Finite Group Databasebeta

← #26123514

Subgroups — #26123514

Order 12032.

868 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
1#20251952C211yesmaximal, characteristic, complemented
2C2×#15689141C211yesmaximal, characteristic
3C2×#15689141C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20243098C211yesmaximal, characteristic, complemented
6#20243097C211yesmaximal, characteristic, complemented
7C47×#1740C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689150—22no—
13#15689150—22no—
14#15689147—22no—
15#15689141—22no—
16#15689141—22no—
17#15689141—22no—
18#15689141—22no—
19C2×#13093952—22no—
20C2×#13093952—22no—
21#15689096—22no—
22#15689096C414yescomplemented
23C2×#13093938—22no—
24#15688883—22no—
25#15688882—22no—
26C2×#117×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C47×#342—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—
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×#10625329D412yes—
57C2×#10625329D412yes—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—21no—
60#13093983—21no—
61#13093983—21no—
62#13093983—21no—
63#13093983C2×C412yes—
64C2×C4⋊Dic47—21no—
65C2×C22⋊Dic47—21no—
66C2×C4×Dic47—21no—
67C4⋊(C47⋊C8)—28no—
68#13093952—28no—
69#13093952—28no—
70#13093938—28no—
71C23×C4×C47C2311yescharacteristic
72#117×C47—21no—
73#117×C47—21no—
74#117×C47—21no—
75#117×C47—21no—
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47C2×C412yes—
82#117×C47D412yes—
83#117×C47D412yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—21no—
87C2×C22⋊C4×C47—21no—
88C2×C22⋊C4×C47—21no—
89C2×C22⋊C4×C47D412yes—
90C2×C42×C47—21no—
91C2×C42×C47—21no—
92C2×C42×C47D412yes—
93C24×C47C2×D411yescharacteristic
94C23×D47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—41no—
98C22×C4×C47C22×C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×Q811yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C2×C22⋊D47—22no—
113C2×C22⋊D47—22no—
114C2×C22⋊D47—41no—
115C2×C22⋊D47—41no—
116C2×C22⋊D47—41no—
117C2×C22⋊D47—41no—
118C22×Dic47—21no—
119C22×Dic47—41no—
120#10625350—22no—
121#10625350—22no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—41no—
129#10625350—41no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188C4⋊C412yes—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—41no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47—41no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94C4⋊C412yes—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—41no—
158C4⋊C4×C47—41no—
159C4⋊C4×C47—41no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—41no—
163C22⋊C4×C47—41no—
164C22⋊C4×C47—41no—
165C22⋊C4×C47C22⋊C414yes—
166C42×C47—22no—
167C42×C47—22no—
168C42×C47—22no—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—41no—
172C22⋊Dic47—22no—
173C22⋊Dic47—41no—
174C4⋊Dic47—22no—
175C4⋊Dic47—41no—
176C4×Dic47—22no—
177C4×Dic47—41no—
178#10625329—22no—
179#10625329—22no—
180#10625329—22no—
181#10625329—22no—
182#10625329—41no—
183#10625329—41no—
184#10625329—41no—
185#10625329—41no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—41no—
189C2×C47⋊C8—41no—
190C2×C47⋊C8—41no—
191C2×C47⋊C8—41no—
192C23×C47—81no—
193C23×C47C2×C22⋊C411yescharacteristic
194C23×C47#12111yescharacteristic
195C23×C47#12111yescharacteristic
196C23×C47#12111yescharacteristic
197C23×C47#12111yescharacteristic
198C23×C47#12111yescharacteristic
199C23×C47C22⋊Q811yescharacteristic
200C22×D47—22no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47—82no—
204C2×C4×C47—24no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—82no—
208C2×C4×C47—82no—
209C2×C4×C47—82no—
210C2×C4×C47C2×C4⋊C411yescharacteristic
211C2×C4×C47#11711yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47C4×D411yescharacteristic
214C2×C4×C47C4×D411yescharacteristic
215C2×C4×C47C4×D411yescharacteristic
216C2×C4×C47#12011yescharacteristic
217C2×C4×C47#12011yescharacteristic
218C2×C4×C47#12011yescharacteristic
219C2×C4×C47#12111yescharacteristic
220C2×C4×C47#12111yescharacteristic
221C2×C4×C47#12111yescharacteristic
222C2×C4×C47C22⋊Q811yescharacteristic
223C2×C4×C47C22⋊Q811yescharacteristic
224C2×C4×C47C22⋊Q811yescharacteristic
225C2×C4×C47C22⋊Q811yescharacteristic
226C2×C4×C47C22⋊Q811yescharacteristic
227C2×C4×C47#12311yescharacteristic
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12311yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12411yescharacteristic
234C2×C4×C47#12411yescharacteristic
235C2×C4×C47#12611yescharacteristic
236C2×C4×C47#12611yescharacteristic
237C2×C4×C47#12711yescharacteristic
238C22⋊D47—42no—
239C22⋊D47—42no—
240C22⋊D47—42no—
241C22⋊D47—42no—
242C2×Dic47—22no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—42no—
246C2×Dic47—82no—
247D188—22no—
248D188—22no—
249D188—41no—
250D188—41no—
251C4×D47—42no—
252C4×D47—42no—
253C4×D47—42no—
254C4×D47—42no—
255Dic94—22no—
256Dic94—22no—
257Dic94—41no—
258Dic94—41no—
259C47⋊C8—42no—
260C47⋊C8—42no—
261#11094—471nomaximal, Sylow
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47#38011yescharacteristic
270C22×C47#38611yescharacteristic
271C22×C47#39011yescharacteristic
272C22×C47#39411yescharacteristic
273C22×C47#39411yescharacteristic
274C22×C47#39611yescharacteristic
275C22×C47#39611yescharacteristic
276C2×D47—42no—
277C2×D47—82no—
278C2×D47—82no—
279C2×D47—82no—
280C4×C47—44no—
281C4×C47—82no—
282C4×C47—82no—
283C4×C47#38511yescharacteristic
284C4×C47#38511yescharacteristic
285C4×C47#38811yescharacteristic
286C4×C47#38811yescharacteristic
287C4×C47#39211yescharacteristic
288C4×C47#39311yescharacteristic
289C4×C47#39711yescharacteristic
290C4×C47#39911yescharacteristic
291Dic47—42no—
292Dic47—82no—
293#2905—471no—
294C2×#421—471no—
295C2×#421—471no—
296C2×#410—471no—
297#1740D4711yescharacteristic, complemented
298#1733—471no—
299#1733—471no—
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47#143911yescharacteristic
308D47—82no—
309C22×#41—471no—
310C22×M16—471no—
311C22×M16—471no—
312C2×#117—471no—
313C2×#117C2×D4711yescharacteristic
314C2×#117C2×D4711yescharacteristic
315C2×#117C2×D4711yescharacteristic
316#424—942no—
317#424—942no—
318#423—942no—
319#421—942no—
320#421—942no—
321#421—942no—
322#421—942no—
323C2×#104—942no—
324C2×#104—942no—
325#410—474no—
326#410—942no—
327C2×#99—942no—
328#342—22no—
329#339—942no—
330#339—942no—
331C47#1109411yescharacteristic, complemented, Sylow
332C2×#41—472no—
333C2×#41—472no—
334C2×#41—941no—
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—
344C22×Q8—471no—
345C22×D4—471no—
346C22×D4—941no—
347C23×C4—941no—
348C23×C4C22×D4711yescharacteristic
349C2×M16—472no—
350C2×M16—472no—
351C2×M16—941no—
352C2×M16—941no—
353C2×M16—941no—
354C2×M16—941no—
355C2×M16—941no—
356C2×M16—941no—
357C2×M16—941no—
358C2×M16—941no—
359C2×M16—941no—
360C2×M16—941no—
361C22×C8—941no—
362C22×C8—941no—
363#117—21no—
364#117—21no—
365#117—21no—
366#117—21no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372#117—472no—
373#117—941no—
374#117—941no—
375#117—941no—
376#117C4×D4712yes—
377#117D18812yes—
378#117D18812yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—941no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4C22⋊D4712yes—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—941no—
390C2×C42D18812yes—
391C4⋊C8—948no—
392#104—948no—
393#104—948no—
394#99—948no—
395C24—941no—
396C24C2×C22⋊D4711yescharacteristic
397#41—942no—
398#41—942no—
399#41—942no—
400#41—942no—
401#41—942no—
402#41—942no—
403#41—942no—
404#41—942no—
405#41—1881no—
406#41—1881no—
407#41—1881no—
408#41—1881no—
409#41—1881no—
410#41—1881no—
411#41—1881no—
412#41—1881no—
413C2×Q8—472no—
414C2×Q8—941no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×Q8—941no—
418C2×Q8—941no—
419C2×D4—472no—
420C2×D4—942no—
421C2×D4—942no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—941no—
427C2×D4—1881no—
428C2×D4—1881no—
429C2×D4—1881no—
430C2×D4—1881no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—941no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—1881no—
438C22×C4—1881no—
439C22×C4—1881no—
440C22×C4—1881no—
441C22×C4—1881no—
442C22×C4C2×C4×D4711yescharacteristic
443C22×C4C2×D18811yescharacteristic
444C22×C4C2×D18811yescharacteristic
445C22×C4C2×D18811yescharacteristic
446C22×C4#1062535011yescharacteristic
447C22×C4#1062535011yescharacteristic
448C22×C4D4×D4711yescharacteristic
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4#1062535211yescharacteristic
452C22×C4#1062535211yescharacteristic
453C22×C4#1062535211yescharacteristic
454C22×C4Q8×D4711yescharacteristic
455C22×C4#1062535411yescharacteristic
456M16—942no—
457M16—942no—
458M16—942no—
459M16—942no—
460M16—1881no—
461M16—1881no—
462M16—1881no—
463M16—1881no—
464C2×C8—942no—
465C2×C8—942no—
466C2×C8—1881no—
467C2×C8—1881no—
468C2×C8—1881no—
469C2×C8—1881no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—942no—
477C4⋊C4—1881no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—942no—
484C22⋊C4—1881no—
485C22⋊C4(C2×D47)⋊C414yes—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—942no—
493C42—1881no—
494C23—81no—
495C23—942no—
496C23—3762no—
497C23—1881no—
498C23—1881no—
499C23C2×(C2×D47)⋊C411yescharacteristic
500C23#1309399511yescharacteristic
501C23#1309399611yescharacteristic
502C23#1309399611yescharacteristic
503C23#1309399811yescharacteristic
504C23#1309399811yescharacteristic
505C23#1309400011yescharacteristic
506Q8—942no—
507Q8—942no—
508Q8—1881no—
509Q8—1881no—
510D4—942no—
511D4—942no—
512D4—1882no—
513D4—1882no—
514D4—1882no—
515D4—1882no—
516D4—1881no—
517D4—1881no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—942no—
525C2×C4—3762no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4C4⋊C4×D4711yescharacteristic
534C2×C4#1309398211yescharacteristic
535C2×C4(C22⋊D47)⋊C411yescharacteristic
536C2×C4(C22⋊D47)⋊C411yescharacteristic
537C2×C4C4×D18811yescharacteristic
538C2×C4C4×D18811yescharacteristic
539C2×C4#1309399411yescharacteristic
540C2×C4#1309399411yescharacteristic
541C2×C4#1309399411yescharacteristic
542C2×C4#1309399911yescharacteristic
543C2×C4#1309399911yescharacteristic
544C2×C4#1309399911yescharacteristic
545C2×C4#1309400211yescharacteristic
546C2×C4#1309400211yescharacteristic
547C2×C4#1309400411yescharacteristic
548C2×C4#1309400411yescharacteristic
549C2×C4#1309400411yescharacteristic
550C2×C4#1309400611yescharacteristic
551C2×C4#1309400611yescharacteristic
552C2×C4#1309400811yescharacteristic
553C2×C4#1309400811yescharacteristic
554C2×C4#1309400811yescharacteristic
555C2×C4#1309401011yescharacteristic
556C2×C4#1309401011yescharacteristic
557C2×C4#1309401211yescharacteristic
558C2×C4#1309401911yescharacteristic
559C2×C4#1309401911yescharacteristic
560C2×C4#1309402111yescharacteristic
561C8—1882no—
562C8—1882no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—3762no—
571C22—3762no—
572C22—3762no—
573C22—1882no—
574C22#1568898011yescharacteristic
575C22#1568900011yescharacteristic
576C22#1568902511yescharacteristic
577C22#1568903911yescharacteristic
578C22#1568903911yescharacteristic
579C22#1568904511yescharacteristic
580C22#1568904511yescharacteristic
581C4—44no—
582C4—82no—
583C4—82no—
584C4—3762no—
585C4—1882no—
586C4#1568899711yescharacteristic
587C4#1568899711yescharacteristic
588C4#1568901511yescharacteristic
589C4#1568901511yescharacteristic
590C4#1568903311yescharacteristic
591C4#1568903611yescharacteristic
592C4#1568905011yescharacteristic
593C4#1568906011yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—3762no—
602C2#2024163411yescharacteristic
subgroupG/H
centerC4#15689033
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1740C2
socleC2×C47#1439
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#1740D47
hypocenterC47#11094
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
1. #20251952C2semidirect
5. #20243098C2semidirect
6. #20243097C2semidirect
7. C47×#1740C2semidirect
22. #15689096C4semidirect
297. #1740D47semidirect
331. C47#11094semidirect

Maximal subgroups

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

subgroupindex
#202519522
C2×#156891412
C2×#156890962
#202430982
#202430972
C47×#17402
#1109447