Finite Group Databasebeta

← #29646901

Subgroups — #29646901

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#21459414C211yesmaximal, characteristic
2#21454775C211yesmaximal, characteristic, complemented
3#21454487C211yesmaximal, characteristic, complemented
4#21444278C212yesmaximal, complemented
5#21444269C211yesmaximal, characteristic, complemented
6#21444252C212yesmaximal, complemented
7#21444226C212yesmaximal, complemented
8#21444219C211yesmaximal, characteristic, complemented
9#21444200C212yesmaximal, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1487C211yesmaximal, characteristic, complemented
12#16840497—21no—
13#16840497C411yescharacteristic
14#16840491C411yescharacteristic
15#16840485—21no—
16#16840485C411yescharacteristic
17#16840460—22no—
18#16840460C412yes—
19#16840434C411yescharacteristic
20#16840423C412yes—
21#16840312—21no—
22#16840312C2211yescharacteristic
23#16840299—22no—
24#16840299C2212yes—
25C4×#11758226C2212yes—
26C4×#11758222—21no—
27C4×#11758222C2211yescharacteristic
28#16839886—21no—
29#16839827—21no—
30#16839825—22no—
31#16839822—22no—
32#16839818C2212yes—
33#16839814C2212yes—
34#16839813C2212yes—
35C2×C4⋊#92418C2211yescharacteristic
36#16839730—21no—
37#16839720—21no—
38C2×C4×#92418C2211yescharacteristic
39C4×C8×D61C2212yes—
40Q8⋊#92418C2211yescharacteristic
41Q8⋊#92418C2211yescharacteristic
42Q8⋊#92418C2211yescharacteristic
43Q8⋊#92418C2211yescharacteristic
44#16839491C2212yes—
45#16839491C2212yes—
46#16839489C2212yes—
47#16839489C2212yes—
48C2×C4×Q8×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C61×#423C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53C2×#11758226C2×C412yes—
54C2×#11758222—21no—
55C2×#11758222D411yescharacteristic
56C22×#92418—21no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777C2×C411yescharacteristic
61#14519774—21no—
62#14519774C2×C411yescharacteristic
63#14519772—22no—
64#14519772—22no—
65#14519772—22no—
66#14519768—22no—
67#14519768C2×C412yes—
68#14519766—22no—
69#14519766C2×C412yes—
70#14519765—21no—
71#14519765—21no—
72#14519765—21no—
73#14519762—22no—
74#14519762—22no—
75#14519762—22no—
76#14519753—22no—
77#14519753C2×C412yes—
78#14519752—22no—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751—22no—
82#14519751—22no—
83#14519751C2×C412yes—
84Q8×Dic61C2×C412yes—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89C4×D244—22no—
90C4×D244—22no—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244—21no—
94C4×D244C2×C411yescharacteristic
95C4×D244D411yescharacteristic
96C4×D244C2312yes—
97#14519739—22no—
98#14519739C2×C412yes—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101C4×C22⋊D61C2×C412yes—
102C4×C22⋊D61D412yes—
103#14519736D411yescharacteristic
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107#14519734D412yes—
108C42×D61—22no—
109C42×D61C2312yes—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—22no—
114#14519689—22no—
115#14519689—22no—
116#14519689—22no—
117#14519687—22no—
118C4×#92418—21no—
119C4×#92418C2311yescharacteristic
120C4×#92418C2311yescharacteristic
121C8×Dic61—22no—
122C8×Dic61—22no—
123C22×Q8×C61C2×C411yescharacteristic
124C2×M16×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C411yescharacteristic
128C4×Q8×C61C2×C411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C22⋊C8×C61—21no—
139C4×C8×C61C2312yes—
140C2×Q8×C61—21no—
141C2×Q8×C61—22no—
142C2×Q8×C61—41no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C2×C22⋊D61—22no—
151C2×C22⋊D61C22⋊C412yes—
152#11758226—24no—
153#11758226—42no—
154#11758222—22no—
155#11758222—22no—
156#11758222—42no—
157#11758222—42no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—22no—
161C2×D244—21no—
162C2×D244C22⋊C411yescharacteristic
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×Dic122—21no—
171C2×Dic122C22⋊C411yescharacteristic
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2411yescharacteristic
190C22⋊Dic61C22⋊C412yes—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198C22⋊C412yes—
200#11758198C22⋊C412yes—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C411yescharacteristic
206C4⋊Dic61C22×C412yes—
207#11758196—22no—
208#11758196—22no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—42no—
221C2×#92418—42no—
222C8×D61—44no—
223C23×C61C2×C22⋊C411yescharacteristic
224C22×D61—22no—
225C22×D61C2×C22⋊C412yes—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—42no—
230Q8×C61#13112yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—81no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
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—24no—
249C22⋊D61—44no—
250C2×Dic61—22no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254D244—22no—
255D244—22no—
256D244—22no—
257D244—44no—
258D244—42no—
259D244—81no—
260D244—82no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—42no—
266C4×D61—44no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61—82no—
270Dic122—22no—
271Dic122—42no—
272C8×C61—42no—
273C8×C61—42no—
274#92418—22no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#17510—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61#41612yes—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61#41612yes—
297C4×C61C2×#13111yescharacteristic
298C4×C61#56811yescharacteristic
299C4×C61C2×#13511yescharacteristic
300C4×C61C2×#13511yescharacteristic
301Dic61—22no—
302Dic61—22no—
303Dic61—22no—
304Dic61—42no—
305#3480—611no—
306C8×#41—611no—
307#2925—611no—
308#1494—612no—
309#1492—611no—
310#1488—612no—
311#1487D6111yescharacteristic, complemented
312#1482—612no—
313#1480—611no—
314#1476—612no—
315C2×Q8⋊C8—611no—
316C2×C61—22no—
317C2×C61#288011yescharacteristic
318C2×C61C2×#41611yescharacteristic
319C2×C61#290111yescharacteristic
320D61—42no—
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—611no—
336C4×#41—612no—
337C4×#41—1221no—
338C2×C4×Q8C2×D6111yescharacteristic
339C8×Q8—1221no—
340C8×D4—1222no—
341C8×D4—1221no—
342#433—1222no—
343#432—612no—
344#431—612no—
345#431—612no—
346#431C2×D6111yescharacteristic
347#423C2×D6111yescharacteristic
348C2×C4⋊C8—611no—
349#408—1221no—
350#405—1221no—
351C2×C4×C8—611no—
352C2×C4×C8—612no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8C2×D6112yes—
358Q8⋊C8C2×D6112yes—
359D4⋊C8—612no—
360D4⋊C8—612no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363C61#1751011yescharacteristic, complemented, Sylow
364C2×#41—611no—
365C2×#41—612no—
366C2×#41—1221no—
367C22×Q8C4×D6111yescharacteristic
368C2×M16—21no—
369C22×C8—1222no—
370C22×C8—1221no—
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—1222no—
396#121—1221no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—611no—
400C4×Q8—611no—
401C4×Q8—612no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8C4×D6111yescharacteristic
405C4×Q8C4×D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×Q8C22×D6111yescharacteristic
408C4×D4—612no—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—612no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1221no—
422#117—611no—
423#117—612no—
424#117—612no—
425#117—1222no—
426#117—1222no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4C4×D6111yescharacteristic
429C2×C42—21no—
430C2×C42—612no—
431C2×C42—1222no—
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—21no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—1222no—
438C4⋊C8—1221no—
439C4⋊C8C22×D6112yes—
440C22⋊C8—21no—
441C22⋊C8—1222no—
442C22⋊C8—1222no—
443C22⋊C8—1222no—
444#97—1222no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—1222no—
448C4×C8—1222no—
449C4×C8—1221no—
450C4×C8C22×D6112yes—
451#41—1222no—
452#41—1222no—
453#41—1224no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—41no—
460C2×Q8—611no—
461C2×Q8—1221no—
462C2×Q8C2×C4×D6111yescharacteristic
463C2×Q8C2×C4×D6111yescharacteristic
464C2×D4—612no—
465C2×D4—612no—
466C2×D4—611no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1221no—
471C2×D4—1221no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483M16—42no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—2444no—
491C2×C8—2442no—
492C2×C8—2442no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—612no—
499C4⋊C4—612no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—612no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1221no—
508C4⋊C4C2×C4×D6111yescharacteristic
509C4⋊C4C2×C4×D6111yescharacteristic
510C22⋊C4—612no—
511C22⋊C4—612no—
512C22⋊C4—612no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—612no—
526C42—612no—
527C42—612no—
528C42—1222no—
529C42—1222no—
530C42C2×C4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C42D4×D6111yescharacteristic
533C42C23×D6111yescharacteristic
534C23—612no—
535C23—1222no—
536C23C22⋊C4×D6111yescharacteristic
537Q8—21no—
538Q8—21no—
539Q8—21no—
540Q8—42no—
541Q8—1222no—
542Q8—2442no—
543Q8#1451978512yes—
544D4—1224no—
545D4—1222no—
546D4—1222no—
547D4—1222no—
548D4—4881no—
549D4—4882no—
550D4—2444no—
551D4—2444no—
552D4—2442no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4882no—
571C2×C4—2442no—
572C2×C4—2444no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C22⋊C4×D6111yescharacteristic
578C2×C4#1451979812yes—
579C2×C4#1451980012yes—
580C2×C4C22×C4×D6111yescharacteristic
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C8—42no—
584C8—42no—
585C8—1222no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—21no—
590C22—21no—
591C22—1222no—
592C22—1222no—
593C22—1222no—
594C22—2442no—
595C22—2442no—
596C22—2442no—
597C22#1683975712yes—
598C22C2×C22⋊C4×D6111yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—1222no—
607C4—1222no—
608C4—1222no—
609C4—2442no—
610C4#1683975712yes—
611C4C2×#1451978511yescharacteristic
612C4#1684052811yescharacteristic
613C4C2×#1451979811yescharacteristic
614C4C2×#1451980011yescharacteristic
615C2—22no—
616C2—4882no—
617C2—2442no—
618C2#2145426911yescharacteristic
619C2C2×#1683975711yescharacteristic
620C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1487C2
socleC22×C61C22×C22⋊C4
prime radical#21459414C2
cosocleC42×C61C24
hypercenter#1487D61
hypocenterC61#17510
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. #21454775C2semidirect
3. #21454487C2semidirect
4. #21444278C2semidirect
5. #21444269C2semidirect
6. #21444252C2semidirect
7. #21444226C2semidirect
8. #21444219C2semidirect
9. #21444200C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#1487C2semidirect
311. #1487D61semidirect
363. C61#17510semidirect

Maximal subgroups

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

subgroupindex
#214594142
#214547752
#214544872
#214442782
#214442692
#214442522
#214442262
#214442192
#214442002
C2×Q8⋊#924182
C61×#14872
#1751061