Finite Group Databasebeta

← #15676722

Subgroups — #15676722

Order 2944.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10625054C212yesmaximal, complemented
3C2×C22⋊C4×D23C211yesmaximal, characteristic, complemented
4#13092333C212yesmaximal, complemented
5#13092332C212yesmaximal, complemented
6#13092293C212yesmaximal, complemented
7#13092292C211yesmaximal, characteristic, complemented
8#13092289C211yesmaximal, characteristic, complemented
9#13092273C211yesmaximal, characteristic, complemented
10#13092272C211yesmaximal, characteristic, complemented
11C23×#379C211yesmaximal, characteristic, complemented
12C25×C23C2211yescharacteristic
13C24×D23C2211yescharacteristic
14C22×C22⋊D23—24no—
15C22×C22⋊D23—28no—
16C22×C22⋊D23C2212yes—
17C22×C22⋊D23C2212yes—
18C23×Dic23C2211yescharacteristic
19C22×C4×D23C2211yescharacteristic, complemented
20C22×C4×D23C2211yescharacteristic, complemented
21C2×C22⋊C4×C23C2211yescharacteristic
22C2×C22⋊C4×C23C2211yescharacteristic
23C2×C22⋊C4×C23C2212yescomplemented
24C2×C22⋊C4×C23C2212yescomplemented
25C2×C22⋊Dic23C2211yescharacteristic
26C2×C22⋊Dic23C2211yescharacteristic
27C2×C22⋊Dic23C2212yescomplemented
28C2×C22⋊Dic23C2212yescomplemented
29C2×(C2×D23)⋊C4C2212yescomplemented
30C2×(C2×D23)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D23)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D23)⋊C4C2212yescomplemented
33C2×C4⋊Dic23C2212yescomplemented
34C2×Dic23⋊C4C2212yescomplemented
35#10625054—28no—
36C22⋊C4×D23—24no—
37C23⋊#95C2212yescomplemented
38C23⋊#95C2212yescomplemented
39C24×C23—24no—
40C24×C23—22no—
41C24×C23—22no—
42C24×C23—22no—
43C24×C23—42no—
44C24×C23C2312yes—
45C24×C23C2311yescharacteristic
46C23×D23—22no—
47C23×D23—22no—
48C23×D23—24no—
49C23×D23—42no—
50C23×D23D414yes—
51C23×D23C2311yescharacteristic
52C23×D23C2311yescharacteristic
53C22×C4×C23C2311yescharacteristic
54C22×C4×C23C2312yes—
55C22×C4×C23C2311yescharacteristic
56C2×C22⋊D23—232no—
57C2×C22⋊D23—24no—
58C2×C22⋊D23—24no—
59C2×C22⋊D23—24no—
60C2×C22⋊D23—24no—
61C2×C22⋊D23—24no—
62C2×C22⋊D23D418yes—
63C22×Dic23—22no—
64C22×Dic23—24no—
65C22×Dic23C2312yes—
66C22×Dic23C2311yescharacteristic
67C22×Dic23C2311yescharacteristic
68C22×Dic23C2312yes—
69C2×C4×D23—22no—
70C2×C4×D23—22no—
71C2×C4×D23—24no—
72C2×C4×D23—42no—
73C22⋊C4×C23—22no—
74C22⋊C4×C23—24no—
75C22⋊C4×C23—22no—
76C22⋊C4×C23—42no—
77C22⋊Dic23—22no—
78C22⋊Dic23—24no—
79C22⋊Dic23—22no—
80C22⋊Dic23—42no—
81(C2×D23)⋊C4—22no—
82(C2×D23)⋊C4—22no—
83(C2×D23)⋊C4—24no—
84(C2×D23)⋊C4—42no—
85C4⋊Dic23—24no—
86Dic23⋊C4—24no—
87C23×C23—22no—
88C23×C23—22no—
89C23×C23—28no—
90C23×C23—22no—
91C23×C23—22no—
92C23×C23—22no—
93C23×C23—22no—
94C23×C23—22no—
95C23×C23—22no—
96C23×C23—48no—
97C23×C23—48no—
98C23×C23—44no—
99C23×C23—44no—
100C23×C23C2×D414yes—
101C23×C23C2×D412yes—
102C23×C23#4112yes—
103C23×C23#4112yes—
104C23×C23C2411yescharacteristic
105C22×D23—21no—
106C22×D23—21no—
107C22×D23—22no—
108C22×D23—28no—
109C22×D23—22no—
110C22×D23—21no—
111C22×D23—21no—
112C22×D23—22no—
113C22×D23—28no—
114C22×D23—44no—
115C22×D23—48no—
116C22×D23—48no—
117C22×D23C2×D412yes—
118C22×D23C2×D414yes—
119C22×D23C2×D412yes—
120C2×C4×C23—22no—
121C2×C4×C23—22no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—22no—
125C2×C4×C23—22no—
126C2×C4×C23—22no—
127C22⋊D23—216no—
128C22⋊D23—216no—
129C2×Dic23—28no—
130C2×Dic23—22no—
131C2×Dic23—22no—
132C2×Dic23—22no—
133C2×Dic23—22no—
134C2×Dic23—21no—
135C2×Dic23—21no—
136C2×Dic23—22no—
137C2×Dic23—22no—
138C2×Dic23—22no—
139C2×Dic23C2×D414yes—
140C4×D23—44no—
141C4×D23—44no—
142#2405—231nomaximal, Sylow
143C22×C23—22no—
144C22×C23—22no—
145C22×C23—24no—
146C22×C23—24no—
147C22×C23—24no—
148C22×C23—24no—
149C22×C23—24no—
150C22×C23—416no—
151C22×C23—42no—
152C22×C23—42no—
153C22×C23—42no—
154C22×C23—42no—
155C22×C23#12314yescomplemented
156C22×C23C22×D411yescharacteristic
157C22×C23C22×D412yes—
158C22×C23C2×#4111yescharacteristic
159C22×C23C2×#4111yescharacteristic
160C22×C23H(5,𝔽2)12yes—
161C2×D23—22no—
162C2×D23—22no—
163C2×D23—24no—
164C2×D23—24no—
165C2×D23—22no—
166C2×D23—22no—
167C2×D23—22no—
168C2×D23—416no—
169C2×D23—42no—
170C2×D23—42no—
171C2×D23#12014yes—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—42no—
175Dic23—24no—
176Dic23—41no—
177Dic23—41no—
178Dic23—42no—
179C23×D4—231no—
180C2×#123—232no—
181C22×C22⋊C4—231no—
182#394—232no—
183#394—232no—
184#386—232no—
185#385—231no—
186#385—231no—
187#379—231no—
188#379—231no—
189#379D2311yescharacteristic, complemented
190C2×C23—24no—
191C2×C23—44no—
192C2×C23C2×#12011yescharacteristic
193C2×C23C2×#12311yescharacteristic
194C2×C23#53411yescharacteristic
195C2×C23#54612yes—
196C2×C23#54612yes—
197D23—24no—
198D23—42no—
199C25—231no—
200C25C2×D2311yescharacteristic
201C22×D4—232no—
202C22×D4—232no—
203C22×D4—468no—
204C22×D4—464no—
205C23×C4—231no—
206C23×C4—231no—
207C23×C4—231no—
208#123—468no—
209C2×C4⋊C4—232no—
210C2×C4⋊C4—232no—
211C2×C22⋊C4—231no—
212C2×C22⋊C4—232no—
213C2×C22⋊C4—231no—
214C2×C22⋊C4—231no—
215C2×C22⋊C4—231no—
216C2×C22⋊C4—232no—
217C2×C22⋊C4—232no—
218C2×C22⋊C4—232no—
219C2×C22⋊C4—464no—
220C2×C22⋊C4C2×D2311yescharacteristic
221C2×C22⋊C4C2×D2311yescharacteristic
222C2×C22⋊C4C2×D2312yescomplemented
223C2×C22⋊C4C2×D2312yescomplemented
224#95—232no—
225#95—232no—
226C23#240511yescharacteristic, complemented, Sylow
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—231no—
233C24—231no—
234C24—234no—
235C24—464no—
236C24—462no—
237C24—462no—
238C24—922no—
239C24C22×D2312yes—
240C24C22×D2311yescharacteristic
241C2×D4—238no—
242C2×D4—464no—
243C2×D4—464no—
244C2×D4—464no—
245C2×D4—464no—
246C2×D4—4632no—
247C2×D4—464no—
248C22×C4—232no—
249C22×C4—231no—
250C22×C4—232no—
251C22×C4—231no—
252C22×C4—462no—
253C22×C4—462no—
254C22×C4—464no—
255C22×C4—464no—
256C22×C4—462no—
257C22×C4—922no—
258C22×C4C22×D2311yescharacteristic
259C22×C4C22×D2312yes—
260C22×C4C22×D2311yescharacteristic
261C4⋊C4—464no—
262C4⋊C4—464no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—462no—
268C22⋊C4—462no—
269C22⋊C4—464no—
270C22⋊C4—462no—
271C22⋊C4—462no—
272C22⋊C4—464no—
273C22⋊C4—922no—
274C22⋊C4—922no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—234no—
289C23—232no—
290C23—232no—
291C23—468no—
292C23—461no—
293C23—461no—
294C23—462no—
295C23—468no—
296C23—461no—
297C23—462no—
298C23—461no—
299C23—462no—
300C23—924no—
301C23—928no—
302C23—928no—
303C23#6728812yes—
304C23D4×D2314yes—
305C23D4×D2312yes—
306C23#6729012yes—
307C23C23×D2311yescharacteristic
308D4—4616no—
309D4—4616no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—234no—
318C2×C4—468no—
319C2×C4—462no—
320C2×C4—462no—
321C2×C4—462no—
322C2×C4—462no—
323C2×C4—461no—
324C2×C4—461no—
325C2×C4—462no—
326C2×C4—462no—
327C2×C4—462no—
328C2×C4—924no—
329C2×C4—924no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—234no—
343C22—462no—
344C22—462no—
345C22—464no—
346C22—464no—
347C22—462no—
348C22—462no—
349C22—462no—
350C22—9216no—
351C22—922no—
352C22—922no—
353C22#1062505414yescomplemented
354C22C2×#6728811yescharacteristic
355C22C2×D4×D2311yescharacteristic
356C22C2×D4×D2312yes—
357C22C2×#6729011yescharacteristic
358C22#1062515512yes—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—464no—
363C4—921no—
364C4—921no—
365C4—922no—
366C2—24no—
367C2—44no—
368C2—464no—
369C2—922no—
370C2#120×D2311yescharacteristic
371C2C2×#1062505411yescharacteristic
372C2#1309297711yescharacteristic
373C2#1309305412yes—
374C2#1309305712yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#379C2
prime radicalC23×C22⋊D23C2
hypercenter#379D23
hypocenterC23#2405
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#10625054C2semidirect
3. C2×C22⋊C4×D23C2semidirect
4. #13092333C2semidirect
5. #13092332C2semidirect
6. #13092293C2semidirect
7. #13092292C2semidirect
8. #13092289C2semidirect
9. #13092273C2semidirect
10. #13092272C2semidirect
11. C23×#379C2semidirect
19. C22×C4×D23C22semidirect
20. C22×C4×D23C22semidirect
23. C2×C22⋊C4×C23C22semidirect
24. C2×C22⋊C4×C23C22semidirect
27. C2×C22⋊Dic23C22semidirect
28. C2×C22⋊Dic23C22semidirect
29. C2×(C2×D23)⋊C4C22semidirect
30. C2×(C2×D23)⋊C4C22semidirect
31. C2×(C2×D23)⋊C4C22semidirect
32. C2×(C2×D23)⋊C4C22semidirect
33. C2×C4⋊Dic23C22semidirect
34. C2×Dic23⋊C4C22semidirect
37. C23⋊#95C22semidirect
38. C23⋊#95C22semidirect
155. C22×C23#123semidirect
189. #379D23semidirect
222. C2×C22⋊C4C2×D23semidirect
223. C2×C22⋊C4C2×D23semidirect
226. C23#2405semidirect
353. C22#10625054semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D232
C2×#106250542
C2×C22⋊C4×D232
#130923332
#130923322
#130922932
#130922922
#130922892
#130922732
#130922722
C23×#3792
#240523