Finite Group Databasebeta

← #29646999

Subgroups — #29646999

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459434C211yesmaximal, characteristic
2C4×#14519784C211yesmaximal, characteristic, complemented
3#21444277C212yesmaximal, complemented
4#21444273C211yesmaximal, characteristic, complemented
5#21444265C212yesmaximal, complemented
6#21444256C211yesmaximal, characteristic, complemented
7#21444239C211yesmaximal, characteristic, complemented
8#21444230C212yesmaximal, complemented
9#21444199C211yesmaximal, characteristic, complemented
10C2×#16839492C212yesmaximal, complemented
11C61×#2990C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840490C411yescharacteristic
14#16840485C411yescharacteristic
15#16840480—21no—
16#16840452C412yes—
17#16840445—22no—
18D4×Dic122C411yescharacteristic
19#16840421C412yes—
20#16840416C411yescharacteristic
21#16840411—21no—
22#16840319C2211yescharacteristic
23#16840316—21no—
24#16840296C2212yes—
25#16840292—22no—
26C4×#11758222C2211yescharacteristic
27C4×#11758224—21no—
28C2×C4×Dic122C2212yes—
29#16839718—21no—
30#16839718—22no—
31C4×#11758194C2212yes—
32C4×#11758194C2211yescharacteristic
33C2×C4×#92418C2212yes—
34C2×C4×#92418C2211yescharacteristic
35#16839492C2211yescharacteristic
36#16839492C2212yes—
37#16839492C2212yes—
38#16839492C2212yes—
39#16839492C2211yescharacteristic
40#16839492C2212yes—
41#16839492C2212yes—
42#16839489C2211yescharacteristic
43#16839489C2211yescharacteristic
44#16839489C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2212yes—
50C61×#431C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2212yes—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758224—21no—
55C22×Dic122C2×C412yes—
56#14519784—28no—
57C2×#11758194—22no—
58C2×#11758194—21no—
59C22×#92418—22no—
60C22×#92418—21no—
61#14519777C2×C411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519776—21no—
64#14519775—21no—
65#14519772—22no—
66#14519771—22no—
67#14519770—22no—
68#14519769—21no—
69#14519769C2×C412yes—
70#14519767—22no—
71#14519767—22no—
72#14519765C2×C412yes—
73#14519763—22no—
74#14519761—22no—
75#14519760—22no—
76#14519758—22no—
77#14519757—21no—
78#14519756—21no—
79#14519756C2×C412yes—
80#14519756C2×C412yes—
81#14519755—22no—
82#14519754—22no—
83#14519753—21no—
84#14519751C2×C412yes—
85#14519749—22no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88C4×Dic122—21no—
89C4×Dic122C2×C412yes—
90C4×Dic122C2×C412yes—
91C4×Dic122C2×C411yescharacteristic
92C4×Dic122C2311yescharacteristic
93C4×Dic122C2312yes—
94#14519742—21no—
95C4×D244C2×C411yescharacteristic
96C4×D244C2311yescharacteristic
97D4×Dic61—21no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—21no—
101#14519737—22no—
102#14519736—21no—
103#14519736C2×C411yescharacteristic
104#14519735—21no—
105#14519734—21no—
106#14519734—22no—
107#14519733—22no—
108C2×C4⋊Dic61C2×C412yes—
109C2×#11758196—22no—
110C2×C4×Dic61—22no—
111C42×D61—21no—
112#14519688—21no—
113#14519688—22no—
114C4×#92418—21no—
115C4×#92418—22no—
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C4×#92418C2311yescharacteristic
119C2×#41×C61D411yescharacteristic
120C2×M16×C61—21no—
121C22×C8×C61—22no—
122C4×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61C2×C411yescharacteristic
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128#117×C61D411yescharacteristic
129#117×C61D412yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133C4⋊C8×C61—21no—
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2312yes—
136C4⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61—21no—
138C22⋊C8×C61—22no—
139#97×C61—22no—
140C4×C8×C61—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×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758224—44no—
156#11758222—22no—
157#11758222—41no—
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×Dic122—24no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122C22×C412yes—
167C2×Dic122C22×C411yescharacteristic
168M16×C61—81no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—22no—
172C2×C8×C61—82no—
173C4⋊C4×C61C22⋊C412yes—
174C4⋊C4×C61C22⋊C411yescharacteristic
175C4⋊C4×C61C22⋊C411yescharacteristic
176C4⋊C4×C61C22⋊C412yes—
177C22⋊C4×C61C22⋊C412yes—
178C22⋊C4×C61C22⋊C412yes—
179C22⋊C4×C61C22⋊C412yes—
180C42×C61C22×C411yescharacteristic
181C42×C61C22×C412yes—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D412yes—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—21no—
190#11758198—21no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C412yes—
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—22no—
209#11758196—21no—
210#11758196—22no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—21no—
216C4×Dic61—22no—
217#11758194—28no—
218#11758194—24no—
219C2×#92418—28no—
220C2×#92418—24no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—22no—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C2×C22⋊C412yes—
226C22×D61—21no—
227Q8×C61—22no—
228D4×C61—24no—
229D4×C61#10414yes—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—44no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258D244—21no—
259D244—21no—
260C4×D61—42no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—42no—
264Dic122—21no—
265Dic122—22no—
266Dic122—22no—
267Dic122—44no—
268Dic122—42no—
269Dic122#13112yescomplemented
270C8×C61—41no—
271C8×C61—42no—
272C8×C61—41no—
273#92418—22no—
274#92418—24no—
275#92418—22no—
276#17519—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61C2×#10414yes—
283C22×C61C22×C22⋊C411yescharacteristic
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—22no—
293C4×C61C2×#10412yes—
294C4×C61C2×#13111yescharacteristic
295C4×C61#56811yescharacteristic
296C4×C61#57711yescharacteristic
297C4×C61#57811yescharacteristic
298Dic61—22no—
299Dic61—42no—
300Dic61—41no—
301Dic61—42no—
302Dic61—41no—
303#3482—611no—
304#2990D6111yescharacteristic, complemented
305C4×#131—611no—
306#1494—612no—
307#1493—611no—
308#1491—612no—
309#1489—611no—
310#1485—611no—
311#1483—612no—
312#1476—611no—
313C2×Q8⋊C8—612no—
314C2×C61—24no—
315C2×C61—41no—
316C2×C61#288011yescharacteristic
317C2×C61#289811yescharacteristic
318C2×C61C22×#10411yescharacteristic
319D61—81no—
320C2×#131—1221no—
321#557—611no—
322#556—611no—
323#556—1221no—
324#552—612no—
325#552—1222no—
326D4×Q8—611no—
327#548—612no—
328#547—611no—
329#547—1221no—
330#533—611no—
331#533—1221no—
332#529—612no—
333#529—1222no—
334C4×#41—611no—
335C4×#41—1221no—
336C4×#41C2×D6111yescharacteristic
337C2×C4×Q8—612no—
338#446—21no—
339#435—22no—
340C8×D4—21no—
341#432C2×D6112yes—
342#431C2×D6111yescharacteristic
343#423C2×D6111yescharacteristic
344C2×C4⋊C8C2×D6112yes—
345#405—1222no—
346#405—1221no—
347C4×M16—612no—
348C4×M16—611no—
349C2×C4×C8—612no—
350C2×C4×C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—611no—
356Q8⋊C8—612no—
357Q8⋊C8—612no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—612no—
361C61#1751911yescharacteristic, complemented, Sylow
362C2×#41—611no—
363C2×#41—1221no—
364C2×#41D24411yescharacteristic
365C22×Q8—612no—
366#131—1228no—
367C2×M16—21no—
368C2×M16—1222no—
369C2×M16—1221no—
370C22×C8—22no—
371C22×C8—1222no—
372C22×C8—1221no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377#126—1222no—
378#126—1222no—
379#126—1222no—
380#125—612no—
381#125—1222no—
382#125—1222no—
383#125—1221no—
384#124—612no—
385#124—1222no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—612no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396#121—612no—
397#121—1222no—
398C4×Q8—612no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—612no—
403C4×Q8—1221no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8C4×D6111yescharacteristic
408C4×Q8C22⋊D6111yescharacteristic
409C4×D4—611no—
410C4×D4—611no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4C4×D6111yescharacteristic
417C4×D4C4×D6112yes—
418C4×D4C22⋊D6111yescharacteristic
419C4×D4C22⋊D6112yes—
420#117—611no—
421#117—1221no—
422#117—1221no—
423#117—1222no—
424#117—1221no—
425#117—1222no—
426#117D24412yes—
427#117D24411yescharacteristic
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C42—1222no—
431C2×C42—1221no—
432C2×C42C22×D6112yes—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—21no—
436C4⋊C8C22×D6111yescharacteristic
437C4⋊C8C22×D6112yes—
438C4⋊C8C22×D6111yescharacteristic
439C22⋊C8—21no—
440C22⋊C8—22no—
441#97—22no—
442#97—1221no—
443#97—1222no—
444C4×C8—21no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—612no—
448C4×C8—1221no—
449C4×C8—1222no—
450#41—24no—
451#41—1222no—
452#41—2441no—
453#41—2444no—
454C2×Q8—611no—
455C2×Q8—612no—
456C2×Q8—1224no—
457C2×Q8—1222no—
458C2×Q8—1221no—
459C2×Q8—2442no—
460C2×Q8#1175819811yescharacteristic
461C2×D4—611no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4#1175819812yes—
465C2×D4#1175819811yescharacteristic
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×C4×D6111yescharacteristic
473C22×C4C2×D24412yes—
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×C22⋊D6112yes—
476C22×C4C2×C22⋊D6111yescharacteristic
477M16—81no—
478M16—1228no—
479M16—1224no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—82no—
484C2×C8—1228no—
485C2×C8—1224no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1222no—
489C4⋊C4—611no—
490C4⋊C4—612no—
491C4⋊C4—611no—
492C4⋊C4—1221no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4#1175819811yescharacteristic
506C4⋊C4#1175819812yes—
507C4⋊C4#1175819811yescharacteristic
508C4⋊C4#1175819812yes—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4#1175819812yes—
518C22⋊C4#1175819812yes—
519C22⋊C4#1175819812yes—
520C42—1221no—
521C42—1222no—
522C42—1221no—
523C42—1222no—
524C42—1221no—
525C42—1222no—
526C42C2×C4×D6111yescharacteristic
527C42C2×C4×D6112yes—
528C42C2×D24411yescharacteristic
529C42C2×D24412yes—
530C42C2×C22⋊D6111yescharacteristic
531C42C2×C22⋊D6112yes—
532C42C23×D6111yescharacteristic
533C23—1221no—
534C23C2×#1175819811yescharacteristic
535C23C2×#1175819812yes—
536Q8—22no—
537Q8—612no—
538Q8—1221no—
539Q8—1222no—
540Q8—1222no—
541Q8—2444no—
542Q8—2442no—
543D4—24no—
544D4—1221no—
545D4—1221no—
546D4—2442no—
547D4—2444no—
548D4#1451970314yes—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2444no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C2×#1175819812yes—
571C2×C4C2×#1175819811yescharacteristic
572C2×C4C2×#1175819811yescharacteristic
573C2×C4C2×#1175819812yes—
574C2×C4C2×#1175819812yes—
575C2×C4C2×#1175819811yescharacteristic
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4C22×D24411yescharacteristic
579C8—41no—
580C8—42no—
581C8—41no—
582C8—1222no—
583C8—1224no—
584C8—1222no—
585C22—22no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—2441no—
591C22—2441no—
592C22—2441no—
593C22C2×#1451970314yes—
594C22C22×#1175819811yescharacteristic
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—21no—
600C4—22no—
601C4—1222no—
602C4—2441no—
603C4—2441no—
604C4—2442no—
605C4—2442no—
606C4C2×#1451970312yes—
607C4#131×D6111yescharacteristic
608C4#1684052711yescharacteristic
609C4#1684057311yescharacteristic
610C4#1684057711yescharacteristic
611C2—24no—
612C2—41no—
613C2—4881no—
614C2#2145426611yescharacteristic
615C2#2145434911yescharacteristic
616C2C22×#1451970311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2990C2
socleC22×C61C22×C22⋊C4
prime radical#21459434C2
cosocleC42×C61C24
hypercenter#2990D61
hypocenterC61#17519
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2990

Complements

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

NH ≅ G/Ntype
2. C4×#14519784C2semidirect
3. #21444277C2semidirect
4. #21444273C2semidirect
5. #21444265C2semidirect
6. #21444256C2semidirect
7. #21444239C2semidirect
8. #21444230C2semidirect
9. #21444199C2semidirect
10. C2×#16839492C2semidirect
11. C61×#2990C2semidirect
269. Dic122#131semidirect
304. #2990D61semidirect
361. C61#17519semidirect

Maximal subgroups

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

subgroupindex
#214594342
C4×#145197842
#214442772
#214442732
#214442652
#214442562
#214442392
#214442302
#214441992
C2×#168394922
C61×#29902
#1751961