Finite Group Databasebeta

← #44560489

Subgroups — #44560489

Order 32128.

424 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 121 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#30632238C211yesmaximal, characteristic
2C4×Q8×D251C211yesmaximal, characteristic, complemented
3#30632002C212yesmaximal, complemented
4#30631979C211yesmaximal, characteristic, complemented
5#30631888C211yesmaximal, characteristic, complemented
6#30631887C211yesmaximal, characteristic, complemented
7#30631827C212yesmaximal, complemented
8#30631790C212yesmaximal, complemented
9#30631786C212yesmaximal, complemented
10#30631743C211yesmaximal, characteristic, complemented
11#477×C251C211yesmaximal, characteristic, complemented
12C2×#16860811—21no—
13C2×Q8×D251C2211yescharacteristic
14C2×#16860799C2212yes—
15C2×#16860797C2212yes—
16#22548712C2211yescharacteristic
17#22548704—21no—
18#22548690—21no—
19#22548690C2212yes—
20Q8×Dic251C2211yescharacteristic
21C4×Dic502—21no—
22C4×Dic502C2211yescharacteristic
23#22548680—21no—
24#22548680C2211yescharacteristic
25#22548677—21no—
26#22548677C2212yes—
27C4⋊C4×D251—21no—
28C4⋊C4×D251C2211yescharacteristic
29C42×D251—21no—
30C42×D251C2211yescharacteristic
31#22548648C2212yes—
32C4⋊(C251⋊C8)C2211yescharacteristic
33#22548638C2212yes—
34#22548637C2212yes—
35#22548635C2212yes—
36(C2×D251)⋊C8C2212yes—
37#22548625C2212yes—
38C4⋊Q8×C251C2211yescharacteristic
39C4×Q8×C251C2211yescharacteristic
40#107×C251C2212yes—
41C4⋊C8×C251C2211yescharacteristic
42#103×C251C2212yes—
43#16860811—44no—
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46Q8×D251—22no—
47Q8×D251—22no—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004—21no—
51C2×D1004C2312yes—
52C2×C4×D251—21no—
53C2×C4×D251—21no—
54C2×C4×D251—22no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58#16860799—24no—
59#16860797—24no—
60C2×C251⋊C8C2312yes—
61C4⋊Dic251—21no—
62C4⋊Dic251C2311yescharacteristic
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65Dic251⋊C4—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68C4×Dic251—21no—
69C4×Dic251—21no—
70C4×Dic251—21no—
71C4×Dic251D411yescharacteristic
72C4×Dic251D411yescharacteristic
73C4×Dic251C2311yescharacteristic
74C2×Q8×C251—21no—
75C2×Q8×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2312yes—
80C4⋊C4×C251C2311yescharacteristic
81C42×C251—21no—
82C42×C251C2311yescharacteristic
83C22×D251—22no—
84C22×D251C2×D411yescharacteristic
85Dic502—24no—
86Dic502—21no—
87Dic502#4112yescomplemented
88D1004—22no—
89D1004—22no—
90D1004—42no—
91D1004—44no—
92D1004—42no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C4×D251—24no—
99C4×D251—22no—
100C4×D251—22no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—44no—
104C4×D251—42no—
105C251⋊C8—22no—
106Q8×C251—21no—
107Q8×C251—42no—
108Q8×C251#4112yescomplemented
109C2×C4×C251—21no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C8×C251—22no—
116C22×C251C22×D411yescharacteristic
117C2×D251—21no—
118C2×D251—21no—
119C2×D251—42no—
120C2×D251—42no—
121C2×D251—42no—
122C2×D251AGL(1,ℤ/8)12yes—
123C4×C251—22no—
124C4×C251—21no—
125C4×C251—42no—
126C4×C251—41no—
127C4×C251#13712yes—
128C4×C251C2×#4111yescharacteristic
129C4×C251C2×#4111yescharacteristic
130Dic251—22no—
131Dic251—21no—
132Dic251—21no—
133Dic251—21no—
134Dic251—41no—
135C2×C251#52511yescharacteristic
136C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C251C2×#13711yescharacteristic
138D251—22no—
139D251—82no—
140C251#311211yescharacteristic, complemented, Sylow
141#3112—2511nomaximal, Sylow
142#532—2511no—
143C2×C4×Q8—2511no—
144#480—2512no—
145#477—2511no—
146#477D25111yescharacteristic, complemented
147#460—2511no—
148#460—2511no—
149#448—2512no—
150#440—2512no—
151#440—2512no—
152#432—2511no—
153C2×#41—5021no—
154C22×Q8—2511no—
155C2×SD16—2512no—
156C2×SD16—2512no—
157C4⋊Q8C2×D25111yescharacteristic
158#127—2511no—
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#107—2512no—
175#107C2×D25112yes—
176C4⋊C8—2511no—
177C4⋊C8C2×D25111yescharacteristic
178#103—2512no—
179#103C2×D25112yes—
180#102—2512no—
181#102—2512no—
182C22⋊C8—2512no—
183#97—2512no—
184#41—10044no—
185C2×Q8—21no—
186C2×Q8—5021no—
187C2×Q8—5022no—
188C2×Q8—5022no—
189C2×Q8—2511no—
190C2×Q8C22×D25111yescharacteristic
191C2×D4—5022no—
192C2×D4—5021no—
193C2×D4—5021no—
194C2×D4—2512no—
195C22×C4—5021no—
196C22×C4—5022no—
197C22×C4—5021no—
198C22×C4—2511no—
199C22×C4—2511no—
200C22×C4—2511no—
201SD16—5024no—
202SD16—5024no—
203C2×C8—2512no—
204C2×C8C22×D25112yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—5021no—
208C4⋊C4—5021no—
209C4⋊C4—5021no—
210C4⋊C4—5021no—
211C4⋊C4—2511no—
212C4⋊C4C22×D25111yescharacteristic
213C4⋊C4C22×D25112yes—
214C22⋊C4—5022no—
215C22⋊C4—5022no—
216C42—21no—
217C42—5021no—
218C42—5021no—
219C42—5021no—
220C42—2511no—
221C42—2511no—
222C42—2511no—
223C42C22×D25111yescharacteristic
224C23—5022no—
225C23—2511no—
226Q8—21no—
227Q8—42no—
228Q8—5024no—
229Q8—5021no—
230Q8—2512no—
231Q8#1686081112yescomplemented
232D4—10042no—
233D4—10044no—
234D4—10042no—
235D4—5022no—
236D4—5022no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—10042no—
241C2×C4—10044no—
242C2×C4—5024no—
243C2×C4—5022no—
244C2×C4—5021no—
245C2×C4—5021no—
246C2×C4—5022no—
247C2×C4—5021no—
248C2×C4—5021no—
249C2×C4—2511no—
250C2×C4—2511no—
251C2×C4—2511no—
252C2×C4D4×D25111yescharacteristic
253C2×C4D4×D25111yescharacteristic
254C2×C4C23×D25111yescharacteristic
255C8—22no—
256C8—5022no—
257C22—10042no—
258C22—10042no—
259C22—10042no—
260C22—5021no—
261C22—5021no—
262C22—2512no—
263C22C2×D4×D25111yescharacteristic
264C4—22no—
265C4—21no—
266C4—42no—
267C4—41no—
268C4—10041no—
269C4—5022no—
270C4—5021no—
271C4—5021no—
272C4—5021no—
273C4#2254874712yes—
274C4#41×D25111yescharacteristic
275C4C2×#1686081111yescharacteristic
276C2—20082no—
277C2—5022no—
278C2#3063220011yescharacteristic
279C2AGL(1,ℤ/8)×D25111yescharacteristic
280C2C2×#2254874711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#477×C251C2
socleC22×C251C22×D4
prime radical#30632238C2
hypercenter#477D251
hypocenterC251#3112
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D251C2semidirect
3. #30632002C2semidirect
4. #30631979C2semidirect
5. #30631888C2semidirect
6. #30631887C2semidirect
7. #30631827C2semidirect
8. #30631790C2semidirect
9. #30631786C2semidirect
10. #30631743C2semidirect
11. #477×C251C2semidirect
87. Dic502#41semidirect
108. Q8×C251#41semidirect
140. C251#3112semidirect
146. #477D251semidirect
231. Q8#16860811semidirect

Maximal subgroups

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

subgroupindex
#306322382
C4×Q8×D2512
#306320022
#306319792
#306318882
#306318872
#306318272
#306317902
#306317862
#306317432
#477×C2512
#3112251