Finite Group Databasebeta

← #26123873

Subgroups — #26123873

Order 12032.

826 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689141C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic
3C2×#15689105C211yesmaximal, characteristic
4C2×#15688893C211yesmaximal, characteristic, complemented
5C2×#15688892C211yesmaximal, characteristic, complemented
6#20243114C211yesmaximal, characteristic, complemented
7C47×#1736C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689141—22no—
13#15689141—22no—
14C2×#13093952—22no—
15#15689122—22no—
16#15689122—22no—
17C2×#13093947—21no—
18C2×#13093944—21no—
19#15689105—22no—
20#15689105C414yescomplemented
21C2×#13093942—21no—
22C2×#13093940—21no—
23#15688893—22no—
24#15688892—22no—
25#15688887—22no—
26C22×M16×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2211yescharacteristic
29C47×#340—22no—
30C2×#10625350—21no—
31C2×#10625350—21no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350C2×C412yes—
41C2×#10625350D412yes—
42C22×Dic94D411yescharacteristic
43C22×D188D411yescharacteristic
44C22×C22⋊D47—21no—
45C22×C4×D47—21no—
46C2×#10625329—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329—21no—
60C2×#10625329—21no—
61C2×#10625329C2×C412yes—
62C2×#10625329D412yes—
63C2×#10625329D412yes—
64C22×C47⋊C8—21no—
65C22×C47⋊C8—21no—
66C22×C47⋊C8—21no—
67#13093952—28no—
68#13093947—24no—
69#13093944—24no—
70#13093942—24no—
71#13093940—24no—
72C23×C4×C47—21no—
73C23×C4×C47C2311yescharacteristic
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47C2×C412yes—
80C22×C8×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47D412yes—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47D412yes—
92C2×C22⋊C4×C47—21no—
93C2×C42×C47—21no—
94C24×C47#4111yescharacteristic
95C23×D47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—41no—
98C22×C4×C47—41no—
99C22×C4×C47—41no—
100C22×C4×C47—82no—
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C2×C22⋊D47—22no—
116C2×C22⋊D47—22no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121C22×Dic47—21no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188C22⋊C412yes—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—41no—
147C2×C4×D47—41no—
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94C22⋊C412yes—
156M16×C47—22no—
157M16×C47—22no—
158M16×C47—41no—
159M16×C47—41no—
160C2×C8×C47—22no—
161C2×C8×C47—41no—
162C2×C8×C47—41no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—41no—
167C4⋊C4×C47—41no—
168C4⋊C4×C47—41no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—41no—
171C42×C47—22no—
172C42×C47—41no—
173#10625329—22no—
174#10625329—22no—
175#10625329—22no—
176#10625329—22no—
177#10625329—22no—
178#10625329—22no—
179#10625329—41no—
180#10625329—41no—
181#10625329—41no—
182#10625329—41no—
183#10625329—41no—
184#10625329—41no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—41no—
189C2×C47⋊C8—41no—
190C2×C47⋊C8—41no—
191C2×C47⋊C8—41no—
192C2×C47⋊C8—41no—
193C2×C47⋊C8—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47#11711yescharacteristic
197C23×C47#12111yescharacteristic
198C23×C47#12311yescharacteristic
199C23×C47#12311yescharacteristic
200C23×C47#12311yescharacteristic
201C23×C47#12411yescharacteristic
202C23×C47#12611yescharacteristic
203C22×D47—22no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—81no—
207C22×D47—81no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—81no—
211C2×C4×C47—81no—
212C2×C4×C47—82no—
213C2×C4×C47—82no—
214C2×C4×C47—82no—
215C2×C4×C47C2×C22⋊C411yescharacteristic
216C2×C4×C47#11711yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C4×D411yescharacteristic
221C2×C4×C47#12011yescharacteristic
222C2×C4×C47#12011yescharacteristic
223C2×C4×C47#12111yescharacteristic
224C2×C4×C47#12111yescharacteristic
225C2×C4×C47#12111yescharacteristic
226C2×C4×C47#12111yescharacteristic
227C2×C4×C47#12111yescharacteristic
228C2×C4×C47#12111yescharacteristic
229C2×C4×C47#12111yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12311yescharacteristic
232C2×C4×C47#12311yescharacteristic
233C2×C4×C47#12311yescharacteristic
234C2×C4×C47#12311yescharacteristic
235C2×C4×C47#12311yescharacteristic
236C2×C4×C47#12311yescharacteristic
237C2×C4×C47#12411yescharacteristic
238C2×C4×C47#12411yescharacteristic
239C2×C4×C47#12411yescharacteristic
240C2×C4×C47#12611yescharacteristic
241C2×C4×C47#12611yescharacteristic
242C2×C4×C47#12611yescharacteristic
243C22⋊D47—42no—
244C22⋊D47—42no—
245C22⋊D47—42no—
246C22⋊D47—42no—
247C2×Dic47—22no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250D188—22no—
251D188—22no—
252D188—41no—
253D188—41no—
254C4×D47—42no—
255C4×D47—42no—
256C4×D47—42no—
257C4×D47—42no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C8×C47—42no—
263C47⋊C8—42no—
264C47⋊C8—42no—
265C47⋊C8—42no—
266#11152—471nomaximal, Sylow
267C22×C47—21no—
268C22×C47—21no—
269C22×C47—21no—
270C22×C47—21no—
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47#38111yescharacteristic
278C22×C47#38811yescharacteristic
279C22×C47#38811yescharacteristic
280C22×C47#39411yescharacteristic
281C22×C47#39411yescharacteristic
282C22×C47#39711yescharacteristic
283C22×C47#39711yescharacteristic
284C2×D47—42no—
285C2×D47—82no—
286C2×D47—81no—
287C2×D47—81no—
288C2×D47—82no—
289C4×C47—82no—
290C4×C47—82no—
291C4×C47#38611yescharacteristic
292C4×C47#38611yescharacteristic
293C4×C47#38811yescharacteristic
294C4×C47#38811yescharacteristic
295C4×C47#39411yescharacteristic
296C4×C47#39411yescharacteristic
297C4×C47#39711yescharacteristic
298C4×C47#39711yescharacteristic
299Dic47—42no—
300C2×#421—471no—
301C2×#417—471no—
302C2×#413—471no—
303C2×#344—471no—
304C2×#343—471no—
305#1738—471no—
306#1736D4711yescharacteristic, complemented
307C2×C47—21no—
308C2×C47—21no—
309C2×C47—21no—
310C2×C47—21no—
311C2×C47—21no—
312C2×C47—21no—
313C2×C47#144011yescharacteristic
314C2×C47#204811yescharacteristic
315C2×C47#204911yescharacteristic
316D47—82no—
317C22×#41—471no—
318C22×M16—471no—
319C22×M16—471no—
320C22×M16—471no—
321C22×M16C2×D4711yescharacteristic
322C2×#117C2×D4711yescharacteristic
323C22×C4⋊C4C2×D4711yescharacteristic
324#421—942no—
325#421—942no—
326C2×#104—942no—
327#417—942no—
328#417—942no—
329C2×#103—941no—
330C2×#102—941no—
331#413—474no—
332#413—942no—
333C2×#101—941no—
334C2×#100—941no—
335#344—942no—
336#343—942no—
337#341—942no—
338#340—22no—
339C47#1115211yescharacteristic, complemented, Sylow
340C2×#41—472no—
341C2×#41—472no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—941no—
347C2×#41—941no—
348C2×#41—941no—
349C2×#41—941no—
350C2×#41—941no—
351C2×#41—941no—
352C22×Q8—471no—
353C22×D4—471no—
354C22×D4—941no—
355C23×C4—21no—
356C23×C4—941no—
357C23×C4C22×D4711yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—472no—
364C2×M16—472no—
365C2×M16—472no—
366C2×M16—941no—
367C2×M16—941no—
368C2×M16—941no—
369C2×M16—941no—
370C2×M16—941no—
371C2×M16—941no—
372C2×M16—941no—
373C2×M16—941no—
374C2×M16—941no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C2×M16—941no—
381C2×M16C4×D4712yes—
382C22×C8—21no—
383C22×C8—941no—
384C22×C8—941no—
385C22×C8—941no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117D18812yes—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4C22⋊D4712yes—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399#104—948no—
400#103—944no—
401#102—944no—
402#101—944no—
403#100—944no—
404C24—941no—
405C24#1062535211yescharacteristic
406#41—942no—
407#41—942no—
408#41—942no—
409#41—942no—
410#41—942no—
411#41—942no—
412#41—942no—
413#41—942no—
414#41—1881no—
415#41—1881no—
416#41—1881no—
417#41—1881no—
418#41—1881no—
419#41—1881no—
420#41—1881no—
421#41—1881no—
422C2×Q8—472no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×D4—472no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—1881no—
437C2×D4—1881no—
438C2×D4—1881no—
439C2×D4—1881no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—82no—
445C22×C4—941no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—1881no—
449C22×C4—1881no—
450C22×C4—1881no—
451C22×C4—1881no—
452C22×C4C2×C4×D4711yescharacteristic
453C22×C4C2×D18811yescharacteristic
454C22×C4#1062535011yescharacteristic
455C22×C4#1062535011yescharacteristic
456C22×C4#1062535011yescharacteristic
457C22×C4#1062535011yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4D4×D4711yescharacteristic
461C22×C4D4×D4711yescharacteristic
462C22×C4#1062535211yescharacteristic
463C22×C4#1062535411yescharacteristic
464C22×C4#1062535411yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—942no—
471M16—942no—
472M16—942no—
473M16—942no—
474M16—942no—
475M16—942no—
476M16—1881no—
477M16—1881no—
478M16—1881no—
479M16—1881no—
480M16—1881no—
481M16—1881no—
482C2×C8—22no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—942no—
486C2×C8—942no—
487C2×C8—942no—
488C2×C8—1881no—
489C2×C8—1881no—
490C2×C8—1881no—
491C2×C8—1881no—
492C2×C8—1881no—
493C2×C8—1881no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C22⋊C4—22no—
501C22⋊C4—41no—
502C42—22no—
503C42—41no—
504C23—41no—
505C23—41no—
506C23—942no—
507C23—3761no—
508C23—3761no—
509C23—1881no—
510C23—1881no—
511C23#1309398211yescharacteristic
512C23#1309399611yescharacteristic
513C23#1309400611yescharacteristic
514C23#1309400611yescharacteristic
515C23#1309400811yescharacteristic
516C23#1309401011yescharacteristic
517C23#1309401911yescharacteristic
518Q8—942no—
519Q8—942no—
520Q8—1881no—
521Q8—1881no—
522D4—942no—
523D4—942no—
524D4—1882no—
525D4—1882no—
526D4—1882no—
527D4—1882no—
528D4—1881no—
529D4—1881no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—942no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4C22⋊C4×D4711yescharacteristic
545C2×C4#1309398111yescharacteristic
546C2×C4(C22⋊D47)⋊C411yescharacteristic
547C2×C4C4×C22⋊D4711yescharacteristic
548C2×C4#1309398611yescharacteristic
549C2×C4C4×D18811yescharacteristic
550C2×C4#1309399211yescharacteristic
551C2×C4#1309399411yescharacteristic
552C2×C4#1309399511yescharacteristic
553C2×C4#1309399711yescharacteristic
554C2×C4#1309399711yescharacteristic
555C2×C4#1309399711yescharacteristic
556C2×C4#1309399811yescharacteristic
557C2×C4#1309399911yescharacteristic
558C2×C4#1309399911yescharacteristic
559C2×C4#1309400611yescharacteristic
560C2×C4#1309400711yescharacteristic
561C2×C4#1309400911yescharacteristic
562C2×C4#1309400911yescharacteristic
563C2×C4#1309400911yescharacteristic
564C2×C4#1309400911yescharacteristic
565C2×C4#1309400911yescharacteristic
566C2×C4#1309401011yescharacteristic
567C2×C4#1309401211yescharacteristic
568C2×C4#1309401311yescharacteristic
569C2×C4#1309401811yescharacteristic
570C2×C4#1309401911yescharacteristic
571C2×C4#1309401911yescharacteristic
572C8—42no—
573C8—1882no—
574C8—1882no—
575C8—1882no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—3762no—
587C22—3762no—
588C22—3761no—
589C22—3761no—
590C22—1882no—
591C22#1568898511yescharacteristic
592C22#1568901311yescharacteristic
593C22#1568901511yescharacteristic
594C22#1568903811yescharacteristic
595C22#1568903911yescharacteristic
596C22#1568905011yescharacteristic
597C22#1568905211yescharacteristic
598C4—82no—
599C4—82no—
600C4—1882no—
601C4#1568899911yescharacteristic
602C4#1568900111yescharacteristic
603C4#1568901211yescharacteristic
604C4#1568901411yescharacteristic
605C4#1568904011yescharacteristic
606C4#1568904111yescharacteristic
607C4#1568904911yescharacteristic
608C4#1568905211yescharacteristic
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—3762no—
616C2#2024163611yescharacteristic
617C2#2024465911yescharacteristic
618C2#2024466611yescharacteristic
subgroupG/H
centerC22#15689038
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1736C2
socleC22×C47#394
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#1736D47
hypocenterC47#11152
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. C2×#15688893C2semidirect
5. C2×#15688892C2semidirect
6. #20243114C2semidirect
7. C47×#1736C2semidirect
20. #15689105C4semidirect
306. #1736D47semidirect
339. C47#11152semidirect

Maximal subgroups

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

subgroupindex
C2×#156891412
C2×#156891222
C2×#156891052
C2×#156888932
C2×#156888922
#202431142
C47×#17362
#1115247