Finite Group Databasebeta

← #15682655

Subgroups — #15682655

Order 2944.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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×#67288C211yesmaximal, characteristic
2C2×#10625113C212yesmaximal, complemented
3#13092577C214yesmaximal, complemented
4#13092572C212yesmaximal, complemented
5#13092567C212yesmaximal, complemented
6#13092416C212yesmaximal, complemented
7#13092381C211yesmaximal, characteristic, complemented
8C2×C23×C22⋊Q8C211yesmaximal, characteristic, complemented
9C22×Q8×C23C2211yescharacteristic, complemented
10C23×C4×C23C2211yescharacteristic
11C22×C22⋊D23—21no—
12C2×#67288—21no—
13C2×#67288—21no—
14C2×#67288—28no—
15C2×#67288C2212yes—
16C2×#67288C2212yes—
17C22×D92C2211yescharacteristic
18C22×C4×D23—21no—
19C22×Dic46C2211yescharacteristic
20C23×C22⋊Q8—21no—
21C23×C22⋊Q8—21no—
22C23×C22⋊Q8C2212yes—
23C23×C22⋊Q8C2212yescomplemented
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23C2211yescharacteristic, complemented
26C2×C22⋊C4×C23—21no—
27C2×#67275C2214yescomplemented
28#10625113—28no—
29C2×#67274C2214yescomplemented
30C2×#67267C2212yescomplemented
31C22⋊(C23⋊C8)C2212yes—
32C22⋊(C23⋊C8)C2212yescomplemented
33#10624992C2214yescomplemented
34#10624991C2214yescomplemented
35C24×C23D411yescharacteristic
36C23×D23—21no—
37C2×Q8×C23—21no—
38C2×Q8×C23—24no—
39C2×Q8×C23C2312yes—
40C22×C4×C23—24no—
41C22×C4×C23—21no—
42C22×C4×C23—21no—
43C22×C4×C23D411yescharacteristic
44C22×C4×C23D411yescharacteristic
45C22×C4×C23D411yescharacteristic
46C22×C4×C23C2311yescharacteristic
47C22×C4×C23C2311yescharacteristic
48C22×C4×C23C2311yescharacteristic
49C2×C22⋊D23—28no—
50C2×C22⋊D23—22no—
51C2×C22⋊D23—22no—
52C22×Dic23—21no—
53#67288—28no—
54#67288—28no—
55#67288—24no—
56#67288—24no—
57#67288—24no—
58#67288D418yes—
59C2×D92—21no—
60C2×D92—24no—
61C2×D92C2312yes—
62C2×C4×D23—28no—
63C2×C4×D23—22no—
64C2×C4×D23—22no—
65C2×Dic46—21no—
66C2×Dic46—24no—
67C2×Dic46C2312yes—
68C4⋊C4×C23—22no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—21no—
71C4⋊C4×C23C2312yes—
72C22⋊C4×C23—22no—
73C22⋊C4×C23—22no—
74#67275—28no—
75#67274—28no—
76#67267—24no—
77C2×C23⋊C8C2314yes—
78C23×C23—22no—
79C23×C23—22no—
80C23×C23—41no—
81C23×C23C2×D411yescharacteristic
82C23×C23C2×D411yescharacteristic
83C23×C23C2×D411yescharacteristic
84C22×D23—24no—
85C22×D23—22no—
86C22×D23—42no—
87C22×D23—42no—
88Q8×C23—24no—
89Q8×C23—42no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23C2×D414yes—
100C2×C4×C23C2×D411yescharacteristic
101C2×C4×C23C2×D411yescharacteristic
102C2×C4×C23C2×D411yescharacteristic
103C2×C4×C23C2411yescharacteristic
104C22⋊D23—28no—
105C22⋊D23—28no—
106C2×Dic23—24no—
107C2×Dic23—22no—
108D92—24no—
109D92—22no—
110D92C2×D414yes—
111C4×D23—28no—
112C4×D23—28no—
113Dic46—24no—
114Dic46—22no—
115Dic46C2×D414yes—
116C23⋊C8—24no—
117#3007—231nomaximal, Sylow
118C22×C23—21no—
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—44no—
126C22×C23#12012yes—
127C22×C23#13712yescomplemented
128C22×C23C22×D411yescharacteristic
129C2×D23—24no—
130C2×D23—48no—
131C2×D23—42no—
132C2×D23—42no—
133C4×C23—21no—
134C4×C23—21no—
135C4×C23—42no—
136C4×C23—42no—
137C4×C23#12012yes—
138C4×C23C22×D412yes—
139Dic23—24no—
140C22×#41—231no—
141C2×#137—232no—
142C2×C22⋊Q8D2311yescharacteristic, complemented
143#452—234no—
144#451—232no—
145#450—232no—
146#417—232no—
147#407—231no—
148C2×C23—22no—
149C2×C23—22no—
150C2×C23—41no—
151C2×C23C2×#12011yescharacteristic
152C2×C23C2×#13711yescharacteristic
153C2×C23#57511yescharacteristic
154D23—44no—
155C2×#41—232no—
156C2×#41—232no—
157C2×#41—468no—
158C2×#41—461no—
159C2×#41—461no—
160C22×Q8—231no—
161C22×Q8C2×D2311yescharacteristic, complemented
162C22×D4—231no—
163C22×D4—461no—
164C23×C4—461no—
165C23×C4C2×D2311yescharacteristic
166#137—468no—
167C2×Q16—234no—
168C2×SD16—234no—
169C2×M16—232no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8C2×D2312yes—
173C22⋊Q8C2×D2312yescomplemented
174C2×C4⋊C4—21no—
175C2×C4⋊C4C2×D2311yescharacteristic, complemented
176C2×C22⋊C4—21no—
177#103—234no—
178#102—234no—
179C22⋊C8—232no—
180C22⋊C8—232no—
181C23#300711yescharacteristic, complemented, Sylow
182C24—461no—
183C24C22⋊D2311yescharacteristic
184#41—238no—
185#41—464no—
186#41—464no—
187#41—468no—
188#41—468no—
189#41—464no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—232no—
193C2×Q8—464no—
194C2×Q8—461no—
195C2×Q8C22×D2312yes—
196C2×D4—232no—
197C2×D4—464no—
198C2×D4—468no—
199C2×D4—462no—
200C2×D4—462no—
201C2×D4—461no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—468no—
206C22×C4—461no—
207C22×C4—462no—
208C22×C4—462no—
209C22×C4C22⋊D2311yescharacteristic
210C22×C4C22⋊D2311yescharacteristic
211C22×C4C22⋊D2311yescharacteristic
212C22×C4C22×D2311yescharacteristic
213C22×C4C22×D2311yescharacteristic
214C22×C4C22×D2311yescharacteristic
215Q16—468no—
216SD16—468no—
217M16—464no—
218C2×C8—234no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D2312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—464no—
229C23—462no—
230C23—922no—
231C23—922no—
232C23C2×C22⋊D2311yescharacteristic
233C23C2×C22⋊D2311yescharacteristic
234C23C2×C22⋊D2311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—234no—
238Q8—464no—
239Q8—462no—
240D4—234no—
241D4—464no—
242D4—468no—
243D4—468no—
244D4—462no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—468no—
255C2×C4—464no—
256C2×C4—462no—
257C2×C4—468no—
258C2×C4D4×D2314yes—
259C2×C4C2×C22⋊D2311yescharacteristic
260C2×C4C2×C22⋊D2311yescharacteristic
261C2×C4C2×C22⋊D2311yescharacteristic
262C2×C4C23×D2311yescharacteristic
263C8—464no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—464no—
273C22—928no—
274C22—922no—
275C22—922no—
276C22#1062510812yes—
277C22#1062511312yescomplemented
278C22C22×C22⋊D2311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—464no—
284C4#1062510812yes—
285C4C2×D4×D2312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—924no—
290C2C2×#1062510811yescharacteristic
291C2C2×#1062511311yescharacteristic
292C2#1309321511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×C22⋊Q8C2
socleC22×C23C22×D4
prime radicalC22×#67288C2
hypercenterC2×C22⋊Q8D23
hypocenterC23#3007
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10625113C2semidirect
3. #13092577C2semidirect
4. #13092572C2semidirect
5. #13092567C2semidirect
6. #13092416C2semidirect
7. #13092381C2semidirect
8. C2×C23×C22⋊Q8C2semidirect
9. C22×Q8×C23C22semidirect
23. C23×C22⋊Q8C22semidirect
25. C2×C4⋊C4×C23C22semidirect
27. C2×#67275C22semidirect
29. C2×#67274C22semidirect
30. C2×#67267C22semidirect
32. C22⋊(C23⋊C8)C22semidirect
33. #10624992C22semidirect
34. #10624991C22semidirect
127. C22×C23#137semidirect
142. C2×C22⋊Q8D23semidirect
161. C22×Q8C2×D23semidirect
173. C22⋊Q8C2×D23semidirect
175. C2×C4⋊C4C2×D23semidirect
181. C23#3007semidirect
277. C22#10625113semidirect

Maximal subgroups

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

subgroupindex
C22×#672882
C2×#106251132
#130925772
#130925722
#130925672
#130924162
#130923812
C2×C23×C22⋊Q82
#300723