Finite Group Databasebeta

← #30617846

Subgroups — #30617846

Order 15616.

1352 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic, complemented
2#21462266C211yesmaximal, characteristic, complemented
3#21461984C214yesmaximal, complemented
4#21461957C212yesmaximal, complemented
5#21461955C212yesmaximal, complemented
6#21461661C214yesmaximal, complemented
7#532×D61C211yesmaximal, characteristic
8#16840785—21no—
9#16840782—21no—
10#16840780—21no—
11#16840779—21no—
12#16840767C2212yescomplemented
13C2×#14519842C2218yescomplemented
14C2×#14519841C2218yescomplemented
15#16840733C2212yes—
16C4×#11758213C2212yescomplemented
17#16840707—21no—
18#16840706—21no—
19#16840705—21no—
20#16840704—21no—
21C4×C4⋊#6923C2212yescomplemented
22C4×C22⋊#6923C2212yes—
23#16840698C2212yescomplemented
24C2×#41×D61C2212yes—
25#16840313C2211yescharacteristic
26#16840312C411yescharacteristic, complemented
27#16840311C2212yes—
28#16840310C2211yescharacteristic
29#16840309—21no—
30#16840309—21no—
31#16840308C412yescomplemented
32#16840307—21no—
33#16840307—21no—
34C4⋊Q8×D61C411yescharacteristic, complemented
35#127×D61C411yescharacteristic, complemented
36#124×D61C412yescomplemented
37#121×D61—21no—
38#121×D61—21no—
39C4×D4×D61—21no—
40C4×D4×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#532C411yescharacteristic, complemented
43C2×#11758213C2314yes—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923C2314yes—
47#14519844—22no—
48#14519844—22no—
49#14519842—48no—
50#14519841—48no—
51#14519834—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54C4×#92420—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C2×#11758226C2312yes—
60C2×#11758222—22no—
61C2×#11758222C2×C412yes—
62C2×#11758224—22no—
63C2×#11758224C2312yes—
64#41×D61—48no—
65C2×Q8×D61C2×C412yes—
66C2×D4×D61—22no—
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61C2×C412yes—
70C22×C4×D61—22no—
71C22×C4×D61D411yescharacteristic
72C22×C4×D61D411yescharacteristic
73C22×C4×D61C2311yescharacteristic
74#14519777C2×C411yescharacteristic
75#14519776—21no—
76#14519776—21no—
77#14519775C2×C412yes—
78#14519774C2×C411yescharacteristic
79#14519773—21no—
80#14519773—21no—
81#14519773C2×C412yes—
82#14519766C2×C412yes—
83#14519765C2×C412yes—
84#14519764C2×C412yes—
85#14519763—22no—
86#14519763—22no—
87#14519752—21no—
88#14519752—21no—
89#14519751—21no—
90#14519751—21no—
91#14519750—22no—
92#14519750—22no—
93#14519749—22no—
94#14519749—22no—
95#14519748—21no—
96#14519748—21no—
97C4×D244—21no—
98C4×D244—21no—
99D4×Dic61—21no—
100D4×Dic61—21no—
101#14519739—21no—
102#14519739—21no—
103C4×C22⋊D61—22no—
104C4×C22⋊D61—22no—
105#14519737—22no—
106#14519737—22no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61—21no—
109C22⋊C4×D61—22no—
110C22⋊C4×D61—22no—
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C2×#41×C61C2×C412yes—
123C4⋊Q8×C61C2×C411yescharacteristic
124#127×C61C2×C411yescharacteristic
125#124×C61C2×C412yes—
126#121×C61—21no—
127#121×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C2×C42×C61C2×C411yescharacteristic
131C23×D61—22no—
132C23×D61C2×D411yescharacteristic
133C22×#6923—24no—
134#41×C61—44no—
135C2×Q8×C61C22×C412yes—
136C2×D4×C61—22no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—22no—
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C22×Dic61—22no—
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C22⋊C411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—28no—
154Q8×D61—44no—
155#11758224—28no—
156#11758224—48no—
157D4×D61—44no—
158D4×D61—44no—
159D4×D61—44no—
160D4×D61—48no—
161#11758222—44no—
162#11758222—48no—
163C2×D244—22no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C22×C412yes—
167C2×C4×D61—24no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—48no—
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22×C411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2411yescharacteristic
186C2×Dic122—22no—
187C2×Dic122C22×C412yes—
188C22⋊#6923—44no—
189C2×#92420—24no—
190C4⋊#6923—44no—
191C4×#6923—24no—
192C4×#6923—24no—
193#11758213—28no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C42×C61C22⋊C411yescharacteristic
199C42×C61C22⋊C411yescharacteristic
200C42×C61C22×C411yescharacteristic
201C42×C61C22×C411yescharacteristic
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C23×C61—22no—
225C23×C61C2×C22⋊C411yescharacteristic
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—44no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61C2×C22⋊C411yescharacteristic
234C22×D61C22×D411yescharacteristic
235C2×#6923—44no—
236C2×#6923—44no—
237C2×#6923—44no—
238Q8×C61—24no—
239D4×C61—24no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—44no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—44no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C22⋊D61—44no—
255C22⋊D61—44no—
256C22⋊D61—44no—
257C22⋊D61—44no—
258C22⋊D61—44no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—44no—
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C22×D411yescharacteristic
270D244—24no—
271D244—42no—
272D244—42no—
273D244—44no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—44no—
286C4×D61—44no—
287C4×D61—44no—
288C4×D61H(5,𝔽2)14yes—
289Dic122—24no—
290Dic122—44no—
291#92420—44no—
292#21816—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61C22×C22⋊C411yescharacteristic
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—44no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—88no—
309C2×D61#42112yes—
310C2×D61#53512yes—
311#6923—84no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—42no—
318C4×C61#51812yes—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—21no—
323Dic61—42no—
324Dic61#42112yes—
325Dic61#53512yes—
326C2×#532—611no—
327#2920—611no—
328#2283—611no—
329#1980—614no—
330#1963—612no—
331#1963—612no—
332#1789—614no—
333C2×C61—41no—
334C2×C61—82no—
335C2×C61#230911yescharacteristic
336C2×C61C2×#42111yescharacteristic
337C2×C61C2×#42111yescharacteristic
338D61—22no—
339D61—22no—
340D61—82no—
341C22×#41—612no—
342#532—612no—
343#532—611no—
344#532—611no—
345#532—612no—
346#532—611no—
347#532—1221no—
348#532—1221no—
349#532—1221no—
350#532—1221no—
351#532#692311yescharacteristic, complemented
352C2×C4⋊Q8—611no—
353C2×#127—611no—
354C2×#124—612no—
355C2×#121—1221no—
356C2×#121—1221no—
357C2×C4×D4—1221no—
358C2×C4×D4—1221no—
359C22×C42—611no—
360#432—1221no—
361#432—1221no—
362#431—1221no—
363#431—1221no—
364#423—612no—
365C2×#104—618no—
366C2×#104—618no—
367#407—612no—
368C4×M16—612no—
369#383—1221no—
370#382—1221no—
371#381—1221no—
372#380—1221no—
373C4×C4⋊C4—612no—
374C4×C22⋊C4—612no—
375#376—612no—
376C61#2181611yescharacteristic, complemented, Sylow
377C2×#41—612no—
378C2×#41—612no—
379C2×#41—612no—
380C2×#41—1222no—
381C2×#41—1222no—
382C2×#41—2448no—
383C2×#41C2×#692312yes—
384C22×Q8—612no—
385C22×D4—612no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4—1222no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1222no—
393C2×M16—614no—
394C4⋊Q8—612no—
395C4⋊Q8—611no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398C4⋊Q8C2×#692311yescharacteristic
399#127—612no—
400#127—611no—
401#127—1221no—
402#127—1221no—
403#127C2×#692311yescharacteristic
404#124—612no—
405#124—612no—
406#124—612no—
407#124—1222no—
408#124—1222no—
409#124C2×#692312yes—
410#121—21no—
411#121—21no—
412#121—1222no—
413#121—1222no—
414#121—1221no—
415#121—1221no—
416#121—1221no—
417#121—1221no—
418#121—1222no—
419#121—1222no—
420#121—1221no—
421#121—1221no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4—1221no—
433C4×D4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1222no—
437C2×C22⋊C4—1222no—
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1222no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—614no—
452C2×C42C2×#692311yescharacteristic
453C4⋊C8—1222no—
454C4⋊C8—1222no—
455#104—2448no—
456#104—2448no—
457C22⋊C8—1222no—
458C22⋊C8—1222no—
459#97—1222no—
460C4×C8—1222no—
461#95—1222no—
462#95—1222no—
463#95—1222no—
464#95—1222no—
465C24—611no—
466C24—1222no—
467#41—44no—
468#41—1228no—
469#41—1228no—
470#41—2444no—
471#41—2448no—
472#41—2448no—
473C2×Q8—612no—
474C2×Q8—1222no—
475C2×Q8—2444no—
476C2×Q8C22×#692312yes—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—612no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—2444no—
490C2×D4—2444no—
491C2×D4—2444no—
492C2×D4—2448no—
493C2×D4C22×#692312yes—
494C22×C4—22no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1224no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2448no—
519C22×C4C22⋊#692311yescharacteristic
520C22×C4C22⋊#692311yescharacteristic
521C22×C4C22×#692311yescharacteristic
522M16—1228no—
523C2×C8—1224no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—2444no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—2444no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—1224no—
553C42—1224no—
554C42C22⋊#692311yescharacteristic
555C42C22⋊#692311yescharacteristic
556C42C22×#692311yescharacteristic
557C42C22×#692311yescharacteristic
558C23—22no—
559C23—611no—
560C23—611no—
561C23—1222no—
562C23—1222no—
563C23—1222no—
564C23—2441no—
565C23—2444no—
566C23—2444no—
567C23—2444no—
568C23C2×C22⋊#692311yescharacteristic
569Q8—24no—
570Q8—1224no—
571Q8—2444no—
572D4—24no—
573D4—42no—
574D4—42no—
575D4—44no—
576D4—1224no—
577D4—2442no—
578D4—2442no—
579D4—2444no—
580D4—2444no—
581D4—2444no—
582D4—2444no—
583D4—2444no—
584D4—2444no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—614no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2444no—
612C2×C4—2444no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4—2441no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4C2×C22⋊#692311yescharacteristic
620C2×C4C2×C22⋊#692311yescharacteristic
621C2×C4C2×C22⋊#692311yescharacteristic
622C2×C4C2×C22⋊#692311yescharacteristic
623C2×C4C2×C22⋊#692311yescharacteristic
624C2×C4C23×#692311yescharacteristic
625C8—2444no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—612no—
633C22—612no—
634C22—1221no—
635C22—1221no—
636C22—1221no—
637C22—1221no—
638C22—4888no—
639C22—2444no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22C22×C22⋊#692311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—42no—
650C4—612no—
651C4—612no—
652C4—1221no—
653C4—1221no—
654C4—1221no—
655C4—1221no—
656C4—4884no—
657C4—2442no—
658C4#1684082912yes—
659C2—41no—
660C2—82no—
661C2—1222no—
662C2—1222no—
663C2—4882no—
664C2#2146229211yescharacteristic
665C2D61⋊#42111yescharacteristic
666C2D61⋊#42111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC22×C61C22×C22⋊C4
prime radical#532×D61C2
cosocleC2×C4×D61C24
hypercenter#532#6923
hypocenterC61#21816
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #21462498C2semidirect
2. #21462266C2semidirect
3. #21461984C2semidirect
4. #21461957C2semidirect
5. #21461955C2semidirect
6. #21461661C2semidirect
12. #16840767C22semidirect
13. C2×#14519842C22semidirect
14. C2×#14519841C22semidirect
16. C4×#11758213C22semidirect
21. C4×C4⋊#6923C22semidirect
23. #16840698C22semidirect
26. #16840312C4semidirect
31. #16840308C4semidirect
34. C4⋊Q8×D61C4semidirect
35. #127×D61C4semidirect
36. #124×D61C4semidirect
42. C61×#532C4semidirect
351. #532#6923semidirect
376. C61#21816semidirect

Maximal subgroups

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

subgroupindex
#214624982
#214622662
#214619842
#214619572
#214619552
#214616612
#532×D612
#2181661