Finite Group Databasebeta

← #44559861

Subgroups — #44559861

Order 32128.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#22548740C211yesmaximal, characteristic
2C2×#22548704C211yesmaximal, characteristic, complemented
3#41×Dic251C211yesmaximal, characteristic, complemented
4#30631911C214yesmaximal, complemented
5#30631904C214yesmaximal, complemented
6#30631710C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631693C211yesmaximal, characteristic, complemented
9C2×#137×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11#22548740—24no—
12C2×#16860797C2212yes—
13C2×#16860795C2212yes—
14C2×#16860794C2211yescharacteristic
15#22548704—21no—
16#22548704—21no—
17#22548704C2214yes—
18Q8×Dic251C2212yescomplemented
19D4×Dic251—22no—
20D4×Dic251C2212yescomplemented
21#22548674—21no—
22#22548674C2211yescharacteristic
23C2×(C2×D251)⋊C4—22no—
24C2×C4×Dic251—21no—
25C2×C4×Dic251C2211yescharacteristic
26Dic251⋊C8C2214yescomplemented
27#22548639C2214yescomplemented
28#22548637C2214yes—
29C2×#41×C251C2211yescharacteristic
30C22×Q8×C251C2211yescharacteristic
31#137×C251—24no—
32C2×Q16×C251C2212yescomplemented
33C2×SD16×C251C2212yescomplemented
34C2×M16×C251C2211yescharacteristic
35C23×D251—21no—
36C2×D1004—22no—
37C2×D1004—21no—
38C2×D1004—22no—
39C2×D1004C2312yes—
40C22×Dic251—22no—
41C22×Dic251D412yes—
42#16860797—24no—
43#16860795—24no—
44#16860794—22no—
45C2×C251⋊C8C2312yes—
46C4⋊Dic251—22no—
47C4⋊Dic251C2312yes—
48C22⋊Dic251—22no—
49C22⋊Dic251—21no—
50(C2×D251)⋊C4—24no—
51(C2×D251)⋊C4—24no—
52C4×Dic251—22no—
53C4×Dic251—21no—
54C4×Dic251D411yescharacteristic
55C4×Dic251D411yescharacteristic
56C4×Dic251C2312yes—
57#41×C251—22no—
58#41×C251D414yes—
59C2×Q8×C251—21no—
60C2×Q8×C251—22no—
61C2×Q8×C251—22no—
62C2×Q8×C251C2312yes—
63C2×Q8×C251C2311yescharacteristic
64C2×D4×C251—21no—
65C2×D4×C251C2311yescharacteristic
66C22×C4×C251—21no—
67C22×C4×C251—21no—
68C22×C4×C251C2311yescharacteristic
69Q16×C251—24no—
70SD16×C251—24no—
71M16×C251—22no—
72C2×C8×C251C2312yes—
73C22×D251—22no—
74C22×D251—44no—
75C22×D251—42no—
76D1004—22no—
77D1004—22no—
78D1004—42no—
79C2×Dic251—22no—
80C2×Dic251—42no—
81C2×Dic251—44no—
82C2×Dic251C2×D412yes—
83C2×Dic251C2×D412yes—
84C251⋊C8—22no—
85C23×C251—21no—
86C23×C251C2×D411yescharacteristic
87Q8×C251—21no—
88Q8×C251—22no—
89Q8×C251—22no—
90Q8×C251—42no—
91Q8×C251C2×D412yes—
92D4×C251—24no—
93D4×C251—21no—
94D4×C251C2×D412yes—
95C2×C4×C251—21no—
96C2×C4×C251—24no—
97C2×C4×C251—22no—
98C2×C4×C251—42no—
99C2×C4×C251C2×D411yescharacteristic
100C2×C4×C251C2×D412yes—
101C2×C4×C251#4112yes—
102C2×C4×C251C2411yescharacteristic
103C8×C251—22no—
104C22×C251—22no—
105C22×C251—22no—
106C22×C251—42no—
107C22×C251#12112yes—
108C22×C251C22×D411yescharacteristic
109C2×D251—42no—
110C2×D251—44no—
111C2×D251—84no—
112C4×C251—22no—
113C4×C251—42no—
114C4×C251#12112yes—
115C4×C251C22×D411yescharacteristic
116C4×C251C2×#4111yescharacteristic
117Dic251—22no—
118Dic251—22no—
119Dic251—42no—
120C2×C251—22no—
121C2×C251—42no—
122C2×C251C2×#12111yescharacteristic
123C2×C251#57512yes—
124D251—82no—
125C251#304011yescharacteristic, complemented, Sylow
126#3040—2511nomaximal, Sylow
127C2×#137D25111yescharacteristic, complemented
128C2×AGL(1,ℤ/8)—2511no—
129C2×#124—2511no—
130C4×#41—2511no—
131#464—2514no—
132#463—2514no—
133#423—2511no—
134#419—2511no—
135#418—2511no—
136C2×#41C2×D25111yescharacteristic
137C22×Q8C2×D25111yescharacteristic
138C22×D4—2511no—
139#137—24no—
140AGL(1,ℤ/8)—5024no—
141C2×Q16C2×D25112yescomplemented
142C2×SD16—2512no—
143C2×SD16C2×D25112yescomplemented
144C2×D8—2512no—
145C2×M16—2511no—
146C2×M16C2×D25111yescharacteristic
147#124—5021no—
148#124—5021no—
149#124—2514no—
150C4×Q8—2512no—
151C4×D4—5022no—
152C4×D4—2512no—
153#117—5021no—
154#117—2511no—
155C2×C22⋊C4—5022no—
156C2×C42—5021no—
157C2×C42—2511no—
158C4⋊C8—2514no—
159#103—2514no—
160#102—2514no—
161C24—5021no—
162#41—22no—
163#41C22⋊D25114yes—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8C22×D25112yes—
168C2×Q8C22×D25111yescharacteristic
169C2×D4—21no—
170C2×D4—5021no—
171C2×D4—5022no—
172C2×D4—5022no—
173C2×D4—2512no—
174C2×D4C22×D25111yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—5022no—
178C22×C4—2512no—
179C22×C4C22×D25111yescharacteristic
180Q16—24no—
181SD16—24no—
182SD16—5024no—
183D8—5024no—
184M16—22no—
185M16—5022no—
186C2×C8—2512no—
187C2×C8C22×D25112yes—
188C4⋊C4—5022no—
189C4⋊C4—2512no—
190C22⋊C4—5024no—
191C22⋊C4—5024no—
192C22⋊C4—5021no—
193C22⋊C4—5022no—
194C42—5022no—
195C42—5021no—
196C42—2511no—
197C42—2511no—
198C42—2512no—
199C23—21no—
200C23—10044no—
201C23—10042no—
202C23—5022no—
203C23D4×D25111yescharacteristic
204Q8—21no—
205Q8—22no—
206Q8—22no—
207Q8—42no—
208Q8C2×C22⋊D25112yes—
209D4—24no—
210D4—21no—
211D4—10042no—
212D4—5022no—
213D4—5022no—
214D4C2×C22⋊D25112yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—10044no—
220C2×C4—10042no—
221C2×C4—5022no—
222C2×C4—2512no—
223C2×C4—2512no—
224C2×C4D4×D25111yescharacteristic
225C2×C4C2×C22⋊D25112yes—
226C2×C4#1686080912yes—
227C2×C4C23×D25111yescharacteristic
228C8—22no—
229C8—5022no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—20084no—
234C22—10044no—
235C22—10042no—
236C22#2254868712yes—
237C22C2×D4×D25111yescharacteristic
238C4—22no—
239C4—42no—
240C4—10042no—
241C4—5022no—
242C4—5022no—
243C4#2254868712yes—
244C4C22×C22⋊D25111yescharacteristic
245C4C2×#1686080911yescharacteristic
246C2—22no—
247C2—42no—
248C2—20082no—
249C2C2×#2254868711yescharacteristic
250C2#3063248612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548740C2
hypercenterC2×#137D251
hypocenterC251#3040
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22548704C2semidirect
3. #41×Dic251C2semidirect
4. #30631911C2semidirect
5. #30631904C2semidirect
6. #30631710C2semidirect
7. #30631696C2semidirect
8. #30631693C2semidirect
9. C2×#137×C251C2semidirect
18. Q8×Dic251C22semidirect
20. D4×Dic251C22semidirect
26. Dic251⋊C8C22semidirect
27. #22548639C22semidirect
32. C2×Q16×C251C22semidirect
33. C2×SD16×C251C22semidirect
125. C251#3040semidirect
127. C2×#137D251semidirect
141. C2×Q16C2×D251semidirect
143. C2×SD16C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C2×#225487042
#41×Dic2512
#306319112
#306319042
#306317102
#306316962
#306316932
C2×#137×C2512
#3040251