Finite Group Databasebeta

← #29647007

Subgroups — #29647007

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#21459437C211yesmaximal, characteristic
2#21454945C211yesmaximal, characteristic, complemented
3C4×#14519783C211yesmaximal, characteristic, complemented
4#21444278C212yesmaximal, complemented
5#21444264C212yesmaximal, complemented
6#21444256C211yesmaximal, characteristic, complemented
7#21444240C211yesmaximal, characteristic, complemented
8#21444231C212yesmaximal, complemented
9#21444202C211yesmaximal, characteristic, complemented
10Q8⋊C8×D61C212yesmaximal, complemented
11C61×#1493C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840487C411yescharacteristic
14#16840486C411yescharacteristic
15#16840483—21no—
16#16840446C412yes—
17#16840444—22no—
18#16840427C411yescharacteristic
19#16840422C412yes—
20#16840415C411yescharacteristic
21#16840414—21no—
22#16840316—21no—
23#16840299C2212yes—
24#16840290—22no—
25C4×#11758226—21no—
26C4×#11758222C2211yescharacteristic
27C4×#11758222C2211yescharacteristic
28C4×Q8×D61C2212yes—
29#16839889—21no—
30#16839831—22no—
31#16839827—21no—
32#16839818C2212yes—
33#16839815C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839717—21no—
36#16839716—22no—
37C4×#11758194C2211yescharacteristic
38C4×#11758189C2212yes—
39C4×C8×D61C2212yes—
40#16839494C2211yescharacteristic
41#16839494C2211yescharacteristic
42Q8⋊#92418C2212yes—
43Q8⋊#92418C2212yes—
44#16839492C2212yes—
45#16839491C2211yescharacteristic
46#16839491C2211yescharacteristic
47#16839489C2212yes—
48C61×#533C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C2×C4×C8×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53C2×#11758226—21no—
54C2×#11758222C2×C411yescharacteristic
55C2×#11758222D411yescharacteristic
56C2×Q8×D61C2×C412yes—
57#14519783—28no—
58C2×#11758194—21no—
59C2×#11758189—22no—
60C2×C8×D61—22no—
61C2×C8×D61—22no—
62#14519776—21no—
63#14519775—21no—
64#14519772—22no—
65#14519772—22no—
66#14519770—22no—
67#14519768C2×C412yes—
68#14519767—22no—
69#14519767—22no—
70#14519766C2×C412yes—
71#14519763—22no—
72#14519762—22no—
73#14519762—22no—
74#14519759—22no—
75#14519757—21no—
76#14519755—22no—
77#14519755—22no—
78#14519753—21no—
79#14519753C2×C412yes—
80#14519752C2×C412yes—
81#14519750—22no—
82Q8×Dic61—21no—
83Q8×Dic61C2×C412yes—
84C4×Dic122—22no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122D411yescharacteristic
88C4×Dic122C2311yescharacteristic
89#14519742—22no—
90#14519742C2×C412yes—
91C4×D244—22no—
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C411yescharacteristic
94C4×D244D411yescharacteristic
95C4×D244C2311yescharacteristic
96#14519739—21no—
97#14519739—22no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—21no—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519736C2×C411yescharacteristic
103#14519736D411yescharacteristic
104#14519735—22no—
105#14519735—21no—
106#14519734—21no—
107#14519734—22no—
108#14519734D412yes—
109C4⋊C4×D61—22no—
110C4⋊C4×D61C2×C412yes—
111C42×D61—22no—
112C42×D61—21no—
113C42×D61C2312yes—
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116#14519706—22no—
117C22⋊#92418—21no—
118#14519689—22no—
119#14519688—21no—
120#14519687—22no—
121#14519687—22no—
122C4×#92418—21no—
123C4×#92418C2312yes—
124C8×Dic61—22no—
125C2×M16×C61—21no—
126C22×C8×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128C4⋊Q8×C61C2×C411yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61C2×C412yes—
132C4×Q8×C61—21no—
133C4×Q8×C61C2312yes—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×C22⋊D61—21no—
147C2×C22⋊D61C22⋊C412yes—
148#11758226—44no—
149Q8×D61—24no—
150Q8×D61—42no—
151#11758222—24no—
152#11758222—22no—
153#11758222—41no—
154C2×D244—22no—
155C2×D244C22⋊C411yescharacteristic
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61C22×C412yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122C22⋊C411yescharacteristic
167C2×Dic122C22×C411yescharacteristic
168M16×C61—81no—
169C2×C8×C61—24no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—21no—
186C22⋊Dic61C22⋊C412yes—
187#11758198—22no—
188#11758198—21no—
189#11758198—21no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198C22⋊C412yes—
194#11758198C22⋊C412yes—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217#11758194—24no—
218C2×#92418—22no—
219C2×#92418—21no—
220C2×#92418—21no—
221#11758189—28no—
222C8×D61—28no—
223C8×D61—82no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—21no—
226C22×D61C2×C22⋊C412yes—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—42no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—24no—
244C22⋊D61—42no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C412yes—
251D244—21no—
252D244—21no—
253D244—44no—
254D244#10414yes—
255C4×D61—24no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—42no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—42no—
266Dic122—21no—
267Dic122—22no—
268Dic122—44no—
269Dic122#13112yescomplemented
270C8×C61—22no—
271C8×C61—22no—
272C8×C61—42no—
273#92418—24no—
274#92418—41no—
275#92418—41no—
276#17519—611nomaximal, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61C2×#10414yes—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61C2×#10412yes—
293C4×C61C2×#13111yescharacteristic
294C4×C61#56811yescharacteristic
295C4×C61#57711yescharacteristic
296C4×C61#57811yescharacteristic
297Dic61—22no—
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—41no—
302Dic61—41no—
303#3482—611no—
304#2990—611no—
305C4×#131—611no—
306#1494—612no—
307#1493D6111yescharacteristic, complemented
308#1491—612no—
309#1489—611no—
310#1485—611no—
311#1483—612no—
312#1476—611no—
313C2×Q8⋊C8—612no—
314C2×C61—41no—
315C2×C61#288011yescharacteristic
316C2×C61#289811yescharacteristic
317C2×C61C22×#10411yescharacteristic
318D61—24no—
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—1221no—
331#533C2×D6111yescharacteristic
332#529—612no—
333#529—1222no—
334C4×#41—611no—
335C4×#41—611no—
336C4×#41—1221no—
337C2×C4×Q8—612no—
338#446—1221no—
339#435—1222no—
340C8×D4—1221no—
341#432—612no—
342#431—611no—
343#423C2×D6111yescharacteristic
344C2×C4⋊C8—612no—
345#405—1222no—
346#405—1221no—
347C4×M16—611no—
348C4×M16—612no—
349C2×C4×C8—612no—
350C2×C4×C8C2×D6111yescharacteristic
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8C2×D6112yes—
357Q8⋊C8C2×D6112yes—
358D4⋊C8—612no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361C61#1751911yescharacteristic, complemented, Sylow
362C2×#41—611no—
363C2×#41—611no—
364C2×#41—1221no—
365C22×Q8—612no—
366#131—1228no—
367C2×M16—21no—
368C2×M16—1222no—
369C2×M16—1221no—
370C22×C8—21no—
371C22×C8—1222no—
372C22×C8—1222no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8C4×D6111yescharacteristic
376C4⋊Q8C4×D6111yescharacteristic
377#126—1222no—
378#126—1222no—
379#126—1222no—
380#125—21no—
381#125—612no—
382#125—1222no—
383#125—1222no—
384#124—612no—
385#124—1222no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—21no—
390C22⋊Q8—612no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395C22⋊Q8C4×D6112yes—
396#121—612no—
397#121—1222no—
398C4×Q8—21no—
399C4×Q8—611no—
400C4×Q8—611no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—611no—
404C4×Q8—611no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1221no—
408C4×Q8C22×D6112yes—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—611no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—21no—
421#117—611no—
422#117—612no—
423#117—611no—
424#117—1222no—
425#117—1222no—
426#117—1221no—
427#117—1221no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C42—612no—
431C2×C42—1222no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—21no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—1222no—
438C4⋊C8C22×D6112yes—
439C22⋊C8—1222no—
440C22⋊C8—1221no—
441#97—1222no—
442#97—1222no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—612no—
446C4×C8—1222no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450#41—1224no—
451#41—1222no—
452#41—2441no—
453#41—2444no—
454C2×Q8—21no—
455C2×Q8—611no—
456C2×Q8—611no—
457C2×Q8—1222no—
458C2×Q8—1224no—
459C2×Q8—2442no—
460C2×Q8C2×C4×D6112yes—
461C2×D4—612no—
462C2×D4—611no—
463C2×D4—611no—
464C2×D4—1222no—
465C2×D4—1221no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477M16—81no—
478M16—1228no—
479M16—1224no—
480C2×C8—24no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—1228no—
485C2×C8—1222no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—4882no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4C2×C4×D6112yes—
509C22⋊C4—22no—
510C22⋊C4—612no—
511C22⋊C4—612no—
512C22⋊C4—612no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C42—22no—
521C42—612no—
522C42—612no—
523C42—612no—
524C42—1222no—
525C42—1222no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42C2×C4×D6111yescharacteristic
530C42D4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C42C23×D6111yescharacteristic
533C23—612no—
534C23—1221no—
535C23C22⋊C4×D6111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—42no—
539Q8—612no—
540Q8—1221no—
541Q8—1222no—
542Q8—2444no—
543D4—614no—
544D4—1224no—
545D4—1221no—
546D4—1221no—
547D4—2444no—
548D4—2442no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—1224no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—2442no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C8—22no—
580C8—22no—
581C8—42no—
582C8—1224no—
583C8—2441no—
584C8—2441no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—614no—
589C22—1222no—
590C22—1222no—
591C22—2441no—
592C22—2441no—
593C22—2441no—
594C22C2×C22⋊C4×D6111yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—1222no—
601C4—1222no—
602C4—1222no—
603C4—1222no—
604C4—2441no—
605C4—2441no—
606C4#104×D6112yes—
607C4#131×D6111yescharacteristic
608C4#1684052711yescharacteristic
609C4#1684056811yescharacteristic
610C4#1684057511yescharacteristic
611C2—41no—
612C2—1224no—
613C2—4881no—
614C2#2145426711yescharacteristic
615C2#2145434711yescharacteristic
616C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1493C2
socleC22×C61C22×C22⋊C4
prime radical#21459437C2
cosocleC42×C61C24
hypercenter#1493D61
hypocenterC61#17519
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
2. #21454945C2semidirect
3. C4×#14519783C2semidirect
4. #21444278C2semidirect
5. #21444264C2semidirect
6. #21444256C2semidirect
7. #21444240C2semidirect
8. #21444231C2semidirect
9. #21444202C2semidirect
10. Q8⋊C8×D61C2semidirect
11. C61×#1493C2semidirect
269. Dic122#131semidirect
307. #1493D61semidirect
361. C61#17519semidirect

Maximal subgroups

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

subgroupindex
#214594372
#214549452
C4×#145197832
#214442782
#214442642
#214442562
#214442402
#214442312
#214442022
Q8⋊C8×D612
C61×#14932
#1751961