Finite Group Databasebeta

← #29712235

Subgroups — #29712235

Order 15616.

1436 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840454C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21449169C211yesmaximal, characteristic
4#21446963C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446176C212yesmaximal, complemented
8#21446138C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445906C211yesmaximal, characteristic, complemented
11C2×C61×#450C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13#16840454—24no—
14#16840454—24no—
15C2×#14519772—21no—
16C2×#14519768—21no—
17C2×#14519768C2211yescharacteristic, complemented
18C2×#14519761—21no—
19C2×#14519753C2211yescharacteristic, complemented
20C2×#14519748C2212yes—
21C2×#14519746—21no—
22C2×Q8×Dic61C2211yescharacteristic, complemented
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519735C2211yescharacteristic, complemented
25C2×#14519733—21no—
26C22×C4⋊Dic61C2211yescharacteristic
27C2×#14519701C2212yescomplemented
28C2×#14519698—24no—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519689C2212yescomplemented
32#16839676—21no—
33#16839670—21no—
34#16839665C2212yes—
35#16839622C2211yescharacteristic
36#16839618—21no—
37#16839525C2212yescomplemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#450—24no—
45C2×#102×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×C22⋊D61—22no—
48C22×C22⋊D61D411yescharacteristic
49C23×Dic61—21no—
50C22×C4×D61C2311yescharacteristic
51C22×#92418—22no—
52C22×#92418—42no—
53C22×#92418C2312yes—
54#14519772—28no—
55#14519768—28no—
56#14519768—22no—
57#14519768—22no—
58#14519761—28no—
59#14519753—22no—
60#14519753—22no—
61#14519748—24no—
62#14519748—42no—
63#14519746—44no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66C4×C22⋊D61—22no—
67C4×C22⋊D61—22no—
68#14519735—22no—
69#14519735—22no—
70#14519733—28no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D414yescomplemented
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—22no—
84C2×C22⋊Dic61—22no—
85C2×C22⋊Dic61—22no—
86C2×C22⋊Dic61D411yescharacteristic
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519701—42no—
94#14519698—24no—
95#14519698—24no—
96#14519689—42no—
97#14519684—21no—
98C25×C61—21no—
99C22×Q8×C61C2311yescharacteristic
100C22×D4×C61—24no—
101C22×D4×C61—22no—
102C22×D4×C61—22no—
103C22×D4×C61—21no—
104C22×D4×C61—42no—
105C22×D4×C61C2312yes—
106C23×C4×C61C2311yescharacteristic
107C2×SD16×C61—24no—
108C2×SD16×C61—24no—
109C2×SD16×C61—42no—
110C22×C8×C61C2312yes—
111C22⋊Q8×C61—22no—
112C22⋊Q8×C61D414yescomplemented
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61—21no—
116#102×C61—24no—
117C22⋊C8×C61—22no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—42no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—82no—
124C24×C61C2×D411yescharacteristic
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D412yes—
130C2×D4×C61—28no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—48no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61—42no—
140C2×D4×C61#4114yes—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—82no—
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168SD16×C61—48no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D412yes—
175C22⋊C4×C61—24no—
176C22⋊Dic61—22no—
177C22⋊Dic61—44no—
178C22⋊Dic61—44no—
179C22⋊Dic61—42no—
180C22⋊Dic61—82no—
181#11758198—24no—
182#11758198—24no—
183#11758198—22no—
184C4⋊Dic61—24no—
185C4⋊Dic61—24no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C2×D412yes—
192C4⋊Dic61C2×D412yes—
193#11758196—24no—
194#11758196—24no—
195#11758196—22no—
196C4×Dic61—24no—
197C4×Dic61—24no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—44no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—24no—
207C23×C61—48no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—88no—
215C23×C61—88no—
216C23×C61—84no—
217C23×C61—84no—
218C23×C61#12012yes—
219C23×C61#13512yes—
220C23×C61AGL(1,ℤ/8)12yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—22no—
227Q8×C61—24no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61—48no—
233D4×C61—42no—
234D4×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61#12012yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4112yes—
247C2×C4×C61H(5,𝔽2)12yes—
248C22⋊D61—44no—
249C22⋊D61—88no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—48no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—42no—
270C8×C61—42no—
271#92418—42no—
272#23966—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—816no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61—82no—
284C22×C61—82no—
285C22×C61#44914yes—
286C22×C61C2×#12011yescharacteristic
287C22×C61#54612yes—
288C22×C61C2×#13511yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61#57712yes—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—42no—
295C2×D61—42no—
296C4×C61—22no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61#52511yescharacteristic
300C4×C61#53511yescharacteristic
301C4×C61#54612yes—
302Dic61—41no—
303Dic61—41no—
304Dic61—42no—
305Dic61—42no—
306Dic61—41no—
307Dic61—82no—
308C2×#552—611no—
309C2×#450D6111yescharacteristic, complemented
310C22×#102—611no—
311#2408—611no—
312#2025—612no—
313#2000—612no—
314#1892—611no—
315#1880—612no—
316#1875—612no—
317#1869—611no—
318#1823—611no—
319C2×C61—24no—
320C2×C61—84no—
321C2×C61#240411yescharacteristic
322C2×C61C2×#44911yescharacteristic
323C2×C61#301011yescharacteristic
324C2×C61#329612yes—
325C2×C61#331012yes—
326D61—81no—
327C23×D4C2×D6111yescharacteristic
328C22×SD16C2×D6112yes—
329C23×C8—611no—
330#552—1224no—
331#552—1224no—
332C2×#126—1221no—
333C2×#125—611no—
334C2×#125—1221no—
335C2×#123—1221no—
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8C2×D6111yescharacteristic
338C2×#121—612no—
339C2×#120—1221no—
340C2×C4×Q8—611no—
341C2×C4×D4—611no—
342C2×#117—611no—
343C2×#117—1221no—
344C22×C4⋊C4—611no—
345#450—24no—
346C2×#103—612no—
347C2×#102—612no—
348C2×#102—612no—
349C2×#102—1224no—
350C2×#102C2×D6112yes—
351C2×C22⋊C8—612no—
352C2×C22⋊C8C2×D6111yescharacteristic
353#394—1221no—
354#392—1221no—
355#390—612no—
356#380—611no—
357#379—1221no—
358#340—612no—
359#340—612no—
360#340—612no—
361#336—612no—
362C61#2396611yescharacteristic, complemented, Sylow
363C25—21no—
364C22×Q8C22×D6111yescharacteristic
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—42no—
370C22×D4—611no—
371C22×D4—1222no—
372C22×D4C22×D6112yes—
373C23×C4—611no—
374C23×C4—1221no—
375C23×C4C22×D6111yescharacteristic
376C2×SD16—24no—
377C2×SD16—24no—
378C2×SD16—42no—
379C22×C8—612no—
380C22×C8—1222no—
381C22×C8—2442no—
382C22×C8C22×D6112yes—
383#126—1228no—
384#125—1222no—
385#125—1228no—
386#125—1222no—
387#123—1228no—
388C22⋊Q8—22no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8C22⋊D6114yescomplemented
392#121—1224no—
393#121—2442no—
394#120—2444no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399#117—1228no—
400#117—1222no—
401#117—1222no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—614no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—1221no—
426C2×C42—1221no—
427#103—2442no—
428#102—24no—
429#102—1224no—
430#102—1224no—
431C22⋊C8—22no—
432C22⋊C8—2442no—
433#95—1221no—
434C24—22no—
435C24—44no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24—611no—
441C24D4×D6111yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8C2×C22⋊D6112yes—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—48no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—4882no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4#1175822414yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C23×D6111yescharacteristic
484SD16—48no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—2444no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1224no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—24no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—4882no—
513C22⋊C4—2444no—
514C22⋊C4—2444no—
515C22⋊C4—2442no—
516C42—1224no—
517C42—1224no—
518C42—1221no—
519C42—1221no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—88no—
531C23—88no—
532C23—84no—
533C23—84no—
534C23—1221no—
535C23—1221no—
536C23—1221no—
537C23—1222no—
538C23—1222no—
539C23#1451974512yes—
540C23#1451979812yes—
541C23#1451980412yes—
542C23C2×D4×D6111yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—4888no—
552D4—2444no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2448no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2442no—
581C2×C4#1451974512yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C2×#1175822412yes—
585C2×C4#1451982212yes—
586C8—42no—
587C8—2442no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—816no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—2441no—
601C22—2441no—
602C22—2441no—
603C22—2442no—
604C22—2442no—
605C22#1683990614yes—
606C22C2×#1451974511yescharacteristic
607C22#1684039912yes—
608C22C2×#1451979811yescharacteristic
609C22C2×#1451980411yescharacteristic
610C22#1684057212yes—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—4882no—
615C4—2441no—
616C4—2441no—
617C4—2442no—
618C4—2442no—
619C4—2441no—
620C4#1684026811yescharacteristic
621C4#1684032711yescharacteristic
622C4#1684040612yes—
623C2—24no—
624C2—84no—
625C2—4881no—
626C2#2144913511yescharacteristic
627C2C2×#1683990611yescharacteristic
628C2#2145510411yescharacteristic
629C2#2145765812yes—
630C2#2145784412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#450C2
socleC23×C61C22×D4
prime radical#21449169C2
hypercenterC2×#450D61
hypocenterC61#23966
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C2×#16840454C2semidirect
2. C22×#14519698C2semidirect
4. #21446963C2semidirect
5. #21446819C2semidirect
6. #21446243C2semidirect
7. #21446176C2semidirect
8. #21446138C2semidirect
9. #21446109C2semidirect
10. #21445906C2semidirect
11. C2×C61×#450C2semidirect
17. C2×#14519768C22semidirect
19. C2×#14519753C22semidirect
22. C2×Q8×Dic61C22semidirect
24. C2×#14519735C22semidirect
27. C2×#14519701C22semidirect
31. C2×#14519689C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
75. C2×C4⋊Dic61D4semidirect
112. C22⋊Q8×C61D4semidirect
309. C2×#450D61semidirect
362. C61#23966semidirect
391. C22⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404542
C22×#145196982
#214491692
#214469632
#214468192
#214462432
#214461762
#214461382
#214461092
#214459062
C2×C61×#4502
#2396661