Finite Group Databasebeta

← #44559599

Subgroups — #44559599

Order 32128.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22548746C212yesmaximal, complemented
2#30632198C211yesmaximal, characteristic
3#30631843C214yesmaximal, complemented
4#30631826C214yesmaximal, complemented
5#30631696C212yesmaximal, complemented
6#30631651C211yesmaximal, characteristic, complemented
7C23×Q8×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9#22548746—28no—
10C2×#16860800C2214yescomplemented
11C2×#16860797C2214yescomplemented
12C2×#16860794C2212yes—
13#22548698—21no—
14#22548694C2212yes—
15#22548689C2212yes—
16#22548684—21no—
17C4×C22⋊D251—22no—
18#22548674C2211yescharacteristic
19#22548639C2218yescomplemented
20C22⋊(C251⋊C8)C2214yes—
21C22×Q8×C251—28no—
22C22×Q8×C251—22no—
23C22×Q8×C251—21no—
24C22×Q8×C251C2214yes—
25C22×Q8×C251C2212yes—
26C23×C4×C251—21no—
27C23×C4×C251C2211yescharacteristic
28#16860810—24no—
29C2×Dic502C2311yescharacteristic
30C2×D1004C2311yescharacteristic
31C2×C22⋊D251—21no—
32C2×C22⋊D251—22no—
33C2×C4×D251—21no—
34#16860800—28no—
35#16860797—28no—
36#16860794—24no—
37C2×C251⋊C8C2314yes—
38C4⋊Dic251C2312yes—
39Dic251⋊C4—22no—
40C22⋊Dic251—21no—
41C22⋊Dic251—22no—
42(C2×D251)⋊C4—22no—
43C4×Dic251—21no—
44C24×C251D411yescharacteristic
45C2×Q8×C251—216no—
46C2×Q8×C251—28no—
47C2×Q8×C251—216no—
48C2×Q8×C251—24no—
49C2×Q8×C251—24no—
50C2×Q8×C251—22no—
51C2×Q8×C251D418yescomplemented
52C2×Q8×C251C2314yes—
53C22×C4×C251—28no—
54C22×C4×C251—24no—
55C22×C4×C251—22no—
56C22×C4×C251—24no—
57C22×C4×C251D412yes—
58C22×C4×C251D411yescharacteristic
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C22×D251—21no—
62Dic502—22no—
63D1004—22no—
64C22⋊D251—42no—
65C22⋊D251—44no—
66C2×Dic251—22no—
67C2×Dic251—21no—
68C4×D251—42no—
69C251⋊C8—24no—
70C23×C251—22no—
71C23×C251—24no—
72C23×C251C2×D411yescharacteristic
73C23×C251C2×D412yes—
74Q8×C251—216no—
75Q8×C251—28no—
76Q8×C251—24no—
77Q8×C251C2×D418yes—
78C2×C4×C251—216no—
79C2×C4×C251—28no—
80C2×C4×C251—22no—
81C2×C4×C251—24no—
82C2×C4×C251—24no—
83C2×C4×C251—24no—
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D414yes—
87C2×C4×C251C2411yescharacteristic
88C22×C251—28no—
89C22×C251—22no—
90C22×C251—22no—
91C22×C251—22no—
92C22×C251#12012yes—
93C22×C251#13714yescomplemented
94C22×C251C22×D411yescharacteristic
95C2×D251—41no—
96C2×D251—42no—
97C4×C251—28no—
98C4×C251—22no—
99C4×C251#12012yes—
100C4×C251C22×D412yes—
101Dic251—41no—
102Dic251—42no—
103C2×C251—22no—
104C2×C251—24no—
105C2×C251C2×#12011yescharacteristic
106C2×C251C2×#13712yes—
107D251—81no—
108C251#300411yescharacteristic, complemented, Sylow
109#3004—2511nomaximal, Sylow
110C23×Q8D25111yescharacteristic, complemented
111C2×#137—2512no—
112#525—2511no—
113#451—2514no—
114#448—2514no—
115#419—2512no—
116#407—2511no—
117C2×#41—2511no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D25114yes—
122C22×Q8C2×D25112yes—
123C23×C4—21no—
124C23×C4C2×D25111yescharacteristic
125#137—5028no—
126C2×Q16—2514no—
127C2×SD16—2514no—
128C2×M16—2512no—
129#123—5021no—
130C22⋊Q8—2512no—
131#121—2512no—
132#120—5021no—
133C4×D4—5022no—
134#117—2511no—
135#103—2518no—
136C22⋊C8—2514no—
137C24C22⋊D25111yescharacteristic
138#41—5024no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—24no—
145C2×Q8—2511no—
146C2×Q8C22⋊D25118yescomplemented
147C2×Q8C22×D25114yes—
148C2×D4—5022no—
149C2×D4—5021no—
150C2×D4—2511no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—5021no—
156C22×C4C22⋊D25112yes—
157C22×C4C22⋊D25111yescharacteristic
158C22×C4C22×D25112yes—
159C22×C4C22×D25111yescharacteristic
160Q16—5028no—
161SD16—5028no—
162M16—5024no—
163C2×C8—2514no—
164C4⋊C4—5022no—
165C4⋊C4—2512no—
166C22⋊C4—5021no—
167C22⋊C4—5022no—
168C22⋊C4—5022no—
169C42—5021no—
170C23—24no—
171C23—22no—
172C23—5021no—
173C23C2×C22⋊D25112yes—
174C23C2×C22⋊D25111yescharacteristic
175Q8—216no—
176Q8—24no—
177Q8—28no—
178Q8—5022no—
179Q8C2×C22⋊D25118yes—
180D4—10044no—
181D4—10042no—
182D4—5022no—
183C2×C4—216no—
184C2×C4—24no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—10042no—
190C2×C4—5022no—
191C2×C4—5021no—
192C2×C4C2×C22⋊D25112yes—
193C2×C4C2×C22⋊D25114yes—
194C2×C4C2×C22⋊D25111yescharacteristic
195C2×C4C23×D25111yescharacteristic
196C8—5024no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—10042no—
202C22—10041no—
203C22#2254868412yes—
204C22#2254874614yescomplemented
205C22C22×C22⋊D25111yescharacteristic
206C4—28no—
207C4—22no—
208C4—10042no—
209C4—10041no—
210C4#2254868412yes—
211C4C22×C22⋊D25112yes—
212C2—24no—
213C2—22no—
214C2—20081no—
215C2C2×#2254868411yescharacteristic
216C2C2×#2254874612yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC23×Q8×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenterC23×Q8D251
hypocenterC251#3004
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
3. #30631843C2semidirect
4. #30631826C2semidirect
5. #30631696C2semidirect
6. #30631651C2semidirect
7. C23×Q8×C251C2semidirect
10. C2×#16860800C22semidirect
11. C2×#16860797C22semidirect
19. #22548639C22semidirect
51. C2×Q8×C251D4semidirect
93. C22×C251#137semidirect
108. C251#3004semidirect
110. C23×Q8D251semidirect
146. C2×Q8C22⋊D251semidirect
204. C22#22548746semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
#306321982
#306318432
#306318262
#306316962
#306316512
C23×Q8×C2512
#3004251