Finite Group Databasebeta

← #30571038

Subgroups — #30571038

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
1#21459684C211yesmaximal, characteristic, complemented
2#21458815C214yesmaximal, complemented
3#21458803C214yesmaximal, complemented
4#21458500C214yesmaximal, complemented
5#21458464C214yesmaximal, complemented
6#21456784C214yesmaximal, complemented
7#21456767C211yesmaximal, characteristic, complemented
8#21456744C211yesmaximal, characteristic, complemented
9#21455899C214yesmaximal, complemented
10#21455827C211yesmaximal, characteristic, complemented
11#21455819C211yesmaximal, characteristic
12#21454900C211yesmaximal, characteristic, complemented
13C2×C61×#556C211yesmaximal, characteristic, complemented
14#16840497C2212yescomplemented
15#16840490C2212yescomplemented
16#16840485C2212yescomplemented
17#16840484—22no—
18#16840472C2212yescomplemented
19#16840453—22no—
20#16840359—22no—
21#16840336—21no—
22#16840319C2211yescharacteristic
23#16840312C2211yescharacteristic
24#16840296C2214yes—
25#16840243—21no—
26#16840144C2218yes—
27#16840141C2212yescomplemented
28#16840136C2212yescomplemented
29#16840116C2218yes—
30#16840112C2212yescomplemented
31#16840112C2212yescomplemented
32#16840092C2212yes—
33#16840075C2212yes—
34#16840069C2218yescomplemented
35#16840052C2212yes—
36#16840042C2212yes—
37#16839984C2218yescomplemented
38#16839981C2218yes—
39#16839974C2218yes—
40#16839969C2214yescomplemented
41#16839967C2212yes—
42#16839964C2214yescomplemented
43#16839957C2212yes—
44#16839918C2214yes—
45#16839913C2212yes—
46#16839901C2214yes—
47#16839896C2212yes—
48Q8×#92418C2214yescomplemented
49Q8×#92418C2214yescomplemented
50C4×#11758202C2218yescomplemented
51C4×#11758201C2218yescomplemented
52C4×#11758200C2214yes—
53C4×#11758199C2214yes—
54#16839815C2214yes—
55#16839815C2211yescharacteristic
56#16839815C2211yescharacteristic
57C61×#556—22no—
58C61×#556—22no—
59C61×#556C2214yes—
60C61×#556C2214yes—
61C2×C4⋊Q8×C61C2211yescharacteristic
62C2×#126×C61—22no—
63C2×#124×C61C2212yes—
64C2×C22⋊Q8×C61—21no—
65C2×C4×Q8×C61C2212yes—
66C2×C4×Q8×C61C2211yescharacteristic
67C2×C4×D4×C61C2211yescharacteristic
68C2×#117×C61—21no—
69C2×#11758222—21no—
70C2×#11758202C2318yes—
71C2×#11758200C2314yes—
72C2×#11758201C2318yes—
73C2×#11758199C2314yes—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519774C2311yescharacteristic
79#14519772—24no—
80#14519771—24no—
81#14519769—21no—
82#14519769C2314yes—
83#14519767—24no—
84#14519766—22no—
85#14519765—21no—
86#14519765C2314yes—
87#14519762—24no—
88#14519760—24no—
89#14519756—21no—
90#14519756C2312yes—
91#14519755—24no—
92#14519753—22no—
93#14519753—22no—
94#14519751—21no—
95#14519751C2312yes—
96C4×Dic122—21no—
97C4×Dic122C2312yes—
98C4×Dic122C2314yes—
99#14519742—22no—
100C4×D244—21no—
101C4×D244C2312yes—
102C4×D244C2314yes—
103#14519739—22no—
104C4×C22⋊D61—24no—
105#14519736—21no—
106#14519735—22no—
107#14519734—24no—
108C4⋊C4×D61—22no—
109#14519710C2314yes—
110#14519708C2314yes—
111C4⋊#92418C2312yes—
112C4⋊#92418C2312yes—
113C4⋊#92418C2318yes—
114#14519701C2318yes—
115#14519701C2318yes—
116#14519700C2314yes—
117#14519698C2314yes—
118#14519698C2314yes—
119#14519697C2314yes—
120C22⋊#92418C2312yes—
121C22⋊#92418C2312yes—
122C4×#92418C2312yes—
123C4×#92418C2312yes—
124C4×#92418C2318yes—
125C22×Q8×C61D411yescharacteristic
126C22×Q8×C61C2312yes—
127C22×D4×C61C2311yescharacteristic
128C23×C4×C61—21no—
129C4⋊Q8×C61—21no—
130C4⋊Q8×C61—21no—
131C4⋊Q8×C61C2312yes—
132C4⋊Q8×C61C2312yes—
133#126×C61—28no—
134#126×C61—28no—
135#124×C61—24no—
136#124×C61C2318yes—
137C22⋊Q8×C61—24no—
138C22⋊Q8×C61—24no—
139C4×Q8×C61—24no—
140C4×Q8×C61D412yes—
141C4×Q8×C61D412yes—
142C4×Q8×C61C2318yes—
143C4×Q8×C61C2312yes—
144C4×Q8×C61C2312yes—
145C4×D4×C61—21no—
146C4×D4×C61—21no—
147C4×D4×C61C2312yes—
148C4×D4×C61C2312yes—
149#117×C61—24no—
150#117×C61—24no—
151C2×C4⋊C4×C61—21no—
152C2×C4⋊C4×C61—21no—
153C2×C4⋊C4×C61—22no—
154C2×C4⋊C4×C61D411yescharacteristic
155C2×C4⋊C4×C61D412yes—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C22⋊C4×C61—22no—
158C2×C22⋊C4×C61—21no—
159C2×C22⋊C4×C61—22no—
160C2×C42×C61—22no—
161C2×C42×C61C2311yescharacteristic
162C2×C42×C61C2312yes—
163C24×C61—21no—
164C2×Q8×C61—24no—
165C2×Q8×C61—22no—
166C2×Q8×C61—44no—
167C2×Q8×C61C2×D412yes—
168C2×Q8×C61C2×D412yes—
169C2×Q8×C61C2414yes—
170C2×D4×C61—21no—
171C2×D4×C61—42no—
172C2×D4×C61C2412yes—
173C22×C4×C61—21no—
174C22×C4×C61—22no—
175C22×C4×C61—22no—
176C22×C4×C61—22no—
177C22×C4×C61—44no—
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2411yescharacteristic
183C2×C22⋊D61—22no—
184#11758222—44no—
185C2×D244—21no—
186C2×D244—21no—
187C2×D244C2412yes—
188C2×C4×D61—22no—
189C2×C4×D61—24no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122C2412yes—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—28no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C2×D412yes—
200C4⋊C4×C61C2×D412yes—
201C4⋊C4×C61C2×D414yes—
202C4⋊C4×C61C2×D414yes—
203C4⋊C4×C61C2412yes—
204C22⋊C4×C61—28no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—28no—
208C42×C61—28no—
209C42×C61C2×D411yescharacteristic
210C42×C61C2×D411yescharacteristic
211C42×C61C2×D414yes—
212C42×C61C2411yescharacteristic
213C42×C61C2411yescharacteristic
214C42×C61C2414yes—
215C22⋊Dic61—22no—
216#11758202—48no—
217#11758201—48no—
218#11758200—44no—
219#11758199—44no—
220#11758198—22no—
221#11758198—24no—
222#11758198—24no—
223#11758198—22no—
224#11758198—24no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61—21no—
228C4⋊Dic61—21no—
229C4⋊Dic61C2412yes—
230C4⋊Dic61C2412yes—
231#11758196—24no—
232#11758196—22no—
233#11758196—24no—
234#11758196—24no—
235#11758196—22no—
236C4×Dic61—24no—
237C4×Dic61—22no—
238C2×#92418C2414yes—
239C2×#92418C2414yes—
240C23×C61—22no—
241C23×C61—42no—
242C23×C61—42no—
243C23×C61—42no—
244C23×C61C22×D411yescharacteristic
245C22×D61—22no—
246Q8×C61—24no—
247Q8×C61—24no—
248Q8×C61—28no—
249Q8×C61—44no—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—42no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—44no—
260C2×C4×C61—42no—
261C2×C4×C61—44no—
262C2×C4×C61—44no—
263C2×C4×C61#13514yes—
264C2×C4×C61C22×D414yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D414yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2511yescharacteristic
272C22⋊D61—44no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276D244—24no—
277D244—42no—
278D244—81no—
279C4×D61—44no—
280C4×D61—84no—
281Dic122—24no—
282Dic122—42no—
283Dic122—81no—
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×C61—84no—
293C22×C61C2×#13512yes—
294C22×C61C23×D411yescharacteristic
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C4×C61—21no—
299C4×C61—24no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—24no—
303C4×C61—44no—
304C4×C61—42no—
305C4×C61C2×#13512yes—
306C4×C61C2×#13512yes—
307C4×C61#58512yes—
308Dic61—42no—
309Dic61—42no—
310Dic61—42no—
311#3507—611no—
312C2×#556D6111yescharacteristic, complemented
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#559—612no—
331#557—612no—
332#556—22no—
333#556—22no—
334#556—612no—
335#556—1222no—
336#556C2×D6114yes—
337#556C2×D6114yes—
338#554—612no—
339#552—1222no—
340#539—1222no—
341#536—1221no—
342#533—611no—
343#532—611no—
344C2×C4⋊Q8C2×D6111yescharacteristic
345#529—614no—
346C2×#126—22no—
347C2×#124C2×D6112yes—
348C2×C22⋊Q8—21no—
349#520—1221no—
350C2×C4×Q8C2×D6112yes—
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×D4C2×D6111yescharacteristic
353C2×#117—21no—
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—614no—
370#461—612no—
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—618no—
380C4×SD16—614no—
381C4×D8—614no—
382#431—611no—
383#431—611no—
384#431—614no—
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8C22⋊D6111yescharacteristic
388C22×Q8C22×D6112yes—
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C2×Q16—618no—
392C2×SD16—618no—
393C2×SD16—614no—
394C2×D8—614no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—611no—
400C4⋊Q8—1222no—
401C4⋊Q8C22×D6112yes—
402C4⋊Q8C22×D6112yes—
403#127—611no—
404#126—28no—
405#126—28no—
406#126—1224no—
407#126—1224no—
408#125—614no—
409#125—1224no—
410#125—1221no—
411#124—24no—
412#124—614no—
413#124—1222no—
414#124—1221no—
415#124C22×D6118yes—
416#123—1224no—
417#123—1224no—
418C22⋊Q8—24no—
419C22⋊Q8—24no—
420C22⋊Q8—612no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1221no—
424C22⋊Q8—1224no—
425#121—612no—
426#121—1221no—
427C4×Q8—24no—
428C4×Q8—614no—
429C4×Q8—612no—
430C4×Q8—1222no—
431C4×Q8—1221no—
432C4×Q8C22⋊D6112yes—
433C4×Q8C22⋊D6112yes—
434C4×Q8C22×D6118yes—
435C4×Q8C22×D6112yes—
436C4×Q8C22×D6112yes—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—614no—
440C4×D4—612no—
441C4×D4—1224no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4C22×D6112yes—
445C4×D4C22×D6112yes—
446#117—24no—
447#117—24no—
448#117—1221no—
449#117—1222no—
450#117—1224no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—22no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—1222no—
455C2×C4⋊C4C22⋊D6111yescharacteristic
456C2×C4⋊C4C22⋊D6112yes—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C22⋊C4—22no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—22no—
461C2×C42—22no—
462C2×C42C22×D6111yescharacteristic
463C2×C42C22×D6112yes—
464#107—614no—
465#106—614no—
466C4⋊C8—612no—
467C4⋊C8—612no—
468C4⋊C8—618no—
469#103—618no—
470#103—618no—
471#103—614no—
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—21no—
481#41—2444no—
482C2×Q8—22no—
483C2×Q8—24no—
484C2×Q8—44no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8C2×C22⋊D6112yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×Q8C23×D6114yes—
491C2×D4—21no—
492C2×D4—42no—
493C2×D4—612no—
494C2×D4—1222no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4C23×D6112yes—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—44no—
503C22×C4—1224no—
504C22×C4—1222no—
505C22×C4C2×C22⋊D6112yes—
506C22×C4C2×C22⋊D6112yes—
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2448no—
511SD16—2444no—
512SD16—2448no—
513D8—2444no—
514C2×C8—614no—
515C2×C8—614no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1221no—
525C4⋊C4—1224no—
526C4⋊C4—1222no—
527C4⋊C4—1224no—
528C4⋊C4—1222no—
529C4⋊C4—1224no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4C2×C22⋊D6112yes—
534C4⋊C4C2×C22⋊D6112yes—
535C4⋊C4C2×C22⋊D6114yes—
536C4⋊C4C2×C22⋊D6114yes—
537C4⋊C4C23×D6112yes—
538C22⋊C4—28no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—28no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—1222no—
547C22⋊C4—1224no—
548C42—28no—
549C42—1224no—
550C42—1222no—
551C42C2×C22⋊D6114yes—
552C42C2×C22⋊D6111yescharacteristic
553C42C2×C22⋊D6111yescharacteristic
554C42C23×D6111yescharacteristic
555C42C23×D6114yes—
556C42C23×D6111yescharacteristic
557C23—22no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—1222no—
562C23C22×C22⋊D6111yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—28no—
566Q8—44no—
567Q8—1224no—
568Q8—4881no—
569Q8—2442no—
570D4—22no—
571D4—22no—
572D4—42no—
573D4—1224no—
574D4—4881no—
575D4—2444no—
576D4—2442no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
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—84no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451979712yes—
613C22C23×C22⋊D6111yescharacteristic
614C4—21no—
615C4—24no—
616C4—24no—
617C4—21no—
618C4—21no—
619C4—44no—
620C4—42no—
621C4—2442no—
622C4—2442no—
623C4—2442no—
624C4C2×#1451979712yes—
625C4C2×#1451979712yes—
626C4#1684059812yes—
627C2—22no—
628C2—82no—
629C2—4882no—
630C2#2145947711yescharacteristic
631C2C22×#1451979711yescharacteristic
632C2#2146060611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#556C2
socleC22×C61C23×D4
prime radical#21455819C2
hypercenterC2×#556D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #21459684C2semidirect
2. #21458815C2semidirect
3. #21458803C2semidirect
4. #21458500C2semidirect
5. #21458464C2semidirect
6. #21456784C2semidirect
7. #21456767C2semidirect
8. #21456744C2semidirect
9. #21455899C2semidirect
10. #21455827C2semidirect
12. #21454900C2semidirect
13. C2×C61×#556C2semidirect
14. #16840497C22semidirect
15. #16840490C22semidirect
16. #16840485C22semidirect
18. #16840472C22semidirect
27. #16840141C22semidirect
28. #16840136C22semidirect
30. #16840112C22semidirect
31. #16840112C22semidirect
34. #16840069C22semidirect
37. #16839984C22semidirect
40. #16839969C22semidirect
42. #16839964C22semidirect
48. Q8×#92418C22semidirect
49. Q8×#92418C22semidirect
50. C4×#11758202C22semidirect
51. C4×#11758201C22semidirect
312. C2×#556D61semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214588152
#214588032
#214585002
#214584642
#214567842
#214567672
#214567442
#214558992
#214558272
#214558192
#214549002
C2×C61×#5562
#6188261