Finite Group Databasebeta

← #26218088

Subgroups — #26218088

Order 12032.

1718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689694C211yesmaximal, characteristic
2C2×#15689600C211yesmaximal, characteristic, complemented
3C2×#15689097C211yesmaximal, characteristic
4#20245972C211yesmaximal, characteristic, complemented
5#20243843C218yesmaximal, complemented
6#20243120C211yesmaximal, characteristic, complemented
7#20243120C211yesmaximal, characteristic, complemented
8C2×C47×#410C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10#15689694—22no—
11#15689694—22no—
12#15689694—22no—
13#15689694—22no—
14C2×#13094022C412yescomplemented
15C2×#13094011C412yescomplemented
16C2×#13094005C2214yes—
17C2×#13094003C414yescomplemented
18#15689600—24no—
19#15689600—24no—
20C2×C4×Dic94—22no—
21C2×#13093989—22no—
22C2×#13093983C2211yescharacteristic
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×#13093980C2214yes—
26#15689097—24no—
27C2×#13093939C2214yes—
28#15689029—22no—
29#15689028—22no—
30#15688984C2214yes—
31#15688889C2214yes—
32#15688889C2214yes—
33C22×#41×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#410—24no—
37C2×#99×C47C2214yes—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94C2×C412yes—
43C22×Dic94C2×C412yes—
44C22×Dic94D412yes—
45C22×Dic94D412yes—
46C22×Dic94D412yes—
47C22×Dic94D412yes—
48C23×Dic47C2312yes—
49#13094022—24no—
50#13094022—24no—
51#13094011—24no—
52#13094011—24no—
53#13094005—28no—
54#13094005—28no—
55#13094003—28no—
56#13094003—28no—
57C4×Dic94—216no—
58#13093989—216no—
59#13093983—22no—
60#13093983—21no—
61#13093983—21no—
62#13093983—21no—
63#13093983—21no—
64#13093983—22no—
65#13093983—22no—
66#13093983D414yes—
67#13093980—28no—
68#13093980—28no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2×C412yes—
72C2×Dic47⋊C4—24no—
73C2×Dic47⋊C4C2×C418yes—
74C2×C22⋊Dic47C2×C418yes—
75C2×C22⋊Dic47C2312yes—
76C2×C22⋊Dic47C2312yes—
77C2×C22⋊Dic47C2312yes—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—24no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47C2×C412yes—
82#13093939—28no—
83C47⋊#95—28no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—21no—
87C2×#41×C47—21no—
88C2×#41×C47—21no—
89C2×#41×C47—21no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C22×Q8×C47C2×C411yescharacteristic
93C22×D4×C47C2×C411yescharacteristic
94C22×D4×C47C2312yes—
95C23×C4×C47C2×C412yes—
96C23×C4×C47C2311yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99#117×C47—22no—
100#117×C47D414yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C22⋊C4×C47C2312yes—
104C2×C22⋊C4×C47C2312yes—
105C2×C42×C47—21no—
106C2×C42×C47—21no—
107#99×C47—28no—
108C24×C47C22×C412yes—
109C24×C47C2411yescharacteristic
110#41×C47—42no—
111#41×C47—42no—
112#41×C47—42no—
113#41×C47—42no—
114#41×C47—42no—
115#41×C47—42no—
116#41×C47—42no—
117#41×C47—42no—
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47—22no—
121C2×Q8×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47—22no—
126C2×D4×C47—22no—
127C2×D4×C47—22no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—22no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C22×C411yescharacteristic
140C22×C4×C47C22×C411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47#4112yes—
148C22×C4×C47#4112yes—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47C22×C414yes—
156C22×Dic47C22×C414yes—
157C22×Dic47C2×D414yes—
158C22×Dic47C2×D414yes—
159C2×Dic94—24no—
160C2×Dic94—24no—
161C2×Dic94—24no—
162C2×Dic94—24no—
163C2×Dic94—24no—
164C2×Dic94—24no—
165C2×Dic94—24no—
166C2×Dic94—24no—
167C2×Dic94—24no—
168C2×Dic94—24no—
169C2×Dic94—24no—
170C2×Dic94—24no—
171C2×Dic94C22⋊C414yes—
172C2×Dic94C22⋊C414yes—
173C2×Dic94C22⋊C414yes—
174C2×Dic94C22⋊C414yes—
175C4⋊C4×C47—24no—
176C4⋊C4×C47—42no—
177C22⋊C4×C47—22no—
178C22⋊C4×C47—22no—
179C22⋊C4×C47—22no—
180C22⋊C4×C47C2×D414yes—
181C42×C47—24no—
182C42×C47—42no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—216no—
187C22⋊Dic47—22no—
188C22⋊Dic47—22no—
189C22⋊Dic47C2×D414yes—
190C4⋊Dic47—22no—
191C4⋊Dic47—22no—
192C4⋊Dic47—24no—
193C4⋊Dic47—42no—
194Dic47⋊C4—216no—
195Dic47⋊C4—216no—
196C4×Dic47—216no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47—24no—
200C4×Dic47—42no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—42no—
205C23×C47—44no—
206C23×C47C23×C411yescharacteristic
207C23×C47C22×D411yescharacteristic
208C23×C47C22×D411yescharacteristic
209C23×C47C22×D411yescharacteristic
210C23×C47C22×D411yescharacteristic
211C23×C47C2×#4111yescharacteristic
212C23×C47C2×#4111yescharacteristic
213Q8×C47—42no—
214Q8×C47—42no—
215D4×C47—42no—
216D4×C47—42no—
217D4×C47—42no—
218D4×C47—42no—
219D4×C47—42no—
220D4×C47—42no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—44no—
226C2×C4×C47—44no—
227C2×C4×C47—44no—
228C2×C4×C47—44no—
229C2×C4×C47—42no—
230C2×C4×C47C2×C22⋊C411yescharacteristic
231C2×C4×C47C2×C22⋊C411yescharacteristic
232C2×C4×C47C2×C22⋊C411yescharacteristic
233C2×C4×C47C2×C22⋊C411yescharacteristic
234C2×C4×C47C4×D412yes—
235C2×C4×C47C4×D412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47#12012yes—
239C2×C4×C47#12012yes—
240C2×C4×C47#12112yes—
241C2×C4×C47#12112yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12112yes—
244C2×C4×C47#12312yes—
245C2×C4×C47#12312yes—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—44no—
255C2×Dic47—44no—
256C2×Dic47—44no—
257C2×Dic47—42no—
258C2×Dic47C2×C22⋊C418yes—
259Dic94—28no—
260Dic94—28no—
261Dic94—28no—
262Dic94—28no—
263#20986—471nomaximal, Sylow
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—44no—
272C22×C47—42no—
273C22×C47—82no—
274C22×C47—82no—
275C22×C47C22×C22⋊C411yescharacteristic
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×C4×D411yescharacteristic
278C22×C47C2×#12011yescharacteristic
279C22×C47C2×#12111yescharacteristic
280C22×C47C2×#12111yescharacteristic
281C22×C47C2×#12311yescharacteristic
282C4×C47—44no—
283C4×C47—42no—
284C4×C47—82no—
285C4×C47#38612yes—
286C4×C47#38612yes—
287C4×C47#38612yes—
288C4×C47#38612yes—
289Dic47—28no—
290Dic47—44no—
291Dic47—44no—
292Dic47—82no—
293C2×#536—471no—
294C2×#519—471no—
295C2×#410—471no—
296C2×#410D4711yescharacteristic
297#2310—471no—
298#1899—478no—
299#1740—471no—
300#1740—471no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47—82no—
308C2×C47C2×#38611yescharacteristic
309C2×C47#288412yes—
310C22×#41C2×D4711yescharacteristic
311C23×Q8—471no—
312#536—942no—
313#536—942no—
314#536—942no—
315#536—942no—
316C2×C4⋊Q8—472no—
317C2×#124—472no—
318C2×#123—474no—
319C2×C22⋊Q8—474no—
320#519—944no—
321#519—944no—
322C2×C4×Q8—942no—
323C2×C4×Q8—942no—
324C2×#117—471no—
325C2×#117—471no—
326C2×#117—471no—
327C2×#117—474no—
328C2×#117C2×D4711yescharacteristic
329C2×#117C2×D4711yescharacteristic
330#410—24no—
331#410—944no—
332C2×#99—474no—
333C2×#99C2×D4714yes—
334#391—942no—
335#391—942no—
336#381—474no—
337#342—474no—
338#342—474no—
339C47#2098611yescharacteristic, complemented, Sylow
340C2×#41—22no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C22×Q8—472no—
349C22×Q8—472no—
350C22×Q8—472no—
351C22×Q8—472no—
352C22×Q8—472no—
353C22×Q8—472no—
354C22×Q8—942no—
355C22×Q8—942no—
356C22×Q8—942no—
357C22×Q8—942no—
358C22×Q8C4×D4711yescharacteristic
359C22×D4C4×D4711yescharacteristic
360C22×D4C22×D4712yes—
361C23×C4—472no—
362C23×C4C4×D4712yes—
363C23×C4C22×D4711yescharacteristic
364C4⋊Q8—944no—
365C4⋊Q8—944no—
366#124—944no—
367#124—944no—
368#123—948no—
369#123—948no—
370C22⋊Q8—948no—
371C22⋊Q8—948no—
372C4×Q8—9416no—
373C4×Q8—9416no—
374#117—22no—
375#117—22no—
376#117—22no—
377#117—474no—
378#117—948no—
379#117—941no—
380#117—941no—
381#117—941no—
382#117—941no—
383#117—942no—
384#117—948no—
385#117—942no—
386#117—942no—
387#117C22⋊D4714yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—472no—
391C2×C4⋊C4—478no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C22⋊C4—472no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—472no—
398C2×C22⋊C4—472no—
399C2×C22⋊C4C22×D4712yes—
400C2×C22⋊C4C22×D4712yes—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—472no—
404C2×C42—944no—
405C2×C42—941no—
406C2×C42—941no—
407#99—28no—
408#99—948no—
409#95—948no—
410C24C2×C4×D4712yes—
411C24C23×D4711yescharacteristic
412#41—42no—
413#41—42no—
414#41—42no—
415#41—42no—
416#41—42no—
417#41—42no—
418#41—42no—
419#41—42no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—474no—
425C2×Q8—474no—
426C2×Q8—474no—
427C2×Q8—474no—
428C2×Q8—944no—
429C2×Q8—944no—
430C2×Q8—944no—
431C2×Q8—944no—
432C2×Q8—944no—
433C2×Q8—944no—
434C2×Q8—944no—
435C2×Q8—944no—
436C2×Q8—944no—
437C2×Q8—944no—
438C2×Q8—944no—
439C2×Q8—944no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—474no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4C2×C4×D4712yes—
467C22×C4C2×C4×D4711yescharacteristic
468C22×C4C2×C4×D4711yescharacteristic
469C22×C4#1062535012yes—
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4711yescharacteristic
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4711yescharacteristic
474C22×C4D4×D4712yes—
475C22×C4#1062535212yes—
476C22×C4C2×C22⋊D4712yes—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—9416no—
480C4⋊C4—9416no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—944no—
484C4⋊C4—1882no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—474no—
489C22⋊C4—9416no—
490C22⋊C4—942no—
491C22⋊C4—942no—
492C22⋊C4—942no—
493C22⋊C4—942no—
494C22⋊C4—942no—
495C22⋊C4C2×C22⋊D4714yes—
496C42—24no—
497C42—42no—
498C42—9416no—
499C42—942no—
500C42—942no—
501C42—944no—
502C42—1882no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—42no—
507C23—44no—
508C23C22×C4×D4711yescharacteristic
509C23C2×D4×D4711yescharacteristic
510C23C2×D4×D4711yescharacteristic
511C23C2×D4×D4711yescharacteristic
512C23C22×C22⋊D4711yescharacteristic
513C23C2×#1062535211yescharacteristic
514C23C2×#1062535011yescharacteristic
515Q8—42no—
516Q8—42no—
517Q8—948no—
518Q8—948no—
519Q8—948no—
520Q8—948no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—478no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—944no—
543C2×C4—944no—
544C2×C4—944no—
545C2×C4—1884no—
546C2×C4—1882no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4C22⋊C4×D4711yescharacteristic
550C2×C4C22⋊C4×D4711yescharacteristic
551C2×C4C22⋊C4×D4711yescharacteristic
552C2×C4C22⋊C4×D4711yescharacteristic
553C2×C4(C22⋊D47)⋊C412yes—
554C2×C4(C22⋊D47)⋊C412yes—
555C2×C4C4×C22⋊D4712yes—
556C2×C4C4×C22⋊D4712yes—
557C2×C4#1309399212yes—
558C2×C4#1309399212yes—
559C2×C4#1309399512yes—
560C2×C4#1309399512yes—
561C2×C4#1309399712yes—
562C2×C4#1309399712yes—
563C2×C4#1309400612yes—
564C2×C4#1309400612yes—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22C2×C22⋊C4×D4711yescharacteristic
577C22C2×(C22⋊D47)⋊C411yescharacteristic
578C22C2×C4×C22⋊D4711yescharacteristic
579C22C2×#1309399211yescharacteristic
580C22C2×#1309399511yescharacteristic
581C22C2×#1309399711yescharacteristic
582C22C2×#1309400611yescharacteristic
583C4—44no—
584C4—42no—
585C4—82no—
586C4—948no—
587C4—3762no—
588C4—1884no—
589C4—1884no—
590C4#1568899912yes—
591C4#1568899912yes—
592C4#1568899912yes—
593C4#1568899912yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—82no—
601C2C2×#1568899911yescharacteristic
602C2#2025185012yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#410C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#41×C47C22
cosocleC24×C47C24
hypercenterC2×#410D47
hypocenterC47#20986
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#15689600C2semidirect
4. #20245972C2semidirect
5. #20243843C2semidirect
6. #20243120C2semidirect
7. #20243120C2semidirect
14. C2×#13094022C4semidirect
15. C2×#13094011C4semidirect
17. C2×#13094003C4semidirect
339. C47#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#156896942
C2×#156896002
C2×#156890972
#202459722
#202438432
#202431202
C2×C47×#4102
#2098647