Finite Group Databasebeta

← #26221947

Subgroups — #26221947

Order 12032.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic94C211yesmaximal, characteristic, complemented
2C2×#15689264C211yesmaximal, characteristic
3C2×#15689225C212yesmaximal, complemented
4C4×#13093998C211yesmaximal, characteristic
5#20243742C212yesmaximal, complemented
6#20243703C211yesmaximal, characteristic
7#20243645C212yesmaximal, complemented
8#20243578C211yesmaximal, characteristic, complemented
9#20243324C211yesmaximal, characteristic, complemented
10#20243219C212yesmaximal, complemented
11C47×#1749C211yesmaximal, characteristic, complemented
12C23×Dic94C2211yescharacteristic
13C22×#10625325C2212yes—
14C2×#13093998C2211yescharacteristic
15C2×C4×Dic94—28no—
16C2×C4×Dic94—21no—
17C2×C4×Dic94—21no—
18C2×C4×Dic94C2212yes—
19C2×C4×Dic94C2212yes—
20C2×C4×D188C2211yescharacteristic
21C2×C4×C22⋊D47—21no—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4—21no—
24C22×C4×Dic47—21no—
25#15689264C418yescomplemented
26#15689225—28no—
27C2×#13093958C2212yes—
28C2×#13093949C2212yes—
29C2×#13093949C2212yes—
30C2×#13093949C2212yes—
31C2×#13093946C2212yes—
32#15689025C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689014—21no—
35#15689002—21no—
36#15688983C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
38C4×(C2×D47)⋊C4—21no—
39#15688886C2212yes—
40#15688886C2212yes—
41C22×C42×C47C2211yescharacteristic
42C2×C22⋊C8×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2211yescharacteristic
44C2×#97×C47C2212yes—
45C47×#336C2212yes—
46C22×Dic94—21no—
47C22×Dic94—22no—
48C22×Dic94—24no—
49C22×Dic94—24no—
50C22×Dic94C2×C414yescomplemented
51C22×Dic94C2312yes—
52C22×D188C2311yescharacteristic
53C22×C22⋊D47—21no—
54C23×Dic47—21no—
55C23×Dic47—21no—
56C22×C4×D47—21no—
57C2×#10625325—24no—
58C2×#10625325—24no—
59C2×#10625325C2×C418yescomplemented
60#13093998—22no—
61#13093998C2×C414yes—
62C4×Dic94—28no—
63C4×Dic94—28no—
64C4×Dic94—24no—
65C4×Dic94—24no—
66C4×Dic94—24no—
67C4×Dic94D418yescomplemented
68C4×D188—24no—
69C4×C22⋊D47—44no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×C4×Dic47—28no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47—22no—
88C2×C4×Dic47—21no—
89#13093958—24no—
90#13093949—24no—
91#13093949—24no—
92#13093949C2×C418yescomplemented
93#13093946—24no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C23×C4×C47D411yescharacteristic
98C23×C4×C47D411yescharacteristic
99C23×C4×C47C2311yescharacteristic
100C22×C8×C47C2312yes—
101C22×C8×C47C2312yes—
102C2×C42×C47—24no—
103C2×C42×C47D411yescharacteristic
104C2×C42×C47D411yescharacteristic
105C2×C42×C47C2311yescharacteristic
106C2×C42×C47C2311yescharacteristic
107C22⋊C8×C47—22no—
108C22⋊C8×C47C2×C414yescomplemented
109#97×C47—24no—
110C24×C47C2×D411yescharacteristic
111C23×D47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—41no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47#4112yes—
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—42no—
130C2×C22⋊D47—42no—
131C22×Dic47—24no—
132C22×Dic47—24no—
133C22×Dic47—24no—
134C22×Dic47—22no—
135C22×Dic47—24no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—21no—
145C2×D188C22×C412yes—
146C2×C4×D47—42no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×Dic94—24no—
150C2×Dic94—24no—
151C2×Dic94—216no—
152C2×Dic94—24no—
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94—22no—
156C2×Dic94—216no—
157C2×Dic94C22×C414yes—
158C2×Dic94C2×D414yes—
159C2×Dic94#4114yes—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47C22×C414yes—
166C42×C47—22no—
167C42×C47—22no—
168C42×C47—22no—
169C42×C47C2×D414yes—
170C22⋊Dic47—42no—
171(C2×D47)⋊C4—24no—
172(C2×D47)⋊C4—42no—
173(C2×D47)⋊C4—42no—
174(C2×D47)⋊C4—42no—
175C4⋊Dic47—24no—
176C4⋊Dic47—22no—
177C4⋊Dic47—21no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C2×D414yes—
182Dic47⋊C4—28no—
183Dic47⋊C4—28no—
184Dic47⋊C4—42no—
185C4×Dic47—28no—
186C4×Dic47—28no—
187C4×Dic47—42no—
188#10625325—216no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47C4×D412yes—
194C23×C47AGL(1,ℤ/8)12yes—
195C23×C47#13712yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—22no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47—41no—
204C2×C4×C47—28no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—44no—
219C2×C4×C47C4×D412yes—
220C2×C4×C47#12012yes—
221C2×C4×C47#12012yes—
222C2×C4×C47#13712yes—
223C2×C4×C47C23×C411yescharacteristic
224C2×C4×C47C22×D412yes—
225C2×C4×C47C2×#4112yes—
226C2×C4×C47C2×#4111yescharacteristic
227C22⋊D47—48no—
228C2×Dic47—216no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—48no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245D188—24no—
246D188—42no—
247C4×D47—84no—
248Dic94—216no—
249Dic94—24no—
250Dic94—24no—
251Dic94—24no—
252Dic94C4×D418yescomplemented
253C8×C47—24no—
254C8×C47—42no—
255#21454—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47#44014yescomplemented
263C22×C47C2×C4×D411yescharacteristic
264C22×C47C2×#12011yescharacteristic
265C22×C47#52511yescharacteristic
266C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C47C2×#13711yescharacteristic
268C22×C47C2×#13711yescharacteristic
269C22×C47#57511yescharacteristic
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—81no—
274C2×D47—81no—
275C2×D47—81no—
276C2×D47—81no—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—41no—
282C4×C47C2×C4×D412yes—
283C4×C47#52512yes—
284Dic47—28no—
285Dic47—42no—
286Dic47—44no—
287Dic47—81no—
288Dic47—81no—
289C22×C4×Q8—471no—
290C2×#448—471no—
291C2×#440—472no—
292C4×#121—471no—
293#1880—472no—
294#1874—471no—
295#1863—472no—
296#1852—471no—
297#1787—471no—
298#1763—472no—
299#1749D4711yescharacteristic, complemented
300C2×C47—24no—
301C2×C47C4×#12011yescharacteristic
302C2×C47C2×#44011yescharacteristic
303C2×C47C4×#13711yescharacteristic
304C2×C47#300411yescharacteristic
305C2×C47#300611yescharacteristic
306C2×C47#311211yescharacteristic
307C2×C47#311711yescharacteristic
308D47—82no—
309C23×Q8—471no—
310C22×SD16—472no—
311C2×#121—471no—
312C2×C4×Q8—472no—
313C2×C4×Q8—472no—
314C2×C4×Q8—948no—
315C2×C4×Q8—941no—
316C2×C4×Q8—941no—
317C2×C4×D4—471no—
318C2×C4×D4—941no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—941no—
321C22×C42—941no—
322C22×C42C2×D4711yescharacteristic
323#448—478no—
324#440—948no—
325C2×#107—472no—
326C2×#103—472no—
327C2×#103—472no—
328C2×#103—472no—
329C2×#102—472no—
330C2×C22⋊C8C2×D4711yescharacteristic
331C2×C22⋊C8C2×D4711yescharacteristic
332C2×#97C2×D4712yes—
333#390—471no—
334#389—471no—
335#388—941no—
336#386—941no—
337#380—471no—
338C4×C4⋊C4—471no—
339C4×C22⋊C4—941no—
340#340—472no—
341#340—472no—
342#336C2×D4712yes—
343C47#2145411yescharacteristic, complemented, Sylow
344C22×Q8—474no—
345C22×Q8—472no—
346C22×Q8—944no—
347C22×Q8—944no—
348C22×Q8—941no—
349C22×Q8—942no—
350C22×D4—471no—
351C22×D4—941no—
352C23×C4—941no—
353C23×C4—941no—
354C23×C4—941no—
355C23×C4D18811yescharacteristic
356C23×C4D18811yescharacteristic
357C23×C4C22×D4711yescharacteristic
358C2×SD16—478no—
359C2×SD16—944no—
360C2×SD16—944no—
361C22×C8C22×D4712yes—
362C22×C8C22×D4712yes—
363#121—474no—
364#121—942no—
365C4×Q8—478no—
366C4×Q8—944no—
367C4×Q8—944no—
368C4×Q8—948no—
369C4×Q8—948no—
370C4×Q8—944no—
371C4×D4—944no—
372C4×D4—1884no—
373C2×C4⋊C4—472no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—944no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—942no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—941no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C42—24no—
389C2×C42—948no—
390C2×C42—942no—
391C2×C42—942no—
392C2×C42—941no—
393C2×C42D18811yescharacteristic
394C2×C42D18811yescharacteristic
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397#107—944no—
398#103—478no—
399#103—944no—
400#103—944no—
401#102—944no—
402C22⋊C8—22no—
403C22⋊C8C4×D4714yescomplemented
404#97—24no—
405#95—941no—
406#95—941no—
407#95—941no—
408C24—941no—
409C24C2×D18811yescharacteristic
410C2×Q8—474no—
411C2×Q8—474no—
412C2×Q8—474no—
413C2×Q8—9416no—
414C2×Q8—944no—
415C2×Q8—944no—
416C2×Q8—9416no—
417C2×Q8—944no—
418C2×Q8—942no—
419C2×Q8—942no—
420C2×Q8—942no—
421C2×D4—472no—
422C2×D4—944no—
423C2×D4—942no—
424C2×D4—942no—
425C2×D4—941no—
426C2×D4—1882no—
427C2×D4—1882no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—944no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4—941no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—1882no—
446C22×C4—1882no—
447C22×C4—1882no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4C2×C4×D4712yes—
451C22×C4C2×D18811yescharacteristic
452C22×C4C2×D18811yescharacteristic
453C22×C4C2×D18811yescharacteristic
454C22×C4C2×D18811yescharacteristic
455C22×C4C2×D18811yescharacteristic
456C22×C4#1062535012yes—
457C22×C4C23×D4711yescharacteristic
458SD16—9416no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8C2×C4×D4714yes—
465C4⋊C4—472no—
466C4⋊C4—474no—
467C4⋊C4—948no—
468C4⋊C4—944no—
469C4⋊C4—948no—
470C4⋊C4—942no—
471C4⋊C4—941no—
472C4⋊C4—942no—
473C4⋊C4—942no—
474C4⋊C4—1882no—
475C22⋊C4—944no—
476C22⋊C4—1882no—
477C22⋊C4—1882no—
478C22⋊C4—1882no—
479C22⋊C4—1882no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—948no—
484C42—948no—
485C42—1882no—
486C42D4×D4714yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—942no—
492C23—1881no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23—1881no—
497C23—1881no—
498C23C4×D18812yes—
499C23#1309405212yes—
500C23#1309405712yes—
501C23C22×D18811yescharacteristic
502Q8—478no—
503Q8—9416no—
504Q8—944no—
505Q8—944no—
506Q8—944no—
507D4—944no—
508D4—1888no—
509D4—1882no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—9416no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—3764no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1888no—
539C2×C4—1881no—
540C2×C4—1882no—
541C2×C4—1881no—
542C2×C4—1882no—
543C2×C4C4×D18812yes—
544C2×C4#1309399412yes—
545C2×C4#1309399412yes—
546C2×C4#1309405712yes—
547C2×C4C22×C4×D4711yescharacteristic
548C2×C4C2×D4×D4712yes—
549C2×C4#41×D4712yes—
550C2×C4C2×#1062535011yescharacteristic
551C8—24no—
552C8—42no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—3761no—
560C22—3761no—
561C22—3761no—
562C22—3761no—
563C22—1882no—
564C22—1882no—
565C22—1882no—
566C22#1568922514yescomplemented
567C22C2×C4×D18811yescharacteristic
568C22C2×#1309399411yescharacteristic
569C22#1568963511yescharacteristic
570C22C2×#1309405211yescharacteristic
571C22C2×#1309405711yescharacteristic
572C22C2×#1309405711yescharacteristic
573C22#1568992311yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—948no—
580C4—3761no—
581C4—3761no—
582C4—1882no—
583C4—1884no—
584C4C4×D4×D4712yes—
585C4#1568963212yes—
586C2—24no—
587C2—3762no—
588C2C4×#1309399411yescharacteristic
589C2C2×#1568922511yescharacteristic
590C2C4×#1309405711yescharacteristic
591C2#2025262511yescharacteristic
592C2#2025264211yescharacteristic
593C2#2025351111yescharacteristic
594C2#2025356111yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1749C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1749D47
hypocenterC47#21454
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic94C2semidirect
3. C2×#15689225C2semidirect
5. #20243742C2semidirect
7. #20243645C2semidirect
8. #20243578C2semidirect
9. #20243324C2semidirect
10. #20243219C2semidirect
11. C47×#1749C2semidirect
25. #15689264C4semidirect
50. C22×Dic94C2×C4semidirect
59. C2×#10625325C2×C4semidirect
67. C4×Dic94D4semidirect
92. #13093949C2×C4semidirect
108. C22⋊C8×C47C2×C4semidirect
252. Dic94C4×D4semidirect
262. C22×C47#440semidirect
299. #1749D47semidirect
343. C47#21454semidirect
403. C22⋊C8C4×D47semidirect
566. C22#15689225semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic942
C2×#156892642
C2×#156892252
C4×#130939982
#202437422
#202437032
#202436452
#202435782
#202433242
#202432192
C47×#17492
#2145447