—5022no—
198C22×C4—2511no—
199C22×C4—2511no—
200C22×C4—2511no—
201SD16—24no—
202SD16—5024no—
203C2×C8—2512no—
204C2×C8C22×D25112yes—
205C4⋊C4—21no—
206C4⋊C4—5021no—
207C4⋊C4—5021no—
208C4⋊C4—5021no—
209C4⋊C4—5021no—
210C4⋊C4—5021no—
211C4⋊C4—2511no—
212C4⋊C4—2512no—
213C4⋊C4C22×D25111yescharacteristic
214C22⋊C4—5022no—
215C22⋊C4—5022no—
216C42—21no—
217C42—5021no—
218C42—5021no—
219C42—5021no—
220C42—2511no—
221C42—2511no—
222C42—2511no—
223C42C22×D25111yescharacteristic
224C23—22no—
225C23—2511no—
226Q8—21no—
227Q8—10042no—
228Q8—5024no—
229Q8—5021no—
230Q8—2512no—
231Q8#1686080912yescomplemented
232D4—22no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—10044no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—10041no—
240C2×C4—10042no—
241C2×C4—10044no—
242C2×C4—5024no—
243C2×C4—5021no—
244C2×C4—5022no—
245C2×C4—5021no—
246C2×C4—5021no—
247C2×C4—5021no—
248C2×C4—2511no—
249C2×C4—2511no—
250C2×C4—2511no—
251C2×C4—2514no—
252C2×C4D4×D25111yescharacteristic
253C2×C4D4×D25111yescharacteristic
254C2×C4C23×D25111yescharacteristic
255C8—22no—
256C8—5022no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—5021no—
261C22—5021no—
262C22—2512no—
263C22C2×D4×D25111yescharacteristic
264C4—22no—
265C4—21no—
266C4—41no—
267C4—10042no—
268C4—10041no—
269C4—5022no—
270C4—5021no—
271C4—5021no—
272C4—2512no—
273C4SD16×D25112yes—
274C4#41×D25111yescharacteristic
275C4C2×#1686080911yescharacteristic
276C2—82no—
277C2—5022no—
278C2#3063219711yescharacteristic
279C2C2×SD16×D25111yescharacteristic
280C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#460×C251C2
socleC22×C251C22×D4
prime radical#30632235C2
hypercenter#460D251
hypocenterC251#3103
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D251C2semidirect
3. #30632004C2semidirect
4. #30631973C2semidirect
5. #30631972C2semidirect
6. #30631884C2semidirect
7. #30631824C2semidirect
8. #30631778C2semidirect
9. SD16×Dic251C2semidirect
10. #30631739C2semidirect
11. #460×C251C2semidirect
87. Dic502#41semidirect
101. C4×D251SD16semidirect
105. Q8×C251#41semidirect
140. C251#3103semidirect
148. #460D251semidirect
231. Q8#16860809semidirect

Maximal subgroups

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

subgroupindex
#306322352
C4×Q8×D2512
#306320042
#306319732
#306319722
#306318842
#306318242
#306317782
SD16×Dic2512
#306317392
#460×C2512
#3103251