Finite Group Databasebeta

← #26226231

Subgroups — #26226231

Order 12032.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×#15689145C211yesmaximal, characteristic
3#20245793C211yesmaximal, characteristic
4#20244338C212yesmaximal, complemented
5#20244309C212yesmaximal, complemented
6#20243820C212yesmaximal
7#20243818C212yesmaximal
8#20243394C211yesmaximal, characteristic, complemented
9#20243393C211yesmaximal, characteristic, complemented
10#20243390C211yesmaximal, characteristic, complemented
11C47×#1803C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13#15689677—24no—
14#15689677C418yescomplemented
15C2×#13094023C2211yescharacteristic
16C2×#13094022C2211yescharacteristic
17C2×#13094014—21no—
18C2×#13094004C2211yescharacteristic
19C2×#13094002—21no—
20C2×#13094000C2211yescharacteristic
21C2×C4×Dic94—21no—
22C2×C4×Dic94C2211yescharacteristic
23C2×#13093982—21no—
24C2×#13093981—21no—
25C2×C42×D47C2211yescharacteristic
26#15689145—24no—
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×Dic47⋊C8C2212yes—
29C2×#13093949C2212yes—
30C2×#13093948C2212yes—
31C2×#13093947C2212yes—
32#15688993C2211yescharacteristic
33#15688988—21no—
34#15688980—21no—
35#15688980C2212yes—
36C4⋊C4×Dic47—21no—
37C4⋊C4×Dic47C2212yes—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#384C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C2×#10625324—24no—
52C2×#10625324—42no—
53C2×#10625324—42no—
54C22×C47⋊C8C2312yes—
55#13094023—22no—
56#13094023C2×C414yes—
57#13094022—22no—
58#13094022C2×C414yes—
59#13094014—28no—
60#13094004—22no—
61#13094004C2×C414yes—
62#13094002—28no—
63#13094000—22no—
64#13094000C2×C414yes—
65C4×Dic94—22no—
66C4×Dic94—28no—
67C4×Dic94C2×C414yes—
68#13093982—44no—
69#13093981—28no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2312yes—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C42×D47C2×C414yes—
89C42×D47D414yes—
90C4⋊(C47⋊C8)—22no—
91Dic47⋊C8—24no—
92#13093949—42no—
93#13093948—42no—
94#13093947—42no—
95C47⋊#95—22no—
96C47⋊#95—22no—
97C22×Q8×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99C4×Q8×C47—22no—
100C4×Q8×C47C2×C414yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47C2312yes—
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C42×C47—21no—
107C2×C42×C47C2311yescharacteristic
108C4⋊C8×C47—22no—
109#103×C47—42no—
110C23×D47C2×D411yescharacteristic
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C22×C412yes—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2411yescharacteristic
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—21no—
132C2×C4×D47—21no—
133C2×C4×D47—21no—
134C2×C4×D47—21no—
135C2×C4×D47—22no—
136C2×C4×D47—22no—
137C2×C4×D47C22⋊C414yes—
138C2×C4×D47C22⋊C414yes—
139C2×C4×D47C22×C412yes—
140C2×C4×D47C2×D412yes—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—24no—
147C2×Dic94—42no—
148C2×Dic94—42no—
149C2×Dic94C22×C412yes—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—42no—
158C4⋊C4×C47—42no—
159C4⋊C4×C47C22×C412yes—
160C42×C47—24no—
161C42×C47C22×C412yes—
162C42×C47C2×D412yes—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—44no—
166C4⋊Dic47—21no—
167C4⋊Dic47—24no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—42no—
171C4⋊Dic47—42no—
172C4⋊Dic47C22×C412yes—
173Dic47⋊C4—24no—
174Dic47⋊C4—24no—
175Dic47⋊C4—24no—
176Dic47⋊C4—24no—
177Dic47⋊C4—24no—
178C4×Dic47—24no—
179C4×Dic47—44no—
180C4×Dic47C22⋊C414yes—
181C4×Dic47C22⋊C414yes—
182C4×Dic47C22×C412yes—
183C4×Dic47C2×D412yes—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187#10625324—48no—
188C23×C47C22×D411yescharacteristic
189C22×D47—21no—
190C22×D47—21no—
191C22×D47—21no—
192C22×D47—21no—
193C22×D47—21no—
194C22×D47—21no—
195C22×D47C2×C22⋊C412yes—
196Q8×C47—22no—
197Q8×C47—22no—
198Q8×C47—41no—
199Q8×C47—41no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47C2×C22⋊C412yes—
212C2×C4×C47C2×C22⋊C412yes—
213C2×C4×C47#13712yes—
214C2×C4×C47#13712yes—
215C2×C4×C47C23×C411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47H(5,𝔽2)11yescharacteristic
218C2×C4×C47H(5,𝔽2)11yescharacteristic
219C2×C4×C472−1+411yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47C2×C22⋊C412yes—
235C2×Dic47C2×C22⋊C412yes—
236C2×Dic47C2×C22⋊C412yes—
237C2×Dic47AGL(1,ℤ/8)12yes—
238C2×Dic47#13712yes—
239C4×D47—22no—
240C4×D47—22no—
241C4×D47—22no—
242C4×D47—22no—
243C4×D47—28no—
244C4×D47—28no—
245Dic94—22no—
246Dic94—22no—
247Dic94—41no—
248Dic94—41no—
249Dic94—48no—
250C8×C47—42no—
251C47⋊C8—42no—
252#21894—471nomaximal, Sylow
253C22×C47C22×C22⋊C411yescharacteristic
254C22×C47#53711yescharacteristic
255C22×C47#53711yescharacteristic
256C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C47C2×#13711yescharacteristic
258C22×C47C2×#13711yescharacteristic
259C22×C47C2×#13711yescharacteristic
260C2×D47—22no—
261C2×D47—22no—
262C2×D47—22no—
263C2×D47—41no—
264C2×D47—41no—
265C2×D47—41no—
266C2×D47—41no—
267C4×C47—22no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—82no—
271C4×C47#41914yes—
272C4×C47#52011yescharacteristic
273C4×C47#52011yescharacteristic
274C4×C47#53511yescharacteristic
275C4×C47#53611yescharacteristic
276Dic47—22no—
277Dic47—22no—
278Dic47—42no—
279Dic47—42no—
280Dic47—82no—
281Dic47#41714yes—
282C2×#533—471no—
283C2×#423—471no—
284#2285—471no—
285#1988—472no—
286#1981—472no—
287#1894—472no—
288#1894—472no—
289#1803—471no—
290#1803—471no—
291#1803—471no—
292#1803D4711yescharacteristic, complemented
293C2×C47#231111yescharacteristic
294C2×C47C2×#41911yescharacteristic
295C2×C47C2×#41711yescharacteristic
296C2×C47#325411yescharacteristic
297C2×C47#325411yescharacteristic
298C2×C47#325511yescharacteristic
299C2×C47#325511yescharacteristic
300D47—42no—
301C22×M16—472no—
302#533—478no—
303#533—944no—
304C2×C4⋊Q8—471no—
305C2×C4⋊Q8—471no—
306C2×#125—941no—
307C2×C22⋊Q8—471no—
308C2×C22⋊Q8—471no—
309C2×C22⋊Q8—941no—
310C2×C4×Q8—471no—
311C2×C4×Q8—941no—
312C2×C4×Q8C2×D4711yescharacteristic
313C2×#117—941no—
314C2×#117—941no—
315C22×C42—471no—
316#423—944no—
317C2×C4⋊C8—472no—
318C2×C4⋊C8—471no—
319C2×C4⋊C8C2×D4711yescharacteristic
320C2×#103—472no—
321C2×#103—472no—
322C2×#103—472no—
323C2×#103C2×D4712yes—
324#384—471no—
325#384C2×D4711yescharacteristic
326#382—941no—
327#380—472no—
328#380—941no—
329C4×C4⋊C4—472no—
330C4×C4⋊C4—941no—
331#340—472no—
332#340—472no—
333#340—472no—
334#340C2×D4712yes—
335C47#2189411yescharacteristic, complemented, Sylow
336C22×Q8—471no—
337C22×Q8—941no—
338C22×Q8C22×D4711yescharacteristic
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—471no—
342C2×M16—944no—
343C2×M16—1882no—
344C2×M16—1882no—
345C22×C8—472no—
346C22×C8C22×D4712yes—
347C4⋊Q8—474no—
348C4⋊Q8—474no—
349C4⋊Q8—942no—
350C4⋊Q8—942no—
351#125—948no—
352C22⋊Q8—474no—
353C22⋊Q8—474no—
354C22⋊Q8—942no—
355C22⋊Q8—948no—
356C22⋊Q8—942no—
357C4×Q8—22no—
358C4×Q8—474no—
359C4×Q8—948no—
360C4×Q8—942no—
361C4×Q8C4×D4714yes—
362#117—948no—
363#117—1884no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4712yes—
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—942no—
382C2×C42—21no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—474no—
388C2×C42—941no—
389C2×C42—942no—
390C2×C42C22×D4711yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—944no—
393C4⋊C8—942no—
394#103—42no—
395#103—1882no—
396#103—1882no—
397#103—1882no—
398#95—942no—
399#95—942no—
400C24—471no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—472no—
407C2×Q8—944no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×Q8—941no—
411C2×Q8—941no—
412C2×Q8—941no—
413C2×Q8—1882no—
414C2×Q8—1882no—
415C2×Q8C2×C4×D4712yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—942no—
437C22×C4D4×D4711yescharacteristic
438C22×C4D4×D4711yescharacteristic
439C22×C4C23×D4711yescharacteristic
440M16—1888no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—942no—
445C2×C8—942no—
446C2×C8—942no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—472no—
454C4⋊C4—944no—
455C4⋊C4—944no—
456C4⋊C4—944no—
457C4⋊C4—944no—
458C4⋊C4—944no—
459C4⋊C4—944no—
460C4⋊C4—941no—
461C4⋊C4—942no—
462C4⋊C4—942no—
463C4⋊C4—1882no—
464C4⋊C4—1882no—
465C4⋊C4C2×C4×D4712yes—
466C22⋊C4—944no—
467C22⋊C4—944no—
468C22⋊C4—1884no—
469C42—24no—
470C42—474no—
471C42—474no—
472C42—472no—
473C42—472no—
474C42—944no—
475C42—1884no—
476C42C2×C4×D4712yes—
477C42D4×D4712yes—
478C23—472no—
479C23—941no—
480C23—941no—
481C23—941no—
482C23—941no—
483C23—941no—
484C23—941no—
485C23C2×D4×D4711yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—942no—
491Q8—942no—
492Q8—1888no—
493Q8—1881no—
494Q8—1881no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—472no—
507C2×C4—472no—
508C2×C4—472no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—948no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—948no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22⋊C4×D4712yes—
532C2×C4#1309405312yes—
533C2×C4#1309405612yes—
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4#1309406911yescharacteristic
537C2×C4#1309407011yescharacteristic
538C2×C4#1309407111yescharacteristic
539C2×C4#1309407211yescharacteristic
540C8—42no—
541C8—1882no—
542C22—942no—
543C22—942no—
544C22—942no—
545C22—1881no—
546C22—1881no—
547C22—1881no—
548C22—1881no—
549C22C2×C22⋊C4×D4711yescharacteristic
550C22#1568970211yescharacteristic
551C22#1568970311yescharacteristic
552C22AGL(1,ℤ/8)×D4711yescharacteristic
553C22#137×D4711yescharacteristic
554C22C2×#1309405311yescharacteristic
555C22C2×#1309405611yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—474no—
561C4—942no—
562C4—942no—
563C4—3762no—
564C4—1882no—
565C4—1882no—
566C4#1568913014yes—
567C4#1568960311yescharacteristic
568C4#1568960411yescharacteristic
569C4#1568968911yescharacteristic
570C4#1568969611yescharacteristic
571C2—1882no—
572C2#2024597711yescharacteristic
573C2C2×#1568913011yescharacteristic
574C2#417×D4711yescharacteristic
575C2#2025477311yescharacteristic
576C2#2025477411yescharacteristic
577C2#2025478011yescharacteristic
578C2#2025478111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1803C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1803D47
hypocenterC47#21894
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #20244338C2semidirect
5. #20244309C2semidirect
8. #20243394C2semidirect
9. #20243393C2semidirect
10. #20243390C2semidirect
11. C47×#1803C2semidirect
14. #15689677C4semidirect
292. #1803D47semidirect
335. C47#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×#156891452
#202457932
#202443382
#202443092
#202438202
#202438182
#202433942
#202433932
#202433902
C47×#18032
#2189447