Finite Group Databasebeta

← #26123625

Subgroups — #26123625

Order 12032.

864 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251978C211yesmaximal, characteristic, complemented
2C2×#15689139C211yesmaximal, characteristic
3C2×#15689139C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20243120C211yesmaximal, characteristic, complemented
6C2×#15688880C211yesmaximal, characteristic, complemented
7C47×#1742C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C2×#13093983C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689165—22no—
13#15689165—22no—
14#15689161—22no—
15#15689139—22no—
16#15689139—22no—
17#15689139—22no—
18#15689139—22no—
19C2×#13093950—22no—
20C2×#13093950C414yescomplemented
21#15689096—22no—
22#15689096—22no—
23C2×#13093938—22no—
24#15688889—22no—
25#15688880—22no—
26C22×M16×C47C2211yescharacteristic
27C22×M16×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C47×#343—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×Dic94Q811yescharacteristic
43C22×D188Q811yescharacteristic
44C22×C22⋊D47—21no—
45C22×C4×D47—21no—
46#13093983—21no—
47#13093983—21no—
48#13093983—21no—
49#13093983—21no—
50#13093983—21no—
51#13093983—21no—
52#13093983—21no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983D412yes—
57#13093983D412yes—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C22⋊Dic47—21no—
62C2×C22⋊Dic47—21no—
63C2×C22⋊Dic47—21no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—21no—
66C2×C4×Dic47C2×C412yes—
67#13093958—24no—
68#13093956—24no—
69#13093950—28no—
70#13093950—28no—
71#13093938—28no—
72C23×C4×C47C2311yescharacteristic
73C2×M16×C47—21no—
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47—21no—
81C2×M16×C47C2×C412yes—
82C2×M16×C47D412yes—
83C2×M16×C47D412yes—
84C2×M16×C47D412yes—
85C22×C8×C47—21no—
86C22×C8×C47—21no—
87#117×C47—21no—
88#117×C47—21no—
89#117×C47—21no—
90#117×C47—21no—
91C2×C4⋊C4×C47—21no—
92C2×C22⋊C4×C47D412yes—
93C2×C42×C47—21no—
94C24×C47C2×D411yescharacteristic
95C23×D47—21no—
96C22×C4×C47—22no—
97C22×C4×C47C22×C411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×Q811yescharacteristic
105C22×C4×C47#4111yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C2×C22⋊D47—22no—
112C2×C22⋊D47—22no—
113C2×C22⋊D47—41no—
114C2×C22⋊D47—41no—
115C2×C22⋊D47—41no—
116C2×C22⋊D47—41no—
117C22×Dic47—21no—
118C22×Dic47—22no—
119C22×Dic47—41no—
120C22×Dic47—41no—
121#10625350—22no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—41no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188C4⋊C412yes—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47—41no—
148C2×C4×D47—41no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94C4⋊C412yes—
155M16×C47—22no—
156M16×C47—22no—
157M16×C47—22no—
158M16×C47—22no—
159M16×C47—22no—
160M16×C47—41no—
161M16×C47—41no—
162M16×C47C22⋊C414yes—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—41no—
168C2×C8×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47—41no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C42×C47—41no—
174C42×C47—41no—
175C22⋊Dic47—22no—
176C22⋊Dic47—22no—
177C22⋊Dic47—41no—
178C22⋊Dic47—41no—
179C22⋊Dic47—41no—
180C22⋊Dic47—41no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—41no—
184C4⋊Dic47—41no—
185C4⋊Dic47—41no—
186C4⋊Dic47—41no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191C4×Dic47—41no—
192C4×Dic47—41no—
193C23×C47—81no—
194C23×C47C4×D411yescharacteristic
195C23×C47#12011yescharacteristic
196C23×C47#12111yescharacteristic
197C23×C47#12111yescharacteristic
198C23×C47C22⋊Q811yescharacteristic
199C23×C47#12311yescharacteristic
200C23×C47#12711yescharacteristic
201C22×D47—22no—
202C22×D47—41no—
203C22×D47—41no—
204C22×D47—82no—
205C2×C4×C47—42no—
206C2×C4×C47—82no—
207C2×C4×C47C2×C22⋊C411yescharacteristic
208C2×C4×C47C2×C4⋊C411yescharacteristic
209C2×C4×C47#11711yescharacteristic
210C2×C4×C47C4×D411yescharacteristic
211C2×C4×C47C4×D411yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47#12011yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47#12111yescharacteristic
216C2×C4×C47#12111yescharacteristic
217C2×C4×C47#12111yescharacteristic
218C2×C4×C47#12111yescharacteristic
219C2×C4×C47#12111yescharacteristic
220C2×C4×C47#12111yescharacteristic
221C2×C4×C47C22⋊Q811yescharacteristic
222C2×C4×C47C22⋊Q811yescharacteristic
223C2×C4×C47C22⋊Q811yescharacteristic
224C2×C4×C47C22⋊Q811yescharacteristic
225C2×C4×C47C22⋊Q811yescharacteristic
226C2×C4×C47#12311yescharacteristic
227C2×C4×C47#12311yescharacteristic
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12411yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12611yescharacteristic
234C2×C4×C47#12611yescharacteristic
235C22⋊D47—42no—
236C22⋊D47—42no—
237C22⋊D47—42no—
238C22⋊D47—42no—
239C2×Dic47—22no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—82no—
246C2×Dic47—82no—
247C2×Dic47—82no—
248D188—22no—
249D188—22no—
250D188—41no—
251D188—41no—
252C4×D47—42no—
253C4×D47—42no—
254C4×D47—42no—
255C4×D47—42no—
256Dic94—22no—
257Dic94—22no—
258Dic94—41no—
259Dic94—41no—
260C8×C47—24no—
261C8×C47—42no—
262#11115—471nomaximal, Sylow
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47#38511yescharacteristic
271C22×C47#38811yescharacteristic
272C22×C47#39011yescharacteristic
273C22×C47#39211yescharacteristic
274C22×C47#39411yescharacteristic
275C22×C47#39611yescharacteristic
276C22×C47#39911yescharacteristic
277C2×D47—42no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47—82no—
281C4×C47—82no—
282C4×C47#38011yescharacteristic
283C4×C47#38511yescharacteristic
284C4×C47#38611yescharacteristic
285C4×C47#38811yescharacteristic
286C4×C47#39311yescharacteristic
287C4×C47#39411yescharacteristic
288C4×C47#39611yescharacteristic
289C4×C47#39711yescharacteristic
290Dic47—42no—
291Dic47—82no—
292Dic47—82no—
293Dic47—82no—
294#2911—471no—
295C2×#421—471no—
296C2×#421—471no—
297C2×#410—471no—
298#1742D4711yescharacteristic, complemented
299#1740—471no—
300C2×H(3,ℤ/4)—471no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47—21no—
308C2×C47#143911yescharacteristic
309D47—82no—
310C22×#41—471no—
311C22×M16C2×D4711yescharacteristic
312C22×M16C2×D4711yescharacteristic
313C2×#117—471no—
314C2×#117—471no—
315C2×#117—471no—
316C2×#117C2×D4711yescharacteristic
317#428—942no—
318#428—942no—
319#427—942no—
320#421—942no—
321#421—942no—
322#421—942no—
323#421—942no—
324C2×#104—474no—
325C2×#104—942no—
326#410—942no—
327#410—942no—
328C2×#99—942no—
329#343—22no—
330#342—942no—
331H(3,ℤ/4)—942no—
332C47#1111511yescharacteristic, complemented, Sylow
333C2×#41—472no—
334C2×#41—472no—
335C2×#41—941no—
336C2×#41—941no—
337C2×#41—941no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C22×Q8—471no—
346C22×D4—471no—
347C22×D4—941no—
348C23×C4—941no—
349C23×C4C22×D4711yescharacteristic
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16C4×D4712yes—
359C2×M16D18812yes—
360C2×M16D18812yes—
361C2×M16C22⋊D4712yes—
362C22×C8—21no—
363C22×C8—21no—
364#117—21no—
365#117—21no—
366#117—21no—
367#117—21no—
368#117—472no—
369#117—472no—
370#117—941no—
371#117—941no—
372#117—941no—
373#117—941no—
374#117—941no—
375#117—941no—
376#117—941no—
377#117—941no—
378#117—941no—
379#117—941no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4D18812yes—
388C2×C42—21no—
389C2×C42—472no—
390C2×C42—941no—
391C2×C42—941no—
392#107—944no—
393#106—944no—
394#104—948no—
395#104—948no—
396#99—948no—
397C24—941no—
398C24C2×D18811yescharacteristic
399#41—942no—
400#41—942no—
401#41—942no—
402#41—942no—
403#41—942no—
404#41—942no—
405#41—942no—
406#41—942no—
407#41—1881no—
408#41—1881no—
409#41—1881no—
410#41—1881no—
411#41—1881no—
412#41—1881no—
413#41—1881no—
414#41—1881no—
415C2×Q8—472no—
416C2×Q8—941no—
417C2×Q8—941no—
418C2×Q8—941no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×D4—472no—
422C2×D4—942no—
423C2×D4—942no—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—941no—
427C2×D4—941no—
428C2×D4—941no—
429C2×D4—1881no—
430C2×D4—1881no—
431C2×D4—1881no—
432C2×D4—1881no—
433C22×C4—22no—
434C22×C4—941no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—1881no—
439C22×C4—1881no—
440C22×C4—1881no—
441C22×C4—1881no—
442C22×C4—1881no—
443C22×C4—1881no—
444C22×C4C2×C4×D4711yescharacteristic
445C22×C4C2×D18811yescharacteristic
446C22×C4C2×D18811yescharacteristic
447C22×C4#1062535011yescharacteristic
448C22×C4#1062535011yescharacteristic
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4D4×D4711yescharacteristic
452C22×C4#1062535211yescharacteristic
453C22×C4#1062535211yescharacteristic
454C22×C4#1062535211yescharacteristic
455C22×C4Q8×D4711yescharacteristic
456C22×C4#1062535411yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—41no—
464M16—41no—
465M16(C2×D47)⋊C414yes—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—942no—
475C4⋊C4—942no—
476C4⋊C4—1881no—
477C4⋊C4—1881no—
478C4⋊C4—1881no—
479C4⋊C4—1881no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—942no—
483C22⋊C4—942no—
484C22⋊C4—1881no—
485C22⋊C4—1881no—
486C22⋊C4—1881no—
487C22⋊C4—1881no—
488C42—41no—
489C42—41no—
490C42—942no—
491C42—942no—
492C42—942no—
493C42—942no—
494C42—1881no—
495C42—1881no—
496C23—81no—
497C23—942no—
498C23—3762no—
499C23—1881no—
500C23—1881no—
501C23C4×D18811yescharacteristic
502C23#1309399411yescharacteristic
503C23#1309399811yescharacteristic
504C23#1309399911yescharacteristic
505C23#1309400411yescharacteristic
506C23#1309400811yescharacteristic
507C23#1309402111yescharacteristic
508Q8—942no—
509Q8—942no—
510Q8—1881no—
511Q8—1881no—
512D4—942no—
513D4—942no—
514D4—1882no—
515D4—1882no—
516D4—1882no—
517D4—1882no—
518D4—1881no—
519D4—1881no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—942no—
523C2×C4—3762no—
524C2×C4—3762no—
525C2×C4—3762no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4C2×(C2×D47)⋊C411yescharacteristic
536C2×C4C4⋊C4×D4711yescharacteristic
537C2×C4#1309398211yescharacteristic
538C2×C4(C22⋊D47)⋊C411yescharacteristic
539C2×C4(C22⋊D47)⋊C411yescharacteristic
540C2×C4C4×D18811yescharacteristic
541C2×C4#1309399411yescharacteristic
542C2×C4#1309399411yescharacteristic
543C2×C4#1309399511yescharacteristic
544C2×C4#1309399611yescharacteristic
545C2×C4#1309399611yescharacteristic
546C2×C4#1309399811yescharacteristic
547C2×C4#1309399911yescharacteristic
548C2×C4#1309399911yescharacteristic
549C2×C4#1309400011yescharacteristic
550C2×C4#1309400211yescharacteristic
551C2×C4#1309400211yescharacteristic
552C2×C4#1309400411yescharacteristic
553C2×C4#1309400411yescharacteristic
554C2×C4#1309400611yescharacteristic
555C2×C4#1309400611yescharacteristic
556C2×C4#1309400811yescharacteristic
557C2×C4#1309400811yescharacteristic
558C2×C4#1309401011yescharacteristic
559C2×C4#1309401011yescharacteristic
560C2×C4#1309401211yescharacteristic
561C2×C4#1309401911yescharacteristic
562C2×C4#1309401911yescharacteristic
563C8—24no—
564C8—42no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3762no—
573C22—3762no—
574C22—3762no—
575C22—1882no—
576C22#1568899711yescharacteristic
577C22#1568901511yescharacteristic
578C22#1568902511yescharacteristic
579C22#1568903311yescharacteristic
580C22#1568903911yescharacteristic
581C22#1568904511yescharacteristic
582C22#1568906011yescharacteristic
583C4—82no—
584C4—3762no—
585C4—3762no—
586C4—3762no—
587C4—1882no—
588C4#1568898011yescharacteristic
589C4#1568899711yescharacteristic
590C4#1568900011yescharacteristic
591C4#1568901511yescharacteristic
592C4#1568903611yescharacteristic
593C4#1568903911yescharacteristic
594C4#1568904511yescharacteristic
595C4#1568905011yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—3762no—
604C2#2024163411yescharacteristic
subgroupG/H
centerC4#15689015
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1742C2
socleC2×C47#1439
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#1742D47
hypocenterC47#11115
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
1. #20251978C2semidirect
5. #20243120C2semidirect
6. C2×#15688880C2semidirect
7. C47×#1742C2semidirect
20. C2×#13093950C4semidirect
298. #1742D47semidirect
332. C47#11115semidirect

Maximal subgroups

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

subgroupindex
#202519782
C2×#156891392
C2×#156890962
#202431202
C2×#156888802
C47×#17422
#1111547