Finite Group Databasebeta

← #44548691

Subgroups — #44548691

Order 32128.

440 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×(C2×D251)⋊C4C211yesmaximal, characteristic
2#30631651C211yesmaximal, characteristic, complemented
3#30631410C214yesmaximal, complemented
4#407×C251C211yesmaximal, characteristic, complemented
5C24×D251C411yescharacteristic
6C23×Dic251C411yescharacteristic
7C2×#16860794—21no—
8C2×(C2×D251)⋊C4—21no—
9C2×(C2×D251)⋊C4—21no—
10C2×(C2×D251)⋊C4—21no—
11C2×(C2×D251)⋊C4—21no—
12C2×(C2×D251)⋊C4C412yes—
13C2×(C2×D251)⋊C4C2212yes—
14C22⋊(C251⋊C8)—21no—
15C22⋊(C251⋊C8)C2212yescomplemented
16C23×C4×C251C2211yescharacteristic
17C2×M16×C251—21no—
18C22⋊C8×C251—21no—
19C22⋊C8×C251C2212yescomplemented
20C23×D251—24no—
21C23×D251—21no—
22C23×D251—21no—
23C23×D251—42no—
24C23×D251—42no—
25C23×D251C2×C412yes—
26C22×Dic251—21no—
27C22×Dic251—21no—
28C22×Dic251—42no—
29C22×Dic251C2×C412yes—
30#16860794—42no—
31C2×C251⋊C8—22no—
32(C2×D251)⋊C4—22no—
33(C2×D251)⋊C4—22no—
34(C2×D251)⋊C4—22no—
35(C2×D251)⋊C4—22no—
36C24×C251C2×C411yescharacteristic
37C22×C4×C251—42no—
38C22×C4×C251C2×C411yescharacteristic
39C22×C4×C251D411yescharacteristic
40C22×C4×C251D411yescharacteristic
41C22×C4×C251D411yescharacteristic
42C22×C4×C251D411yescharacteristic
43C22×C4×C251C2311yescharacteristic
44M16×C251—42no—
45C2×C8×C251—22no—
46C22×D251—24no—
47C22×D251—24no—
48C22×D251—22no—
49C22×D251—22no—
50C22×D251—416no—
51C22×D251—42no—
52C22×D251—42no—
53C22×D251—42no—
54C22×D251—42no—
55C22×D251—42no—
56C22×D251—42no—
57C22×D251M1614yescomplemented
58C2×Dic251—22no—
59C2×Dic251—42no—
60C2×Dic251—42no—
61C2×Dic251—42no—
62C251⋊C8—42no—
63C23×C251—22no—
64C23×C251—22no—
65C23×C251—22no—
66C23×C251C22⋊C411yescharacteristic
67C23×C251C22⋊C411yescharacteristic
68C23×C251C22×C411yescharacteristic
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251—41no—
75C2×C4×C251C22⋊C411yescharacteristic
76C2×C4×C251C22⋊C411yescharacteristic
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C8×C251—42no—
80C22×C251—28no—
81C22×C251—21no—
82C22×C251—21no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251—21no—
87C22×C251#9912yes—
88C22×C251#10012yes—
89C22×C251C2×C22⋊C411yescharacteristic
90C22×C251C2×M1612yes—
91C2×D251—24no—
92C2×D251—48no—
93C2×D251—48no—
94C2×D251—44no—
95C2×D251—42no—
96C2×D251—42no—
97C2×D251—44no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—21no—
101C4×C251—21no—
102Dic251—42no—
103C2×C251—22no—
104C2×C251—22no—
105C2×C251—22no—
106C2×C251#40711yescharacteristic
107C2×C251C2×#9911yescharacteristic
108C2×C251C2×#10011yescharacteristic
109D251—44no—
110C251#146211yescharacteristic, complemented, Sylow
111#1462—2511nomaximal, Sylow
112C22×C22⋊C4—2511no—
113#407—2511no—
114#407D25111yescharacteristic, complemented
115#323—2514no—
116C25—2511no—
117C23×C4—2511no—
118C23×C4C2×D25111yescharacteristic
119C2×M16—21no—
120C2×M16—5021no—
121C2×C22⋊C4—5021no—
122C2×C22⋊C4—5021no—
123C2×C22⋊C4—5021no—
124C2×C22⋊C4—5021no—
125C2×C22⋊C4—2512no—
126C2×C22⋊C4—2512no—
127C22⋊C8—21no—
128C22⋊C8—5021no—
129C22⋊C8—2512no—
130C22⋊C8C2×D25112yescomplemented
131C24—10042no—
132C24—10042no—
133C24—5024no—
134C24—5021no—
135C24—5021no—
136C24—2512no—
137C24C4×D25111yescharacteristic
138C22×C4—42no—
139C22×C4—10042no—
140C22×C4—5021no—
141C22×C4—5021no—
142C22×C4—2512no—
143C22×C4C4×D25111yescharacteristic
144C22×C4C22⋊D25111yescharacteristic
145C22×C4C22⋊D25111yescharacteristic
146C22×C4D100411yescharacteristic
147C22×C4D100411yescharacteristic
148C22×C4C22×D25111yescharacteristic
149M16—42no—
150M16—10042no—
151C2×C8—22no—
152C2×C8—5022no—
153C22⋊C4—5022no—
154C22⋊C4—5022no—
155C22⋊C4—5022no—
156C22⋊C4—5022no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—10042no—
161C23—10042no—
162C23—100416no—
163C23—10042no—
164C23—10042no—
165C23—10042no—
166C23—10042no—
167C23—5024no—
168C23—5024no—
169C23—5022no—
170C23—5022no—
171C23—2514no—
172C23(C2×D251)⋊C411yescharacteristic
173C23(C2×D251)⋊C411yescharacteristic
174C23C2×C4×D25111yescharacteristic
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—10042no—
182C2×C4—10042no—
183C2×C4—10042no—
184C2×C4—5022no—
185C2×C4(C2×D251)⋊C411yescharacteristic
186C2×C4(C2×D251)⋊C411yescharacteristic
187C2×C4C2×C22⋊D25111yescharacteristic
188C2×C4C2×D100411yescharacteristic
189C8—42no—
190C8—10042no—
191C22—28no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—10044no—
199C22—10048no—
200C22—10048no—
201C22—10044no—
202C22—10042no—
203C22—10042no—
204C22—5024no—
205C22#2254862912yes—
206C22#2254863112yes—
207C22C2×(C2×D251)⋊C411yescharacteristic
208C22M16×D25112yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—10042no—
214C2—22no—
215C2—22no—
216C2—22no—
217C2—10044no—
218C2#3063165011yescharacteristic
219C2C2×#2254862911yescharacteristic
220C2C2×#2254863111yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC24×D251C4
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1462
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #30631651C2semidirect
3. #30631410C2semidirect
4. #407×C251C2semidirect
15. C22⋊(C251⋊C8)C22semidirect
19. C22⋊C8×C251C22semidirect
57. C22×D251M16semidirect
110. C251#1462semidirect
114. #407D251semidirect
130. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D251)⋊C42
#306316512
#306314102
#407×C2512
#1462251