Finite Group Databasebeta

← #26214073

Subgroups — #26214073

Order 12032.

1706 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625352C211yesmaximal, characteristic
2C2×#15689121C212yesmaximal
3C2×#15689075C211yesmaximal, characteristic, complemented
4C4×C4⋊C4×D47C211yesmaximal, characteristic, complemented
5#20243221C211yesmaximal, characteristic, complemented
6#20243219C211yesmaximal, characteristic, complemented
7#20243213C211yesmaximal, characteristic, complemented
8#20243207C212yesmaximal, complemented
9#102×Dic47C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1762C211yesmaximal, characteristic, complemented
12C22×#10625352C2211yescharacteristic
13C22×#10625324C2212yes—
14C4×#10625352—24no—
15C4×#10625352C418yescomplemented
16C2×C4×Dic94C2211yescharacteristic
17C2×#13093989C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×C4×C22⋊D47—21no—
20C2×(C22⋊D47)⋊C4—21no—
21C2×#13093982C2211yescharacteristic
22C2×#13093980—21no—
23C2×C4⋊C4×D47C2212yes—
24C22×C4×Dic47—21no—
25C2×C42×D47—21no—
26C2×C42×D47C2211yescharacteristic
27#15689121C4116yescomplemented
28C2×#13093949C2212yes—
29C2×#13093947C2212yes—
30C2×#13093945C2212yes—
31#15689075—24no—
32C2×#13093935C2211yescharacteristic
33C2×C8×Dic47C2212yes—
34C4⋊C4×Dic47C2212yes—
35C4×Dic47⋊C4—21no—
36C4×C4⋊Dic47C2211yescharacteristic
37#15688974—21no—
38#95×D47—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×#102×C47C2212yes—
44C2×#97×C47C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625352—22no—
48C2×#10625352—24no—
49C2×#10625352C2×C414yes—
50C2×#10625352C2×C418yes—
51C22×Dic94C2311yescharacteristic
52C22×C22⋊D47—21no—
53C23×Dic47—21no—
54C23×Dic47—21no—
55C22×C4×D47—22no—
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C2×#10625324—24no—
60C2×#10625324—24no—
61C2×#10625324C2×C418yescomplemented
62C22×C47⋊C8C2312yes—
63C4×Dic94—22no—
64C4×Dic94C2×C414yescomplemented
65#13093989—22no—
66#13093989C2×C414yescomplemented
67D4×Dic47—22no—
68D4×Dic47C2×C414yescomplemented
69C4×C22⋊D47—28no—
70(C22⋊D47)⋊C4—28no—
71#13093982—22no—
72#13093982C2×C414yes—
73#13093980—28no—
74C2×C4⋊Dic47C2312yes—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—21no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47C2×C418yescomplemented
81C2×C22⋊Dic47—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—28no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—22no—
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C42×D47—44no—
93C42×D47C2×C414yes—
94C42×D47D414yes—
95#13093949C2×C418yescomplemented
96#13093947C2×C418yescomplemented
97#13093945C2×C418yescomplemented
98#13093935—22no—
99C8×Dic47—24no—
100C47⋊#95—21no—
101C47⋊#95—22no—
102C22×D4×C47C2311yescharacteristic
103C23×C4×C47—21no—
104C22×C8×C47C2312yes—
105C4×D4×C47—22no—
106C4×D4×C47C2×C414yescomplemented
107C2×C4⋊C4×C47C2312yes—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112#102×C47C2×C418yescomplemented
113#97×C47—22no—
114#95×C47—21no—
115C24×C47—21no—
116C23×D47C2×D411yescharacteristic
117C2×D4×C47—21no—
118C2×D4×C47—22no—
119C2×D4×C47C22×C412yes—
120C2×D4×C47C22×C414yes—
121C22×C4×C47—22no—
122C22×C4×C47—21no—
123C22×C4×C47—24no—
124C22×C4×C47—44no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—28no—
130C22×Dic47—21no—
131C22×Dic47—24no—
132C22×Dic47—28no—
133C22×Dic47—22no—
134C22×Dic47—21no—
135C22×Dic47—24no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141#10625352—28no—
142#10625352—216no—
143#10625352C42116yescomplemented
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—28no—
147C2×C4×D47—42no—
148C2×C4×D47—44no—
149C2×C4×D47—44no—
150C2×C4×D47C22⋊C414yes—
151C2×C4×D47C22⋊C414yes—
152C2×C4×D47C22×C412yes—
153C2×C4×D47C22×C414yes—
154C2×C4×D47C2×D412yes—
155C2×C4×D47#4114yes—
156C2×Dic94—21no—
157C2×Dic94—22no—
158C2×Dic94C22×C414yes—
159C2×Dic94C22×C412yes—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47C22×C414yes—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47C22×C412yes—
166C4⋊C4×C47C22×C414yes—
167C22⋊C4×C47—24no—
168C42×C47—42no—
169C42×C47C22×C412yes—
170C42×C47C2×D412yes—
171C22⋊Dic47—24no—
172(C2×D47)⋊C4—24no—
173C4⋊Dic47—21no—
174C4⋊Dic47—22no—
175C4⋊Dic47C22×C414yes—
176C4⋊Dic47C22×C412yes—
177Dic47⋊C4—24no—
178Dic47⋊C4—24no—
179Dic47⋊C4—28no—
180C4×Dic47—24no—
181C4×Dic47—24no—
182C4×Dic47—24no—
183C4×Dic47—42no—
184C4×Dic47—44no—
185C4×Dic47C22⋊C414yes—
186C4×Dic47C22⋊C414yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C2×D412yes—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8C22×C414yes—
192#10625324—216no—
193C23×C47—22no—
194C23×C47—24no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—42no—
198C23×C47C22×D411yescharacteristic
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—22no—
202C22×D47C2×C22⋊C412yes—
203C22×D47C4×D414yes—
204D4×C47—22no—
205D4×C47—24no—
206D4×C47C2×C4214yescomplemented
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—48no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C23×C412yes—
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4112yes—
222C22⋊D47—216no—
223C2×Dic47—22no—
224C2×Dic47—24no—
225C2×Dic47—22no—
226C2×Dic47—216no—
227C2×Dic47—22no—
228C2×Dic47—24no—
229C2×Dic47—42no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—48no—
234C2×Dic47C2×C22⋊C412yes—
235C2×Dic47C2×C22⋊C412yes—
236C2×Dic47C2×C22⋊C412yes—
237C2×Dic47C4×D414yes—
238C2×Dic47AGL(1,ℤ/8)12yes—
239C2×Dic47#13712yes—
240C4×D47—28no—
241C4×D47—44no—
242C4×D47—416no—
243C4×D47C2×C4214yes—
244C4×D47#11714yes—
245C4×D47C4×D418yes—
246Dic94—22no—
247Dic94—24no—
248Dic94C2×C4214yescomplemented
249C8×C47—24no—
250C47⋊C8—24no—
251C4×#417—471nomaximal, Sylow
252C22×C47—24no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47C22×C22⋊C411yescharacteristic
258C22×C47C2×C4×D412yes—
259C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C47C2×#13711yescharacteristic
261C22×C47#57512yes—
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—24no—
265C2×D47—24no—
266C2×D47C4×C22⋊C414yes—
267C4×C47—22no—
268C4×C47—22no—
269C4×C47—42no—
270C4×C47—44no—
271C4×C47C22×C4211yescharacteristic
272C4×C47C2×#11711yescharacteristic
273C4×C47C2×C4×D412yes—
274Dic47—22no—
275Dic47—24no—
276Dic47—42no—
277Dic47—44no—
278Dic47C4×C22⋊C414yes—
279Dic47#41714yescomplemented
280C2×C4×#41—471no—
281C2×#417—472no—
282C2×C4×M16—471no—
283C2×C4×C4⋊C4—471no—
284#1763—471no—
285#1763—471no—
286#1762—471no—
287#1762D4711yescharacteristic, complemented
288C4×#103—472no—
289C4×#102—472no—
290#1736—472no—
291C2×C47—44no—
292C2×C47C2×C4×C22⋊C411yescharacteristic
293C2×C47C2×#41711yescharacteristic
294C2×C47#289611yescharacteristic
295C2×C47C4×AGL(1,ℤ/8)12yes—
296C2×C47C4×#13712yes—
297D47—24no—
298C22×#41—471no—
299C22×M16—472no—
300C4×#41—478no—
301C4×#41—944no—
302C2×C4×Q8—471no—
303C2×C4×Q8—471no—
304C2×C4×D4—471no—
305C2×C4×D4—941no—
306C2×C4×D4—941no—
307C2×C4×D4C2×D4711yescharacteristic
308C2×#117—471no—
309C2×#117—941no—
310C22×C4⋊C4—472no—
311C22×C42—471no—
312C22×C42—941no—
313C22×C42—941no—
314#417—4716no—
315C2×#103—472no—
316C2×#103—472no—
317C2×#102—472no—
318C2×#102C2×D4712yes—
319C4×M16—944no—
320C2×#97—471no—
321C2×#97C2×D4711yescharacteristic
322C2×C4×C8—472no—
323C4×C4⋊C4—472no—
324C4×C4⋊C4—471no—
325C4×C4⋊C4—941no—
326C4×C4⋊C4—941no—
327C4×C4⋊C4C2×D4711yescharacteristic
328C2×#95—941no—
329#340—472no—
330#340—472no—
331#340—472no—
332#340C2×D4712yes—
333C47C4×#41711yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—478no—
336C2×#41—942no—
337C2×#41—944no—
338C22×Q8—471no—
339C22×D4—941no—
340C22×D4C22×D4711yescharacteristic
341C23×C4—21no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—942no—
346C23×C4—941no—
347C23×C4—941no—
348C2×M16—478no—
349C2×M16—944no—
350C2×M16—944no—
351C22×C8—472no—
352C22×C8C22×D4712yes—
353C4×Q8—474no—
354C4×Q8—474no—
355C4×Q8—942no—
356C4×Q8—942no—
357C4×D4—22no—
358C4×D4—474no—
359C4×D4—942no—
360C4×D4—948no—
361C4×D4—948no—
362C4×D4C4×D4714yescomplemented
363#117—474no—
364#117—948no—
365#117—942no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—478no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C42—21no—
379C2×C42—471no—
380C2×C42—471no—
381C2×C42—471no—
382C2×C42—474no—
383C2×C42—474no—
384C2×C42—948no—
385C2×C42—941no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42—942no—
389C2×C42—941no—
390C2×C42—1884no—
391C2×C42C22×D4711yescharacteristic
392#103—478no—
393#103—478no—
394#102—478no—
395#102C4×D4718yescomplemented
396#97—22no—
397#97—942no—
398C4×C8—944no—
399#95—21no—
400#95—942no—
401#95—941no—
402C24—21no—
403C24—471no—
404#41—4716no—
405#41—948no—
406#41—9416no—
407C2×Q8—474no—
408C2×Q8—472no—
409C2×Q8—941no—
410C2×Q8—942no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—944no—
414C2×D4—948no—
415C2×D4C2×C4×D4714yes—
416C2×D4C2×C4×D4712yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—472no—
424C22×C4—474no—
425C22×C4—472no—
426C22×C4—474no—
427C22×C4—471no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—948no—
433C22×C4—944no—
434C22×C4—948no—
435C22×C4—944no—
436C22×C4—941no—
437C22×C4—942no—
438C22×C4—941no—
439C22×C4—1882no—
440C22×C4—1884no—
441C22×C4—1884no—
442C22×C4—1882no—
443C22×C4—1882no—
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4711yescharacteristic
446C22×C4C23×D4711yescharacteristic
447M16—9416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—474no—
451C2×C8—942no—
452C2×C8—942no—
453C2×C8C2×C4×D4714yes—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—474no—
457C4⋊C4—472no—
458C4⋊C4—948no—
459C4⋊C4—944no—
460C4⋊C4—944no—
461C4⋊C4—941no—
462C4⋊C4—942no—
463C4⋊C4C2×C4×D4714yes—
464C4⋊C4C2×C4×D4712yes—
465C22⋊C4—24no—
466C22⋊C4—944no—
467C22⋊C4—944no—
468C42—42no—
469C42—474no—
470C42—474no—
471C42—472no—
472C42—472no—
473C42—944no—
474C42—944no—
475C42—944no—
476C42—1882no—
477C42—1884no—
478C42C2×C4×D4712yes—
479C42D4×D4712yes—
480C23—22no—
481C23—24no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—472no—
486C23—474no—
487C23—941no—
488C23—941no—
489C23—942no—
490C23C2×D4×D4711yescharacteristic
491Q8—474no—
492Q8—942no—
493Q8—944no—
494D4—22no—
495D4—24no—
496D4—9416no—
497D4C42×D4714yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—472no—
508C2×C4—472no—
509C2×C4—474no—
510C2×C4—472no—
511C2×C4—472no—
512C2×C4—474no—
513C2×C4—474no—
514C2×C4—478no—
515C2×C4—472no—
516C2×C4—948no—
517C2×C4—9416no—
518C2×C4—942no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—944no—
523C2×C4—1884no—
524C2×C4—18816no—
525C2×C4—1888no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1882no—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22⋊C4×D4712yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C22×C4×D4712yes—
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4#41×D4712yes—
536C8—24no—
537C8—944no—
538C22—24no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—474no—
544C22—944no—
545C22—944no—
546C22—942no—
547C22—942no—
548C22C2×C22⋊C4×D4711yescharacteristic
549C22C4×D4×D4712yes—
550C22AGL(1,ℤ/8)×D4711yescharacteristic
551C22#137×D4711yescharacteristic
552C22#1568991912yes—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—42no—
557C4—474no—
558C4—474no—
559C4—944no—
560C4—942no—
561C4—1884no—
562C4—1882no—
563C4C2×C42×D4711yescharacteristic
564C4#117×D4711yescharacteristic
565C4C4×D4×D4712yes—
566C2—44no—
567C2—944no—
568C2C4×C22⋊C4×D4711yescharacteristic
569C2#417×D4711yescharacteristic
570C2#2025190211yescharacteristic
571C2#2025219912yes—
572C2#2025221012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×#10625352C22
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47C4×#417
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C2×#15689075C2semidirect
4. C4×C4⋊C4×D47C2semidirect
5. #20243221C2semidirect
6. #20243219C2semidirect
7. #20243213C2semidirect
8. #20243207C2semidirect
9. #102×Dic47C2semidirect
10. #20243106C2semidirect
11. C47×#1762C2semidirect
15. C4×#10625352C4semidirect
27. #15689121C4semidirect
61. C2×#10625324C2×C4semidirect
64. C4×Dic94C2×C4semidirect
66. #13093989C2×C4semidirect
68. D4×Dic47C2×C4semidirect
80. C4⋊C4×D47C2×C4semidirect
95. #13093949C2×C4semidirect
96. #13093947C2×C4semidirect
97. #13093945C2×C4semidirect
106. C4×D4×C47C2×C4semidirect
112. #102×C47C2×C4semidirect
143. #10625352C42semidirect
206. D4×C47C2×C42semidirect
248. Dic94C2×C42semidirect
279. Dic47#417semidirect
287. #1762D47semidirect
333. C47C4×#417semidirect
362. C4×D4C4×D47semidirect
395. #102C4×D47semidirect
497. D4C42×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253522
C2×#156891212
C2×#156890752
C4×C4⋊C4×D472
#202432212
#202432192
#202432132
#202432072
#102×Dic472
#202431062
C47×#17622
C4×#41747