Finite Group Databasebeta

← C22×#5125 #11658880

Subgroups — C22×#5125 #11658880

Order 768.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic
2C23×#208C212yesmaximal
3C2×#5125C2124yesmaximal, direct product factor
4C22×#819C214yesmaximal, complemented
5C22×#430—31nomaximal, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16C414yes—
8C22×C3×M16C4112yes—
9C22×C3×M16C22112yes—
10C23×C3×C8C2212yes—
11C22×#208—22no—
12C22×#208—212no—
13C22×#208C2214yes—
14C22×#208C22124yes—
15C23×C3⋊C8—22no—
16#5125C22164yesdirect product factor
17C2×#819C22148yescomplemented
18C23×M16—32no—
19C23×M16D311yescharacteristic
20C2×#430—324no—
21C22×#108—34no—
22C25×C3Q811yescharacteristic
23C23×C3×C4C2×C412yes—
24C23×C3×C4C2×C416yes—
25C23×C3×C4D412yes—
26C23×C3×C4D416yes—
27C23×C3×C4Q811yescharacteristic
28C23×C3×C4Q816yes—
29C23×C3×C4C2311yescharacteristic
30C23×C3×C4C2316yes—
31C2×C3×M16C2×C4148yes—
32C2×C3×M16C2×C4148yes—
33C2×C3×M16C23116yes—
34C22×C3×C8C2×C4112yes—
35C22×C3×C8C2×C414yes—
36C22×C3×C8C23112yes—
37C2×#208—224no—
38C2×#208—212no—
39C2×#208—248no—
40C2×#208C23132yes—
41C2×#208C23124yes—
42C22×C3⋊C8—224no—
43C22×C3⋊C8—24no—
44#819C23164yescomplemented
45C24×C4C2×D311yescharacteristic
46C22×M16—34no—
47C22×M16—324no—
48C22×M16—612no—
49C22×M16—62no—
50C22×M16Dic314yes—
51C22×M16Dic3112yes—
52C22×M16C2×D3112yes—
53C23×C8—62no—
54C23×C8C2×D312yes—
55#430—364no—
56C2×#108—348no—
57C24×C3—44no—
58C24×C3C4⋊C412yes—
59C24×C3C4⋊C416yes—
60C24×C3C2×Q811yescharacteristic
61C24×C3C2×Q816yes—
62C22×C3×C4C4⋊C4112yes—
63C22×C3×C4C4⋊C4112yes—
64C22×C3×C4C4⋊C416yes—
65C22×C3×C4C4⋊C4124yes—
66C22×C3×C4C4⋊C412yes—
67C22×C3×C4C22×C413yes—
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C22×C4112yes—
70C22×C3×C4C22×C4112yes—
71C22×C3×C4C2×D4112yes—
72C22×C3×C4C2×D413yes—
73C22×C3×C4C2×D4112yes—
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4C2×Q813yes—
76C22×C3×C4C2×Q816yes—
77C22×C3×C4C2×Q8112yes—
78C22×C3×C4C2413yes—
79C22×C3×C4C2414yes—
80C3×M16C22×C4164yes—
81C2×C3×C8C22×C4124yes—
82C2×C3×C8C22×C4124yes—
83C2×C3×C8C2418yes—
84#208—232no—
85#208—224no—
86#208C24116yes—
87C2×C3⋊C8—224no—
88C2×C3⋊C8—232no—
89C25Dic611yescharacteristic
90C23×C4Dic611yescharacteristic
91C23×C4Dic616yes—
92C23×C4D1216yes—
93C23×C4D1212yes—
94C23×C4C2×Dic312yes—
95C23×C4C2×Dic316yes—
96C23×C4C22×D316yes—
97C23×C4C22×D311yescharacteristic
98C2×M16—324no—
99C2×M16—332no—
100C2×M16—624no—
101C2×M16—612no—
102C2×M16—648no—
103C2×M16C2×Dic3148yes—
104C2×M16C2×Dic3148yes—
105C2×M16C22×D3116yes—
106C22×C8—624no—
107C22×C8—64no—
108C22×C8C2×Dic314yes—
109C22×C8C2×Dic3112yes—
110C22×C8C22×D3112yes—
111#108—364no—
112C23×C3—28no—
113C23×C3—24no—
114C23×C3—412no—
115C23×C3—412no—
116C23×C3C2×C4⋊C413yes—
117C23×C3C2×C4⋊C4112yes—
118C23×C3C2×C4⋊C4112yes—
119C23×C3C2×C4⋊C411yescharacteristic
120C23×C3C22×Q813yes—
121C23×C3C22×Q814yes—
122C2×C3×C4C2×C4⋊C416yes—
123C2×C3×C4C2×C4⋊C4124yes—
124C2×C3×C4C2×C4⋊C4124yes—
125C2×C3×C4C2×C4⋊C4112yes—
126C2×C3×C4C2×C4⋊C413yes—
127C2×C3×C4C2×C4⋊C4112yes—
128C2×C3×C4C2×C4⋊C413yes—
129C2×C3×C4C23×C413yes—
130C2×C3×C4C23×C413yes—
131C2×C3×C4C23×C418yes—
132C2×C3×C4C22×D413yes—
133C2×C3×C4C22×D418yes—
134C2×C3×C4C22×D413yes—
135C2×C3×C4C22×Q813yes—
136C2×C3×C4C22×Q814yes—
137C2×C3×C4C2511yescharacteristic
138C3×C8C23×C4116yes—
139C3⋊C8—216no—
140C24—44no—
141C24C4⋊Dic312yes—
142C24C4⋊Dic316yes—
143C24C2×Dic611yescharacteristic
144C24C2×Dic616yes—
145C22×C4C4⋊Dic316yes—
146C22×C4C4⋊Dic3112yes—
147C22×C4C4⋊Dic3124yes—
148C22×C4C4⋊Dic312yes—
149C22×C4C4⋊Dic3112yes—
150C22×C4C2×Dic616yes—
151C22×C4C2×Dic6112yes—
152C22×C4C2×Dic613yes—
153C22×C4C2×D12112yes—
154C22×C4C2×D1213yes—
155C22×C4C2×D12112yes—
156C22×C4C2×D1211yescharacteristic
157C22×C4C22×Dic311yescharacteristic
158C22×C4C22×Dic313yes—
159C22×C4C22×Dic3112yes—
160C22×C4C22×Dic3112yes—
161C22×C4C23×D313yes—
162C22×C4C23×D314yes—
163M16—316no—
164M16—632no—
165M16—624no—
166M16C22×Dic3164yes—
167C2×C8—624no—
168C2×C8—632no—
169C2×C8C22×Dic3124yes—
170C2×C8C22×Dic3124yes—
171C2×C8C23×D318yes—
172C22×C3—224no—
173C22×C3—26no—
174C22×C3—26no—
175C22×C3—412no—
176C22×C3—44no—
177C22×C3#43014yescomplemented
178C22×C3C22×C4⋊C413yes—
179C22×C3C22×C4⋊C418yes—
180C22×C3C22×C4⋊C413yes—
181C22×C3C23×Q811yescharacteristic
182C3×C4C22×C4⋊C413yes—
183C3×C4C22×C4⋊C413yes—
184C3×C4C22×C4⋊C418yes—
185C3×C4C24×C411yescharacteristic
186C3×C4C23×D411yescharacteristic
187C23—28no—
188C23—24no—
189C23—412no—
190C23—412no—
191C23C2×C4⋊Dic3112yes—
192C23C2×C4⋊Dic313yes—
193C23C2×C4⋊Dic3112yes—
194C23C2×C4⋊Dic311yescharacteristic
195C23C22×Dic613yes—
196C23C22×Dic614yes—
197C2×C4C2×C4⋊Dic3124yes—
198C2×C4C2×C4⋊Dic316yes—
199C2×C4C2×C4⋊Dic3112yes—
200C2×C4C2×C4⋊Dic3124yes—
201C2×C4C2×C4⋊Dic3112yes—
202C2×C4C2×C4⋊Dic313yes—
203C2×C4C2×C4⋊Dic313yes—
204C2×C4C22×Dic613yes—
205C2×C4C22×Dic614yes—
206C2×C4C22×D1213yes—
207C2×C4C22×D1218yes—
208C2×C4C22×D1213yes—
209C2×C4C23×Dic313yes—
210C2×C4C23×Dic313yes—
211C2×C4C23×Dic318yes—
212C2×C4C24×D311yescharacteristic
213C8—616no—
214C8C23×Dic3116yes—
215C2×C3—28no—
216C2×C3—23no—
217C2×C3—21no—
218C2×C3C2×#43016yescomplemented
219C2×C3C23×C4⋊C411yescharacteristic
220C22—224no—
221C22—26no—
222C22—26no—
223C22—412no—
224C22—44no—
225C22#512514yesdirect product factor
226C22C22×C4⋊Dic318yes—
227C22C22×C4⋊Dic313yes—
228C22C22×C4⋊Dic313yes—
229C22C23×Dic611yescharacteristic
230C4C22×C4⋊Dic318yes—
231C4C22×C4⋊Dic313yes—
232C4C22×C4⋊Dic313yes—
233C4C23×D1211yescharacteristic
234C4C24×Dic311yescharacteristic
235C3C22×#43011yescharacteristic, complemented, Sylow
236C2—21no—
237C2—28no—
238C2—23no—
239C2C2×#512516yesdirect product factor
240C2C23×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC23×C3×M16C2
socleC23×C3C2×C4⋊C4
prime radicalC25×C3Q8
cosocleC2×C3×C4C25
hypercenterC23×M16D3
hypocenterC3C22×#430
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#5125C2direct (N × H)
4. C22×#819C2semidirect
16. #5125C22direct (N × H)
17. C2×#819C22semidirect
44. #819C23semidirect
177. C22×C3#430semidirect
218. C2×C3C2×#430semidirect
225. C22#5125direct (N × H)
235. C3C22×#430semidirect
239. C2C2×#5125direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C23×#2082
C2×#51252
C22×#8192
C22×#4303