Finite Group Databasebeta

← #29681548

Subgroups — #29681548

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519803C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4C2×#16839767C211yesmaximal, characteristic, complemented
5#21446198C218yesmaximal, complemented
6#21445558C211yesmaximal, characteristic, complemented
7#21445529C211yesmaximal, characteristic, complemented
8C61×#2309C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519803—24no—
11C2×#14519803—22no—
12C2×#14519803C418yescomplemented
13C2×#14519803C2214yes—
14C22×#11758200—21no—
15C22×#11758199—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18#16840243C2214yescomplemented
19#16840242—22no—
20#16840239—21no—
21#16840236—21no—
22#16840235—22no—
23C4×#11758222—22no—
24#16840230—22no—
25C2×#14519736C2211yescharacteristic
26#117×D61—22no—
27#16839780—24no—
28C2×#14519705C2214yescomplemented
29#16839767—22no—
30#16839767C2214yescomplemented
31C2×#14519698—22no—
32#16839531C2214yescomplemented
33#16839522C2214yescomplemented
34C23×D4×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#390—22no—
38C61×#390C2214yescomplemented
39C61×#379—22no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C2×#11758222D412yes—
47C2×#11758222C2312yes—
48C22×Dic122D411yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52#14519803—216no—
53#14519803—28no—
54#14519803C2×C4116yescomplemented
55C2×#11758200—28no—
56C2×#11758200—24no—
57C2×#11758199—28no—
58C2×#11758199—24no—
59C2×#11758194—22no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194C2×C414yescomplemented
65C2×#11758194C2312yes—
66C2×#11758194C2312yes—
67C22×#92418—21no—
68C22×#92418—21no—
69C4×Dic122—24no—
70#14519742—24no—
71C4×D244—24no—
72#14519739—24no—
73C4×C22⋊D61—28no—
74#14519737—28no—
75#14519736—21no—
76#14519736—21no—
77#14519736D412yes—
78#14519736C2312yes—
79#14519735—24no—
80#14519734—24no—
81#14519733—24no—
82C2×C4⋊Dic61D412yes—
83C4⋊C4×D61—24no—
84C2×C22⋊Dic61—21no—
85C22⋊C4×D61—24no—
86C2×C4×Dic61—21no—
87C42×D61—24no—
88#14519705—28no—
89#14519698—28no—
90C25×C61—21no—
91C22×D4×C61—21no—
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61—22no—
96C22×D4×C61—21no—
97C22×D4×C61—22no—
98C22×D4×C61—22no—
99C22×D4×C61C2×C414yescomplemented
100C22×D4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102#117×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—28no—
112C2×C22⋊C4×C61D412yes—
113C2×C42×C61D412yes—
114C2×C42×C61C2312yes—
115C24×C61—22no—
116C24×C61—24no—
117C24×C61—41no—
118C24×C61—41no—
119C24×C61—42no—
120C24×C61—84no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C2×D4×C61—24no—
124C2×D4×C61—24no—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61C22×C414yes—
141C22×C4×C61—22no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—24no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—22no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159#11758222—24no—
160#11758222—24no—
161#11758222—24no—
162#11758222—24no—
163#11758222—24no—
164#11758222—22no—
165#11758222—22no—
166#11758222C22⋊C418yescomplemented
167#11758222C22×C414yes—
168#11758222C2×D414yes—
169C2×D244—21no—
170C2×D244—22no—
171C2×D244C22⋊C414yescomplemented
172C2×D244C2×D412yes—
173C2×C4×D61—24no—
174C2×C4×D61—22no—
175C2×C4×D61—24no—
176C2×C4×D61—22no—
177C2×C4×D61—28no—
178C2×Dic122—21no—
179C2×Dic122—22no—
180C2×Dic122C22⋊C414yescomplemented
181C2×Dic122C2×D412yes—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—42no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—88no—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758200—216no—
196#11758199—216no—
197#11758198—28no—
198C4⋊Dic61—22no—
199C4⋊Dic61C2×D414yes—
200#11758196—28no—
201C4×Dic61—28no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204#11758194—24no—
205#11758194—22no—
206#11758194—22no—
207#11758194—22no—
208#11758194—42no—
209#11758194C22×C414yes—
210C2×#92418—24no—
211C2×#92418—22no—
212C2×#92418—22no—
213C2×#92418—42no—
214C23×C61—24no—
215C23×C61—42no—
216C23×C61—48no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—44no—
220C23×C61—44no—
221C23×C61—84no—
222C23×C61—84no—
223C23×C61—88no—
224C23×C61—88no—
225C23×C61C4×D412yes—
226C23×C61#12011yescharacteristic
227C23×C61#12011yescharacteristic
228C23×C61#12112yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—24no—
232C22×D61—82no—
233D4×C61—24no—
234D4×C61—24no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—44no—
238D4×C61—44no—
239C2×C4×C61—24no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61C4×D412yes—
247C2×C4×C61C4×D412yes—
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12312yes—
254C2×C4×C61#12312yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C22⋊D61—24no—
262C22⋊D61—24no—
263C22⋊D61—28no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—82no—
268D244—22no—
269D244—24no—
270D244C2×C22⋊C414yes—
271C4×D61—24no—
272C4×D61—24no—
273C4×D61—28no—
274C4×D61—88no—
275Dic122—22no—
276Dic122—24no—
277Dic122C2×C22⋊C414yes—
278#92418—24no—
279#92418—42no—
280#20945—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—48no—
287C22×C61—44no—
288C22×C61—84no—
289C22×C61—88no—
290C22×C61—88no—
291C22×C61—84no—
292C22×C61#38612yes—
293C22×C61#38612yes—
294C22×C61C2×C4×D411yescharacteristic
295C22×C61C2×#12011yescharacteristic
296C22×C61C2×#12111yescharacteristic
297C2×D61—24no—
298C2×D61—44no—
299C2×D61—84no—
300C4×C61—44no—
301C4×C61—82no—
302C4×C61#38612yes—
303C4×C61#38612yes—
304C4×C61C22×C22⋊C411yescharacteristic
305C4×C61C2×C4×D411yescharacteristic
306C4×C61C2×#12111yescharacteristic
307C4×C61C2×#12311yescharacteristic
308Dic61—24no—
309Dic61—44no—
310C22×AGL(1,ℤ/8)—611no—
311#3428—611no—
312C2×#421—611no—
313C2×#418—611no—
314#2309D6111yescharacteristic, complemented
315#1883—618no—
316#1742—611no—
317#1733—611no—
318C2×C61—21no—
319C2×C61—22no—
320C2×C61—21no—
321C2×C61—22no—
322C2×C61—84no—
323C2×C61C2×#38611yescharacteristic
324C2×C61#302112yes—
325D61—44no—
326C22×#41—611no—
327C23×D4C2×D6111yescharacteristic
328C2×AGL(1,ℤ/8)—618no—
329C2×AGL(1,ℤ/8)—614no—
330C2×AGL(1,ℤ/8)—1222no—
331C2×AGL(1,ℤ/8)—1224no—
332C22×SD16—1221no—
333C22×D8—1221no—
334C22×M16—611no—
335C22×M16—611no—
336#520—614no—
337#520—1222no—
338#519—1221no—
339#518—1221no—
340#518—1222no—
341C4×#41—1222no—
342C4×#41—1222no—
343C2×#117—611no—
344C2×#117—1222no—
345C2×#117C2×D6111yescharacteristic
346C2×#117C2×D6111yescharacteristic
347#421—1224no—
348C2×#104—614no—
349#418—614no—
350#418—1222no—
351C2×#102—1222no—
352#390—22no—
353#390C2×D6114yescomplemented
354#379—22no—
355#343—614no—
356#339—614no—
357C61#2094511yescharacteristic, complemented, Sylow
358C25—21no—
359C2×#41—614no—
360C2×#41—612no—
361C2×#41—614no—
362C2×#41—612no—
363C2×#41—1222no—
364C2×#41—1221no—
365C2×#41—1222no—
366C2×#41—1221no—
367C22×Q8—611no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—22no—
372C22×D4—21no—
373C22×D4—21no—
374C22×D4—22no—
375C22×D4—21no—
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4C4×D6114yescomplemented
379C22×D4C22×D6112yes—
380C23×C4—1221no—
381C23×C4C22×D6111yescharacteristic
382AGL(1,ℤ/8)—6116no—
383AGL(1,ℤ/8)—1228no—
384AGL(1,ℤ/8)—12216no—
385C2×SD16—1224no—
386C2×SD16—1228no—
387C2×D8—1224no—
388C2×D8—1228no—
389C2×M16—614no—
390C2×M16—612no—
391C2×M16—612no—
392C2×M16—1221no—
393C2×M16—1222no—
394C2×M16—1221no—
395C2×M16—1222no—
396C2×M16—1222no—
397C22×C8—1221no—
398C22×C8—1221no—
399C4×Q8—1224no—
400C4×Q8—1224no—
401C4×D4—1224no—
402C4×D4—1228no—
403C4×D4—1224no—
404C4×D4—1228no—
405#117—21no—
406#117—21no—
407#117—21no—
408#117—21no—
409#117—22no—
410#117—22no—
411#117—612no—
412#117—612no—
413#117—1224no—
414#117—1224no—
415#117—1224no—
416#117—1221no—
417#117—1221no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—1224no—
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—28no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4C22⋊D6112yes—
427C2×C42—1224no—
428C2×C42—1221no—
429C2×C42C22⋊D6112yes—
430C2×C42C22×D6112yes—
431#104—1228no—
432#102—1228no—
433C24—22no—
434C24—24no—
435C24—41no—
436C24—41no—
437C24—42no—
438C24—84no—
439C24—1221no—
440C24C2×C22⋊D6111yescharacteristic
441#41—614no—
442#41—614no—
443#41—618no—
444#41—1222no—
445#41—1222no—
446#41—1224no—
447#41—1224no—
448#41—1224no—
449#41—1224no—
450#41—1224no—
451C2×Q8—612no—
452C2×Q8—614no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—612no—
473C2×D4—614no—
474C2×D4—1222no—
475C2×D4—1224no—
476C2×D4—1222no—
477C2×D4—1224no—
478C2×D4—1221no—
479C2×D4—1222no—
480C2×D4C2×C4×D6114yes—
481C22×C4—22no—
482C22×C4—1228no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1224no—
486C22×C4—1222no—
487C22×C4—1224no—
488C22×C4—1222no—
489C22×C4C2×C4×D6112yes—
490C22×C4#1175822212yes—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4D4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4#1175822412yes—
497C22×C4C2×C22⋊D6114yes—
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500SD16—12216no—
501D8—12216no—
502M16—614no—
503M16—1222no—
504M16—1224no—
505M16—1222no—
506M16—1222no—
507M16—2442no—
508C2×C8—1222no—
509C2×C8—1224no—
510C2×C8—1222no—
511C2×C8—2442no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—614no—
516C4⋊C4—1228no—
517C4⋊C4—1222no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—42no—
521C22⋊C4—88no—
522C22⋊C4—1228no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—1228no—
530C42—1222no—
531C42—1222no—
532C23—24no—
533C23—42no—
534C23—48no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—84no—
540C23—84no—
541C23—88no—
542C23—88no—
543C23—1224no—
544C23—1222no—
545C23—4882no—
546C23C4×C22⋊D6112yes—
547C23#1451974511yescharacteristic
548C23#1451974511yescharacteristic
549C23#1451974812yes—
550C23C22×C22⋊D6111yescharacteristic
551Q8—614no—
552Q8—1222no—
553Q8—1224no—
554D4—24no—
555D4—24no—
556D4—42no—
557D4—42no—
558D4—44no—
559D4—44no—
560D4—614no—
561D4—1224no—
562D4—1224no—
563D4—1228no—
564D4—1222no—
565D4—1224no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1228no—
575C2×C4—1224no—
576C2×C4—4888no—
577C2×C4—4882no—
578C2×C4C22⋊C4×D6112yes—
579C2×C4C22⋊C4×D6112yes—
580C2×C4#1451973712yes—
581C2×C4#1451973712yes—
582C2×C4C4×C22⋊D6112yes—
583C2×C4#1451974511yescharacteristic
584C2×C4#1451974511yescharacteristic
585C2×C4#1451974812yes—
586C2×C4#1451975012yes—
587C2×C4#1451975012yes—
588C2×C4#1451975912yes—
589C2×C4#1451975912yes—
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×#1175822411yescharacteristic
595C2×C4C2×#1175822211yescharacteristic
596C8—1224no—
597C8—2442no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—48no—
604C22—44no—
605C22—84no—
606C22—88no—
607C22—88no—
608C22—84no—
609C22—1224no—
610C22—4884no—
611C22—2444no—
612C22#1683963812yes—
613C22#1683963812yes—
614C22C2×C4×C22⋊D6111yescharacteristic
615C22C2×#1451974511yescharacteristic
616C22C2×#1451974811yescharacteristic
617C4—44no—
618C4—82no—
619C4—1224no—
620C4—2444no—
621C4#1683963812yes—
622C4#1683963812yes—
623C4C2×C22⋊C4×D6111yescharacteristic
624C4C2×#1451973711yescharacteristic
625C4C2×#1451975011yescharacteristic
626C4C2×#1451975911yescharacteristic
627C2—21no—
628C2—21no—
629C2—22no—
630C2—22no—
631C2—84no—
632C2—2444no—
633C2C2×#1683963811yescharacteristic
634C2#2145519012yes—
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC22×C61C2×#120
prime radicalC22×#14519803C2
cosocleC22×C4×C61C24
hypercenter#2309D61
hypocenterC61#20945
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #21459020C2semidirect
3. C2×#16839780C2semidirect
4. C2×#16839767C2semidirect
5. #21446198C2semidirect
6. #21445558C2semidirect
7. #21445529C2semidirect
8. C61×#2309C2semidirect
12. C2×#14519803C4semidirect
18. #16840243C22semidirect
28. C2×#14519705C22semidirect
30. #16839767C22semidirect
32. #16839531C22semidirect
33. #16839522C22semidirect
38. C61×#390C22semidirect
54. #14519803C2×C4semidirect
64. C2×#11758194C2×C4semidirect
99. C22×D4×C61C2×C4semidirect
166. #11758222C22⋊C4semidirect
171. C2×D244C22⋊C4semidirect
180. C2×Dic122C22⋊C4semidirect
314. #2309D61semidirect
353. #390C2×D61semidirect
357. C61#20945semidirect
378. C22×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
#214590202
C2×#168397802
C2×#168397672
#214461982
#214455582
#214455292
C61×#23092
#2094561