Finite Group Databasebeta

← #29646902

Subgroups — #29646902

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459412C211yesmaximal, characteristic
2#21454485C211yesmaximal, characteristic, complemented
3#21444280C212yesmaximal, complemented
4#21444271C211yesmaximal, characteristic, complemented
5#21444254C212yesmaximal, complemented
6#21444249C211yesmaximal, characteristic, complemented
7#21444228C212yesmaximal, complemented
8#21444219C211yesmaximal, characteristic, complemented
9#21444202C212yesmaximal, complemented
10C2×#16839494C211yesmaximal, characteristic, complemented
11C8×#41×C61C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840495—21no—
14#16840490C411yescharacteristic
15#16840484—21no—
16#16840482C411yescharacteristic
17#16840461—22no—
18#16840458C412yes—
19#16840433C411yescharacteristic
20#16840421C412yes—
21#16840312C2211yescharacteristic
22#16840308—21no—
23#16840296C2212yes—
24#16840290—22no—
25C4×#11758222C2212yes—
26#16840230—21no—
27C2×C4×Dic122C2211yescharacteristic
28#16839823—22no—
29#16839819C2212yes—
30#16839815C2212yes—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839729—21no—
34#16839718—21no—
35#16839494C2211yescharacteristic
36#16839494C2211yescharacteristic
37#16839494C2211yescharacteristic
38#16839494C2211yescharacteristic
39#16839494C2212yes—
40#16839494C2212yes—
41#16839491C2212yes—
42#16839491C2212yes—
43#16839491C2212yes—
44#16839491C2212yes—
45C4×#41×C61C2211yescharacteristic
46C8×Q8×C61—21no—
47C8×D4×C61—22no—
48C8×D4×C61—21no—
49C61×#431C2211yescharacteristic
50C61×#431C2212yes—
51C2×C4×C8×C61C2211yescharacteristic
52C2×C4×C8×C61C2212yes—
53C2×#11758222—21no—
54C2×#11758222C2×C412yes—
55C22×Dic122C2×C411yescharacteristic
56C2×#11758194—21no—
57C22×#92418—21no—
58#14519777C2×C411yescharacteristic
59#14519775—21no—
60#14519774C2×C411yescharacteristic
61#14519773—21no—
62#14519772—22no—
63#14519771—22no—
64#14519770—22no—
65#14519769C2×C412yes—
66#14519767—22no—
67#14519766—21no—
68#14519765—21no—
69#14519765C2×C412yes—
70#14519764—21no—
71#14519763—22no—
72#14519762—22no—
73#14519760—22no—
74#14519759—22no—
75#14519756C2×C412yes—
76#14519755—22no—
77#14519751—21no—
78#14519751C2×C412yes—
79#14519751C2×C412yes—
80#14519750—22no—
81#14519749—22no—
82#14519749—22no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C412yes—
88C4×Dic122C2311yescharacteristic
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519742—21no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94C4×D244C2312yes—
95#14519739—21no—
96#14519739—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519737—22no—
102#14519736C2×C412yes—
103#14519734—22no—
104#14519734—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×#11758196—21no—
107C2×C4×Dic61—21no—
108C42×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2312yes—
114C22⋊#92418—22no—
115C22⋊#92418—22no—
116#14519688—22no—
117C4×#92418—22no—
118C2×#41×C61D411yescharacteristic
119C22×C8×C61—21no—
120C22×C8×C61—22no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C412yes—
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61D412yes—
126C4×D4×C61D411yescharacteristic
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—21no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61—22no—
135C4×C8×C61—22no—
136C4×C8×C61—21no—
137C4×C8×C61C2311yescharacteristic
138C4×C8×C61C2312yes—
139C4×C8×C61C2311yescharacteristic
140#41×C61—22no—
141#41×C61—22no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C2×D4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156#11758222—24no—
157#11758222—42no—
158#11758222—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—41no—
170C2×Dic122C22×C411yescharacteristic
171C2×Dic122C22×C411yescharacteristic
172C2×C8×C61—21no—
173C2×C8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61—44no—
176C2×C8×C61—42no—
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C4⋊C4×C61C22⋊C411yescharacteristic
180C4⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C42×C61C22×C412yes—
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D412yes—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D412yes—
189C42×C61C2×D411yescharacteristic
190C42×C61C2411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C412yes—
204#11758196—21no—
205#11758196—21no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—21no—
211#11758196—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218#11758194—42no—
219C2×#92418—21no—
220C2×#92418—22no—
221C2×#92418—21no—
222C2×#92418—42no—
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—22no—
226Q8×C61—22no—
227D4×C61—24no—
228D4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61#13512yes—
244C2×C4×C61#13512yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—44no—
249C22⋊D61—44no—
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—42no—
257C2×Dic61—81no—
258D244—22no—
259D244—22no—
260D244—42no—
261D244—81no—
262C4×D61—44no—
263C4×D61—42no—
264C4×D61—42no—
265C4×D61—82no—
266Dic122—21no—
267Dic122—21no—
268Dic122—21no—
269Dic122—42no—
270Dic122—42no—
271Dic122#13112yes—
272C8×C61—22no—
273C8×C61—42no—
274C8×C61—41no—
275#92418—41no—
276#92418—41no—
277#92418—42no—
278#17510—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61#41612yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C4×C61—21no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—21no—
293C4×C61—22no—
294C4×C61—21no—
295C4×C61#41612yes—
296C4×C61C2×#13111yescharacteristic
297C4×C61#56811yescharacteristic
298C4×C61C2×#13511yescharacteristic
299C4×C61C2×#13511yescharacteristic
300Dic61—22no—
301Dic61—41no—
302Dic61—42no—
303Dic61—41no—
304Dic61—41no—
305#3480—611no—
306C8×#41D6111yescharacteristic, complemented
307#2925—611no—
308#1494—612no—
309#1492—611no—
310#1488—612no—
311#1487—611no—
312#1482—612no—
313#1480—611no—
314#1476—612no—
315C2×Q8⋊C8—611no—
316C2×C61—22no—
317C2×C61—42no—
318C2×C61#288011yescharacteristic
319C2×C61C2×#41611yescharacteristic
320C2×C61#290111yescharacteristic
321D61—82no—
322#559—611no—
323#559—1221no—
324#557—611no—
325#556—611no—
326#556—1221no—
327#553—612no—
328#553—1222no—
329#550—611no—
330#548—612no—
331#532—611no—
332#532—1221no—
333#529—612no—
334#529—1222no—
335C4×#41—612no—
336C4×#41—1221no—
337C4×#41C2×D6111yescharacteristic
338C2×C4×Q8—611no—
339C8×Q8—21no—
340C8×D4—22no—
341C8×D4—21no—
342#433—1222no—
343#432—612no—
344#431—612no—
345#431C2×D6111yescharacteristic
346#431C2×D6112yes—
347#423—611no—
348C2×C4⋊C8—611no—
349#408—1221no—
350#405—1221no—
351C2×C4×C8C2×D6111yescharacteristic
352C2×C4×C8C2×D6112yes—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—612no—
358Q8⋊C8—612no—
359D4⋊C8—612no—
360D4⋊C8—612no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363C61#1751011yescharacteristic, complemented, Sylow
364C2×#41—612no—
365C2×#41—1221no—
366C2×#41D24411yescharacteristic
367C22×Q8—611no—
368C2×M16—1221no—
369C22×C8—21no—
370C22×C8—22no—
371C22×C8—1221no—
372C4⋊Q8—611no—
373C4⋊Q8—1221no—
374#127—611no—
375#127—1221no—
376#126—1222no—
377#126—1222no—
378#126—1222no—
379#125—612no—
380#125—1222no—
381#124—612no—
382#124—1221no—
383#124—1221no—
384#124—1222no—
385#124—1221no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—1222no—
391#121—612no—
392#121—612no—
393#121—1222no—
394#121—1222no—
395#121—1221no—
396#121—1222no—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8C4×D6111yescharacteristic
407C4×Q8C22⋊D6111yescharacteristic
408C4×D4—612no—
409C4×D4—612no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1221no—
414C4×D4—1221no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4C4×D6112yes—
419C4×D4C4×D6111yescharacteristic
420C4×D4C22⋊D6112yes—
421C4×D4C22⋊D6111yescharacteristic
422#117—612no—
423#117—1222no—
424#117—1222no—
425#117D24411yescharacteristic
426#117D24412yes—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C42—1221no—
430C2×C42—1222no—
431C2×C42C22×D6111yescharacteristic
432C2×C42C22×D6112yes—
433C4⋊C8—22no—
434C4⋊C8—21no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—612no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440C22⋊C8—21no—
441C22⋊C8—22no—
442C22⋊C8—1222no—
443C22⋊C8—1222no—
444#97—1222no—
445C4×C8—22no—
446C4×C8—21no—
447C4×C8—1222no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6112yes—
450C4×C8C22×D6111yescharacteristic
451#41—22no—
452#41—22no—
453#41—1224no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457C2×Q8—611no—
458C2×Q8—611no—
459C2×Q8—1222no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—2441no—
463C2×Q8#1175819811yescharacteristic
464C2×D4—612no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4#1175819812yes—
471C2×D4#1175819811yescharacteristic
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4C2×C4×D6112yes—
479C22×C4C2×D24412yes—
480C22×C4C2×D24411yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483M16—2442no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—44no—
488C2×C8—42no—
489C2×C8—1221no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—2442no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—612no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4#1175819811yescharacteristic
507C4⋊C4#1175819811yescharacteristic
508C4⋊C4#1175819812yes—
509C4⋊C4#1175819812yes—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4#1175819812yes—
519C22⋊C4#1175819812yes—
520C22⋊C4#1175819812yes—
521C42—1221no—
522C42—1221no—
523C42—1222no—
524C42—1222no—
525C42—1221no—
526C42—1221no—
527C42C2×C4×D6112yes—
528C42C2×C4×D6111yescharacteristic
529C42C2×D24412yes—
530C42C2×D24411yescharacteristic
531C42C2×C22⋊D6112yes—
532C42C2×C22⋊D6111yescharacteristic
533C42C23×D6111yescharacteristic
534C23—1222no—
535C23C2×#1175819812yes—
536C23C2×#1175819811yescharacteristic
537Q8—22no—
538Q8—612no—
539Q8—1221no—
540Q8—1221no—
541Q8—1221no—
542Q8—2442no—
543Q8—2442no—
544D4—24no—
545D4—22no—
546D4—1222no—
547D4—1222no—
548D4—4881no—
549D4—4882no—
550D4—2444no—
551D4—2444no—
552D4—2442no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—4882no—
567C2×C4—4881no—
568C2×C4—2444no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4C2×#1175819812yes—
573C2×C4C2×#1175819812yes—
574C2×C4C2×#1175819811yescharacteristic
575C2×C4C2×#1175819812yes—
576C2×C4C2×#1175819811yescharacteristic
577C2×C4C2×#1175819811yescharacteristic
578C2×C4#1451979912yes—
579C2×C4#1451979912yes—
580C2×C4C22×C4×D6111yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C22×D24411yescharacteristic
583C8—22no—
584C8—42no—
585C8—41no—
586C8—2441no—
587C8—2441no—
588C8—2442no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—2442no—
595C22—2442no—
596C22—2442no—
597C22#1683975912yes—
598C22C22×#1175819811yescharacteristic
599C4—21no—
600C4—22no—
601C4—21no—
602C4—22no—
603C4—21no—
604C4—22no—
605C4—1222no—
606C4—2441no—
607C4—2441no—
608C4—2441no—
609C4—2442no—
610C4#1683975912yes—
611C4#131×D6111yescharacteristic
612C4#1684052711yescharacteristic
613C4C2×#1451979911yescharacteristic
614C4C2×#1451979911yescharacteristic
615C2—22no—
616C2—42no—
617C2—4882no—
618C2#2145426611yescharacteristic
619C2C2×#1683975911yescharacteristic
620C2#2145436711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459412C2
cosocleC42×C61C24
hypercenterC8×#41D61
hypocenterC61#17510
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #21454485C2semidirect
3. #21444280C2semidirect
4. #21444271C2semidirect
5. #21444254C2semidirect
6. #21444249C2semidirect
7. #21444228C2semidirect
8. #21444219C2semidirect
9. #21444202C2semidirect
10. C2×#16839494C2semidirect
11. C8×#41×C61C2semidirect
306. C8×#41D61semidirect
363. C61#17510semidirect

Maximal subgroups

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

subgroupindex
#214594122
#214544852
#214442802
#214442712
#214442542
#214442492
#214442282
#214442192
#214442022
C2×#168394942
C8×#41×C612
#1751061