Finite Group Databasebeta

← #15676724

Subgroups — #15676724

Order 2944.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D23C211yesmaximal, characteristic
2C2×#10625100C212yesmaximal, complemented
3#13092332C214yesmaximal, complemented
4#13092296C212yesmaximal, complemented
5#13092289C212yesmaximal, complemented
6#13092273C211yesmaximal, characteristic, complemented
7#13092272C212yesmaximal, complemented
8C22×C22⋊C4×C23C211yesmaximal, characteristic, complemented
9C25×C23C2211yescharacteristic
10C24×D23C2211yescharacteristic
11C23×C4×C23C2211yescharacteristic, complemented
12C23×C4×C23C2211yescharacteristic, complemented
13C22×C22⋊D23—24no—
14C22×C22⋊D23—28no—
15C22×C22⋊D23C2212yes—
16C22×C22⋊D23C2212yes—
17C23×Dic23C2211yescharacteristic
18C2×C22⋊C4×C23—24no—
19C2×C22⋊C4×C23C2212yes—
20C2×C22⋊C4×C23C2212yescomplemented
21C2×C22⋊Dic23C2212yes—
22C2×C22⋊Dic23C2212yescomplemented
23C2×C22⋊Dic23C2212yescomplemented
24#10625100—28no—
25C2×(C2×D23)⋊C4C2214yescomplemented
26C2×(C2×D23)⋊C4C2214yescomplemented
27C2×Dic23⋊C4C2214yescomplemented
28C23⋊#95C2214yescomplemented
29C24×C23—24no—
30C24×C23—24no—
31C24×C23—42no—
32C24×C23D414yes—
33C24×C23C2311yescharacteristic
34C24×C23C2311yescharacteristic
35C24×C23C2311yescharacteristic
36C23×D23—22no—
37C23×D23—24no—
38C23×D23—24no—
39C23×D23—42no—
40C23×D23C2312yes—
41C22×C4×C23—24no—
42C22×C4×C23—22no—
43C22×C4×C23—22no—
44C22×C4×C23—42no—
45C22×C4×C23C2312yes—
46C22×C4×C23C2312yes—
47C2×C22⋊D23—232no—
48C2×C22⋊D23—24no—
49C2×C22⋊D23—24no—
50C2×C22⋊D23—24no—
51C2×C22⋊D23—24no—
52C2×C22⋊D23—24no—
53C2×C22⋊D23D418yes—
54C22×Dic23—22no—
55C22×Dic23—24no—
56C22×Dic23C2314yes—
57C22×Dic23C2312yes—
58C22⋊C4×C23—24no—
59C22⋊C4×C23—24no—
60C22⋊Dic23—24no—
61C22⋊Dic23—24no—
62C22⋊Dic23—42no—
63(C2×D23)⋊C4—28no—
64(C2×D23)⋊C4—44no—
65Dic23⋊C4—28no—
66C23×C23—22no—
67C23×C23—22no—
68C23×C23—22no—
69C23×C23—22no—
70C23×C23—28no—
71C23×C23—28no—
72C23×C23—22no—
73C23×C23—22no—
74C23×C23—22no—
75C23×C23—44no—
76C23×C23—416no—
77C23×C23C2×D414yes—
78C23×C23C2×D412yes—
79C23×C23C2×D412yes—
80C23×C23C2×D412yes—
81C23×C23#4114yes—
82C23×C23C2411yescharacteristic
83C22×D23—28no—
84C22×D23—22no—
85C22×D23—22no—
86C22×D23—24no—
87C22×D23—24no—
88C22×D23—416no—
89C22×D23—44no—
90C22×D23—44no—
91C22×D23C2×D414yes—
92C2×C4×C23—24no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—24no—
96C2×C4×C23—44no—
97C2×C4×C23—44no—
98C22⋊D23—216no—
99C22⋊D23—216no—
100C2×Dic23—28no—
101C2×Dic23—22no—
102C2×Dic23—22no—
103C2×Dic23—24no—
104C2×Dic23—24no—
105C2×Dic23—24no—
106C2×Dic23C2×D414yes—
107#2405—231nomaximal, Sylow
108C22×C23—22no—
109C22×C23—22no—
110C22×C23—22no—
111C22×C23—22no—
112C22×C23—28no—
113C22×C23—28no—
114C22×C23—22no—
115C22×C23—22no—
116C22×C23—22no—
117C22×C23—416no—
118C22×C23—44no—
119C22×C23#12014yes—
120C22×C23#12314yescomplemented
121C22×C23C22×D412yes—
122C22×C23C22×D411yescharacteristic
123C22×C23C2×#4112yes—
124C22×C23H(5,𝔽2)12yes—
125C2×D23—24no—
126C2×D23—28no—
127C2×D23—416no—
128C2×D23—42no—
129C2×D23—42no—
130C2×D23—44no—
131C4×C23—42no—
132C4×C23—42no—
133Dic23—24no—
134Dic23—44no—
135C23×D4—231no—
136C2×#123—232no—
137C22×C22⋊C4D2311yescharacteristic, complemented
138#394—234no—
139#386—232no—
140#385—232no—
141#379—231no—
142#379—232no—
143C2×C23—24no—
144C2×C23—24no—
145C2×C23—42no—
146C2×C23C2×#12011yescharacteristic
147C2×C23C2×#12311yescharacteristic
148C2×C23#53411yescharacteristic
149C2×C23#54614yes—
150D23—44no—
151C25—231no—
152C25C2×D2311yescharacteristic
153C22×D4—232no—
154C22×D4—232no—
155C22×D4—468no—
156C22×D4—464no—
157C23×C4—231no—
158C23×C4C2×D2311yescharacteristic, complemented
159C23×C4C2×D2311yescharacteristic, complemented
160#123—468no—
161C2×C4⋊C4—234no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—232no—
164C2×C22⋊C4—232no—
165C2×C22⋊C4—234no—
166C2×C22⋊C4—232no—
167C2×C22⋊C4—234no—
168C2×C22⋊C4C2×D2312yes—
169C2×C22⋊C4C2×D2312yescomplemented
170#95—234no—
171C23#240511yescharacteristic, complemented, Sylow
172C24—24no—
173C24—24no—
174C24—42no—
175C24—232no—
176C24—464no—
177C24—462no—
178C24—464no—
179C24—922no—
180C24C22⋊D2314yes—
181C24C22×D2311yescharacteristic
182C24C22×D2311yescharacteristic
183C24C22×D2311yescharacteristic
184C2×D4—238no—
185C2×D4—464no—
186C2×D4—464no—
187C2×D4—4632no—
188C2×D4—464no—
189C2×D4—464no—
190C2×D4—464no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—232no—
196C22×C4—234no—
197C22×C4—464no—
198C22×C4—462no—
199C22×C4C22×D2312yes—
200C22×C4C22×D2312yes—
201C4⋊C4—468no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—468no—
205C22⋊C4—464no—
206C22⋊C4—464no—
207C22⋊C4—924no—
208C22⋊C4—922no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—234no—
221C23—468no—
222C23—464no—
223C23—462no—
224C23—464no—
225C23—462no—
226C23—9216no—
227C23—924no—
228C23—924no—
229C23#6728814yes—
230C23D4×D2314yes—
231C23C2×C22⋊D2312yes—
232C23C2×C22⋊D2312yes—
233C23C2×C22⋊D2312yes—
234C23C23×D2311yescharacteristic
235D4—4616no—
236D4—4616no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—234no—
244C2×C4—468no—
245C2×C4—462no—
246C2×C4—462no—
247C2×C4—464no—
248C2×C4—464no—
249C2×C4—464no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—464no—
262C22—468no—
263C22—9216no—
264C22—922no—
265C22—922no—
266C22—924no—
267C22#1062510014yescomplemented
268C22#1062510814yes—
269C22C2×#6728812yes—
270C22C2×D4×D2312yes—
271C22#1062515512yes—
272C22C22×C22⋊D2311yescharacteristic
273C4—42no—
274C4—42no—
275C4—464no—
276C4—924no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—924no—
281C2C2×#1062510811yescharacteristic
282C2C2×#1062510011yescharacteristic
283C2#1309297611yescharacteristic
284C2#1309305714yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC22×C22⋊C4×C23C2
prime radicalC23×C22⋊D23C2
hypercenterC22×C22⋊C4D23
hypocenterC23#2405
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#10625100C2semidirect
3. #13092332C2semidirect
4. #13092296C2semidirect
5. #13092289C2semidirect
6. #13092273C2semidirect
7. #13092272C2semidirect
8. C22×C22⋊C4×C23C2semidirect
11. C23×C4×C23C22semidirect
12. C23×C4×C23C22semidirect
20. C2×C22⋊C4×C23C22semidirect
22. C2×C22⋊Dic23C22semidirect
23. C2×C22⋊Dic23C22semidirect
25. C2×(C2×D23)⋊C4C22semidirect
26. C2×(C2×D23)⋊C4C22semidirect
27. C2×Dic23⋊C4C22semidirect
28. C23⋊#95C22semidirect
120. C22×C23#123semidirect
137. C22×C22⋊C4D23semidirect
158. C23×C4C2×D23semidirect
159. C23×C4C2×D23semidirect
169. C2×C22⋊C4C2×D23semidirect
171. C23#2405semidirect
267. C22#10625100semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D232
C2×#106251002
#130923322
#130922962
#130922892
#130922732
#130922722
C22×C22⋊C4×C232
#240523