Finite Group Databasebeta

← D23×#342 #15673125

Subgroups — D23×#342 #15673125

Order 2944.

680 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 145 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×C22⋊C4×D23C211yesmaximal, characteristic
2C2×C22⋊C4×D23C212yesmaximal
3#13092184C211yesmaximal, characteristic, complemented
4#13092183C212yesmaximal, complemented
5C23×#342C211yesmaximal, characteristic, complemented
6C24×D23C2211yescharacteristic
7C22×C4×D23—22no—
8C22×C4×D23C412yescomplemented
9C2×C22⋊C4×C23C2211yescharacteristic
10C2×C22⋊C4×C23C2212yes—
11C2×C22⋊Dic23C2211yescharacteristic
12C2×C22⋊Dic23C2212yes—
13C2×(C2×D23)⋊C4—22no—
14C2×(C2×D23)⋊C4C412yescomplemented
15C22⋊C4×D23—22no—
16C22⋊C4×D23—22no—
17C22⋊C4×D23—24no—
18C22⋊C4×D23C418yescomplemented
19C24×C23C2311yescharacteristic
20C23×D23—44no—
21C23×D23C2×C411yescharacteristic
22C23×D23C2×C414yes—
23C23×D23C2×C411yescharacteristic
24C23×D23D412yes—
25C23×D23D412yes—
26C23×D23D412yes—
27C23×D23Q812yes—
28C22×C4×C23—22no—
29C22×C4×C23C2×C412yescomplemented
30C22×Dic23—22no—
31C22×Dic23C2×C412yescomplemented
32C2×C4×D23—22no—
33C2×C4×D23—22no—
34C2×C4×D23—28no—
35C2×C4×D23—48no—
36C2×C4×D23—44no—
37C22⋊C4×C23—22no—
38C22⋊C4×C23—22no—
39C22⋊C4×C23C2×C414yescomplemented
40C22⋊Dic23—22no—
41C22⋊Dic23—22no—
42C22⋊Dic23C2×C414yescomplemented
43(C2×D23)⋊C4—24no—
44(C2×D23)⋊C4—28no—
45C23×C23—42no—
46C23×C23C22×C411yescharacteristic
47C23×C23C22×C412yes—
48C23×C23C2×D412yes—
49C23×C23C2×D411yescharacteristic
50C23×C23C2×Q811yescharacteristic
51C22×D23—24no—
52C22×D23—28no—
53C22×D23—48no—
54C22×D23—44no—
55C22×D23—44no—
56C22×D23—42no—
57C22×D23—44no—
58C22×D23C4212yes—
59C22×D23C4212yes—
60C22×D23C22⋊C412yes—
61C22×D23C22⋊C414yes—
62C22×D23C22⋊C411yescharacteristic
63C22×D23C22⋊C414yes—
64C22×D23C22⋊C411yescharacteristic
65C22×D23C4⋊C412yes—
66C22×D23C4⋊C414yes—
67C22×D23C4⋊C414yes—
68C22×D23C4⋊C412yes—
69C2×C4×C23—22no—
70C2×C4×C23—24no—
71C2×C4×C23—44no—
72C2×C4×C23—42no—
73C2×Dic23—22no—
74C2×Dic23—24no—
75C2×Dic23—44no—
76C2×Dic23—42no—
77C4×D23—416no—
78C4×D23—44no—
79C4×D23—44no—
80C2×#342—231nomaximal, Sylow
81C22×C23—22no—
82C22×C23—24no—
83C22×C23—42no—
84C22×C23—41no—
85C22×C23—41no—
86C22×C23C2×C4211yescharacteristic
87C22×C23C2×C22⋊C412yes—
88C22×C23C2×C22⋊C411yescharacteristic
89C22×C23C2×C4⋊C411yescharacteristic
90C22×C23C2×C4⋊C412yes—
91C2×D23—22no—
92C2×D23—24no—
93C2×D23—28no—
94C2×D23—24no—
95C2×D23—24no—
96C2×D23—28no—
97C2×D23—42no—
98C2×D23—44no—
99C2×D23—44no—
100C2×D23—42no—
101C2×D23#9512yes—
102C2×D23#9514yes—
103C2×D23#9512yes—
104C2×D23#9914yes—
105C4×C23—44no—
106C4×C23—42no—
107Dic23—44no—
108Dic23—42no—
109C22×C22⋊C4—232no—
110C22×C22⋊C4—231no—
111#342—232no—
112#342—231no—
113#342D2311yescharacteristic, direct product factor
114C2×C23—22no—
115C2×C23—21no—
116C2×C23—21no—
117C2×C23—22no—
118C2×C23C2×#9511yescharacteristic
119C2×C23C2×#9912yes—
120D23—22no—
121D23—24no—
122D23#34214yesdirect product factor
123C25—231no—
124C23×C4—232no—
125C23×C4—462no—
126C2×C22⋊C4—232no—
127C2×C22⋊C4—238no—
128C2×C22⋊C4—231no—
129C2×C22⋊C4—232no—
130C2×C22⋊C4—464no—
131C2×C22⋊C4—462no—
132C2×C22⋊C4—462no—
133C2×C22⋊C4—462no—
134C2×C22⋊C4C2×D2311yescharacteristic
135C2×C22⋊C4C2×D2312yes—
136C23C2×#34211yescharacteristic, complemented, Sylow
137C24—231no—
138C24—232no—
139C24—234no—
140C24—231no—
141C24—232no—
142C24—232no—
143C24—232no—
144C24—924no—
145C24C22×D2311yescharacteristic
146C22×C4—22no—
147C22×C4—232no—
148C22×C4—468no—
149C22×C4—462no—
150C22×C4—462no—
151C22×C4—462no—
152C22×C4—928no—
153C22×C4—924no—
154C22×C4C4×D2312yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—234no—
158C22⋊C4—468no—
159C22⋊C4—464no—
160C22⋊C4—462no—
161C22⋊C4—462no—
162C22⋊C4C4×D2314yescomplemented
163C23—42no—
164C23—231no—
165C23—232no—
166C23—234no—
167C23—232no—
168C23—231no—
169C23—232no—
170C23—234no—
171C23—232no—
172C23—234no—
173C23—234no—
174C23—232no—
175C23—464no—
176C23—468no—
177C23—922no—
178C23—924no—
179C23—928no—
180C23—924no—
181C23—924no—
182C23C2×C4×D2311yescharacteristic
183C23C2×C4×D2312yes—
184C23D4×D2312yes—
185C23D4×D2311yescharacteristic
186C23Q8×D2311yescharacteristic
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—464no—
192C2×C4—462no—
193C2×C4—924no—
194C2×C4—9216no—
195C2×C4—924no—
196C2×C4—924no—
197C2×C4—922no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—234no—
204C22—232no—
205C22—232no—
206C22—234no—
207C22—468no—
208C22—464no—
209C22—464no—
210C22—468no—
211C22—462no—
212C22—464no—
213C22—924no—
214C22—922no—
215C22—922no—
216C22—924no—
217C22C42×D2311yescharacteristic
218C22C22⋊C4×D2312yes—
219C22C22⋊C4×D2311yescharacteristic
220C22C4⋊C4×D2311yescharacteristic
221C22C4⋊C4×D2312yes—
222C4—44no—
223C4—42no—
224C4—924no—
225C4—922no—
226C2—22no—
227C2—21no—
228C2—21no—
229C2—22no—
230C2—234no—
231C2—464no—
232C2—462no—
233C2#95×D2311yescharacteristic
234C2#99×D2312yes—
subgroupG/H
centerC22C22⋊C4×D23
commutatorC22×C23C2×C42
FrattiniC24C22×D23
FittingC23×#342C2
socleC22×C23C2×C22⋊C4
prime radicalC24×D23C22
cosocleC24×C23C23
hypercenter#342D23
hypocenterC23C2×#342
LCSG ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #13092184C2semidirect
4. #13092183C2semidirect
5. C23×#342C2semidirect
8. C22×C4×D23C4semidirect
14. C2×(C2×D23)⋊C4C4semidirect
18. C22⋊C4×D23C4semidirect
29. C22×C4×C23C2×C4semidirect
31. C22×Dic23C2×C4semidirect
39. C22⋊C4×C23C2×C4semidirect
42. C22⋊Dic23C2×C4semidirect
113. #342D23direct (N × H)
122. D23#342direct (N × H)
136. C23C2×#342semidirect
154. C22×C4C4×D23semidirect
162. C22⋊C4C4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D232
#130921842
#130921832
C23×#3422
C2×#34223