Finite Group Databasebeta

← #29646587

Subgroups — #29646587

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459235C211yesmaximal, characteristic
2#21454786C211yesmaximal, characteristic, complemented
3#21444235C212yesmaximal, complemented
4#21444230C212yesmaximal, complemented
5#21444225C212yesmaximal, complemented
6#21444217C212yesmaximal, complemented
7#21444209C211yesmaximal, characteristic, complemented
8#21444204C211yesmaximal, characteristic, complemented
9#21444199C211yesmaximal, characteristic, complemented
10C2×#16839489C211yesmaximal, characteristic, complemented
11C61×#2988C211yesmaximal, characteristic, complemented
12#16840390C411yescharacteristic
13#16840384—21no—
14#16840383C412yes—
15#16840367C412yes—
16#16840363—21no—
17#16840345—22no—
18#16840340C411yescharacteristic
19#16840331C411yescharacteristic
20#16840326C411yescharacteristic
21#16840312C2212yes—
22#16840309—21no—
23#16840299—21no—
24#16840296C2212yes—
25#16840271—22no—
26C4×#11758222C2211yescharacteristic
27C2×C4×D244C2211yescharacteristic
28#16839893—21no—
29#16839840—21no—
30#16839834—22no—
31#16839815C2211yescharacteristic
32#16839815C2212yescomplemented
33C4×#11758194C2212yescomplemented
34C2×C4×#92418C2211yescharacteristic
35#16839492C2211yescharacteristic, complemented
36#16839492C2211yescharacteristic, complemented
37#16839492C2212yescomplemented
38#16839489C2212yes—
39#16839489C2212yes—
40#16839489C2212yes—
41#16839489C2212yes—
42#16839489C2211yescharacteristic, complemented
43#16839489C2211yescharacteristic, complemented
44#16839489C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C8×Q8×C61—21no—
47C8×D4×C61—21no—
48C8×D4×C61—22no—
49C61×#431C2211yescharacteristic
50C61×#431C2212yescomplemented
51C2×C4⋊C8×C61C2211yescharacteristic
52C2×C4⋊C8×C61C2212yescomplemented
53C2×#11758222—22no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×C4×D61—21no—
57C2×#11758194—22no—
58C22×#92418—21no—
59#14519777C2×C412yes—
60#14519776—21no—
61#14519774C2×C412yes—
62#14519773—21no—
63#14519772—22no—
64#14519771—22no—
65#14519769C2×C412yes—
66#14519768—21no—
67#14519766—21no—
68#14519765—22no—
69#14519765C2×C412yes—
70#14519763—22no—
71#14519762—22no—
72#14519762—22no—
73#14519760—22no—
74#14519759—22no—
75#14519756C2×C412yes—
76#14519755—22no—
77#14519754—22no—
78#14519753—21no—
79#14519752—21no—
80#14519752—21no—
81#14519751—22no—
82#14519751C2×C412yes—
83#14519751C2×C412yes—
84#14519751C2×C412yes—
85#14519750—22no—
86#14519750—22no—
87#14519749—22no—
88#14519748—21no—
89#14519747—22no—
90#14519745—22no—
91C4×Dic122C2×C411yescharacteristic
92C4×Dic122C2311yescharacteristic
93C4×D244—21no—
94C4×D244—21no—
95C4×D244—22no—
96C4×D244C2×C412yes—
97C4×D244C2×C411yescharacteristic
98C4×D244C2312yes—
99C4×D244C2311yescharacteristic
100C4×C22⋊D61—21no—
101C4×C22⋊D61—21no—
102C4×C22⋊D61—22no—
103#14519736C2×C411yescharacteristic
104#14519734—21no—
105#14519734—22no—
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×#11758198—21no—
108C42×D61—21no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—21no—
112C22⋊#92418—22no—
113#14519688—22no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C2×#41×C61C2×C411yescharacteristic
119C22×C8×C61—21no—
120C22×C8×C61—22no—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61D412yes—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D412yes—
126C4×D4×C61D411yescharacteristic
127#117×C61C2×C411yescharacteristic
128#117×C61C2×C412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—22no—
133C4⋊C8×C61C2312yes—
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C22⋊C8×C61—22no—
138C4×C8×C61—21no—
139C4×C8×C61—22no—
140C23×D61—21no—
141#41×C61—24no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C2×D4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C22×Dic61—21no—
156#11758222—24no—
157#11758222—82no—
158#11758222—82no—
159C2×D244—24no—
160C2×D244—21no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244C22×C412yes—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×C4×D61—21no—
168C2×C4×D61—22no—
169C2×C4×D61—21no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C2×Dic122—22no—
173C2×Dic122C22×C411yescharacteristic
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—22no—
177C2×C8×C61—81no—
178C2×C8×C61—82no—
179C4⋊C4×C61C22⋊C411yescharacteristic
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22⋊C412yes—
182C4⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C22⋊C4×C61C22⋊C412yes—
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C412yes—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D412yes—
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D412yes—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—21no—
194C22⋊Dic61—22no—
195#11758198—22no—
196#11758198—21no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C411yescharacteristic
209#11758196—21no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217#11758194—44no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—22no—
221C2×#92418—42no—
222C23×C61C2×C22⋊C412yes—
223C23×C61C2×C22⋊C411yescharacteristic
224C22×D61—22no—
225C22×D61—21no—
226C22×D61—42no—
227C22×D61—81no—
228C22×D61—81no—
229Q8×C61—22no—
230D4×C61—24no—
231D4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61#13114yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—82no—
251C22⋊D61—82no—
252C22⋊D61—82no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—81no—
258D244—24no—
259D244—22no—
260D244—42no—
261D244—82no—
262D244—82no—
263C4×D61—42no—
264C4×D61—41no—
265C4×D61—42no—
266C4×D61—41no—
267C4×D61—42no—
268C4×D61—81no—
269C4×D61—82no—
270Dic122—22no—
271Dic122—82no—
272C8×C61—41no—
273C8×C61—42no—
274C8×C61—41no—
275#92418—22no—
276#92418—41no—
277#92418—42no—
278#17468—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61#40812yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—41no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—42no—
292C2×D61—84no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—22no—
299C4×C61#40812yes—
300C4×C61C2×#13112yes—
301C4×C61#57611yescharacteristic
302C4×C61#57711yescharacteristic
303Dic61—41no—
304Dic61—41no—
305Dic61—42no—
306Dic61—41no—
307#3460—611no—
308#2988D6111yescharacteristic, complemented
309#2965—611no—
310#1484—612no—
311#1483—612no—
312#1482—612no—
313#1480—612no—
314#1478—611no—
315#1477—611no—
316#1476—611no—
317C2×D4⋊C8—611no—
318C2×C61—22no—
319C2×C61—42no—
320C2×C61C2×#40811yescharacteristic
321C2×C61#289711yescharacteristic
322C2×C61#290111yescharacteristic
323D61—81no—
324D61—82no—
325#544—611no—
326#542—612no—
327#542—1221no—
328#540—612no—
329#540—1221no—
330#538—1222no—
331#537—611no—
332#535—611no—
333#535—611no—
334#532—612no—
335#532—1221no—
336#529—612no—
337#529—1221no—
338#525—1222no—
339C4×#41—611no—
340C4×#41C2×D6111yescharacteristic
341C2×C4×D4—611no—
342#446—1221no—
343C8×Q8—21no—
344#436—1221no—
345#435—1222no—
346C8×D4—21no—
347C8×D4—22no—
348#431—611no—
349#431—612no—
350#431C2×D6111yescharacteristic
351#431C2×D6112yescomplemented
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×C4⋊C8C2×D6112yescomplemented
354C4×M16—612no—
355C2×C4×C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—612no—
359D4⋊C8—612no—
360D4⋊C8—612no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—612no—
366C61#1746811yescharacteristic, complemented, Sylow
367C2×#41—611no—
368C2×#41—1222no—
369C2×#41C4×D6111yescharacteristic
370C22×D4—611no—
371C23×C4—1221no—
372C2×M16—1222no—
373C22×C8—21no—
374C22×C8—22no—
375C22×C8—1221no—
376C4⋊Q8—612no—
377C4⋊Q8—1221no—
378#127—612no—
379#127—1221no—
380#126—1222no—
381#126—1222no—
382#125—612no—
383#125—1221no—
384#124—612no—
385#124—1221no—
386#124—1222no—
387#124—1222no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—612no—
393C22⋊Q8—1221no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396#121—612no—
397#121—612no—
398#121—612no—
399#121—1222no—
400#121—1221no—
401#121—1222no—
402#121—1221no—
403#121—1221no—
404#121—1222no—
405#121—1222no—
406#120—1222no—
407#120—1222no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8D24411yescharacteristic
411C4×Q8C22⋊D6111yescharacteristic
412C4×D4—612no—
413C4×D4—611no—
414C4×D4—612no—
415C4×D4—611no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1222no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4D24412yes—
423C4×D4D24411yescharacteristic
424C4×D4C22⋊D6112yes—
425C4×D4C22⋊D6111yescharacteristic
426#117—611no—
427#117—1221no—
428#117—1222no—
429#117C4×D6111yescharacteristic
430#117C4×D6112yes—
431C2×C4⋊C4—611no—
432C2×C22⋊C4—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435C2×C42C22×D6112yes—
436C4⋊C8—21no—
437C4⋊C8—22no—
438C4⋊C8—1222no—
439C4⋊C8—1221no—
440C4⋊C8C22×D6112yes—
441C4⋊C8C22×D6111yescharacteristic
442C4⋊C8C22×D6111yescharacteristic
443C22⋊C8—21no—
444C22⋊C8—22no—
445C22⋊C8—1221no—
446C22⋊C8—1222no—
447#97—1222no—
448C4×C8—21no—
449C4×C8—22no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8—612no—
453C4×C8—1221no—
454C24—1221no—
455#41—24no—
456#41—1224no—
457#41—4882no—
458#41—4882no—
459C2×Q8—611no—
460C2×Q8—1222no—
461C2×Q8#1175819811yescharacteristic
462C2×D4—612no—
463C2×D4—611no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1221no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1221no—
471C2×D4—1224no—
472C2×D4#1175819812yes—
473C2×D4#1175819811yescharacteristic
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—4881no—
482C22×C4C2×C4×D6112yes—
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24412yes—
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6112yes—
488M16—2444no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—81no—
493C2×C8—82no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1222no—
497C2×C8—2442no—
498C4⋊C4—612no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4#1175819811yescharacteristic
509C4⋊C4#1175819812yes—
510C4⋊C4#1175819811yescharacteristic
511C4⋊C4#1175819812yes—
512C22⋊C4—1221no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1221no—
516C22⋊C4—1222no—
517C22⋊C4—1221no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4#1175819812yes—
524C22⋊C4#1175819812yes—
525C22⋊C4#1175819812yes—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42C2×C4×D6111yescharacteristic
530C42C2×C4×D6112yes—
531C42C2×D24411yescharacteristic
532C42C2×D24412yes—
533C42C2×C22⋊D6111yescharacteristic
534C42C2×C22⋊D6112yes—
535C42C23×D6111yescharacteristic
536C23—1222no—
537C23—1221no—
538C23—4881no—
539C23—4881no—
540C23—2442no—
541C23C2×#1175819812yes—
542C23C2×#1175819811yescharacteristic
543Q8—22no—
544Q8—1222no—
545Q8—4882no—
546D4—24no—
547D4—22no—
548D4—1224no—
549D4—1222no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—4882no—
555D4—2442no—
556D4—2442no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—4881no—
569C2×C4—4882no—
570C2×C4—4881no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2441no—
574C2×C4—2442no—
575C2×C4—2441no—
576C2×C4C2×#1175819811yescharacteristic
577C2×C4C2×#1175819811yescharacteristic
578C2×C4C2×#1175819812yes—
579C2×C4C2×#1175819812yes—
580C2×C4C2×#1175819812yes—
581C2×C4C2×#1175819811yescharacteristic
582C2×C4#1451978314yes—
583C2×C4C22×C4×D6111yescharacteristic
584C2×C4C22×C22⋊D6111yescharacteristic
585C2×C4C22×D24411yescharacteristic
586C8—41no—
587C8—42no—
588C8—41no—
589C8—1222no—
590C8—2441no—
591C8—2442no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—4884no—
598C22—2442no—
599C22—2441no—
600C22—2441no—
601C22—2442no—
602C22—2441no—
603C22—2442no—
604C22#1683972812yes—
605C22C22×#1175819811yescharacteristic
606C4—21no—
607C4—22no—
608C4—21no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—2442no—
613C4—2441no—
614C4—2441no—
615C4—2441no—
616C4#1683972812yes—
617C4C2×#1451978312yes—
618C4#1684056511yescharacteristic
619C4#1684057011yescharacteristic
620C2—22no—
621C2—42no—
622C2—4882no—
623C2—4881no—
624C2C2×#1683972811yescharacteristic
625C2#2145434211yescharacteristic
626C2#2145436411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C22×C22⋊C4
prime radical#21459235C2
cosocleC42×C61C24
hypercenter#2988D61
hypocenterC61#17468
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #21454786C2semidirect
3. #21444235C2semidirect
4. #21444230C2semidirect
5. #21444225C2semidirect
6. #21444217C2semidirect
7. #21444209C2semidirect
8. #21444204C2semidirect
9. #21444199C2semidirect
10. C2×#16839489C2semidirect
11. C61×#2988C2semidirect
32. #16839815C22semidirect
33. C4×#11758194C22semidirect
35. #16839492C22semidirect
36. #16839492C22semidirect
37. #16839492C22semidirect
42. #16839489C22semidirect
43. #16839489C22semidirect
44. #16839489C22semidirect
50. C61×#431C22semidirect
52. C2×C4⋊C8×C61C22semidirect
308. #2988D61semidirect
351. #431C2×D61semidirect
353. C2×C4⋊C8C2×D61semidirect
366. C61#17468semidirect

Maximal subgroups

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

subgroupindex
#214592352
#214547862
#214442352
#214442302
#214442252
#214442172
#214442092
#214442042
#214441992
C2×#168394892
C61×#29882
#1746861