Finite Group Databasebeta

← #30625197

Subgroups — #30625197

Order 15616.

4810 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#568C211yesmaximal, characteristic, complemented
2#21462799C211yesmaximal, characteristic, complemented
3#21462441C2116yesmaximal, complemented
4#21462439C216yesmaximal, complemented
5C2×#16840755C216yesmaximal, complemented
6C2×2−1+4×D61C211yesmaximal, characteristic
7#16840842—21no—
8#16840841—24no—
9#16840840—23no—
10D61⋊#131C2218yescomplemented
11C2×D61⋊M16C2216yes—
12#16840831C2218yescomplemented
13#16840830—21no—
14#16840829—23no—
15#41×#6923—24no—
16C2×#14519856C2216yes—
17#16840755C22112yes—
18#16840755C22124yescomplemented
19#16840754C2214yes—
20#16840753C22124yescomplemented
21#16840752C22124yescomplemented
22C2×#14519840C22124yescomplemented
23C2×#14519825—21no—
24C2×#14519825C413yescomplemented
25C2×#14519824—23no—
26C2×#14519824C411yescharacteristic, complemented
27C2×#14519824C413yescomplemented
282−1+4×D61—24no—
292−1+4×D61—24no—
302−1+4×D61C418yescomplemented
312−1+4×D61C2218yes—
32C2×#41×D61—23no—
33C2×#41×D61C2213yes—
34C2×#41×D61C2211yescharacteristic
35C22×Q8×D61—21no—
36C22×Q8×D61C2213yes—
37C2×2−1+4×C61C411yescharacteristic, complemented
38#14519865—28no—
39#14519864—224no—
40C2×#11758213—26no—
41D61⋊M16—26no—
42D61⋊M16—26no—
43D61⋊M16C23112yes—
44D61⋊M16C2314yes—
45C2×#11758217—26no—
46C2×D61⋊C8C23112yes—
47Q8×#6923—28no—
48D4×#6923—224no—
49#14519856—26no—
50#14519856—26no—
51#14519856C23112yes—
52#14519856C2314yes—
53C2×C4⋊#6923C23112yes—
54C2×C22⋊#6923—26no—
55C2×C4×#6923—26no—
56#14519840C23148yes—
57#14519839C23116yes—
58#14519825—28no—
59#14519825—28no—
60#14519825—212no—
61#14519825C2×C4124yescomplemented
62#14519824—24no—
63#14519824—224no—
64#14519824—224no—
65#14519824—212no—
66#14519824C2×C4124yescomplemented
67#14519824C2×C418yescomplemented
68C2×#11758226—23no—
69C2×#11758226C2×C413yescomplemented
70C2×#11758226D411yescharacteristic
71C2×#11758222—26no—
72C2×#11758222—23no—
73C2×#11758222—23no—
74C2×#11758222C2×C413yes—
75C2×#11758222C2×C413yescomplemented
76C2×#11758224—23no—
77C2×#11758224—26no—
78C2×#11758224—23no—
79C2×#11758224C2×C413yescomplemented
80C2×#11758224D413yes—
81#41×D61—224no—
82#41×D61—224no—
83#41×D61—26no—
84#41×D61—26no—
85#41×D61C2×C414yes—
86#41×D61C2×C4112yescomplemented
87#41×D61D414yes—
88#41×D61D414yes—
89#41×D61C23112yes—
90#41×D61C2314yes—
91C22×Dic122—23no—
92C22×Dic122—23no—
93C22×Dic122C2×C413yescomplemented
94C2×Q8×D61—21no—
95C2×Q8×D61—28no—
96C2×Q8×D61—28no—
97C2×Q8×D61—23no—
98C2×Q8×D61—26no—
99C2×Q8×D61—26no—
100C2×Q8×D61—23no—
101C2×Q8×D61—23no—
102C2×Q8×D61C2×C4112yescomplemented
103C2×Q8×D61C2×C416yescomplemented
104C2×Q8×D61D411yescharacteristic
105C2×Q8×D61C23112yes—
106C2×Q8×D61C2313yes—
107C2×D4×D61—23no—
108C2×D4×D61D413yes—
109C2×D4×D61C2311yescharacteristic
110C22×C4×D61—23no—
111C22×C4×D61—23no—
112C22×C4×D61C2313yes—
1132−1+4×C61—24no—
1142−1+4×C61C2×C418yescomplemented
115C2×#41×C61—23no—
116C2×#41×C61C2×C411yescharacteristic
117C2×#41×C61C2×C413yescomplemented
118C22×Q8×C61—21no—
119C22×Q8×C61C2×C413yescomplemented
120C23×D61—21no—
121C23×D61C2×D413yes—
122C22×#6923—212no—
123#41×C61—224no—
124#41×C61—26no—
125#41×C61C22⋊C414yes—
126#41×C61C22×C4112yes—
127#41×C61C22×C414yes—
128C2×Q8×C61—28no—
129C2×Q8×C61—23no—
130C2×Q8×C61—26no—
131C2×Q8×C61C22⋊C411yescharacteristic
132C2×Q8×C61C22×C413yes—
133C2×Q8×C61C22×C4112yes—
134C2×D4×C61—23no—
135C2×D4×C61C22⋊C413yes—
136C2×D4×C61C22×C411yescharacteristic
137C22×C4×C61—23no—
138C22×C4×C61—23no—
139C22×C4×C61C22×C413yes—
140C2×C22⋊D61—23no—
141C2×C22⋊D61—23no—
142C2×C22⋊D61—21no—
143C2×C22⋊D61C22⋊C416yes—
144C22×Dic61—23no—
145C22×Dic61—23no—
146C22×Dic61C2×D413yes—
147#11758226—224no—
148#11758226—26no—
149#11758226C22⋊C414yes—
150#11758226C22×C4112yes—
151#11758226C2×D414yes—
152Q8×D61—28no—
153Q8×D61—212no—
154Q8×D61—212no—
155Q8×D61—212no—
156Q8×D61—23no—
157Q8×D61—212no—
158Q8×D61—23no—
159Q8×D61—212no—
160Q8×D61—224no—
161Q8×D61C22⋊C414yes—
162Q8×D61C22×C416yes—
163Q8×D61C22×C4124yes—
164Q8×D61C2×D414yes—
165Q8×D61C2416yes—
166#11758224—224no—
167#11758224—248no—
168#11758224—212no—
169#11758224—212no—
170#11758224—26no—
171#11758224C22⋊C4112yes—
172#11758224C22×C4112yes—
173#11758224C2×D4112yes—
174D4×D61—212no—
175D4×D61—212no—
176D4×D61—21no—
177D4×D61—21no—
178D4×D61C22⋊C4112yes—
179D4×D61C22×C412yes—
180D4×D61C2×D4112yes—
181D4×D61C2412yes—
182#11758222—26no—
183#11758222—248no—
184#11758222—224no—
185#11758222—212no—
186#11758222—212no—
187#11758222C22⋊C4112yes—
188#11758222C22×C4112yes—
189#11758222C22×C4112yes—
190C2×D244—23no—
191C2×D244C22⋊C413yes—
192C2×D244C22×C411yescharacteristic
193C2×C4×D61—212no—
194C2×C4×D61—26no—
195C2×C4×D61—23no—
196C2×C4×D61—212no—
197C2×C4×D61—23no—
198C2×C4×D61—23no—
199C2×C4×D61—212no—
200C2×C4×D61—212no—
201C2×C4×D61C22×C416yes—
202C2×C4×D61C22×C413yes—
203C2×C4×D61C2×D413yes—
204C2×C4×D61C2×D416yes—
205C2×C4×D61C2×D416yes—
206C2×C4×D61C2×D413yes—
207C2×C4×D61C2411yescharacteristic
208C2×C4×D61C2416yes—
209C2×Dic122—23no—
210C2×Dic122—26no—
211C2×Dic122—26no—
212C2×Dic122—212no—
213C2×Dic122—23no—
214C2×Dic122—212no—
215C2×Dic122—224no—
216C2×Dic122C22⋊C413yes—
217C2×Dic122C22×C413yes—
218C2×Dic122C22×C4112yes—
219C22⋊#6923—24no—
220C22⋊#6923—212no—
221#11758217—24no—
222#11758217—212no—
223C2×#92420—212no—
224C4⋊#6923—212no—
225C4⋊#6923C2418yes—
226C4×#6923—24no—
227C4×#6923—212no—
228#11758213—24no—
229#11758213—212no—
230D61⋊C8—212no—
231D61⋊C8C2418yes—
232C23×C61—21no—
233C23×C61C2×C22⋊C413yes—
234C22×D61—21no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—46no—
238C22×D61—44no—
239C22×D61C2×C22⋊C413yes—
240C22×D61C2×C22⋊C416yes—
241C22×D61C22×D411yescharacteristic
242C22×D61C22×D416yes—
243C2×#6923—28no—
244C2×#6923—812no—
245Q8×C61—23no—
246Q8×C61—212no—
247Q8×C61C2×C22⋊C414yes—
248Q8×C61C23×C416yes—
249D4×C61—212no—
250D4×C61—21no—
251D4×C61C2×C22⋊C4112yes—
252D4×C61C23×C412yes—
253C2×C4×C61—23no—
254C2×C4×C61—212no—
255C2×C4×C61—212no—
256C2×C4×C61C2×C22⋊C413yes—
257C2×C4×C61C2×C22⋊C416yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C23×C416yes—
260C22⋊D61—212no—
261C22⋊D61—212no—
262C22⋊D61—22no—
263C22⋊D61—22no—
264C22⋊D61C2×C22⋊C4124yes—
265C2×Dic61—23no—
266C2×Dic61—212no—
267C2×Dic61—212no—
268C2×Dic61C2×C22⋊C413yes—
269C2×Dic61C2×C22⋊C416yes—
270C2×Dic61C22×D416yes—
271C2×Dic61C22×D411yescharacteristic
272D244—212no—
273D244—21no—
274D244C2×C22⋊C4112yes—
275D244C23×C412yes—
276C4×D61—26no—
277C4×D61—224no—
278C4×D61—26no—
279C4×D61—212no—
280C4×D61—212no—
281C4×D61C2×C22⋊C416yes—
282C4×D61C2×C22⋊C416yes—
283C4×D61C2×C22⋊C416yes—
284C4×D61C23×C411yescharacteristic
285C4×D61C23×C416yes—
286C4×D61C22×D411yescharacteristic
287C4×D61C22×D416yes—
288C4×D61C22×D416yes—
289C4×D61C22×D411yescharacteristic
290C4×D61C2511yescharacteristic
291Dic122—23no—
292Dic122—212no—
293Dic122—26no—
294Dic122—224no—
295Dic122—26no—
296Dic122C2×C22⋊C4112yes—
297Dic122C23×C416yes—
298#92420—28no—
299#60088—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—43no—
302C22×C61—42no—
303C22×C61C22×C22⋊C411yescharacteristic
304C22×C61C22×C22⋊C416yes—
305C2×D61—212no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—46no—
309C2×D61—43no—
310C2×D61—44no—
311C2×D61—42no—
312C2×D61—44no—
313C2×D61C22×C22⋊C411yescharacteristic
314C2×D61C22×C22⋊C416yes—
315C2×D61C23×D411yescharacteristic
316#6923—48no—
317C4×C61—26no—
318C4×C61C22×C22⋊C411yescharacteristic
319C4×C61C22×C22⋊C416yes—
320C4×C61C24×C411yescharacteristic
321Dic61—26no—
322Dic61C22×C22⋊C411yescharacteristic
323Dic61C22×C22⋊C416yes—
324Dic61C23×D411yescharacteristic
325C22×2−1+4—611no—
326C2×#568—611no—
327#3428—611no—
328#2898—6116no—
329#2896—616no—
330C2×#419—616no—
331C2×C61—21no—
332C2×C61—26no—
333C2×C61—42no—
334C2×C61C23×C22⋊C411yescharacteristic
335D61—21no—
336D61—26no—
337D61—42no—
338D61#358612yes—
339C2×2−1+4—618no—
340C2×2−1+4—618no—
341C2×2−1+4—611no—
342C2×2−1+4—613no—
343C2×2−1+4—613no—
344C2×2−1+4—1221no—
345C2×2−1+4—1223no—
346C2×2−1+4—1224no—
347C2×2−1+4—1224no—
348C2×2−1+4#692311yescharacteristic, complemented
349C22×#41—611no—
350C22×#41—613no—
351C22×#41—1223no—
352C23×Q8—613no—
353C23×Q8—1221no—
354#568—1223no—
355#568—1224no—
356#568—1221no—
357C2×#131—618no—
358C22×M16—616no—
359#520—618no—
360#519—1221no—
361#518—1223no—
362C4×#41—1224no—
363C2×#117—616no—
364#419—6112no—
365#419—6124no—
366#418—614no—
367#417—6124no—
368#416—6124no—
369C2×#103—6124no—
370C61#6008811yescharacteristic, complemented, Sylow
3712−1+4—24no—
3722−1+4—6124no—
3732−1+4—618no—
3742−1+4—6124no—
3752−1+4—1228no—
3762−1+4—1228no—
3772−1+4—12212no—
3782−1+4—12224no—
3792−1+4—12224no—
3802−1+4—12212no—
3812−1+4—1224no—
3822−1+4C2×#692318yescomplemented
383C2×#41—23no—
384C2×#41—614no—
385C2×#41—614no—
386C2×#41—614no—
387C2×#41—614no—
388C2×#41—611no—
389C2×#41—613no—
390C2×#41—613no—
391C2×#41—6112no—
392C2×#41—6112no—
393C2×#41—613no—
394C2×#41—613no—
395C2×#41—613no—
396C2×#41—1223no—
397C2×#41—1226no—
398C2×#41—1223no—
399C2×#41—1223no—
400C2×#41—1226no—
401C2×#41—1226no—
402C2×#41—1226no—
403C2×#41—1223no—
404C2×#41—12224no—
405C2×#41—12224no—
406C2×#41—1223no—
407C2×#41C2×#692311yescharacteristic
408C2×#41C2×#692313yescomplemented
409C22×Q8—21no—
410C22×Q8—611no—
411C22×Q8—6112no—
412C22×Q8—616no—
413C22×Q8—613no—
414C22×Q8—6112no—
415C22×Q8—613no—
416C22×Q8—1223no—
417C22×Q8—1223no—
418C22×Q8—1223no—
419C22×Q8—1226no—
420C22×Q8—1223no—
421C22×Q8—1226no—
422C22×Q8—1228no—
423C22×Q8—1228no—
424C22×Q8—1223no—
425C22×Q8—1221no—
426C22×Q8C2×#692313yescomplemented
427C22×D4—613no—
428C22×D4—611no—
429C22×D4—1223no—
430C23×C4—613no—
431C23×C4—1223no—
432C23×C4—1223no—
433#131—12224no—
434#131—1228no—
435C2×M16—614no—
436C2×M16—6112no—
437C2×M16—1226no—
438C2×M16—1226no—
439C2×M16—1226no—
440C2×M16—1226no—
441C22×C8—6112no—
442C4×Q8—1228no—
443C4×D4—12224no—
444#117—614no—
445#117—6112no—
446#117—1226no—
447#117—1226no—
448C2×C4⋊C4—6112no—
449C2×C22⋊C4—1226no—
450C2×C42—1226no—
451#103—6148no—
452#102—6116no—
453C24—613no—
454C24—1221no—
455#41—224no—
456#41—26no—
457#41—614no—
458#41—614no—
459#41—6112no—
460#41—6112no—
461#41—6112no—
462#41—6112no—
463#41—6112no—
464#41—6112no—
465#41—6112no—
466#41—12212no—
467#41—12212no—
468#41—12248no—
469#41—12224no—
470#41—12212no—
471#41—12212no—
472#41—1226no—
473#41—1226no—
474#41—12248no—
475#41—12224no—
476#41—12224no—
477#41—1226no—
478#41C22⋊#692314yes—
479#41C22×#6923112yes—
480#41C22×#692314yes—
481C2×Q8—28no—
482C2×Q8—23no—
483C2×Q8—26no—
484C2×Q8—614no—
485C2×Q8—614no—
486C2×Q8—613no—
487C2×Q8—616no—
488C2×Q8—616no—
489C2×Q8—6124no—
490C2×Q8—613no—
491C2×Q8—6112no—
492C2×Q8—1226no—
493C2×Q8—1223no—
494C2×Q8—12212no—
495C2×Q8—12212no—
496C2×Q8—12224no—
497C2×Q8—12212no—
498C2×Q8—1223no—
499C2×Q8—12212no—
500C2×Q8—1223no—
501C2×Q8—12212no—
502C2×Q8—12212no—
503C2×Q8—12212no—
504C2×Q8—12224no—
505C2×Q8—1228no—
506C2×Q8—1223no—
507C2×Q8—1226no—
508C2×Q8C22⋊#692311yescharacteristic
509C2×Q8C22×#692313yes—
510C2×Q8C22×#6923112yes—
511C2×D4—23no—
512C2×D4—616no—
513C2×D4—6112no—
514C2×D4—6112no—
515C2×D4—612no—
516C2×D4—612no—
517C2×D4—613no—
518C2×D4—611no—
519C2×D4—1223no—
520C2×D4—1221no—
521C2×D4—1221no—
522C2×D4—1221no—
523C2×D4—12212no—
524C2×D4—12212no—
525C2×D4—1223no—
526C2×D4—1223no—
527C2×D4C22⋊#692313yes—
528C2×D4C22×#692311yescharacteristic
529C22×C4—23no—
530C22×C4—23no—
531C22×C4—613no—
532C22×C4—616no—
533C22×C4—613no—
534C22×C4—616no—
535C22×C4—613no—
536C22×C4—616no—
537C22×C4—611no—
538C22×C4—613no—
539C22×C4—616no—
540C22×C4—1223no—
541C22×C4—12212no—
542C22×C4—12212no—
543C22×C4—1223no—
544C22×C4—1226no—
545C22×C4—12212no—
546C22×C4—12212no—
547C22×C4—1223no—
548C22×C4—1223no—
549C22×C4—1223no—
550C22×C4—12212no—
551C22×C4C22×#692313yes—
552M16—1224no—
553M16—1224no—
554M16—12212no—
555M16—12212no—
556C2×C8—618no—
557C2×C8—12212no—
558C2×C8—12212no—
559C4⋊C4—618no—
560C4⋊C4—12212no—
561C22⋊C4—1224no—
562C22⋊C4—12212no—
563C42—1224no—
564C42—12212no—
565C23—21no—
566C23—611no—
567C23—616no—
568C23—616no—
569C23—613no—
570C23—1222no—
571C23—1222no—
572C23—1221no—
573C23—2446no—
574C23—2444no—
575C23C2×C22⋊#692313yes—
576Q8—212no—
577Q8—23no—
578Q8—6112no—
579Q8—616no—
580Q8—1226no—
581Q8—12224no—
582Q8—1226no—
583Q8—12212no—
584Q8—1223no—
585Q8C2×C22⋊#692314yes—
586Q8C23×#692316yes—
587D4—212no—
588D4—21no—
589D4—6124no—
590D4—6112no—
591D4—612no—
592D4—12212no—
593D4—1222no—
594D4—1222no—
595D4—12212no—
596D4—12212no—
597D4—1221no—
598D4C2×C22⋊#6923112yes—
599D4C23×#692312yes—
600C2×C4—23no—
601C2×C4—212no—
602C2×C4—212no—
603C2×C4—613no—
604C2×C4—616no—
605C2×C4—616no—
606C2×C4—611no—
607C2×C4—616no—
608C2×C4—616no—
609C2×C4—616no—
610C2×C4—611no—
611C2×C4—616no—
612C2×C4—616no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—616no—
616C2×C4—611no—
617C2×C4—12212no—
618C2×C4—1226no—
619C2×C4—1226no—
620C2×C4—12224no—
621C2×C4—12212no—
622C2×C4—1223no—
623C2×C4—12212no—
624C2×C4—12212no—
625C2×C4—1228no—
626C2×C4—48812no—
627C2×C4C2×C22⋊#692313yes—
628C2×C4C2×C22⋊#692316yes—
629C2×C4C23×#692311yescharacteristic
630C2×C4C23×#692316yes—
631C8—1228no—
632C22—22no—
633C22—43no—
634C22—42no—
635C22—611no—
636C22—611no—
637C22—616no—
638C22—12212no—
639C22—1222no—
640C22—1222no—
641C22—2443no—
642C22—2446no—
643C22—2442no—
644C22—2444no—
645C22—2444no—
646C22C22×C22⋊#692311yescharacteristic
647C22C22×C22⋊#692316yes—
648C4—26no—
649C4—611no—
650C4—616no—
651C4—611no—
652C4—1226no—
653C4—2448no—
654C4C22×C22⋊#692316yes—
655C4C22×C22⋊#692311yescharacteristic
656C4C24×#692311yescharacteristic
657C2—21no—
658C2—26no—
659C2—42no—
660C2—612no—
661C2—1221no—
662C2—1226no—
663C2—2442no—
664C2C23×C22⋊#692311yescharacteristic
subgroupG/H
centerC2C23×C22⋊#6923
commutatorC4×C61C24×C4
FrattiniC4C24×#6923
FittingC2×2−1+4×C61C4
socleC2×C61C23×C22⋊C4
prime radicalC2×2−1+4×D61C2
cosocleC4×D61C25
hypercenterC2×2−1+4#6923
hypocenterC61#60088
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. D61⋊#568C2semidirect
2. #21462799C2semidirect
3. #21462441C2semidirect
4. #21462439C2semidirect
5. C2×#16840755C2semidirect
10. D61⋊#131C22semidirect
12. #16840831C22semidirect
18. #16840755C22semidirect
20. #16840753C22semidirect
21. #16840752C22semidirect
22. C2×#14519840C22semidirect
24. C2×#14519825C4semidirect
26. C2×#14519824C4semidirect
27. C2×#14519824C4semidirect
30. 2−1+4×D61C4semidirect
37. C2×2−1+4×C61C4semidirect
61. #14519825C2×C4semidirect
66. #14519824C2×C4semidirect
67. #14519824C2×C4semidirect
69. C2×#11758226C2×C4semidirect
75. C2×#11758222C2×C4semidirect
79. C2×#11758224C2×C4semidirect
86. #41×D61C2×C4semidirect
93. C22×Dic122C2×C4semidirect
102. C2×Q8×D61C2×C4semidirect
103. C2×Q8×D61C2×C4semidirect
114. 2−1+4×C61C2×C4semidirect
117. C2×#41×C61C2×C4semidirect
119. C22×Q8×C61C2×C4semidirect
348. C2×2−1+4#6923semidirect
370. C61#60088semidirect
382. 2−1+4C2×#6923semidirect
408. C2×#41C2×#6923semidirect
426. C22×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#5682
#214627992
#214624412
#214624392
C2×#168407552
C2×2−1+4×D612
#6008861