Finite Group Databasebeta

← #29707095

Subgroups — #29707095

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×#14519802C211yesmaximal, characteristic, complemented
2C23×#11758199C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446896C214yesmaximal, complemented
7#21446890C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C2×C61×#535C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802—24no—
14C22×#11758201C2212yescomplemented
15C22×#11758199—24no—
16C22×#11758199—24no—
17C22×#11758199—24no—
18C22×#11758199C2214yes—
19C22×#11758199C2212yescomplemented
20C22×#11758194C2211yescharacteristic
21C23×#92418C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839767—24no—
24#16839758—24no—
25C2×#14519701C2212yescomplemented
26C2×#14519698C2212yescomplemented
27C2×#14519698C2214yescomplemented
28#16839664C2214yescomplemented
29#16839663—22no—
30#16839617—22no—
31#16839517C2214yescomplemented
32C22×#41×C61C2211yescharacteristic
33C23×D4×C61C2211yescharacteristic
34C61×#535—24no—
35C61×#535—24no—
36C2×#127×C61C2212yescomplemented
37C2×#124×C61C2212yescomplemented
38C2×#121×C61—22no—
39C2×#120×C61—22no—
40C2×#117×C61C2211yescharacteristic
41C24×D61—21no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244D414yescomplemented
49C22×D244C2312yes—
50#14519802—232no—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758199—232no—
55C2×#11758199—28no—
56C2×#11758199—24no—
57C2×#11758199—28no—
58C2×#11758199—24no—
59C2×#11758199—28no—
60C2×#11758199—24no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194D414yescomplemented
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418D414yescomplemented
67C22×#92418C2312yes—
68C22×#92418C2312yes—
69#14519736—22no—
70#14519736—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—21no—
73C2×#11758198—28no—
74C2×C4×Dic61—21no—
75#14519701—24no—
76#14519698—24no—
77#14519698—28no—
78C25×C61—21no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61D414yes—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61—21no—
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61—21no—
92C22×D4×C61—24no—
93C22×D4×C61—22no—
94C22×D4×C61D414yes—
95C22×D4×C61C2311yescharacteristic
96C22×D4×C61C2312yes—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99#127×C61—24no—
100#127×C61—24no—
101#124×C61—24no—
102#124×C61—24no—
103#121×C61—216no—
104#120×C61—216no—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61D412yes—
108C2×C22⋊C4×C61—24no—
109C2×C22⋊C4×C61D412yes—
110C2×C42×C61C2312yes—
111C24×C61—22no—
112C24×C61—21no—
113C24×C61—24no—
114C24×C61—42no—
115C24×C61—42no—
116C24×C61—84no—
117C24×C61C2×D411yescharacteristic
118C23×D61—22no—
119C23×D61—24no—
120C23×D61—42no—
121C23×D61—42no—
122C23×D61—84no—
123#41×C61—28no—
124#41×C61—28no—
125#41×C61—44no—
126#41×C61—44no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—24no—
134C2×D4×C61—216no—
135C2×D4×C61—28no—
136C2×D4×C61—24no—
137C2×D4×C61—28no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—44no—
151C2×D4×C61—44no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—88no—
155C2×D4×C61C2×D414yes—
156C2×D4×C61C2×D412yes—
157C22×C4×C61—21no—
158C22×C4×C61—24no—
159C22×C4×C61—42no—
160C22×C4×C61—42no—
161C22×C4×C61C2×D414yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61#4112yes—
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—22no—
171C2×D244—28no—
172C2×D244—28no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—22no—
177C2×D244—24no—
178C2×D244—22no—
179C2×D244—42no—
180C2×D244—44no—
181C2×D244—44no—
182C2×D244—42no—
183C2×D244C2×D414yes—
184C4⋊C4×C61—24no—
185C22⋊C4×C61—216no—
186C22⋊C4×C61—24no—
187C42×C61—22no—
188C42×C61—22no—
189C22⋊Dic61—42no—
190#11758201—216no—
191#11758199—232no—
192#11758199—216no—
193#11758198—88no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4×Dic61—42no—
197#11758194—28no—
198C2×#92418—28no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418C2×D414yes—
204C2×#92418C2×D414yes—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—44no—
213C23×C61—48no—
214C23×C61—48no—
215C23×C61—88no—
216C23×C61—88no—
217C23×C61—88no—
218C23×C61#12012yes—
219C23×C61#12112yes—
220C23×C61#12712yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—24no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—48no—
226C22×D61—48no—
227C22×D61—88no—
228C22×D61—88no—
229C22×D61—88no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—24no—
235D4×C61—22no—
236D4×C61—24no—
237D4×C61—22no—
238D4×C61—48no—
239D4×C61—42no—
240D4×C61—41no—
241D4×C61—42no—
242D4×C61—41no—
243D4×C61—48no—
244D4×C61—816no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—28no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—48no—
251C2×C4×C61#12014yes—
252C2×C4×C61#12014yes—
253C2×C4×C61#12012yes—
254C2×C4×C61#12114yes—
255C2×C4×C61#12114yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12712yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—42no—
265C2×Dic61—44no—
266D244—24no—
267D244—24no—
268D244—42no—
269D244—42no—
270D244—48no—
271#92418—24no—
272#92418—24no—
273#23422—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—48no—
281C22×C61—816no—
282C22×C61—82no—
283C22×C61—84no—
284C22×C61—82no—
285C22×C61—84no—
286C22×C61#39214yes—
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×#12111yescharacteristic
289C22×C61C2×#12711yescharacteristic
290C22×C61#57614yes—
291C2×D61—44no—
292C2×D61—48no—
293C2×D61—816no—
294C2×D61—84no—
295C2×D61—84no—
296C4×C61—42no—
297C4×C61—44no—
298C4×C61#39214yes—
299C4×C61C2×#12011yescharacteristic
300C4×C61C2×#12011yescharacteristic
301C4×C61C2×#12111yescharacteristic
302C4×C61C2×#12111yescharacteristic
303Dic61—82no—
304C22×AGL(1,ℤ/8)—611no—
305C23×D8—611no—
306C2×#535D6111yescharacteristic, complemented
307C2×#418—611no—
308C2×#416—611no—
309#2309—611no—
310#2012—614no—
311#2011—614no—
312#1729—611no—
313C2×C61—24no—
314C2×C61—82no—
315C2×C61—84no—
316C2×C61C2×#39211yescharacteristic
317C2×C61#300812yes—
318C2×C61#304112yes—
319C2×C61#315512yes—
320D61—84no—
321C22×#41C2×D6111yescharacteristic
322C23×D4—611no—
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—614no—
329C22×D8—612no—
330C22×D8—1224no—
331C22×D8—1224no—
332C22×D8—1224no—
333C22×M16—611no—
334C23×C8—611no—
335#535—24no—
336#535—24no—
337C2×#127C2×D6112yescomplemented
338C2×#124C2×D6112yescomplemented
339C2×#121—22no—
340C2×#120—22no—
341C2×#117—611no—
342C2×#117C2×D6111yescharacteristic
343#418—1224no—
344#416—1224no—
345C2×#103—612no—
346C2×#102—612no—
347C2×#102—614no—
348#390—614no—
349#390—1222no—
350#379—1222no—
351#336—614no—
352C61#2342211yescharacteristic, complemented, Sylow
353C25—21no—
354C25—1221no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C22⋊D6114yes—
361C22×Q8C22×D6111yescharacteristic
362C22×D4—21no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—24no—
370C22×D4—614no—
371C22×D4—612no—
372C22×D4—1222no—
373C22×D4—1222no—
374C22×D4—1222no—
375C22×D4—1222no—
376C22×D4—1222no—
377C22×D4—1221no—
378C22×D4C22⋊D6114yes—
379C22×D4C22×D6112yes—
380C22×D4C22×D6111yescharacteristic
381C23×C4—21no—
382C23×C4C22×D6111yescharacteristic
383AGL(1,ℤ/8)—12232no—
384C2×SD16—1224no—
385C2×SD16—1224no—
386C2×SD16—1224no—
387C2×D8—1228no—
388C2×D8—1224no—
389C2×D8—1228no—
390C2×D8—1224no—
391C2×D8—12232no—
392C2×D8—1228no—
393C2×D8—1224no—
394C2×M16—614no—
395C2×M16—1222no—
396C2×M16—1222no—
397C22×C8—614no—
398C22×C8—612no—
399C22×C8—612no—
400C22×C8—1222no—
401C22×C8—1222no—
402#127—24no—
403#127—24no—
404#124—24no—
405#124—24no—
406#121—216no—
407#120—216no—
408#117—22no—
409#117—22no—
410#117—1222no—
411#117—1222no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4C22⋊D6112yes—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—1228no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4C22⋊D6112yes—
418C2×C42—1221no—
419C2×C42C22×D6112yes—
420#103—1224no—
421#102—1224no—
422#102—1228no—
423C24—22no—
424C24—21no—
425C24—24no—
426C24—42no—
427C24—42no—
428C24—84no—
429C24—1224no—
430C24—1222no—
431C24—4884no—
432C24—2442no—
433C24—2442no—
434C24C2×C22⋊D6111yescharacteristic
435#41—28no—
436#41—28no—
437#41—44no—
438#41—44no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C22⋊D6112yes—
445C2×D4—24no—
446C2×D4—216no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—28no—
458C2×D4—24no—
459C2×D4—28no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—88no—
467C2×D4—614no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1228no—
474C2×D4—1228no—
475C2×D4—1224no—
476C2×D4—2442no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4—2442no—
480C2×D4C2×C22⋊D6114yes—
481C2×D4C2×C22⋊D6112yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—1222no—
487C22×C4D4×D6112yes—
488C22×C4D4×D6112yes—
489C22×C4D4×D6112yes—
490C22×C4#1175822412yes—
491C22×C4C2×C22⋊D6114yes—
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16—12216no—
497D8—12232no—
498D8—12216no—
499M16—1228no—
500C2×C8—614no—
501C2×C8—614no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1228no—
507C4⋊C4—24no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C22⋊C4—216no—
511C22⋊C4—24no—
512C22⋊C4—4888no—
513C22⋊C4—2442no—
514C42—22no—
515C42—22no—
516C42—2442no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—44no—
525C23—48no—
526C23—48no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23—1224no—
531C23—4888no—
532C23—4888no—
533C23—4888no—
534C23—2442no—
535C23—2442no—
536C23—2448no—
537C23—2448no—
538C23#1451974612yes—
539C23#1451974812yes—
540C23#1451977312yes—
541C23C22×C22⋊D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—24no—
547D4—22no—
548D4—24no—
549D4—22no—
550D4—48no—
551D4—42no—
552D4—41no—
553D4—42no—
554D4—41no—
555D4—48no—
556D4—816no—
557D4—1224no—
558D4—1224no—
559D4—2442no—
560D4—2442no—
561D4—2448no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—48no—
568C2×C4—2442no—
569C2×C4—2444no—
570C2×C4#1451974514yes—
571C2×C4#1451974514yes—
572C2×C4#1451974612yes—
573C2×C4#1451974812yes—
574C2×C4#1451974914yes—
575C2×C4#1451974914yes—
576C2×C4#1451977312yes—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C2×#1175822411yescharacteristic
583C8—1224no—
584C8—1224no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—816no—
593C22—82no—
594C22—84no—
595C22—82no—
596C22—84no—
597C22—48816no—
598C22—4884no—
599C22—4884no—
600C22—2444no—
601C22—2448no—
602C22#1683967014yes—
603C22C2×#1451974611yescharacteristic
604C22C2×#1451974811yescharacteristic
605C22C2×#1451977311yescharacteristic
606C22#1684056514yes—
607C4—42no—
608C4—44no—
609C4—4882no—
610C4#1683967014yes—
611C4C2×#1451974511yescharacteristic
612C4C2×#1451974511yescharacteristic
613C4C2×#1451974911yescharacteristic
614C4C2×#1451974911yescharacteristic
615C2—24no—
616C2—84no—
617C2—82no—
618C2—4884no—
619C2C2×#1683967011yescharacteristic
620C2#2145508712yes—
621C2#2145531812yes—
622C2#2145630212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#535C2
socleC23×C61C22×D4
hypercenterC2×#535D61
hypocenterC61#23422
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C22×#14519802C2semidirect
2. C23×#11758199C2semidirect
3. C2×#16839767C2semidirect
4. C2×#16839758C2semidirect
5. #21448399C2semidirect
6. #21446896C2semidirect
7. #21446890C2semidirect
8. #21445516C2semidirect
9. C2×C61×#535C2semidirect
14. C22×#11758201C22semidirect
19. C22×#11758199C22semidirect
25. C2×#14519701C22semidirect
26. C2×#14519698C22semidirect
27. C2×#14519698C22semidirect
28. #16839664C22semidirect
31. #16839517C22semidirect
36. C2×#127×C61C22semidirect
37. C2×#124×C61C22semidirect
48. C22×D244D4semidirect
63. C2×#11758194D4semidirect
66. C22×#92418D4semidirect
306. C2×#535D61semidirect
337. C2×#127C2×D61semidirect
338. C2×#124C2×D61semidirect
352. C61#23422semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C23×#117581992
C2×#168397672
C2×#168397582
#214483992
#214468962
#214468902
#214455162
C2×C61×#5352
#2342261