Finite Group Databasebeta

← #26122746

Subgroups — #26122746

Order 12032.

1526 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C212yesmaximal
2C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
3C2×#15688890C212yesmaximal, complemented
4#20241611C211yesmaximal, characteristic, complemented
5C2×C47×#386C211yesmaximal, characteristic
6C22×C22⋊Dic47C2212yes—
7C22×C22⋊Dic47C2212yes—
8C22×C4×Dic47—21no—
9C2×#13093939—24no—
10C2×#13093939—24no—
11C2×#13093939C418yescomplemented
12C4×C22⋊Dic47—24no—
13C2×C47⋊#95—21no—
14C2×C47⋊#95—22no—
15#15688890—24no—
16C23×D4×C47C2211yescharacteristic
17C24×C4×C47C2211yescharacteristic
18C22×C22⋊C4×C47C412yescomplemented
19C22×C22⋊C4×C47C2211yescharacteristic
20C47×#386—21no—
21C47×#386—22no—
22C47×#386—21no—
23C2×#95×C47C412yescomplemented
24C23×Dic47—22no—
25C23×Dic47—22no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47C2×C414yes—
35C2×C22⋊Dic47C2×C414yes—
36C2×C22⋊Dic47D414yes—
37C2×C22⋊Dic47D414yes—
38C2×C4×Dic47—24no—
39C2×C4×Dic47—44no—
40#13093939—216no—
41C47⋊#95—24no—
42C47⋊#95—44no—
43C25×C47C2×C411yescharacteristic
44C25×C47C2311yescharacteristic
45C22×D4×C47—22no—
46C22×D4×C47—22no—
47C22×D4×C47—21no—
48C22×D4×C47—21no—
49C22×D4×C47—22no—
50C22×D4×C47—22no—
51C22×D4×C47—22no—
52C22×D4×C47C2×C414yes—
53C23×C4×C47—22no—
54C23×C4×C47—22no—
55C23×C4×C47—44no—
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2×C412yes—
58C23×C4×C47C2×C412yes—
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D412yes—
61C23×C4×C47D411yescharacteristic
62C2×C22⋊C4×C47—22no—
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—22no—
66C2×C22⋊C4×C47—22no—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—44no—
69#95×C47—22no—
70#95×C47—22no—
71C24×C47—21no—
72C24×C47—21no—
73C24×C47—22no—
74C24×C47—22no—
75C24×C47—42no—
76C24×C47—42no—
77C24×C47—42no—
78C24×C47—42no—
79C24×C47C4212yescomplemented
80C24×C47C22×C411yescharacteristic
81C24×C47C22×C412yes—
82C24×C47C2×D411yescharacteristic
83C24×C47C2×D411yescharacteristic
84C24×C47C2×D412yes—
85C24×C47C2×D412yes—
86C24×C47#4112yes—
87C24×C47#4112yes—
88C24×C47#4112yes—
89C2×D4×C47—24no—
90C2×D4×C47—24no—
91C2×D4×C47—24no—
92C2×D4×C47—24no—
93C2×D4×C47—24no—
94C2×D4×C47—24no—
95C2×D4×C47—41no—
96C2×D4×C47—42no—
97C2×D4×C47—42no—
98C2×D4×C47—42no—
99C2×D4×C47—41no—
100C2×D4×C47—42no—
101C2×D4×C47—42no—
102C2×D4×C47—41no—
103C2×D4×C47—42no—
104C2×D4×C47—41no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—24no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—24no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—24no—
115C22×C4×C47—21no—
116C22×C4×C47—24no—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—24no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47—42no—
123C22×C4×C47—48no—
124C22×C4×C47—48no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C4212yescomplemented
128C22×C4×C47C22⋊C412yes—
129C22×C4×C47C22⋊C412yes—
130C22×Dic47—24no—
131C22×Dic47—24no—
132C22×Dic47—24no—
133C22×Dic47—24no—
134C22×Dic47—44no—
135C22×Dic47—44no—
136C22×Dic47—44no—
137C22×Dic47—44no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22⋊C4×C47—44no—
141C22⋊C4×C47—42no—
142C22⋊C4×C47—41no—
143C22⋊C4×C47—41no—
144C22⋊C4×C47—44no—
145C22⋊Dic47—24no—
146C22⋊Dic47—24no—
147C22⋊Dic47—24no—
148C22⋊Dic47—24no—
149C22⋊Dic47—24no—
150C22⋊Dic47—42no—
151C22⋊Dic47—42no—
152C22⋊Dic47C22⋊C418yescomplemented
153C4×Dic47—48no—
154C23×C47—22no—
155C23×C47—22no—
156C23×C47—22no—
157C23×C47—22no—
158C23×C47—22no—
159C23×C47—22no—
160C23×C47—22no—
161C23×C47—24no—
162C23×C47—42no—
163C23×C47—42no—
164C23×C47—44no—
165C23×C47—42no—
166C23×C47—44no—
167C23×C47—44no—
168C23×C47—42no—
169C23×C47—44no—
170C23×C47—44no—
171C23×C47—44no—
172C23×C47—41no—
173C23×C47—42no—
174C23×C47—41no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47—82no—
178C23×C47—82no—
179C23×C47—84no—
180C23×C47C2×C4211yescharacteristic
181C23×C47C2×C22⋊C412yes—
182C23×C47C2×C22⋊C411yescharacteristic
183C23×C47#11712yes—
184C23×C47#11711yescharacteristic
185C23×C47C4×D412yes—
186C23×C47C4×D412yes—
187C23×C47C4×D412yes—
188C23×C47C4×D412yes—
189C23×C47C4×D412yes—
190C23×C47C4×D412yes—
191C23×C47#12011yescharacteristic
192C23×C47#12112yes—
193C23×C47#12112yes—
194C23×C47#12112yes—
195C23×C47#12311yescharacteristic
196C23×C47#12312yes—
197C23×C47#12412yes—
198C23×C47#12411yescharacteristic
199C23×C47#12612yes—
200C23×C47#12711yescharacteristic
201D4×C47—44no—
202D4×C47—44no—
203D4×C47—44no—
204D4×C47—44no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—48no—
215C2×C4×C47—48no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—41no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—84no—
225C2×C4×C47#9914yescomplemented
226C2×Dic47—28no—
227C2×Dic47—44no—
228C2×Dic47—44no—
229C2×Dic47—48no—
230C2×Dic47—48no—
231C2×Dic47—48no—
232C2×Dic47—44no—
233C2×Dic47—82no—
234C2×Dic47—82no—
235C2×Dic47—84no—
236#10960—471nomaximal, Sylow
237C22×C47—21no—
238C22×C47—22no—
239C22×C47—21no—
240C22×C47—22no—
241C22×C47—21no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—21no—
245C22×C47—24no—
246C22×C47—22no—
247C22×C47—24no—
248C22×C47—22no—
249C22×C47—24no—
250C22×C47—24no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—44no—
263C22×C47—84no—
264C22×C47—84no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—82no—
268C22×C47C4×C22⋊C411yescharacteristic
269C22×C47C4×C22⋊C412yes—
270C22×C47#38612yes—
271C22×C47#38611yescharacteristic
272C22×C47#38812yes—
273C22×C47#38812yes—
274C22×C47#38812yes—
275C22×C47#39012yes—
276C22×C47#39011yescharacteristic
277C22×C47C2×#9912yes—
278C22×C47#41012yes—
279C4×C47—24no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47—82no—
283Dic47—48no—
284Dic47—84no—
285C22×#99—472no—
286C2×#386D4711yescharacteristic
287C2×C4×C22⋊C4—471no—
288C2×#342—472no—
289C22⋊#95—471no—
290C2×C47—22no—
291C2×C47—22no—
292C2×C47—22no—
293C2×C47—22no—
294C2×C47—22no—
295C2×C47—22no—
296C2×C47—82no—
297C2×C47#143711yescharacteristic
298C2×C47C4×#9912yes—
299C2×C47#189811yescharacteristic
300C2×C47#189911yescharacteristic
301C2×C47#196412yes—
302C23×D4C2×D4711yescharacteristic
303C24×C4C2×D4711yescharacteristic
304C22×C22⋊C4—472no—
305C22×C22⋊C4—472no—
306C22×C22⋊C4Dic4712yescomplemented
307C22×C22⋊C4C2×D4711yescharacteristic
308C22×C42—941no—
309C2×#99—478no—
310C2×#99—944no—
311C2×#99—944no—
312#386—21no—
313#386—22no—
314#386—21no—
315C4×C22⋊C4—944no—
316C2×#95—941no—
317C2×#95—942no—
318C2×#95Dic4712yescomplemented
319#342—944no—
320C47#1096011yescharacteristic, complemented, Sylow
321C25C2×Dic4711yescharacteristic
322C25C22×D4711yescharacteristic
323C22×D4—22no—
324C22×D4—22no—
325C22×D4—22no—
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4C4×D4714yes—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—44no—
334C23×C4—942no—
335C23×C4—942no—
336C23×C4C2×Dic4711yescharacteristic
337C23×C4C2×Dic4712yes—
338C23×C4C2×Dic4712yes—
339C23×C4C22⋊D4711yescharacteristic
340C23×C4C22⋊D4712yes—
341C23×C4C22⋊D4711yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—474no—
350C2×C22⋊C4—474no—
351C2×C22⋊C4—474no—
352C2×C22⋊C4—474no—
353C2×C22⋊C4—942no—
354C2×C22⋊C4—942no—
355C2×C22⋊C4—942no—
356C2×C22⋊C4—942no—
357C2×C22⋊C4—942no—
358C2×C22⋊C4—942no—
359C2×C22⋊C4—942no—
360C2×C22⋊C4—942no—
361C2×C42—944no—
362C2×C42—1884no—
363#99—9416no—
364#95—22no—
365#95—22no—
366#95—944no—
367#95—1884no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24C4×Dic4712yescomplemented
377C24C2×C4×D4712yes—
378C24#1062535012yes—
379C24D4×D4712yes—
380C24D4×D4712yes—
381C24#1062535212yes—
382C24#1062535212yes—
383C24C22×Dic4711yescharacteristic
384C24C2×C22⋊D4711yescharacteristic
385C24C2×C22⋊D4711yescharacteristic
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—41no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—41no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—944no—
425C22×C4—944no—
426C22×C4—944no—
427C22×C4—944no—
428C22×C4—1884no—
429C22×C4—1884no—
430C22×C4—1882no—
431C22×C4—1882no—
432C22×C4—1884no—
433C22×C4—1884no—
434C22×C4C4×Dic4712yescomplemented
435C22×C4C22⋊Dic4712yes—
436C22×C4C22⋊Dic4712yes—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—44no—
442C22⋊C4—478no—
443C22⋊C4—944no—
444C22⋊C4—944no—
445C22⋊C4—944no—
446C22⋊C4—944no—
447C22⋊C4—944no—
448C22⋊C4—1882no—
449C22⋊C4—1882no—
450C42—1888no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—84no—
475C23—82no—
476C23—82no—
477C23C2×C4×Dic4711yescharacteristic
478C23C22⋊C4×D4712yes—
479C23C2×C22⋊Dic4711yescharacteristic
480C23#1309398012yes—
481C23#1309398311yescharacteristic
482C23(C22⋊D47)⋊C412yes—
483C23(C22⋊D47)⋊C412yes—
484C23C4×C22⋊D4712yes—
485C23C4×C22⋊D4712yes—
486C23D4×Dic4712yes—
487C23D4×Dic4712yes—
488C23#1309399211yescharacteristic
489C23#1309399512yes—
490C23#1309399612yes—
491C23#1309399712yes—
492C23#1309400511yescharacteristic
493C23#1309400612yes—
494C23#1309401012yes—
495C23#1309401111yescharacteristic
496C23#1309401712yes—
497C23#1309402011yescharacteristic
498D4—44no—
499D4—44no—
500D4—44no—
501D4—44no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—948no—
523C2×C4—3764no—
524C2×C4—3762no—
525C2×C4—3762no—
526C2×C4—1888no—
527C2×C4—1884no—
528C2×C4—1884no—
529C2×C4—1888no—
530C2×C4—1884no—
531C2×C4—1888no—
532C2×C4#1309393914yescomplemented
533C22—24no—
534C22—22no—
535C22—24no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—24no—
546C22—24no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—84no—
560C22—84no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22C4×C22⋊Dic4711yescharacteristic
565C22C22⋊C4×Dic4712yes—
566C22#1568899912yes—
567C22#1568900311yescharacteristic
568C22#1568901112yes—
569C22#1568901612yes—
570C22#1568901712yes—
571C22#1568902312yes—
572C22#1568902611yescharacteristic
573C22C2×#1309393912yes—
574C22#1568909712yes—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—3764no—
580C4—1888no—
581C2—22no—
582C2—22no—
583C2—22no—
584C2—22no—
585C2—22no—
586C2—22no—
587C2—82no—
588C2#2024162511yescharacteristic
589C2C4×#1309393912yes—
590C2#2024384211yescharacteristic
591C2#2024384711yescharacteristic
592C2#2024422512yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C2×C42
FrattiniC25C22×D47
FittingC2×C47×#386C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C22
cosocleC25×C47C23
hypercenterC2×#386D47
hypocenterC47#10960
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊Dic47C2semidirect
3. C2×#15688890C2semidirect
4. #20241611C2semidirect
11. C2×#13093939C4semidirect
18. C22×C22⋊C4×C47C4semidirect
23. C2×#95×C47C4semidirect
79. C24×C47C42semidirect
127. C22×C4×C47C42semidirect
152. C22⋊Dic47C22⋊C4semidirect
225. C2×C4×C47#99semidirect
306. C22×C22⋊C4Dic47semidirect
318. C2×#95Dic47semidirect
320. C47#10960semidirect
376. C24C4×Dic47semidirect
434. C22×C4C4×Dic47semidirect
532. C2×C4#13093939semidirect

Maximal subgroups

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

subgroupindex
C22×#130939392
C2×C4×C22⋊Dic472
C2×#156888902
#202416112
C2×C47×#3862
#1096047