Finite Group Databasebeta

← #15676746

Subgroups — #15676746

Order 2944.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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×D4×D23C211yesmaximal, characteristic, complemented
2C2×#10625110C212yesmaximal, complemented
3C2×#10625108C212yesmaximal, complemented
4C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
5#13092335C211yesmaximal, characteristic, complemented
6#13092331C211yesmaximal, characteristic, complemented
7#13092326C211yesmaximal, characteristic, complemented
8#13092317C212yesmaximal, complemented
9#13092276C211yesmaximal, characteristic, complemented
10#13092272C212yesmaximal, complemented
11C23×#392C211yesmaximal, characteristic, complemented
12C24×D23C2212yes—
13C22×D4×C23C2211yescharacteristic
14C22×D4×C23C2212yescomplemented
15C22×C22⋊D23C2211yescharacteristic
16C22×C22⋊D23C2212yes—
17C22×C22⋊D23C2212yescomplemented
18C23×Dic23C2211yescharacteristic, complemented
19C2×D4×D23—28no—
20C2×D4×D23—24no—
21C22×D92C2211yescharacteristic
22C22×C4×D23C2211yescharacteristic
23C2×C22⋊C4×C23C2211yescharacteristic, complemented
24C2×C22⋊C4×C23C2212yescomplemented
25C2×C22⋊Dic23C2212yescomplemented
26C2×C22⋊Dic23C2212yescomplemented
27#10625110—28no—
28#10625108—28no—
29C2×(C2×D23)⋊C4C2211yescharacteristic
30C2×(C2×D23)⋊C4C2212yes—
31C2×(C2×D23)⋊C4C2212yescomplemented
32C2×(C2×D23)⋊C4C2212yescomplemented
33C2×(C2×D23)⋊C4C2212yescomplemented
34C2×Dic23⋊C4C2211yescharacteristic, complemented
35C2×Dic23⋊C4C2212yescomplemented
36C2×C4×Dic23C2211yescharacteristic, complemented
37(C22⋊D23)⋊C4—24no—
38C23×#95C2211yescharacteristic, complemented
39C23⋊#95C2211yescharacteristic, complemented
40C24×C23C2312yes—
41C24×C23C2311yescharacteristic
42C23×D23—24no—
43C23×D23—24no—
44C23×D23—24no—
45C23×D23—24no—
46C23×D23—24no—
47C23×D23—42no—
48C23×D23—42no—
49C23×D23C2312yes—
50C23×D23C2311yescharacteristic
51C2×D4×C23—24no—
52C2×D4×C23—24no—
53C2×D4×C23—24no—
54C2×D4×C23—22no—
55C2×D4×C23—42no—
56C22×C4×C23C2311yescharacteristic
57C22×C4×C23C2312yes—
58C22×C4×C23C2311yescharacteristic
59C2×C22⋊D23—22no—
60C2×C22⋊D23—22no—
61C2×C22⋊D23—24no—
62C2×C22⋊D23—24no—
63C2×C22⋊D23—24no—
64C2×C22⋊D23—24no—
65C2×C22⋊D23—42no—
66C2×C22⋊D23D418yescomplemented
67C2×C22⋊D23D414yescomplemented
68C22×Dic23—22no—
69C22×Dic23—24no—
70C22×Dic23C2311yescharacteristic
71C22×Dic23C2312yes—
72C22×Dic23C2311yescharacteristic
73C22×Dic23C2311yescharacteristic
74D4×D23—232no—
75C2×D92—24no—
76C2×D92—22no—
77C2×C4×D23—24no—
78C2×C4×D23D414yescomplemented
79C22⋊C4×C23—22no—
80C22⋊C4×C23—42no—
81C22⋊Dic23—24no—
82C22⋊Dic23—24no—
83(C2×D23)⋊C4—22no—
84(C2×D23)⋊C4—24no—
85(C2×D23)⋊C4—24no—
86(C2×D23)⋊C4—24no—
87(C2×D23)⋊C4—42no—
88Dic23⋊C4—22no—
89Dic23⋊C4—24no—
90C4×Dic23—22no—
91C23×C23—22no—
92C23×C23—21no—
93C23×C23—22no—
94C23×C23—22no—
95C23×C23—22no—
96C23×C23—21no—
97C23×C23—22no—
98C23×C23—22no—
99C23×C23—22no—
100C23×C23—22no—
101C23×C23C2×D414yes—
102C23×C23C2×D412yescomplemented
103C23×C23C2411yescharacteristic
104C22×D23—22no—
105C22×D23—22no—
106C22×D23—216no—
107C22×D23—22no—
108C22×D23—22no—
109C22×D23—22no—
110C22×D23—22no—
111C22×D23—22no—
112C22×D23—22no—
113C22×D23—48no—
114C22×D23—48no—
115C22×D23—48no—
116C22×D23—48no—
117C22×D23—48no—
118C22×D23—48no—
119C22×D23C2×D414yes—
120C22×D23C2×D412yes—
121C22×D23C2×D412yes—
122C22×D23#4112yes—
123D4×C23—28no—
124D4×C23—48no—
125C2×C4×C23—22no—
126C2×C4×C23—21no—
127C2×C4×C23—22no—
128C2×C4×C23—22no—
129C2×C4×C23—22no—
130C2×C4×C23—22no—
131C2×C4×C23C2×D412yes—
132C22⋊D23—28no—
133C22⋊D23—216no—
134C22⋊D23—48no—
135C2×Dic23—21no—
136C2×Dic23—21no—
137C2×Dic23—21no—
138C2×Dic23—22no—
139C2×Dic23—22no—
140C2×Dic23—22no—
141C2×Dic23—22no—
142C2×Dic23—44no—
143C2×Dic23C2×D412yes—
144C2×Dic23C2×D414yes—
145C2×Dic23C2×D412yescomplemented
146D92—28no—
147C4×D23—28no—
148#2408—231nomaximal, Sylow
149C22×C23—24no—
150C22×C23—22no—
151C22×C23—24no—
152C22×C23—24no—
153C22×C23—24no—
154C22×C23—41no—
155C22×C23—42no—
156C22×C23—42no—
157C22×C23—41no—
158C22×C23—42no—
159C22×C23—42no—
160C22×C23—42no—
161C22×C23C22×D411yescharacteristic
162C22×C23C22×D412yes—
163C22×C23C22×D411yescharacteristic
164C22×C23C2×#4111yescharacteristic
165C22×C23H(5,𝔽2)12yes—
166C2×D23—22no—
167C2×D23—22no—
168C2×D23—24no—
169C2×D23—24no—
170C2×D23—24no—
171C2×D23—24no—
172C2×D23—24no—
173C2×D23—432no—
174C2×D23—42no—
175C2×D23—42no—
176C2×D23—42no—
177C2×D23—42no—
178C2×D23#12114yescomplemented
179C4×C23—22no—
180C4×C23—41no—
181C4×C23—42no—
182Dic23—22no—
183Dic23—41no—
184Dic23—42no—
185Dic23#12014yescomplemented
186C23×D4—231no—
187C2×#121—232no—
188C2×#120—232no—
189C2×C4×D4—231no—
190#394—231no—
191#394—231no—
192#392—231no—
193#392D2311yescharacteristic, complemented
194#390—232no—
195#380—231no—
196#379—232no—
197C2×C23—44no—
198C2×C23—42no—
199C2×C23C2×#12011yescharacteristic
200C2×C23C2×#12111yescharacteristic
201C2×C23#53511yescharacteristic
202C2×C23D4212yes—
203C2×C23#54612yes—
204D23—24no—
205D23—44no—
206C25—232no—
207C22×D4—231no—
208C22×D4—232no—
209C22×D4—232no—
210C22×D4—231no—
211C22×D4—464no—
212C22×D4—468no—
213C22×D4C2×D2311yescharacteristic
214C22×D4C2×D2312yescomplemented
215C23×C4—231no—
216C23×C4—231no—
217#121—468no—
218#120—468no—
219C4×D4—464no—
220C2×C4⋊C4—231no—
221C2×C4⋊C4—232no—
222C2×C22⋊C4—232no—
223C2×C22⋊C4—232no—
224C2×C22⋊C4—231no—
225C2×C22⋊C4—232no—
226C2×C22⋊C4—232no—
227C2×C22⋊C4—232no—
228C2×C22⋊C4—232no—
229C2×C22⋊C4C2×D2311yescharacteristic, complemented
230C2×C22⋊C4C2×D2312yescomplemented
231C2×C42—231no—
232#95—231no—
233#95C2×D2311yescharacteristic, complemented
234C23#240811yescharacteristic, complemented, Sylow
235C24—231no—
236C24—232no—
237C24—464no—
238C24—464no—
239C24—464no—
240C24—464no—
241C24—464no—
242C24—922no—
243C24—922no—
244C24C22×D2312yes—
245C24C22×D2311yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—238no—
252C2×D4—234no—
253C2×D4—464no—
254C2×D4—464no—
255C2×D4—464no—
256C2×D4—4632no—
257C2×D4—462no—
258C2×D4—462no—
259C2×D4—464no—
260C2×D4—464no—
261C2×D4—462no—
262C2×D4—922no—
263C22×C4—231no—
264C22×C4—231no—
265C22×C4—234no—
266C22×C4—232no—
267C22×C4—231no—
268C22×C4—464no—
269C22×C4—462no—
270C22×C4—464no—
271C22×C4C22×D2312yes—
272C22×C4C22×D2311yescharacteristic
273C22×C4C22×D2311yescharacteristic
274C4⋊C4—464no—
275C4⋊C4—462no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—464no—
279C22⋊C4—462no—
280C22⋊C4—464no—
281C22⋊C4—464no—
282C22⋊C4—464no—
283C22⋊C4—464no—
284C22⋊C4—922no—
285C42—462no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23—232no—
297C23—232no—
298C23—232no—
299C23—234no—
300C23—4616no—
301C23—462no—
302C23—462no—
303C23—462no—
304C23—462no—
305C23—462no—
306C23—462no—
307C23—462no—
308C23—462no—
309C23—928no—
310C23—928no—
311C23—928no—
312C23—928no—
313C23—928no—
314C23—928no—
315C23D4×D2314yes—
316C23D4×D2312yescomplemented
317C23C23×D2311yescharacteristic
318D4—28no—
319D4—48no—
320D4—468no—
321D4—4616no—
322D4—468no—
323D4—928no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—232no—
331C2×C4—234no—
332C2×C4—232no—
333C2×C4—468no—
334C2×C4—461no—
335C2×C4—461no—
336C2×C4—462no—
337C2×C4—462no—
338C2×C4—461no—
339C2×C4—462no—
340C2×C4—462no—
341C2×C4—924no—
342C2×C4D4×D2312yes—
343C22—24no—
344C22—22no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—234no—
356C22—462no—
357C22—462no—
358C22—464no—
359C22—464no—
360C22—464no—
361C22—464no—
362C22—464no—
363C22—9232no—
364C22—922no—
365C22—922no—
366C22—922no—
367C22—922no—
368C22C2×D4×D2311yescharacteristic
369C22C2×D4×D2312yes—
370C22C2×D4×D2311yescharacteristic
371C22#1062515512yes—
372C22#41×D2311yescharacteristic
373C4—22no—
374C4—41no—
375C4—42no—
376C4—234no—
377C4—462no—
378C4—921no—
379C4—922no—
380C2—44no—
381C2—42no—
382C2—464no—
383C2—924no—
384C2#120×D2311yescharacteristic
385C2#121×D2311yescharacteristic
386C2#1309298011yescharacteristic
387C2#1309304912yes—
388C2#1309305612yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#392C2
hypercenter#392D23
hypocenterC23#2408
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D23C2semidirect
2. C2×#10625110C2semidirect
3. C2×#10625108C2semidirect
4. C2×(C22⋊D23)⋊C4C2semidirect
5. #13092335C2semidirect
6. #13092331C2semidirect
7. #13092326C2semidirect
8. #13092317C2semidirect
9. #13092276C2semidirect
10. #13092272C2semidirect
11. C23×#392C2semidirect
14. C22×D4×C23C22semidirect
17. C22×C22⋊D23C22semidirect
18. C23×Dic23C22semidirect
23. C2×C22⋊C4×C23C22semidirect
24. C2×C22⋊C4×C23C22semidirect
25. C2×C22⋊Dic23C22semidirect
26. C2×C22⋊Dic23C22semidirect
31. C2×(C2×D23)⋊C4C22semidirect
32. C2×(C2×D23)⋊C4C22semidirect
33. C2×(C2×D23)⋊C4C22semidirect
34. C2×Dic23⋊C4C22semidirect
35. C2×Dic23⋊C4C22semidirect
36. C2×C4×Dic23C22semidirect
38. C23×#95C22semidirect
39. C23⋊#95C22semidirect
66. C2×C22⋊D23D4semidirect
67. C2×C22⋊D23D4semidirect
78. C2×C4×D23D4semidirect
102. C23×C23C2×D4semidirect
145. C2×Dic23C2×D4semidirect
178. C2×D23#121semidirect
185. Dic23#120semidirect
193. #392D23semidirect
214. C22×D4C2×D23semidirect
229. C2×C22⋊C4C2×D23semidirect
230. C2×C22⋊C4C2×D23semidirect
233. #95C2×D23semidirect
234. C23#2408semidirect
316. C23D4×D23semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D232
C2×#106251102
C2×#106251082
C2×(C22⋊D23)⋊C42
#130923352
#130923312
#130923262
#130923172
#130922762
#130922722
C23×#3922
#240823