Finite Group Databasebeta

← #29681598

Subgroups — #29681598

Order 15616.

1974 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840331C211yesmaximal, characteristic
2C2×#16840236C211yesmaximal, characteristic, complemented
3#21454339C211yesmaximal, characteristic, complemented
4#21454336C211yesmaximal, characteristic, complemented
5C22×#14519693C211yesmaximal, characteristic, complemented
6#21446216C214yesmaximal, complemented
7#21446206C214yesmaximal, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C61×#1733C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840331—24no—
14#16840331C418yescomplemented
15C2×#14519774C2212yes—
16C2×#14519765C2212yes—
17C2×#14519752—22no—
18C2×#14519747—22no—
19#16840236—22no—
20#16840236—22no—
21#16840236—22no—
22#16840236C2214yes—
23C2×C4×D244—22no—
24C2×#14519739—22no—
25C2×#14519736C2211yescharacteristic
26C2×C22⋊C4×D61—22no—
27#16839772C2212yescomplemented
28#16839770C2212yescomplemented
29#16839768C2212yescomplemented
30#16839766C2212yescomplemented
31#16839764—22no—
32#16839761—22no—
33#16839759—22no—
34#16839756—22no—
35C2×#14519693—24no—
36C2×#14519693C2214yes—
37#16839522C2214yescomplemented
38C22×M16×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#339C2214yescomplemented
42C24×D61D412yes—
43C2×#11758222—22no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222C2×C414yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244—21no—
55C22×D244—21no—
56C22×D244—22no—
57C22×D244—22no—
58C22×D244—24no—
59C22×D244C2×C414yes—
60C22×D244D412yes—
61C22×D244D414yes—
62C22×D244C2311yescharacteristic
63C22×D244C2312yes—
64C22×C22⋊D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—24no—
67C2×#11758194—21no—
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194C2312yes—
71C22×#92418—21no—
72#14519774—24no—
73#14519774C2×C418yescomplemented
74#14519765—24no—
75#14519765C2×C418yescomplemented
76#14519752—216no—
77#14519747—216no—
78C4×D244—28no—
79C4×D244—28no—
80#14519739—28no—
81#14519739—28no—
82#14519736—21no—
83#14519736—21no—
84#14519736D412yescomplemented
85#14519736C2312yes—
86C2×C4⋊Dic61D412yescomplemented
87C2×C22⋊Dic61—21no—
88C2×#11758198—24no—
89C2×#11758198—24no—
90C22⋊C4×D61—28no—
91C22⋊C4×D61—28no—
92C2×C4×Dic61—21no—
93#14519702—24no—
94#14519700—24no—
95#14519699—24no—
96#14519697—24no—
97#14519693—28no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—21no—
100C2×M16×C61—22no—
101C2×M16×C61—22no—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61C2×C414yescomplemented
108#117×C61D412yescomplemented
109#117×C61C2312yes—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61D412yescomplemented
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61C2312yes—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—24no—
120C23×D61—21no—
121C23×D61—84no—
122C23×D61C22⋊C418yes—
123C23×D61C2×D414yes—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—42no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142#11758222—24no—
143#11758222—24no—
144#11758222—24no—
145#11758222—24no—
146#11758222—42no—
147#11758222—42no—
148#11758222—42no—
149#11758222—42no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—24no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—24no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—216no—
171C2×D244—88no—
172C2×D244C22⋊C418yes—
173C2×D244C22×C414yes—
174C2×D244C22×C412yescomplemented
175C2×D244C2×D414yes—
176C2×C4×D61—24no—
177C2×C4×D61—216no—
178C2×C4×D61—41no—
179C2×C4×D61—41no—
180C2×C4×D61—42no—
181C2×C4×D61—88no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122C22×C412yescomplemented
188M16×C61—22no—
189M16×C61—22no—
190M16×C61—42no—
191C2×C8×C61—22no—
192C2×C8×C61—42no—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61C2×D414yes—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C42×C61—22no—
200C42×C61—22no—
201C42×C61—22no—
202C42×C61C22×C414yescomplemented
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205#11758198—216no—
206#11758198—216no—
207C4⋊Dic61—22no—
208C4⋊Dic61C2×D414yes—
209C4×Dic61—22no—
210C4×Dic61—22no—
211#11758194—22no—
212#11758194—22no—
213#11758194—42no—
214C2×#92418—22no—
215C2×#92418—42no—
216C23×C61—42no—
217C23×C61C2×C22⋊C412yes—
218C23×C61#12011yescharacteristic
219C23×C61#12011yescharacteristic
220C23×C61#12312yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—24no—
223C22×D61—28no—
224C22×D61—24no—
225C22×D61—28no—
226C22×D61—22no—
227C22×D61—48no—
228C22×D61—48no—
229C22×D61—44no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—88no—
233C22×D61—84no—
234C22×D61—88no—
235C22×D61C2×C22⋊C418yes—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12312yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—44no—
259C22⋊D61—44no—
260C2×Dic61—22no—
261C2×Dic61—24no—
262C2×Dic61—44no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265D244—24no—
266D244—24no—
267D244—22no—
268D244—22no—
269D244—24no—
270D244—24no—
271D244—28no—
272D244—28no—
273D244—41no—
274D244—41no—
275D244—816no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61—816no—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283C8×C61—42no—
284#92418—42no—
285#20953—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61#38612yes—
292C22×C61#38612yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61C2×#12011yescharacteristic
295C22×C61C2×#12311yescharacteristic
296C2×D61—28no—
297C2×D61—416no—
298C2×D61—48no—
299C2×D61—48no—
300C2×D61—44no—
301C2×D61—42no—
302C2×D61—84no—
303C2×D61—88no—
304C2×D61—82no—
305C2×D61—82no—
306C4×C61—44no—
307C4×C61—44no—
308C4×C61#38612yes—
309C4×C61#38612yes—
310C4×C61C2×C4×D411yescharacteristic
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12111yescharacteristic
314Dic61—44no—
315Dic61—42no—
316C2×#535—611no—
317C2×#518—611no—
318#2896—611no—
319#2896—611no—
320C22×#100—611no—
321#1887—614no—
322#1885—614no—
323#1733—611no—
324#1733D6111yescharacteristic, complemented
325C2×C61—21no—
326C2×C61—22no—
327C2×C61—21no—
328C2×C61—22no—
329C2×C61C2×#38611yescharacteristic
330C2×C61#301912yes—
331D61—48no—
332D61—82no—
333C22×#41—611no—
334C23×D4—611no—
335C22×M16—611no—
336C22×M16C2×D6111yescharacteristic
337#535—618no—
338#535—1224no—
339C2×#127—612no—
340C2×#124—612no—
341C2×#121—1222no—
342C2×#120—1222no—
343#518—614no—
344#518—1222no—
345#518—1222no—
346#518—1222no—
347C2×C4×D4—1222no—
348C2×C4×D4—1222no—
349C2×#117—611no—
350C2×#117C2×D6111yescharacteristic
351C2×#117C2×D6111yescharacteristic
352C22×C22⋊C4—1222no—
353#419—612no—
354#419—612no—
355#418—612no—
356#418—612no—
357#417—1222no—
358#417—1222no—
359#416—1222no—
360#416—1222no—
361C2×#100—614no—
362C2×#100—1224no—
363#339—614no—
364#339C2×D6114yescomplemented
365C61#2095311yescharacteristic, complemented, Sylow
366C25—612no—
367C2×#41—614no—
368C2×#41—1222no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1222no—
372C2×#41—1221no—
373C2×#41—1221no—
374C2×#41—1222no—
375C22×Q8—611no—
376C22×D4—614no—
377C22×D4—612no—
378C22×D4—614no—
379C22×D4—612no—
380C22×D4—611no—
381C22×D4—1224no—
382C22×D4—1221no—
383C22×D4—1221no—
384C22×D4—1221no—
385C22×D4—1222no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4—1222no—
389C23×C4—1224no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C2×M16—21no—
393C2×M16—22no—
394C2×M16—22no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1222no—
398C2×M16—1222no—
399C2×M16C22×D6112yes—
400C22×C8—21no—
401C22×C8—1221no—
402#127—618no—
403#127—1224no—
404#124—618no—
405#124—1224no—
406#121—12216no—
407#120—12216no—
408C4×D4—1228no—
409C4×D4—1228no—
410C4×D4—1228no—
411C4×D4—1228no—
412#117—21no—
413#117—21no—
414#117—22no—
415#117—612no—
416#117—612no—
417#117—1221no—
418#117—1221no—
419#117C4×D6114yescomplemented
420#117D24412yescomplemented
421#117C22×D6112yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4D24412yescomplemented
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4—1228no—
429C2×C22⋊C4—1224no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—1221no—
434C2×C42C22×D6112yes—
435#103—1224no—
436#103—1224no—
437#102—1224no—
438#102—1224no—
439#100—1228no—
440C24—618no—
441C24—614no—
442C24—1222no—
443C24—1224no—
444C24—1222no—
445C24—1221no—
446C24—4884no—
447C24D4×D6111yescharacteristic
448#41—1224no—
449#41—1224no—
450#41—1224no—
451#41—1224no—
452#41—2442no—
453#41—2442no—
454#41—2442no—
455#41—2442no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×D4—614no—
463C2×D4—612no—
464C2×D4—614no—
465C2×D4—618no—
466C2×D4—12216no—
467C2×D4—1224no—
468C2×D4—1222no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4—1222no—
474C2×D4—1221no—
475C2×D4—1222no—
476C2×D4—1221no—
477C2×D4—1224no—
478C2×D4—1224no—
479C2×D4—1224no—
480C2×D4—1224no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1224no—
484C2×D4—1224no—
485C2×D4—1224no—
486C2×D4—1224no—
487C2×D4—1224no—
488C2×D4—4888no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4—2442no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—12216no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1224no—
498C22×C4—4888no—
499C22×C4—2441no—
500C22×C4—2441no—
501C22×C4—2442no—
502C22×C4C2×C4×D6112yes—
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4#1175822212yes—
506C22×C4D4×D6112yes—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4#1175822612yes—
511C22×C4C23×D6111yescharacteristic
512M16—22no—
513M16—22no—
514M16—42no—
515M16—1222no—
516M16—1222no—
517M16—2442no—
518C2×C8—22no—
519C2×C8—42no—
520C2×C8—1222no—
521C2×C8—2442no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—614no—
525C4⋊C4—1222no—
526C4⋊C4C2×D24414yes—
527C22⋊C4—22no—
528C22⋊C4—24no—
529C22⋊C4—22no—
530C22⋊C4—12216no—
531C22⋊C4—12216no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—1222no—
538C42—1222no—
539C42C2×C4×D6114yescomplemented
540C23—42no—
541C23—618no—
542C23—1224no—
543C23—1228no—
544C23—1224no—
545C23—1228no—
546C23—1222no—
547C23—4884no—
548C23—4888no—
549C23—4888no—
550C23—2444no—
551C23—2448no—
552C23—2448no—
553C23—2441no—
554C23—2441no—
555C23C22⋊C4×D6112yes—
556C23#1451974711yescharacteristic
557C23#1451974711yescharacteristic
558C23#1451976212yes—
559C23C2×D4×D6111yescharacteristic
560Q8—1222no—
561Q8—1222no—
562Q8—2441no—
563Q8—2441no—
564D4—1224no—
565D4—1222no—
566D4—1224no—
567D4—1222no—
568D4—1224no—
569D4—1224no—
570D4—1228no—
571D4—1228no—
572D4—48816no—
573D4—2444no—
574D4—2444no—
575D4—2441no—
576D4—2441no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—48816no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4C22⋊C4×D6112yes—
590C2×C4#1451973912yes—
591C2×C4#1451973912yes—
592C2×C4C4×D24412yes—
593C2×C4C4×D24412yes—
594C2×C4#1451974711yescharacteristic
595C2×C4#1451974711yescharacteristic
596C2×C4#1451975012yes—
597C2×C4#1451975012yes—
598C2×C4#1451975212yes—
599C2×C4#1451975212yes—
600C2×C4#1451976212yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C22×D24411yescharacteristic
605C2×C4C2×#1175822211yescharacteristic
606C2×C4C2×#1175822611yescharacteristic
607C8—42no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—1228no—
615C22—4882no—
616C22—4888no—
617C22—4884no—
618C22—4882no—
619C22—2444no—
620C22—2448no—
621C22—24416no—
622C22—2442no—
623C22—2448no—
624C22#1683964012yes—
625C22#1683964012yes—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22C2×#1451974711yescharacteristic
628C22C2×#1451976211yescharacteristic
629C4—44no—
630C4—44no—
631C4—2442no—
632C4—2444no—
633C4#1683964012yes—
634C4#1683964012yes—
635C4C2×#1451973911yescharacteristic
636C4C2×C4×D24411yescharacteristic
637C4C2×#1451975011yescharacteristic
638C4C2×#1451975211yescharacteristic
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—4882no—
644C2—2448no—
645C2C2×#1683964011yescharacteristic
646C2#2145517812yes—
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C2×#120
prime radicalC2×#16840331C2
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20953
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. C2×#16840236C2semidirect
3. #21454339C2semidirect
4. #21454336C2semidirect
5. C22×#14519693C2semidirect
6. #21446216C2semidirect
7. #21446206C2semidirect
8. #21445529C2semidirect
9. C61×#1733C2semidirect
14. #16840331C4semidirect
27. #16839772C22semidirect
28. #16839770C22semidirect
29. #16839768C22semidirect
30. #16839766C22semidirect
37. #16839522C22semidirect
41. C61×#339C22semidirect
50. C2×#11758222C2×C4semidirect
73. #14519774C2×C4semidirect
75. #14519765C2×C4semidirect
84. #14519736D4semidirect
86. C2×C4⋊Dic61D4semidirect
107. #117×C61C2×C4semidirect
108. #117×C61D4semidirect
111. C2×C4⋊C4×C61D4semidirect
174. C2×D244C22×C4semidirect
187. C2×Dic122C22×C4semidirect
202. C42×C61C22×C4semidirect
324. #1733D61semidirect
364. #339C2×D61semidirect
365. C61#20953semidirect
419. #117C4×D61semidirect
420. #117D244semidirect
424. C2×C4⋊C4D244semidirect
539. C42C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168402362
#214543392
#214543362
C22×#145196932
#214462162
#214462062
#214455292
C61×#17332
#2095361