Finite Group Databasebeta

← #15683022

Subgroups — #15683022

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625086C211yesmaximal, characteristic, complemented
2C2×M16×D23C211yesmaximal, characteristic, complemented
3#13092927C211yesmaximal, characteristic
4C22⋊Q8×D23C211yesmaximal, characteristic, complemented
5#13092675C211yesmaximal, characteristic, complemented
6#13092673C211yesmaximal, characteristic, complemented
7#13092661C211yesmaximal, characteristic, complemented
8#13092659C211yesmaximal, characteristic, complemented
9#13092657C211yesmaximal, characteristic, complemented
10#13092645C211yesmaximal, characteristic, complemented
11#13092643C211yesmaximal, characteristic, complemented
12#13092458C211yesmaximal, characteristic, complemented
13#13092424C211yesmaximal, characteristic, complemented
14#13092415C211yesmaximal, characteristic, complemented
15C23×#468C211yesmaximal, characteristic, complemented
16C2×Q8×D23C2211yescharacteristic
17C2×#67290C2211yescharacteristic
18C22×C4×D23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic, complemented
20C2×M16×C23C2211yescharacteristic
21C23×C22⋊Q8C2211yescharacteristic
22C23×#121C2211yescharacteristic
23#10625117C2211yescharacteristic
24C2×#67275C2211yescharacteristic, complemented
25#10625109C2211yescharacteristic
26#10625108—21no—
27D4×Dic23—21no—
28C2×#67273C2211yescharacteristic, complemented
29#10625095C2211yescharacteristic
30#10625095C2211yescharacteristic
31C2×#67267C2211yescharacteristic
32#10625086—24no—
33M16×D23—24no—
34C2×Dic92C2211yescharacteristic, complemented
35C2×#67262C2211yescharacteristic
36C2×#67262C2211yescharacteristic
37C2×C8×D23C2211yescharacteristic
38C2×C8×D23C2211yescharacteristic
39#10625068C2211yescharacteristic
40#10625068C2211yescharacteristic
41#10625067—21no—
42#10625063C2211yescharacteristic
43C4⋊C4×D23—21no—
44C4⋊C4×D23C2211yescharacteristic
45#10625054—21no—
46(C22⋊D23)⋊C4—21no—
47C22⋊C4×D23—21no—
48#10625049—21no—
49C23×#107C2211yescharacteristic, complemented
50C23×#103C2211yescharacteristic, complemented
51C23×#102C2211yescharacteristic, complemented
52#10625017C2211yescharacteristic, complemented
53#10625014C2211yescharacteristic, complemented
54#10624999C2211yescharacteristic, complemented
55#10624998C2211yescharacteristic, complemented
56#10624998C2211yescharacteristic, complemented
57#10624992C2211yescharacteristic, complemented
58#10624992C2211yescharacteristic, complemented
59#10624990C2211yescharacteristic, complemented
60#10624989C2211yescharacteristic, complemented
61C23×D23D411yescharacteristic
62C2×Q8×C23C2311yescharacteristic
63C2×D4×C23—21no—
64C2×D4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C2×C22⋊D23—21no—
67C2×C22⋊D23—21no—
68C22×Dic23—21no—
69C22×Dic23D411yescharacteristic
70Q8×D23—22no—
71Q8×D23—22no—
72#67290—22no—
73#67290—22no—
74C2×C4×D23—22no—
75C2×C4×D23—22no—
76C2×C4×D23—21no—
77C2×C4×D23—21no—
78C2×C4×D23D411yescharacteristic
79C2×C4×D23D411yescharacteristic
80C2×C4×D23C2311yescharacteristic
81C2×C4×D23C2311yescharacteristic
82C2×Dic46—21no—
83C2×Dic46C2311yescharacteristic
84C2×Dic46C2311yescharacteristic
85SD16×C23—22no—
86M16×C23—22no—
87C2×C8×C23C2311yescharacteristic
88C2×C8×C23C2311yescharacteristic
89C4⋊C4×C23—21no—
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C22⋊C4×C23—21no—
93C22⋊C4×C23—21no—
94C22⋊Dic23—21no—
95C22⋊Dic23—21no—
96C22⋊Dic23—21no—
97#67275—22no—
98#67273—22no—
99(C2×D23)⋊C4—21no—
100(C2×D23)⋊C4—21no—
101(C2×D23)⋊C4—21no—
102(C2×D23)⋊C4—21no—
103C4⋊Dic23—21no—
104C4⋊Dic23—21no—
105C4⋊Dic23C2311yescharacteristic
106C4⋊Dic23C2311yescharacteristic
107Dic23⋊C4—21no—
108Dic23⋊C4—21no—
109Dic23⋊C4—21no—
110Dic23⋊C4—21no—
111C4×Dic23—21no—
112#67267—22no—
113C2×C23⋊C8C2311yescharacteristic
114C2×C23⋊C8C2311yescharacteristic
115Dic92—22no—
116#67262—22no—
117#67262D414yescomplemented
118C8×D23—22no—
119C8×D23—22no—
120C23×C23—21no—
121C23×C23C2×D411yescharacteristic
122C22×D23—22no—
123C22×D23—22no—
124C22×D23—41no—
125C22×D23C2×D411yescharacteristic
126C22×D23C2×D411yescharacteristic
127Q8×C23—21no—
128Q8×C23—21no—
129D4×C23—21no—
130D4×C23—21no—
131D4×C23—42no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23C2×D411yescharacteristic
138C2×C4×C23C2411yescharacteristic
139C22⋊D23—42no—
140C22⋊D23—42no—
141C2×Dic23—22no—
142C2×Dic23—21no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—42no—
147C2×Dic23C2×D411yescharacteristic
148C2×Dic23C2×D411yescharacteristic
149C4×D23—22no—
150C4×D23—21no—
151C4×D23—21no—
152C4×D23—22no—
153C4×D23—42no—
154C4×D23—42no—
155C4×D23C2×D412yes—
156C4×D23#4112yes—
157Dic46—21no—
158Dic46—21no—
159Dic46—21no—
160Dic46—21no—
161Dic46—42no—
162C8×C23—21no—
163C8×C23C2×D412yes—
164C23⋊C8—21no—
165C23⋊C8C2×D412yes—
166#3056—231nomaximal, Sylow
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23C22×D411yescharacteristic
174C2×D23—21no—
175C2×D23—21no—
176C2×D23—21no—
177C2×D23—21no—
178C2×D23—44no—
179C2×D23#12112yes—
180C2×D23#13712yescomplemented
181C4×C23—21no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—41no—
185C4×C23C22×D411yescharacteristic
186C4×C23C2×#4111yescharacteristic
187Dic23—21no—
188Dic23—41no—
189Dic23—41no—
190Dic23—41no—
191Dic23—41no—
192Dic23#12112yes—
193C2×#137—231no—
194C22×M16—231no—
195#525—231no—
196C2×C22⋊Q8—231no—
197#470—231no—
198#470—231no—
199#468—231no—
200#468D2311yescharacteristic, complemented
201#467—231no—
202#467—231no—
203#465—231no—
204#465—231no—
205#428—231no—
206#419—231no—
207#417—231no—
208C2×C23—41no—
209C2×C23—81no—
210C2×C23C2×#12111yescharacteristic
211C2×C23C2×#13711yescharacteristic
212C2×C23#57511yescharacteristic
213D23—22no—
214D23—22no—
215C2×#41—231no—
216C22×Q8—231no—
217C23×C4—231no—
218#137—464no—
219C2×Q16—231no—
220C2×Q16—231no—
221C2×SD16—231no—
222C2×SD16C2×D2311yescharacteristic, complemented
223C2×M16—231no—
224C2×M16—231no—
225C2×M16—231no—
226C2×M16—464no—
227C2×M16C2×D2311yescharacteristic
228C22×C8—231no—
229C22×C8—231no—
230#123—461no—
231C22⋊Q8—231no—
232C22⋊Q8—231no—
233C22⋊Q8—231no—
234C22⋊Q8—231no—
235C22⋊Q8—231no—
236C22⋊Q8—461no—
237C22⋊Q8—461no—
238C22⋊Q8C2×D2311yescharacteristic
239#121—231no—
240#121C2×D2311yescharacteristic
241#120—461no—
242C4×D4—461no—
243C4×D4—461no—
244#117—231no—
245C2×C4⋊C4—231no—
246C2×C4⋊C4—461no—
247C2×C22⋊C4—461no—
248#107—231no—
249#107C2×D2311yescharacteristic, complemented
250#106—231no—
251#106—231no—
252#103—231no—
253#103—231no—
254#103—231no—
255#103—231no—
256#103—231no—
257#103C2×D2311yescharacteristic, complemented
258#102—231no—
259#102C2×D2311yescharacteristic, complemented
260C23#305611yescharacteristic, complemented, Sylow
261C24—231no—
262#41—462no—
263#41—462no—
264C2×Q8—231no—
265C2×Q8—231no—
266C2×Q8—461no—
267C2×Q8—462no—
268C2×Q8—462no—
269C2×Q8C22×D2311yescharacteristic
270C2×D4—21no—
271C2×D4—461no—
272C2×D4—461no—
273C2×D4C22×D2311yescharacteristic
274C22×C4—231no—
275C22×C4—231no—
276C22×C4—231no—
277C22×C4—231no—
278C22×C4—231no—
279C22×C4—462no—
280C22×C4—462no—
281C22×C4—461no—
282C22×C4—461no—
283C22×C4—461no—
284C22×C4C22×D2311yescharacteristic
285Q16—462no—
286Q16—462no—
287SD16—22no—
288SD16—462no—
289M16—22no—
290M16—234no—
291M16—462no—
292M16—462no—
293C2×C8—231no—
294C2×C8—231no—
295C2×C8—462no—
296C2×C8—462no—
297C2×C8C22×D2311yescharacteristic
298C2×C8C22×D2311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—231no—
301C4⋊C4—231no—
302C4⋊C4—461no—
303C4⋊C4—461no—
304C4⋊C4—461no—
305C4⋊C4—461no—
306C4⋊C4—461no—
307C4⋊C4—461no—
308C4⋊C4C22×D2311yescharacteristic
309C4⋊C4C22×D2311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—461no—
313C22⋊C4—461no—
314C22⋊C4—461no—
315C22⋊C4—461no—
316C22⋊C4—461no—
317C22⋊C4—461no—
318C22⋊C4—461no—
319C42—461no—
320C23—21no—
321C23—231no—
322C23—231no—
323C23—462no—
324C23—462no—
325C23—921no—
326C23D4×D2311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—461no—
330Q8—461no—
331Q8—461no—
332Q8—461no—
333Q8—922no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—922no—
338D4—922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—231no—
345C2×C4—232no—
346C2×C4—232no—
347C2×C4—231no—
348C2×C4—462no—
349C2×C4—462no—
350C2×C4—462no—
351C2×C4—461no—
352C2×C4—461no—
353C2×C4—461no—
354C2×C4—461no—
355C2×C4—461no—
356C2×C4—461no—
357C2×C4—922no—
358C2×C4—922no—
359C2×C4—922no—
360C2×C4D4×D2311yescharacteristic
361C2×C4C23×D2311yescharacteristic
362C8—21no—
363C8—232no—
364C8—461no—
365C8D4×D2312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—232no—
373C22—232no—
374C22—461no—
375C22—461no—
376C22—461no—
377C22—461no—
378C22—924no—
379C22C2×D4×D2311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—232no—
385C4—461no—
386C4—921no—
387C4—921no—
388C4—921no—
389C4—921no—
390C4C2×D4×D2311yescharacteristic
391C4#41×D2311yescharacteristic
392C2—41no—
393C2—81no—
394C2—462no—
395C2—462no—
396C2#121×D2311yescharacteristic
397C2#137×D2311yescharacteristic
398C2#1309321311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#468C2
socleC22×C23C22×D4
prime radical#13092927C2
hypercenter#468D23
hypocenterC23#3056
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#10625086C2semidirect
2. C2×M16×D23C2semidirect
4. C22⋊Q8×D23C2semidirect
5. #13092675C2semidirect
6. #13092673C2semidirect
7. #13092661C2semidirect
8. #13092659C2semidirect
9. #13092657C2semidirect
10. #13092645C2semidirect
11. #13092643C2semidirect
12. #13092458C2semidirect
13. #13092424C2semidirect
14. #13092415C2semidirect
15. C23×#468C2semidirect
19. C2×SD16×C23C22semidirect
24. C2×#67275C22semidirect
28. C2×#67273C22semidirect
34. C2×Dic92C22semidirect
49. C23×#107C22semidirect
50. C23×#103C22semidirect
51. C23×#102C22semidirect
52. #10625017C22semidirect
53. #10625014C22semidirect
54. #10624999C22semidirect
55. #10624998C22semidirect
56. #10624998C22semidirect
57. #10624992C22semidirect
58. #10624992C22semidirect
59. #10624990C22semidirect
60. #10624989C22semidirect
117. #67262D4semidirect
180. C2×D23#137semidirect
200. #468D23semidirect
222. C2×SD16C2×D23semidirect
249. #107C2×D23semidirect
257. #103C2×D23semidirect
259. #102C2×D23semidirect
260. C23#3056semidirect

Maximal subgroups

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

subgroupindex
C2×#106250862
C2×M16×D232
#130929272
C22⋊Q8×D232
#130926752
#130926732
#130926612
#130926592
#130926572
#130926452
#130926432
#130924582
#130924242
#130924152
C23×#4682
#305623