Finite Group Databasebeta

← #29681546

Subgroups — #29681546

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×#14519805C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446196C218yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C2211yescharacteristic
11C2×#14519805—24no—
12C2×#14519805—22no—
13C2×#14519805C418yescomplemented
14C2×#14519805C2214yes—
15C22×#11758190—21no—
16C22×D488—21no—
17C22×#11758194C2211yescharacteristic
18#16840243—22no—
19#16840242C2214yescomplemented
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#16839766—22no—
30#16839766C2214yescomplemented
31C2×#14519697—22no—
32#16839664C2214yescomplemented
33#16839663—22no—
34#16839617—22no—
35#16839530C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#339C2214yescomplemented
40C24×D61—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C414yes—
46C2×#11758222D414yes—
47C2×#11758222D412yes—
48C2×#11758222C2312yes—
49C22×Dic122D411yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244—22no—
56C22×D244—21no—
57C22×D244—22no—
58C22×D244C2×C414yescomplemented
59C22×D244D411yescharacteristic
60C22×D244C2312yes—
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63#14519805—216no—
64#14519805—28no—
65#14519805C2×C4116yescomplemented
66C2×#11758190—28no—
67C2×#11758190—24no—
68C2×D488—28no—
69C2×D488—24no—
70C2×#11758194—21no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194C2312yes—
74C22×#92418—21no—
75C4×Dic122—24no—
76#14519742—24no—
77C4×D244—24no—
78#14519739—24no—
79C4×C22⋊D61—28no—
80#14519737—28no—
81#14519736—21no—
82#14519736—21no—
83#14519736—21no—
84#14519736—21no—
85#14519735—24no—
86#14519734—24no—
87#14519733—24no—
88C2×C4⋊Dic61C2312yes—
89C4⋊C4×D61—24no—
90C2×C22⋊Dic61D412yes—
91C2×#11758198—28no—
92C22⋊C4×D61—24no—
93C2×C4×Dic61D412yes—
94C42×D61—24no—
95#14519705—28no—
96#14519697—28no—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—22no—
99C2×M16×C61—21no—
100C2×M16×C61C2×C414yescomplemented
101C2×M16×C61C2312yes—
102C22×C8×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—22no—
106#117×C61—22no—
107#117×C61D412yes—
108#117×C61C2312yes—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D412yes—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2312yes—
115C24×C61C2×D411yescharacteristic
116C23×D61—21no—
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—41no—
120C23×D61—41no—
121C23×D61—42no—
122C23×D61—84no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—24no—
138C2×C22⋊D61—22no—
139C22×Dic61—21no—
140C22×Dic61C2×D414yes—
141#11758222—24no—
142#11758222—24no—
143#11758222—24no—
144#11758222—24no—
145#11758222—24no—
146#11758222—22no—
147#11758222—22no—
148#11758222C22⋊C418yescomplemented
149#11758222C22×C414yes—
150#11758222C2×D414yes—
151C2×D244—21no—
152C2×D244—22no—
153C2×D244—24no—
154C2×D244—24no—
155C2×D244—24no—
156C2×D244—24no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—24no—
162C2×D244—22no—
163C2×D244—41no—
164C2×D244—41no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—42no—
170C2×D244C22⋊C414yescomplemented
171C2×D244C22×C414yes—
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—
182M16×C61—24no—
183M16×C61—22no—
184M16×C61C22×C414yes—
185C2×C8×C61—24no—
186C2×C8×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61C2×D414yes—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—42no—
193C42×C61—22no—
194C42×C61—22no—
195C42×C61—22no—
196C42×C61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199#11758198—28no—
200#11758198—88no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203#11758196—28no—
204C4×Dic61—28no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207#11758194—22no—
208#11758194—22no—
209#11758194—42no—
210C2×#92418—22no—
211C2×#92418—42no—
212D488—216no—
213#11758190—216no—
214C23×C61—42no—
215C23×C61C4×D412yes—
216C23×C61#12011yescharacteristic
217C23×C61#12011yescharacteristic
218C23×C61#12112yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—24no—
222C22×D61—24no—
223C22×D61—48no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—44no—
227C22×D61—44no—
228C22×D61—82no—
229C22×D61—88no—
230C22×D61—84no—
231C22×D61—84no—
232C22×D61—88no—
233C2×C4×C61—24no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—82no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12112yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12112yes—
247C2×C4×C61#12312yes—
248C2×C4×C61#12312yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C22⋊D61—24no—
256C22⋊D61—24no—
257C22⋊D61—28no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—44no—
262D244—24no—
263D244—24no—
264D244—22no—
265D244—24no—
266D244—42no—
267D244—42no—
268D244—44no—
269D244—44no—
270D244C2×C22⋊C414yes—
271C4×D61—24no—
272C4×D61—24no—
273C4×D61—28no—
274C4×D61—88no—
275Dic122—22no—
276Dic122—24no—
277Dic122C2×C22⋊C414yes—
278C8×C61—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#38612yes—
287C22×C61#38612yes—
288C22×C61C2×C4×D411yescharacteristic
289C22×C61C2×#12011yescharacteristic
290C22×C61C2×#12111yescharacteristic
291C2×D61—24no—
292C2×D61—44no—
293C2×D61—48no—
294C2×D61—44no—
295C2×D61—84no—
296C2×D61—84no—
297C2×D61—88no—
298C2×D61—88no—
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#2309—611no—
315#1883—618no—
316#1742—611no—
317#1733D6111yescharacteristic, complemented
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—22no—
321C2×C61—22no—
322C2×C61C2×#38611yescharacteristic
323C2×C61#302112yes—
324D61—44no—
325D61—84no—
326C22×#41—611no—
327C23×D4—611no—
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×M16C2×D6111yescharacteristic
336#520—614no—
337#520—1222no—
338#519—1221no—
339#518—1222no—
340#518—1221no—
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—614no—
353#390—1222no—
354#379—1222no—
355#343—614no—
356#339C2×D6114yescomplemented
357C61#2094511yescharacteristic, complemented, Sylow
358C25—1221no—
359C2×#41—614no—
360C2×#41—612no—
361C2×#41—614no—
362C2×#41—612no—
363C2×#41—1221no—
364C2×#41—1222no—
365C2×#41—1221no—
366C2×#41—1222no—
367C22×Q8—611no—
368C22×D4—614no—
369C22×D4—611no—
370C22×D4—612no—
371C22×D4—1221no—
372C22×D4—1221no—
373C22×D4—1221no—
374C22×D4—1222no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—1222no—
378C22×D4—1221no—
379C22×D4—1222no—
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—21no—
390C2×M16—22no—
391C2×M16—612no—
392C2×M16—1221no—
393C2×M16—1222no—
394C2×M16—1222no—
395C2×M16C4×D6114yescomplemented
396C2×M16C22×D6112yes—
397C22×C8—21no—
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—22no—
408#117—22no—
409#117—1224no—
410#117—1224no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414#117—1221no—
415#117—1224no—
416#117D24412yes—
417#117C22×D6112yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—1224no—
421C2×C4⋊C4D24412yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—612no—
425C2×C22⋊C4—1224no—
426C2×C22⋊C4—1228no—
427C2×C42—21no—
428C2×C42—612no—
429C2×C42—1224no—
430C2×C42C22×D6112yes—
431#104—1228no—
432#102—1228no—
433C24—1221no—
434C24—1224no—
435C24—1222no—
436C24—4884no—
437C24—2441no—
438C24—2441no—
439C24—2442no—
440C24D4×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—614no—
456C2×D4—612no—
457C2×D4—614no—
458C2×D4—1222no—
459C2×D4—1224no—
460C2×D4—1222no—
461C2×D4—1224no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1224no—
468C2×D4—1224no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—1221no—
473C2×D4—1222no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4—2442no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—614no—
484C22×C4—1228no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—1224no—
488C22×C4—1222no—
489C22×C4—1224no—
490C22×C4C2×C4×D6112yes—
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×D24411yescharacteristic
493C22×C4#1175822212yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6112yes—
497C22×C4D4×D6111yescharacteristic
498C22×C4#1175822612yes—
499C22×C4C23×D6111yescharacteristic
500SD16—12216no—
501D8—12216no—
502M16—22no—
503M16—24no—
504M16—1222no—
505M16—1222no—
506M16—2442no—
507M16C2×C4×D6114yes—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—1222no—
511C2×C8—2442no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—1228no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4C2×D24414yes—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—42no—
521C22⋊C4—1228no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—4888no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—1228no—
530C42—1222no—
531C42—1222no—
532C23—42no—
533C23—1224no—
534C23—1222no—
535C23—1224no—
536C23—4884no—
537C23—4884no—
538C23—4882no—
539C23—4888no—
540C23—4888no—
541C23—2448no—
542C23—2442no—
543C23—2442no—
544C23—2444no—
545C23—2444no—
546C23#1451973912yes—
547C23#1451974711yescharacteristic
548C23#1451974711yescharacteristic
549C23#1451975012yes—
550C23C2×D4×D6111yescharacteristic
551Q8—614no—
552Q8—1222no—
553Q8—1224no—
554D4—614no—
555D4—1224no—
556D4—1224no—
557D4—1228no—
558D4—1224no—
559D4—1224no—
560D4—1222no—
561D4—1224no—
562D4—2442no—
563D4—2442no—
564D4—2444no—
565D4—2444no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1228no—
575C2×C4—1224no—
576C2×C4—4888no—
577C2×C4—2444no—
578C2×C4C22⋊C4×D6112yes—
579C2×C4C22⋊C4×D6112yes—
580C2×C4#1451973912yes—
581C2×C4C4×D24412yes—
582C2×C4C4×D24412yes—
583C2×C4#1451974711yescharacteristic
584C2×C4#1451974711yescharacteristic
585C2×C4#1451975012yes—
586C2×C4#1451975212yes—
587C2×C4#1451975212yes—
588C2×C4#1451976212yes—
589C2×C4#1451976212yes—
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C22×D24411yescharacteristic
594C2×C4C2×#1175822211yescharacteristic
595C2×C4C2×#1175822611yescharacteristic
596C8—24no—
597C8—2442no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—1224no—
604C22—4888no—
605C22—4888no—
606C22—4884no—
607C22—4884no—
608C22—4884no—
609C22—2448no—
610C22—2444no—
611C22—2444no—
612C22#1683964012yes—
613C22#1683964012yes—
614C22C2×#1451973911yescharacteristic
615C22C2×#1451974711yescharacteristic
616C22C2×#1451975011yescharacteristic
617C4—44no—
618C4—82no—
619C4—1224no—
620C4—2444no—
621C4#1683964012yes—
622C4#1683964012yes—
623C4C2×C22⋊C4×D6111yescharacteristic
624C4C2×C4×D24411yescharacteristic
625C4C2×#1451975211yescharacteristic
626C4C2×#1451976211yescharacteristic
627C2—21no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2—4884no—
632C2—2444no—
633C2C2×#1683964011yescharacteristic
634C2#2145519212yes—
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C2×#120
prime radicalC22×#14519805C2
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20945
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. #21459020C2semidirect
3. C2×#16839780C2semidirect
4. C2×#16839766C2semidirect
5. #21448399C2semidirect
6. #21446196C2semidirect
7. #21445557C2semidirect
8. C61×#1733C2semidirect
13. C2×#14519805C4semidirect
19. #16840242C22semidirect
28. C2×#14519705C22semidirect
30. #16839766C22semidirect
32. #16839664C22semidirect
35. #16839530C22semidirect
39. C61×#339C22semidirect
58. C22×D244C2×C4semidirect
65. #14519805C2×C4semidirect
100. C2×M16×C61C2×C4semidirect
148. #11758222C22⋊C4semidirect
170. C2×D244C22⋊C4semidirect
180. C2×Dic122C22⋊C4semidirect
317. #1733D61semidirect
356. #339C2×D61semidirect
357. C61#20945semidirect
395. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
#214590202
C2×#168397802
C2×#168397662
#214483992
#214461962
#214455572
C61×#17332
#2094561