Finite Group Databasebeta

← #29707092

Subgroups — #29707092

Order 15616.

2290 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519803C211yesmaximal, characteristic, complemented
2C2×#16840329C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5#21448401C211yesmaximal, characteristic, complemented
6#21446894C214yesmaximal, complemented
7#21446888C214yesmaximal, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C23×D8×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519803—24no—
12C2×#14519803—24no—
13C2×#14519803—24no—
14C22×#11758200C2212yescomplemented
15C22×#11758199C2212yescomplemented
16C22×#11758194C2211yescharacteristic
17#16840329—24no—
18#16840329—24no—
19C2×#14519773C2212yescomplemented
20C2×#14519764C2212yescomplemented
21C2×#14519748—22no—
22C2×#14519746—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25#16839767—24no—
26#16839759—24no—
27C2×#14519702C2212yescomplemented
28C2×#14519699C2212yescomplemented
29C2×#14519698C2214yescomplemented
30#16839665—22no—
31#16839665C2214yescomplemented
32#16839618—22no—
33#16839516C2214yescomplemented
34C23×D4×C61C2211yescharacteristic
35C23×D4×C61C2211yescharacteristic
36C22×D8×C61—24no—
37C22×D8×C61—24no—
38C22×D8×C61—24no—
39C22×D8×C61C2214yes—
40C23×C8×C61C2211yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222D414yes—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C22⋊D61—24no—
51C22×C4×D61—21no—
52#14519803—232no—
53C2×#11758200—24no—
54C2×#11758200—24no—
55C2×#11758200—24no—
56C2×#11758199—24no—
57C2×#11758199—24no—
58C2×#11758199—24no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194D414yescomplemented
62C22×#92418C2312yes—
63#14519773—24no—
64#14519773—24no—
65#14519764—24no—
66#14519764—24no—
67#14519748—216no—
68#14519746—216no—
69#14519736—22no—
70#14519736—22no—
71#14519736—22no—
72#14519736—22no—
73C2×C4⋊Dic61D412yes—
74C2×C4⋊Dic61C2312yes—
75C2×C22⋊Dic61—24no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—28no—
78C2×C22⋊Dic61D412yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2312yes—
81#14519702—24no—
82#14519699—24no—
83#14519698—28no—
84C25×C61—21no—
85C25×C61—21no—
86C22×D4×C61—22no—
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61—22no—
92C22×D4×C61—22no—
93C22×D4×C61—21no—
94C22×D4×C61—22no—
95C22×D4×C61—21no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61D414yes—
99C22×D4×C61D414yescomplemented
100C22×D4×C61C2312yes—
101C22×D4×C61C2312yes—
102C23×C4×C61C2311yescharacteristic
103C2×D8×C61—232no—
104C2×D8×C61—28no—
105C2×D8×C61—28no—
106C2×D8×C61—28no—
107C22×C8×C61—22no—
108C22×C8×C61—22no—
109C22×C8×C61D414yescomplemented
110C22×C8×C61C2312yes—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61—24no—
114C24×C61—24no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61—84no—
120C24×C61—84no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C2×D4×C61—28no—
124C2×D4×C61—28no—
125C2×D4×C61—28no—
126C2×D4×C61—24no—
127C2×D4×C61—28no—
128C2×D4×C61—24no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—44no—
140C2×D4×C61—42no—
141C2×D4×C61—44no—
142C2×D4×C61—44no—
143C2×D4×C61—44no—
144C2×D4×C61—42no—
145C2×D4×C61—42no—
146C2×D4×C61—42no—
147C2×D4×C61C2×D414yes—
148C2×D4×C61C2×D414yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61#4112yes—
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—24no—
158C2×C22⋊D61—216no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—42no—
161C2×C22⋊D61—88no—
162C22×Dic61—21no—
163C22×Dic61—22no—
164C22×Dic61C2×D414yes—
165#11758222—28no—
166#11758222—28no—
167#11758222—44no—
168#11758222—44no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244C2×D412yes—
175C2×C4×D61—24no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122C2×D412yes—
184D8×C61—232no—
185C2×C8×C61—28no—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61C2×D414yes—
189C22⋊Dic61—216no—
190C22⋊Dic61—24no—
191C22⋊Dic61—42no—
192C22⋊Dic61—88no—
193#11758200—216no—
194#11758199—216no—
195C4⋊Dic61—24no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—42no—
201#11758194—28no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418C2×D414yes—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—48no—
213C23×C61—48no—
214C23×C61—48no—
215C23×C61—48no—
216C23×C61—88no—
217C23×C61—88no—
218C23×C61—88no—
219C23×C61—88no—
220C23×C61—88no—
221C23×C61—88no—
222C23×C61#12012yes—
223C23×C61#12112yes—
224C23×C61#12712yes—
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—44no—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61—24no—
233D4×C61—24no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—48no—
239D4×C61—48no—
240C2×C4×C61#12014yes—
241C2×C4×C61#12014yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12114yes—
244C2×C4×C61#12114yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12712yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C22⋊D61—48no—
254C22⋊D61—816no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—28no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—44no—
262D244—22no—
263D244—22no—
264D244—41no—
265D244—41no—
266C4×D61—48no—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—24no—
272#92418—24no—
273#23422—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—48no—
280C22×C61—48no—
281C22×C61—816no—
282C22×C61—816no—
283C22×C61—84no—
284C22×C61—84no—
285C22×C61—84no—
286C22×C61—84no—
287C22×C61#39214yes—
288C22×C61C2×#12011yescharacteristic
289C22×C61C2×#12111yescharacteristic
290C22×C61C2×#12711yescharacteristic
291C22×C61#57614yes—
292C2×D61—42no—
293C2×D61—44no—
294C2×D61—82no—
295C2×D61—82no—
296C4×C61#39214yes—
297C4×C61C2×#12011yescharacteristic
298C4×C61C2×#12011yescharacteristic
299C4×C61C2×#12111yescharacteristic
300C4×C61C2×#12111yescharacteristic
301Dic61—42no—
302Dic61—44no—
303Dic61—82no—
304C22×AGL(1,ℤ/8)—611no—
305C23×D8D6111yescharacteristic, complemented
306C2×#535—611no—
307C2×#418—611no—
308C2×#416—611no—
309#2309—611no—
310#2012—614no—
311#2011—614no—
312#1729—611no—
313C2×C61—24no—
314C2×C61—84no—
315C2×C61—84no—
316C2×C61C2×#39211yescharacteristic
317C2×C61#300812yes—
318C2×C61#304112yes—
319C2×C61#315512yes—
320D61—82no—
321C22×#41—611no—
322C23×D4C2×D6111yescharacteristic
323C23×D4C2×D6111yescharacteristic
324C2×AGL(1,ℤ/8)—1224no—
325C2×AGL(1,ℤ/8)—1224no—
326C2×AGL(1,ℤ/8)—1224no—
327C22×SD16—612no—
328C22×D8—24no—
329C22×D8—24no—
330C22×D8—24no—
331C22×D8—612no—
332C22×D8C2×D6114yes—
333C22×M16—611no—
334C23×C8C2×D6111yescharacteristic
335#535—1224no—
336#535—1224no—
337C2×#127—612no—
338C2×#124—612no—
339C2×#121—1222no—
340C2×#120—1222no—
341C2×#117—611no—
342C2×#117—611no—
343#418—1224no—
344#416—1224no—
345C2×#103—612no—
346C2×#102—614no—
347C2×#102—612no—
348#390—614no—
349#390—1222no—
350#379—1222no—
351#336—614no—
352C61#2342211yescharacteristic, complemented, Sylow
353C25—21no—
354C25—21no—
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1222no—
361C22×Q8—611no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—21no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—22no—
370C22×D4—22no—
371C22×D4—22no—
372C22×D4—22no—
373C22×D4—22no—
374C22×D4—611no—
375C22×D4—1224no—
376C22×D4—1221no—
377C22×D4C22⋊D6114yes—
378C22×D4C22⋊D6114yescomplemented
379C22×D4C22×D6112yes—
380C22×D4C22×D6112yes—
381C23×C4—1221no—
382C23×C4C22×D6111yescharacteristic
383AGL(1,ℤ/8)—12232no—
384C2×SD16—1224no—
385C2×SD16—1224no—
386C2×SD16—1224no—
387C2×D8—232no—
388C2×D8—28no—
389C2×D8—28no—
390C2×D8—28no—
391C2×D8—1224no—
392C2×D8—1224no—
393C2×D8—1224no—
394C2×M16—614no—
395C2×M16—1222no—
396C2×M16—1222no—
397C22×C8—22no—
398C22×C8—22no—
399C22×C8—612no—
400C22×C8C22⋊D6114yescomplemented
401C22×C8C22×D6112yes—
402#127—1224no—
403#127—1224no—
404#124—1224no—
405#124—1224no—
406#121—12216no—
407#120—12216no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—612no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—1224no—
416C2×C22⋊C4—1228no—
417C2×C22⋊C4—1221no—
418C2×C42—612no—
419C2×C42—1221no—
420#103—1224no—
421#102—1228no—
422#102—1224no—
423C24—22no—
424C24—22no—
425C24—24no—
426C24—24no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—84no—
432C24—84no—
433C24—1221no—
434C24D4×D6111yescharacteristic
435#41—1228no—
436#41—1228no—
437#41—2444no—
438#41—2444no—
439C2×Q8—612no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—28no—
456C2×D4—24no—
457C2×D4—28no—
458C2×D4—28no—
459C2×D4—28no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—612no—
470C2×D4—12216no—
471C2×D4—1224no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—4888no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4C2×C22⋊D6114yes—
481C2×D4C2×C22⋊D6114yes—
482C22×C4—614no—
483C22×C4—1221no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4D4×D6112yes—
489C22×C4D4×D6112yes—
490C22×C4D4×D6111yescharacteristic
491C22×C4#1175822412yes—
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6112yes—
495C22×C4C23×D6111yescharacteristic
496SD16—12216no—
497D8—232no—
498D8—12216no—
499M16—1228no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—28no—
503C2×C8—614no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8C2×C22⋊D6114yes—
507C4⋊C4—1224no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C22⋊C4—12216no—
511C22⋊C4—1224no—
512C22⋊C4—4888no—
513C22⋊C4—2442no—
514C42—1222no—
515C42—1222no—
516C42—2442no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—88no—
529C23—88no—
530C23—88no—
531C23—88no—
532C23—88no—
533C23—88no—
534C23—1222no—
535C23—2441no—
536C23—2441no—
537C23—2444no—
538C23#1451974512yes—
539C23#1451974912yes—
540C23#1451977312yes—
541C23C2×D4×D6111yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—2441no—
545Q8—2441no—
546D4—24no—
547D4—24no—
548D4—24no—
549D4—24no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—48no—
555D4—48no—
556D4—1222no—
557D4—1222no—
558D4—48816no—
559D4—2448no—
560D4—2441no—
561D4—2441no—
562C2×C4—1224no—
563C2×C4—1222no—
564C2×C4—1228no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2448no—
570C2×C4#1451974514yes—
571C2×C4#1451974512yes—
572C2×C4#1451974614yes—
573C2×C4#1451974814yes—
574C2×C4#1451974914yes—
575C2×C4#1451974912yes—
576C2×C4#1451977312yes—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C2×#1175822411yescharacteristic
583C8—24no—
584C8—1224no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—816no—
593C22—816no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—4882no—
599C22—4882no—
600C22—2442no—
601C22—2444no—
602C22#1683967014yes—
603C22C2×#1451974511yescharacteristic
604C22C2×#1451974911yescharacteristic
605C22C2×#1451977311yescharacteristic
606C22#1684056314yes—
607C4—4882no—
608C4—2442no—
609C4—2444no—
610C4#1683967014yes—
611C4C2×#1451974511yescharacteristic
612C4C2×#1451974611yescharacteristic
613C4C2×#1451974811yescharacteristic
614C4C2×#1451974911yescharacteristic
615C2—24no—
616C2—84no—
617C2—84no—
618C2—4882no—
619C2C2×#1683967011yescharacteristic
620C2#2145508312yes—
621C2#2145531912yes—
622C2#2145630012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC23×D8×C61C2
socleC23×C61C22×D4
hypercenterC23×D8D61
hypocenterC61#23422
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14519803C2semidirect
2. C2×#16840329C2semidirect
3. C2×#16839767C2semidirect
4. C2×#16839759C2semidirect
5. #21448401C2semidirect
6. #21446894C2semidirect
7. #21446888C2semidirect
8. #21445515C2semidirect
9. C23×D8×C61C2semidirect
14. C22×#11758200C22semidirect
15. C22×#11758199C22semidirect
19. C2×#14519773C22semidirect
20. C2×#14519764C22semidirect
27. C2×#14519702C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519698C22semidirect
31. #16839665C22semidirect
33. #16839516C22semidirect
61. C2×#11758194D4semidirect
99. C22×D4×C61D4semidirect
109. C22×C8×C61D4semidirect
305. C23×D8D61semidirect
352. C61#23422semidirect
378. C22×D4C22⋊D61semidirect
400. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
C2×#168403292
C2×#168397672
C2×#168397592
#214484012
#214468942
#214468882
#214455152
C23×D8×C612
#2342261