Finite Group Databasebeta

← #26123202

Subgroups — #26123202

Order 12032.

1454 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C212yesmaximal
2C2×#15688983C211yesmaximal, characteristic, complemented
3C2×#15688890C212yesmaximal, complemented
4#20241611C211yesmaximal, characteristic, complemented
5C2×C47×#386C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C22×C22⋊Dic47C2212yes—
8C22×C22⋊Dic47C2212yes—
9C2×#13093939—24no—
10C2×#13093939—24no—
11C2×#13093939—24no—
12#15688983—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×C4⋊Dic47—24no—
27C2×C4⋊Dic47—44no—
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⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47D414yes—
39C2×C22⋊Dic47D414yes—
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—21no—
50C22×D4×C47—21no—
51C22×D4×C47—22no—
52C22×D4×C47—22no—
53C22×D4×C47D412yes—
54C22×D4×C47Q812yes—
55C23×C4×C47—22no—
56C23×C4×C47—22no—
57C23×C4×C47—44no—
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2×C412yes—
60C23×C4×C47C2×C412yes—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D412yes—
63C23×C4×C47D411yescharacteristic
64C2×C22⋊C4×C47—22no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—44no—
71#95×C47—22no—
72#95×C47—22no—
73C24×C47—21no—
74C24×C47—21no—
75C24×C47—22no—
76C24×C47—22no—
77C24×C47—42no—
78C24×C47—42no—
79C24×C47—42no—
80C24×C47—42no—
81C24×C47C4⋊C412yescomplemented
82C24×C47C22×C411yescharacteristic
83C24×C47C2×D411yescharacteristic
84C24×C47C2×D411yescharacteristic
85C24×C47C2×D411yescharacteristic
86C24×C47C2×D412yes—
87C24×C47C2×D412yes—
88C24×C47C2×Q811yescharacteristic
89C24×C47#4111yescharacteristic
90C24×C47#4111yescharacteristic
91C24×C47#4112yes—
92C24×C47#4112yes—
93C2×D4×C47—24no—
94C2×D4×C47—22no—
95C2×D4×C47—22no—
96C2×D4×C47—22no—
97C2×D4×C47—22no—
98C2×D4×C47—24no—
99C2×D4×C47—24no—
100C2×D4×C47—24no—
101C2×D4×C47—41no—
102C2×D4×C47—42no—
103C2×D4×C47—42no—
104C2×D4×C47—42no—
105C2×D4×C47—41no—
106C2×D4×C47—42no—
107C2×D4×C47—42no—
108C2×D4×C47—41no—
109C2×D4×C47—42no—
110C2×D4×C47—41no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—24no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—24no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—21no—
120C22×C4×C47—24no—
121C22×C4×C47—21no—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—48no—
130C22×C4×C47—48no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C22⋊C412yes—
134C22×C4×C47C22⋊C412yes—
135C22×C4×C47C4⋊C412yescomplemented
136C22×Dic47—24no—
137C22×Dic47—24no—
138C22×Dic47—44no—
139C22×Dic47—44no—
140C22×Dic47—44no—
141C22×Dic47—44no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22⋊C4×C47—44no—
147C22⋊C4×C47—42no—
148C22⋊C4×C47—41no—
149C22⋊C4×C47—41no—
150C22⋊C4×C47—44no—
151C22⋊Dic47—24no—
152C22⋊Dic47—24no—
153C22⋊Dic47—24no—
154C22⋊Dic47—24no—
155C22⋊Dic47—42no—
156C22⋊Dic47—42no—
157C22⋊Dic47—42no—
158C22⋊Dic47—42no—
159C4⋊Dic47—48no—
160C23×C47—22no—
161C23×C47—22no—
162C23×C47—22no—
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—22no—
166C23×C47—22no—
167C23×C47—24no—
168C23×C47—42no—
169C23×C47—42no—
170C23×C47—42no—
171C23×C47—42no—
172C23×C47—44no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47—44no—
178C23×C47—44no—
179C23×C47—44no—
180C23×C47—41no—
181C23×C47—42no—
182C23×C47—41no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—82no—
186C23×C47—82no—
187C23×C47—84no—
188C23×C47C2×C22⋊C411yescharacteristic
189C23×C47C2×C4⋊C411yescharacteristic
190C23×C47#11711yescharacteristic
191C23×C47C4×D412yes—
192C23×C47C4×D412yes—
193C23×C47#12011yescharacteristic
194C23×C47#12012yes—
195C23×C47#12112yes—
196C23×C47#12112yes—
197C23×C47#12112yes—
198C23×C47#12111yescharacteristic
199C23×C47#12111yescharacteristic
200C23×C47C22⋊Q812yes—
201C23×C47C22⋊Q812yes—
202C23×C47C22⋊Q811yescharacteristic
203C23×C47C22⋊Q811yescharacteristic
204C23×C47#12311yescharacteristic
205C23×C47#12312yes—
206C23×C47#12312yes—
207C23×C47#12412yes—
208C23×C47#12411yescharacteristic
209C23×C47#12612yes—
210C23×C47#12711yescharacteristic
211D4×C47—42no—
212D4×C47—42no—
213D4×C47—44no—
214D4×C47—44no—
215D4×C47—42no—
216D4×C47—42no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—48no—
227C2×C4×C47—48no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—41no—
231C2×C4×C47—42no—
232C2×C4×C47—41no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—84no—
237C2×C4×C47#9914yescomplemented
238C2×Dic47—44no—
239C2×Dic47—44no—
240C2×Dic47—48no—
241C2×Dic47—44no—
242C2×Dic47—44no—
243C2×Dic47—84no—
244C2×Dic47—84no—
245C2×Dic47—82no—
246C2×Dic47—82no—
247C2×Dic47—84no—
248#11043—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—21no—
256C22×C47—22no—
257C22×C47—21no—
258C22×C47—22no—
259C22×C47—21no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—21no—
263C22×C47—24no—
264C22×C47—22no—
265C22×C47—24no—
266C22×C47—22no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—44no—
276C22×C47—42no—
277C22×C47—84no—
278C22×C47—84no—
279C22×C47—81no—
280C22×C47—81no—
281C22×C47—82no—
282C22×C47#38011yescharacteristic
283C22×C47#38512yes—
284C22×C47#38611yescharacteristic
285C22×C47#38812yes—
286C22×C47#39011yescharacteristic
287C22×C47#39211yescharacteristic
288C22×C47#39311yescharacteristic
289C22×C47#39412yes—
290C22×C47#39612yes—
291C22×C47#39711yescharacteristic
292C22×C47#39911yescharacteristic
293C22×C47C2×#9912yes—
294C22×C47#41012yes—
295C4×C47—24no—
296C4×C47—42no—
297C4×C47—42no—
298C4×C47—82no—
299Dic47—84no—
300Dic47—84no—
301C22×#99—472no—
302C2×#386D4711yescharacteristic
303C2×#380—471no—
304C2×#342—472no—
305C22⋊#95—471no—
306C2×C47—21no—
307C2×C47—22no—
308C2×C47—22no—
309C2×C47—22no—
310C2×C47—21no—
311C2×C47—22no—
312C2×C47—21no—
313C2×C47—21no—
314C2×C47—82no—
315C2×C47#143911yescharacteristic
316C2×C47#179912yes—
317C2×C47#189811yescharacteristic
318C2×C47#189911yescharacteristic
319C2×C47#196412yes—
320C23×D4C2×D4711yescharacteristic
321C24×C4C2×D4711yescharacteristic
322C22×C4⋊C4—941no—
323C22×C22⋊C4—472no—
324C22×C22⋊C4—472no—
325C22×C22⋊C4Dic4712yescomplemented
326C22×C22⋊C4C2×D4711yescharacteristic
327C2×#99—944no—
328C2×#99—944no—
329C2×#99—944no—
330#386—21no—
331#386—22no—
332#386—21no—
333#380—944no—
334C2×#95—941no—
335C2×#95—942no—
336C2×#95Dic4712yescomplemented
337#342—944no—
338C47#1104311yescharacteristic, complemented, Sylow
339C25C2×Dic4711yescharacteristic
340C25C22×D4711yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4Dic9412yes—
350C22×D4D18812yes—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—44no—
354C23×C4—942no—
355C23×C4—942no—
356C23×C4C2×Dic4711yescharacteristic
357C23×C4C2×Dic4712yes—
358C23×C4C2×Dic4712yes—
359C23×C4C22⋊D4711yescharacteristic
360C23×C4C22⋊D4712yes—
361C23×C4C22⋊D4711yescharacteristic
362C2×C4⋊C4—944no—
363C2×C4⋊C4—1884no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—474no—
372C2×C22⋊C4—474no—
373C2×C22⋊C4—942no—
374C2×C22⋊C4—942no—
375C2×C22⋊C4—942no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4—942no—
378C2×C22⋊C4—942no—
379C2×C22⋊C4—942no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383#99—9416no—
384#95—22no—
385#95—22no—
386#95—944no—
387#95—1884no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C4⋊Dic4712yescomplemented
397C24C2×Dic9411yescharacteristic
398C24C2×D18811yescharacteristic
399C24#1062535011yescharacteristic
400C24#1062535011yescharacteristic
401C24D4×D4712yes—
402C24D4×D4712yes—
403C24#1062535212yes—
404C24#1062535212yes—
405C24C22×Dic4711yescharacteristic
406C24C2×C22⋊D4711yescharacteristic
407C24C2×C22⋊D4711yescharacteristic
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—41no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—1884no—
451C22×C4—1884no—
452C22×C4—1884no—
453C22×C4—1884no—
454C22×C4—1882no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4C4⋊Dic4712yescomplemented
459C22×C4C22⋊Dic4712yes—
460C22×C4C22⋊Dic4712yes—
461C4⋊C4—1888no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—44no—
467C22⋊C4—944no—
468C22⋊C4—944no—
469C22⋊C4—944no—
470C22⋊C4—944no—
471C22⋊C4—1882no—
472C22⋊C4—1882no—
473C22⋊C4—1882no—
474C22⋊C4—1882no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—84no—
501C23—82no—
502C23—82no—
503C23C2×C22⋊Dic4711yescharacteristic
504C23C2×C4⋊Dic4711yescharacteristic
505C23#1309398311yescharacteristic
506C23D4×Dic4712yes—
507C23D4×Dic4712yes—
508C23#1309399211yescharacteristic
509C23#1309399412yes—
510C23#1309399512yes—
511C23#1309399612yes—
512C23#1309399712yes—
513C23#1309399811yescharacteristic
514C23#1309399811yescharacteristic
515C23#1309400112yes—
516C23#1309400112yes—
517C23#1309400311yescharacteristic
518C23#1309400311yescharacteristic
519C23#1309400511yescharacteristic
520C23#1309400612yes—
521C23#1309400812yes—
522C23#1309401012yes—
523C23#1309401111yescharacteristic
524C23#1309401712yes—
525C23#1309402011yescharacteristic
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—3764no—
553C2×C4—3762no—
554C2×C4—3762no—
555C2×C4—3764no—
556C2×C4—3764no—
557C2×C4—1884no—
558C2×C4—1888no—
559C2×C4—1884no—
560C2×C4—1884no—
561C2×C4—1884no—
562C2×C4#1309393914yescomplemented
563C22—24no—
564C22—22no—
565C22—24no—
566C22—22no—
567C22—21no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—22no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22#1568898311yescharacteristic
597C22#1568899812yes—
598C22#1568900311yescharacteristic
599C22#1568901612yes—
600C22#1568902611yescharacteristic
601C22#1568903211yescharacteristic
602C22#1568903511yescharacteristic
603C22#1568903912yes—
604C22#1568904612yes—
605C22#1568904811yescharacteristic
606C22#1568905911yescharacteristic
607C22C2×#1309393912yes—
608C22#1568909712yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—3764no—
614C4—3764no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—82no—
624C2#2024163311yescharacteristic
625C2#2024337612yes—
626C2#2024384211yescharacteristic
627C2#2024384711yescharacteristic
628C2#2024422512yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC24×C47C22×C4
FrattiniC25C22×D47
FittingC2×C47×#386C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C22
cosocleC25×C47C23
hypercenterC2×#386D47
hypocenterC47#11043
LCSG ▷ C24×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×#15688983C2semidirect
3. C2×#15688890C2semidirect
4. #20241611C2semidirect
18. C22×C22⋊C4×C47C4semidirect
23. C2×#95×C47C4semidirect
81. C24×C47C4⋊C4semidirect
135. C22×C4×C47C4⋊C4semidirect
237. C2×C4×C47#99semidirect
325. C22×C22⋊C4Dic47semidirect
336. C2×#95Dic47semidirect
338. C47#11043semidirect
396. C24C4⋊Dic47semidirect
458. C22×C4C4⋊Dic47semidirect
562. 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×#156889832
C2×#156888902
#202416112
C2×C47×#3862
#1104347