Finite Group Databasebeta

← #29707096

Subgroups — #29707096

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×D488C211yesmaximal, characteristic, complemented
3C2×#16840326C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446897C214yesmaximal, complemented
7#21446891C214yesmaximal, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519802—24no—
13C2×#14519802—24no—
14C2×#14519802—24no—
15C22×#11758201C2212yescomplemented
16C22×D488—24no—
17C22×D488—24no—
18C22×D488—24no—
19C22×D488C2214yes—
20C22×#11758199C2212yescomplemented
21C22×#11758194C2211yescharacteristic
22#16840326—24no—
23#16840326—24no—
24C2×#14519773C2212yescomplemented
25C2×#14519766C2212yescomplemented
26C2×#14519751—22no—
27C2×#14519745—22no—
28C2×#14519736C2211yescharacteristic
29#16839766—24no—
30C2×#14519697C2214yescomplemented
31#16839664—22no—
32#16839663C2214yescomplemented
33#16839617—22no—
34#16839516C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C23×C8×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#416—24no—
39C2×#103×C61C2212yescomplemented
40C2×#102×C61C2212yescomplemented
41C24×D61—21no—
42C24×D61—21no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244—21no—
54C22×D244—22no—
55C22×D244D414yes—
56C22×D244D414yescomplemented
57C22×D244C2312yes—
58C22×D244C2312yes—
59C22×C22⋊D61—24no—
60#14519802—232no—
61C2×#11758201—24no—
62C2×#11758201—24no—
63C2×#11758201—24no—
64C2×D488—232no—
65C2×D488—28no—
66C2×D488—28no—
67C2×D488—28no—
68C2×#11758199—24no—
69C2×#11758199—24no—
70C2×#11758199—24no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194D414yescomplemented
74C22×#92418C2312yes—
75#14519773—24no—
76#14519773—24no—
77#14519766—24no—
78#14519766—24no—
79#14519751—216no—
80#14519745—216no—
81#14519736—22no—
82#14519736—22no—
83C2×C4⋊Dic61D412yes—
84C2×C22⋊Dic61D412yes—
85C2×#11758198—24no—
86C2×#11758198—28no—
87C2×C4×Dic61C2312yes—
88#14519697—28no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61D414yes—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61D414yescomplemented
103C22×C8×C61C2312yes—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109#103×C61—24no—
110#102×C61—24no—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—22no—
114C23×D61—22no—
115C23×D61—24no—
116C23×D61—24no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—84no—
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—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—22no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—216no—
156C2×C22⋊D61—88no—
157C22×Dic61C2×D414yes—
158C2×D244—28no—
159C2×D244—28no—
160C2×D244—28no—
161C2×D244—28no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—24no—
171C2×D244—24no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—44no—
175C2×D244—42no—
176C2×D244—44no—
177C2×D244—44no—
178C2×D244—44no—
179C2×D244—42no—
180C2×D244—42no—
181C2×D244—42no—
182C2×D244C2×D414yes—
183C2×D244C2×D414yes—
184C2×C8×C61—28no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C2×C8×C61C2×D414yes—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C22⋊C4×C61—42no—
191C42×C61—42no—
192C22⋊Dic61—24no—
193#11758201—216no—
194#11758199—216no—
195#11758198—216no—
196#11758198—88no—
197C4⋊Dic61—24no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200#11758194—28no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418C2×D414yes—
204D488—232no—
205C23×C61—24no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—44no—
210C23×C61#12012yes—
211C23×C61#12112yes—
212C23×C61#12712yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—48no—
223C22×D61—48no—
224C22×D61—88no—
225C22×D61—88no—
226C22×D61—88no—
227C22×D61—88no—
228C22×D61—88no—
229C22×D61—88no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—48no—
237D4×C61—41no—
238D4×C61—41no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—48no—
243C2×C4×C61—42no—
244C2×C4×C61—44no—
245C2×C4×C61#12014yes—
246C2×C4×C61#12014yes—
247C2×C4×C61#12012yes—
248C2×C4×C61#12114yes—
249C2×C4×C61#12114yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12712yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—816no—
259C2×Dic61—24no—
260C2×Dic61—28no—
261D244—24no—
262D244—24no—
263D244—24no—
264D244—24no—
265D244—42no—
266D244—42no—
267D244—42no—
268D244—42no—
269D244—48no—
270D244—48no—
271C8×C61—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—82no—
280C22×C61—82no—
281C22×C61#39214yes—
282C22×C61C2×#12011yescharacteristic
283C22×C61C2×#12111yescharacteristic
284C22×C61C2×#12711yescharacteristic
285C22×C61#57614yes—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—48no—
289C2×D61—48no—
290C2×D61—816no—
291C2×D61—816no—
292C2×D61—84no—
293C2×D61—84no—
294C2×D61—84no—
295C2×D61—84no—
296C4×C61—42no—
297C4×C61—82no—
298C4×C61#39214yes—
299C4×C61C2×#12011yescharacteristic
300C4×C61C2×#12011yescharacteristic
301C4×C61C2×#12111yescharacteristic
302C4×C61C2×#12111yescharacteristic
303Dic61—44no—
304C22×AGL(1,ℤ/8)—611no—
305C23×D8—611no—
306C2×#535—611no—
307C2×#418—611no—
308C2×#416D6111yescharacteristic, complemented
309#2309—611no—
310#2012—614no—
311#2011—614no—
312#1729—611no—
313C2×C61—24no—
314C2×C61—82no—
315C2×C61C2×#39211yescharacteristic
316C2×C61#300812yes—
317C2×C61#304112yes—
318C2×C61#315512yes—
319D61—84no—
320D61—84no—
321C22×#41C2×D6111yescharacteristic
322C23×D4—611no—
323C23×D4—611no—
324C2×AGL(1,ℤ/8)—1224no—
325C2×AGL(1,ℤ/8)—1224no—
326C2×AGL(1,ℤ/8)—1224no—
327C22×SD16—612no—
328C22×D8—612no—
329C22×D8—614no—
330C22×D8—1224no—
331C22×D8—1224no—
332C22×D8—1224no—
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×#117C2×D6111yescharacteristic
343#418—1224no—
344#416—24no—
345C2×#103C2×D6112yescomplemented
346C2×#102—614no—
347C2×#102C2×D6112yescomplemented
348#390—614no—
349#390—1222no—
350#379—1222no—
351#336—614no—
352C61#2342211yescharacteristic, complemented, Sylow
353C25—1221no—
354C25—1221no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41D24414yes—
361C22×Q8C22×D6111yescharacteristic
362C22×D4—21no—
363C22×D4—614no—
364C22×D4—614no—
365C22×D4—612no—
366C22×D4—612no—
367C22×D4—1224no—
368C22×D4—1222no—
369C22×D4—1222no—
370C22×D4—1222no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4—1222no—
374C22×D4—1222no—
375C22×D4—1222no—
376C22×D4—1221no—
377C22×D4—1221no—
378C22×D4—1222no—
379C22×D4—1222no—
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—1228no—
389C2×D8—12232no—
390C2×D8—1224no—
391C2×D8—1224no—
392C2×D8—1224no—
393C2×D8—1228no—
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—22no—
409#117—22no—
410#117—1222no—
411#117—1222no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4C22×D6112yes—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1228no—
418C2×C42—21no—
419C2×C42—612no—
420#103—24no—
421#102—24no—
422#102—1228no—
423C24—21no—
424C24—1224no—
425C24—1224no—
426C24—1222no—
427C24—1222no—
428C24—4884no—
429C24—4884no—
430C24—2442no—
431C24—2442no—
432C24—2442no—
433C24—2442no—
434C24D4×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×D24412yes—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—614no—
454C2×D4—614no—
455C2×D4—12216no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1228no—
467C2×D4—1228no—
468C2×D4—1228no—
469C2×D4—1224no—
470C2×D4—1228no—
471C2×D4—1224no—
472C2×D4—4888no—
473C2×D4—2444no—
474C2×D4—2442no—
475C2×D4—2444no—
476C2×D4—2442no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4—2442no—
480C2×D4—2442no—
481C2×D4C2×D24412yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—614no—
488C22×C4C2×D24412yes—
489C22×C4#1175822212yes—
490C22×C4D4×D6112yes—
491C22×C4D4×D6112yes—
492C22×C4D4×D6112yes—
493C22×C4D4×D6111yescharacteristic
494C22×C4C2×C22⋊D6112yes—
495C22×C4C23×D6111yescharacteristic
496SD16—12216no—
497D8—12232no—
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—22no—
508C4⋊C4—22no—
509C4⋊C4—1224no—
510C22⋊C4—42no—
511C22⋊C4—12216no—
512C22⋊C4—1224no—
513C22⋊C4—4888no—
514C42—42no—
515C42—1222no—
516C42—1222no—
517C23—24no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—44no—
522C23—1224no—
523C23—1224no—
524C23—4888no—
525C23—4888no—
526C23—4888no—
527C23—4888no—
528C23—4888no—
529C23—4888no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2448no—
535C23—2448no—
536C23—2448no—
537C23—2448no—
538C23#1451974712yes—
539C23#1451975012yes—
540C23#1451977312yes—
541C23C2×D4×D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—22no—
547D4—22no—
548D4—48no—
549D4—41no—
550D4—41no—
551D4—1224no—
552D4—1224no—
553D4—1224no—
554D4—1224no—
555D4—48816no—
556D4—2442no—
557D4—2442no—
558D4—2442no—
559D4—2442no—
560D4—2448no—
561D4—2448no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—1224no—
569C2×C4—1228no—
570C2×C4#1451974514yes—
571C2×C4#1451974714yes—
572C2×C4#1451974712yes—
573C2×C4#1451975014yes—
574C2×C4#1451975012yes—
575C2×C4#1451975114yes—
576C2×C4#1451977312yes—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C22×D24411yescharacteristic
582C2×C4C2×#1175822211yescharacteristic
583C8—24no—
584C8—1224no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—48816no—
593C22—48816no—
594C22—4884no—
595C22—4884no—
596C22—4884no—
597C22—4884no—
598C22—2444no—
599C22—2444no—
600C22—2448no—
601C22—2448no—
602C22#1683967114yes—
603C22C2×#1451974711yescharacteristic
604C22C2×#1451975011yescharacteristic
605C22C2×#1451977311yescharacteristic
606C22#1684056414yes—
607C4—42no—
608C4—82no—
609C4—2444no—
610C4#1683967114yes—
611C4C2×#1451974511yescharacteristic
612C4C2×#1451974711yescharacteristic
613C4C2×#1451975011yescharacteristic
614C4C2×#1451975111yescharacteristic
615C2—24no—
616C2—82no—
617C2—4884no—
618C2—4884no—
619C2C2×#1683967111yescharacteristic
620C2#2145509012yes—
621C2#2145532512yes—
622C2#2145630612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
hypercenterC2×#416D61
hypocenterC61#23422
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C22×#14519802C2semidirect
2. C23×D488C2semidirect
3. C2×#16840326C2semidirect
4. C2×#16839766C2semidirect
5. #21448399C2semidirect
6. #21446897C2semidirect
7. #21446891C2semidirect
8. #21445515C2semidirect
9. C2×C61×#416C2semidirect
15. C22×#11758201C22semidirect
20. C22×#11758199C22semidirect
24. C2×#14519773C22semidirect
25. C2×#14519766C22semidirect
30. C2×#14519697C22semidirect
32. #16839663C22semidirect
34. #16839516C22semidirect
39. C2×#103×C61C22semidirect
40. C2×#102×C61C22semidirect
56. C22×D244D4semidirect
73. C2×#11758194D4semidirect
102. C22×C8×C61D4semidirect
308. C2×#416D61semidirect
345. C2×#103C2×D61semidirect
347. C2×#102C2×D61semidirect
352. C61#23422semidirect
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×#145198022
C23×D4882
C2×#168403262
C2×#168397662
#214483992
#214468972
#214468912
#214455152
C2×C61×#4162
#2342261