Finite Group Databasebeta

← #30571033

Subgroups — #30571033

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic, complemented
2#21458811C214yesmaximal, complemented
3#21458803C214yesmaximal, complemented
4#21458504C214yesmaximal, complemented
5#21458468C214yesmaximal, complemented
6#21456780C214yesmaximal, complemented
7#21456764C211yesmaximal, characteristic, complemented
8#21456741C211yesmaximal, characteristic, complemented
9#21455895C214yesmaximal, complemented
10#21455823C211yesmaximal, characteristic, complemented
11#21455815C211yesmaximal, characteristic
12#21454900C211yesmaximal, characteristic, complemented
13C61×#3507C211yesmaximal, characteristic, complemented
14#16840485—22no—
15#16840485—22no—
16#16840485C2214yes—
17#16840485C2214yes—
18C2×#14519777C2211yescharacteristic
19C2×#14519772—22no—
20C2×#14519765C2212yes—
21C2×#14519753—21no—
22C2×C4×Dic122C2212yes—
23C2×C4×D244C2211yescharacteristic
24C2×#14519735—21no—
25#16840144C2218yes—
26#16840141C2212yescomplemented
27#16840134C2212yescomplemented
28#16840119C2218yes—
29#16840114C2212yescomplemented
30#16840114C2212yescomplemented
31#16840090C2212yes—
32#16840073C2212yes—
33#16840066C2218yescomplemented
34#16840049C2212yes—
35#16840039C2212yes—
36#16839984C2218yes—
37#16839981C2218yescomplemented
38#16839977C2218yes—
39#16839972C2214yescomplemented
40#16839967C2214yescomplemented
41#16839964C2212yes—
42#16839954C2212yes—
43#16839917C2214yes—
44#16839912C2212yes—
45#16839900C2214yes—
46#16839895C2212yes—
47Q8×#92418C2214yescomplemented
48Q8×#92418C2214yescomplemented
49C4×#11758202C2218yescomplemented
50C4×#11758201C2214yes—
51C4×#11758200C2218yescomplemented
52C4×#11758199C2214yes—
53#16839815C2214yes—
54#16839815C2211yescharacteristic
55#16839815C2211yescharacteristic
56C61×#559C2212yescomplemented
57C61×#557C2212yescomplemented
58C61×#556—22no—
59C61×#556C2212yescomplemented
60C61×#554C2212yescomplemented
61C61×#552—22no—
62C61×#539—22no—
63C61×#536—21no—
64C61×#533C2211yescharacteristic
65C61×#532C2211yescharacteristic
66C61×#529C2214yes—
67C61×#520—21no—
68C2×C4×Q8×C61C2211yescharacteristic
69C22×Dic122C2312yes—
70C22×D244C2311yescharacteristic
71C22×C4×D61—21no—
72C2×#11758202C2318yes—
73C2×#11758200C2318yes—
74C2×#11758201C2314yes—
75C2×#11758199C2314yes—
76#14519777—21no—
77#14519777—21no—
78#14519777C2312yes—
79#14519777C2312yes—
80#14519772—28no—
81#14519772—28no—
82#14519765—24no—
83#14519765C2318yes—
84#14519753—24no—
85#14519753—24no—
86C4×Dic122—24no—
87C4×Dic122C2318yes—
88C4×D244—21no—
89C4×D244—21no—
90C4×D244C2312yes—
91C4×D244C2312yes—
92#14519735—24no—
93#14519735—24no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—22no—
98C2×#11758198—22no—
99C2×#11758198—21no—
100C2×#11758198—22no—
101C2×C4×Dic61—22no—
102#14519710C2314yes—
103#14519708C2314yes—
104C4⋊#92418C2312yes—
105C4⋊#92418C2312yes—
106C4⋊#92418C2318yes—
107#14519701C2314yes—
108#14519700C2318yes—
109#14519700C2318yes—
110#14519698C2314yes—
111#14519697C2314yes—
112#14519697C2314yes—
113C22⋊#92418C2312yes—
114C22⋊#92418C2312yes—
115C4×#92418C2312yes—
116C4×#92418C2318yes—
117C4×#92418C2312yes—
118C2×#41×C61—21no—
119C22×Q8×C61D411yescharacteristic
120C4⋊Q8×C61—22no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#126×C61—24no—
126#126×C61—24no—
127#125×C61—24no—
128#125×C61—21no—
129#125×C61C2314yes—
130#124×C61—22no—
131#124×C61—21no—
132#124×C61C2314yes—
133#123×C61—24no—
134#123×C61—24no—
135C22⋊Q8×C61—22no—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—21no—
138C22⋊Q8×C61—24no—
139C22⋊Q8×C61C2312yes—
140#121×C61—21no—
141#121×C61C2312yes—
142C4×Q8×C61—22no—
143C4×Q8×C61—21no—
144C4×Q8×C61D412yes—
145C4×Q8×C61D412yes—
146C4×Q8×C61C2312yes—
147C4×Q8×C61C2314yes—
148C4×Q8×C61C2312yes—
149C4×Q8×C61C2312yes—
150C4×D4×C61—24no—
151C4×D4×C61—22no—
152C4×D4×C61—21no—
153C4×D4×C61C2312yes—
154C4×D4×C61C2314yes—
155#117×C61—21no—
156#117×C61—22no—
157#117×C61—24no—
158C2×C4⋊C4×C61—22no—
159C2×C4⋊C4×C61D412yes—
160C2×C4⋊C4×C61D411yescharacteristic
161C2×C42×C61C2311yescharacteristic
162C2×C42×C61C2312yes—
163C23×D61—21no—
164#41×C61—44no—
165C2×Q8×C61—24no—
166C2×Q8×C61—21no—
167C2×Q8×C61—21no—
168C2×Q8×C61C2×D412yes—
169C2×Q8×C61C2×D412yes—
170C2×Q8×C61C2412yes—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C2412yes—
175C22×C4×C61—22no—
176C22×C4×C61—24no—
177C22×C4×C61C2×D412yes—
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2411yescharacteristic
182C22×Dic61—21no—
183C22×Dic61—22no—
184C2×D244—21no—
185C2×D244—42no—
186C2×D244C2412yes—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—44no—
190C2×Dic122—22no—
191C2×Dic122—44no—
192C2×Dic122C2414yes—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—24no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—24no—
202C4⋊C4×C61C2×D412yes—
203C4⋊C4×C61C2×D412yes—
204C4⋊C4×C61C2×D414yes—
205C4⋊C4×C61C2×D414yes—
206C4⋊C4×C61C2412yes—
207C4⋊C4×C61C2412yes—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—24no—
211C22⋊C4×C61—24no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—24no—
214C42×C61—24no—
215C42×C61—22no—
216C42×C61C2×D414yes—
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219C42×C61C2411yescharacteristic
220C42×C61C2414yes—
221C42×C61C2411yescharacteristic
222#11758202—48no—
223#11758201—44no—
224#11758200—48no—
225#11758199—44no—
226#11758198—28no—
227#11758198—22no—
228#11758198—22no—
229#11758198—28no—
230C4⋊Dic61—22no—
231C4⋊Dic61—22no—
232C4⋊Dic61—22no—
233C4⋊Dic61—22no—
234C4⋊Dic61—21no—
235C4⋊Dic61C2412yes—
236#11758196—28no—
237C4×Dic61—28no—
238C2×#92418C2414yes—
239C2×#92418C2414yes—
240C23×C61—22no—
241C23×C61C22×D411yescharacteristic
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246Q8×C61—24no—
247Q8×C61—24no—
248Q8×C61—24no—
249Q8×C61—42no—
250Q8×C61—81no—
251D4×C61—24no—
252D4×C61—44no—
253D4×C61—42no—
254D4×C61—81no—
255C2×C4×C61—24no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—42no—
261C2×C4×C61—41no—
262C2×C4×C61—44no—
263C2×C4×C61—84no—
264C2×C4×C61#13514yes—
265C2×C4×C61C22×D414yes—
266C2×C4×C61C22×D414yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2511yescharacteristic
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—44no—
276C2×Dic61—42no—
277D244—22no—
278D244—22no—
279D244—42no—
280C4×D61—44no—
281C4×D61—44no—
282Dic122—28no—
283Dic122—44no—
284#92418—24no—
285#92418—24no—
286#61882—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61C2×#13512yes—
293C22×C61C23×D411yescharacteristic
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—84no—
298C4×C61—24no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—24no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61C2×#13512yes—
307C4×C61C2×#13512yes—
308C4×C61#58512yes—
309Dic61—44no—
310Dic61—42no—
311#3507D6111yescharacteristic, complemented
312C2×#556—611no—
313#3394—614no—
314#3393—614no—
315#3364—614no—
316#3360—614no—
317#3211—614no—
318#3209—611no—
319#3206—611no—
320#3108—614no—
321#3101—611no—
322#3100—611no—
323#2982—611no—
324C2×C61—22no—
325C2×C61—82no—
326C2×C61#348711yescharacteristic
327C2×C61C22×#13511yescharacteristic
328C2×C61#358611yescharacteristic
329D61—82no—
330#559C2×D6112yescomplemented
331#557C2×D6112yescomplemented
332#556—22no—
333#556—614no—
334#556—614no—
335#556—1222no—
336#556—1222no—
337#556C2×D6112yescomplemented
338#554C2×D6112yescomplemented
339#552—22no—
340#539—22no—
341#536—21no—
342#533C2×D6111yescharacteristic
343#532C2×D6111yescharacteristic
344C2×C4⋊Q8—611no—
345#529C2×D6114yes—
346C2×#126—1222no—
347C2×#124—612no—
348C2×C22⋊Q8—1221no—
349#520—21no—
350C2×C4×Q8—612no—
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×D4—611no—
353C2×#117—1221no—
354#495—618no—
355#494—612no—
356#492—612no—
357#488—618no—
358#487—612no—
359#487—612no—
360#483—612no—
361#480—612no—
362#479—618no—
363#476—612no—
364#474—612no—
365#464—618no—
366#464—618no—
367#463—618no—
368#462—614no—
369#461—612no—
370#461—614no—
371#459—612no—
372#451—614no—
373#450—612no—
374#448—614no—
375#447—612no—
376C8×Q8—614no—
377C8×Q8—614no—
378C4×Q16—618no—
379C4×SD16—614no—
380C4×SD16—618no—
381C4×D8—614no—
382#431—611no—
383#431—611no—
384#431—614no—
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C22×Q8—612no—
388C22×Q8C22⋊D6111yescharacteristic
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16—618no—
392C2×SD16—614no—
393C2×SD16—618no—
394C2×D8—614no—
395C4⋊Q8—22no—
396C4⋊Q8—612no—
397C4⋊Q8—612no—
398C4⋊Q8—1221no—
399C4⋊Q8—1221no—
400C4⋊Q8C22×D6111yescharacteristic
401C4⋊Q8C22×D6111yescharacteristic
402C4⋊Q8C22×D6111yescharacteristic
403#127C22×D6111yescharacteristic
404#126—24no—
405#126—24no—
406#126—1228no—
407#126—1228no—
408#125—24no—
409#125—21no—
410#125C22×D6114yes—
411#124—22no—
412#124—21no—
413#124—618no—
414#124—1224no—
415#124C22×D6114yes—
416#123—24no—
417#123—24no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—24no—
422C22⋊Q8—1224no—
423C22⋊Q8—1224no—
424C22⋊Q8C22×D6112yes—
425#121—21no—
426#121C22×D6112yes—
427C4×Q8—22no—
428C4×Q8—21no—
429C4×Q8—618no—
430C4×Q8—1224no—
431C4×Q8C22⋊D6112yes—
432C4×Q8C22⋊D6112yes—
433C4×Q8C22×D6114yes—
434C4×Q8C22×D6112yes—
435C4×Q8C22×D6112yes—
436C4×Q8C22×D6112yes—
437C4×D4—24no—
438C4×D4—22no—
439C4×D4—21no—
440C4×D4—612no—
441C4×D4—612no—
442C4×D4—1221no—
443C4×D4—1221no—
444C4×D4C22×D6114yes—
445C4×D4C22×D6112yes—
446#117—21no—
447#117—22no—
448#117—24no—
449#117—1224no—
450#117—1224no—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1222no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4C22⋊D6111yescharacteristic
457C2×C4⋊C4C22⋊D6112yes—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1222no—
461C2×C42—1222no—
462C2×C42C22×D6111yescharacteristic
463C2×C42C22×D6112yes—
464#107—614no—
465#106—614no—
466C4⋊C8—612no—
467C4⋊C8—612no—
468C4⋊C8—618no—
469#103—614no—
470#103—618no—
471#103—618no—
472#102—614no—
473#102—614no—
474#102—614no—
475C22⋊C8—612no—
476C22⋊C8—612no—
477C4×C8—612no—
478C4×C8—612no—
479C4×C8—618no—
480C24—1221no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—24no—
485C2×Q8—614no—
486C2×Q8—1222no—
487C2×Q8—2444no—
488C2×Q8C2×C22⋊D6112yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×Q8C23×D6112yes—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—612no—
495C2×D4—1221no—
496C2×D4—2442no—
497C2×D4C23×D6112yes—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—2444no—
505C22×C4C2×C22⋊D6112yes—
506C22×C4C2×C22⋊D6112yes—
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2448no—
511SD16—2448no—
512SD16—2444no—
513D8—2444no—
514C2×C8—614no—
515C2×C8—614no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—612no—
526C4⋊C4—1221no—
527C4⋊C4—1228no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4C2×C22⋊D6112yes—
533C4⋊C4C2×C22⋊D6112yes—
534C4⋊C4C2×C22⋊D6114yes—
535C4⋊C4C2×C22⋊D6114yes—
536C4⋊C4C23×D6112yes—
537C4⋊C4C23×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—22no—
543C22⋊C4—24no—
544C22⋊C4—1228no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1228no—
548C42—24no—
549C42—22no—
550C42—1228no—
551C42C2×C22⋊D6114yes—
552C42C2×C22⋊D6111yescharacteristic
553C42C2×C22⋊D6111yescharacteristic
554C42C23×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C42C23×D6114yes—
557C23—22no—
558C23—1222no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23C22×C22⋊D6111yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—24no—
566Q8—42no—
567Q8—81no—
568Q8—1228no—
569Q8—2444no—
570D4—24no—
571D4—44no—
572D4—42no—
573D4—81no—
574D4—1222no—
575D4—1222no—
576D4—2442no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—84no—
586C2×C4—1222no—
587C2×C4—1224no—
588C2×C4—2444no—
589C2×C4—2442no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4#1451979714yes—
593C2×C4C22×C22⋊D6114yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C22×C22⋊D6114yes—
596C2×C4C22×C22⋊D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C22×C22⋊D6111yescharacteristic
599C2×C4C22×C22⋊D6111yescharacteristic
600C2×C4C24×D6111yescharacteristic
601C8—1224no—
602C8—1224no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—4884no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451979712yes—
613C22C23×C22⋊D6111yescharacteristic
614C4—21no—
615C4—24no—
616C4—24no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—2444no—
623C4—2442no—
624C4C2×#1451979712yes—
625C4C2×#1451979712yes—
626C4#1684059912yes—
627C2—22no—
628C2—82no—
629C2—4882no—
630C2#2145947411yescharacteristic
631C2C22×#1451979711yescharacteristic
632C2#2146060511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3507C2
socleC22×C61C23×D4
prime radical#21455815C2
hypercenter#3507D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. C2×#16840485C2semidirect
2. #21458811C2semidirect
3. #21458803C2semidirect
4. #21458504C2semidirect
5. #21458468C2semidirect
6. #21456780C2semidirect
7. #21456764C2semidirect
8. #21456741C2semidirect
9. #21455895C2semidirect
10. #21455823C2semidirect
12. #21454900C2semidirect
13. C61×#3507C2semidirect
26. #16840141C22semidirect
27. #16840134C22semidirect
29. #16840114C22semidirect
30. #16840114C22semidirect
33. #16840066C22semidirect
37. #16839981C22semidirect
39. #16839972C22semidirect
40. #16839967C22semidirect
47. Q8×#92418C22semidirect
48. Q8×#92418C22semidirect
49. C4×#11758202C22semidirect
51. C4×#11758200C22semidirect
56. C61×#559C22semidirect
57. C61×#557C22semidirect
59. C61×#556C22semidirect
60. C61×#554C22semidirect
311. #3507D61semidirect
330. #559C2×D61semidirect
331. #557C2×D61semidirect
337. #556C2×D61semidirect
338. #554C2×D61semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214588112
#214588032
#214585042
#214584682
#214567802
#214567642
#214567412
#214558952
#214558232
#214558152
#214549002
C61×#35072
#6188261