Finite Group Databasebeta

← #30547721

Subgroups — #30547721

Order 15616.

1970 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519824C211yesmaximal, characteristic
2#21455677C213yesmaximal, complemented
3#21455666C213yesmaximal, complemented
4#21455655C213yesmaximal, complemented
5#21455655C213yesmaximal, complemented
6#21455637C213yesmaximal, complemented
7#21455620C213yesmaximal, complemented
8#21455601C213yesmaximal, complemented
9C2×#16840066C213yesmaximal, complemented
10#21454417C211yesmaximal, characteristic, complemented
11#21454412C211yesmaximal, characteristic, complemented
12#21454387C211yesmaximal, characteristic, complemented
13#21454349C211yesmaximal, characteristic
14#21454349C211yesmaximal, characteristic
15C61×#3504C211yesmaximal, characteristic, complemented
16C2×#14519824C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840242C2213yes—
21#16840241—23no—
22#16840239C2213yes—
23#16840238—21no—
24C4×#11758222C2213yes—
25C4×#11758224—23no—
26C2×C4×Dic122C2213yes—
27C4×Q8×D61—21no—
28#16840066C2213yes—
29#16840066C2216yes—
30#16840066C2216yes—
31#16840066C2216yes—
32#16840066C2216yes—
33#16840066C2213yes—
34#16840059C2213yes—
35#16840059C2216yes—
36#16840059C2211yescharacteristic
37#16840054C2216yes—
38#16840054C2213yes—
39#16840049C2213yes—
40#16840044C2216yes—
41#16840044C2213yes—
42#16840039C2213yes—
43#16839805C2213yes—
44#16839805C2213yes—
45#16839802C2213yes—
46#16839802C2213yes—
47#16839798C2213yes—
48#16839798C2213yes—
49C2×#14519710C2213yes—
50C2×#14519708C2213yes—
51#16839789C2213yes—
52#16839789C2213yes—
53#16839786C2213yes—
54C2×C4⋊#92418C2213yes—
55#16839770C2213yes—
56#16839770C2213yes—
57#16839761C2213yes—
58#16839761C2213yes—
59#16839756C2213yes—
60#16839756C2213yes—
61C2×#14519700C2213yes—
62C2×#14519700C2213yes—
63C61×#557—21no—
64C61×#557—21no—
65C61×#554—23no—
66C61×#554—23no—
67C61×#544C2213yes—
68C61×#544C2213yes—
69C61×#533C2213yes—
70C2×#125×C61C2213yes—
71C61×#520C2211yescharacteristic
72C61×#520C2211yescharacteristic
73C4×#41×C61C2211yescharacteristic
74#14519824—24no—
75#14519824Q818yescomplemented
76C2×#11758222C2313yes—
77C2×#11758224—23no—
78C22×Dic122C2313yes—
79C2×Q8×D61—21no—
80#14519784—24no—
81#14519784—24no—
82C2×#11758194C2313yes—
83C2×#11758194C2313yes—
84C22×#92418C2313yes—
85C22×#92418C2313yes—
86Q8×Dic61—21no—
87C4×Dic122—23no—
88C4×Dic122—23no—
89C4×Dic122C2313yes—
90C4×Dic122C2316yes—
91C4×Dic122C2313yes—
92#14519742—23no—
93#14519742—23no—
94C4×D244C2311yescharacteristic
95C4×D244C2313yes—
96D4×Dic61—23no—
97C4×C22⋊D61—23no—
98C4×C22⋊D61—23no—
99#14519737—26no—
100#14519736C2313yes—
101#14519735—23no—
102#14519734—23no—
103#14519733—26no—
104C2×C4⋊Dic61C2313yes—
105C2×#11758196—23no—
106C4⋊C4×D61—23no—
107C2×C4×Dic61—23no—
108C42×D61—23no—
109#14519710C2313yes—
110#14519710C2313yes—
111#14519710C2316yes—
112#14519710C2311yescharacteristic
113#14519710C2313yes—
114#14519708C2311yescharacteristic
115#14519708C2313yes—
116#14519708C2316yes—
117#14519708C2313yes—
118#14519708C2313yes—
119C4⋊#92418C2311yescharacteristic
120C4⋊#92418C2313yes—
121C4⋊#92418C2313yes—
122C4⋊#92418C2313yes—
123C4⋊#92418C2316yes—
124#14519700C2313yes—
125#14519700C2313yes—
126#14519700C2316yes—
127#14519700C2316yes—
128#14519700C2313yes—
129#14519700C2313yes—
130#14519697C2311yescharacteristic
131#14519697C2311yescharacteristic
132#14519697C2313yes—
133#14519697C2313yes—
134C2×#41×C61C2311yescharacteristic
135C4⋊Q8×C61—23no—
136C4⋊Q8×C61—23no—
137C4⋊Q8×C61C2313yes—
138C4⋊Q8×C61C2313yes—
139#125×C61—23no—
140#125×C61—23no—
141#125×C61—23no—
142#125×C61—23no—
143#125×C61C2311yescharacteristic
144#125×C61C2313yes—
145#125×C61C2316yes—
146C22⋊Q8×C61—23no—
147C22⋊Q8×C61—23no—
148C22⋊Q8×C61—26no—
149C22⋊Q8×C61—26no—
150C4×Q8×C61—21no—
151C4×Q8×C61—21no—
152C4×Q8×C61—23no—
153C4×Q8×C61—23no—
154C4×Q8×C61D411yescharacteristic
155C4×Q8×C61D411yescharacteristic
156C4×D4×C61—23no—
157C4×D4×C61—23no—
158C4×D4×C61D413yes—
159C4×D4×C61D413yes—
160#117×C61C2313yes—
161#117×C61C2313yes—
162#117×C61C2313yes—
163C2×C4⋊C4×C61—23no—
164C2×C4⋊C4×C61—23no—
165C2×C4⋊C4×C61C2313yes—
166C2×C4⋊C4×C61C2313yes—
167C2×C42×C61C2313yes—
168#41×C61C2×Q814yes—
169#41×C61#4114yes—
170C2×Q8×C61—23no—
171C2×Q8×C61—23no—
172C2×Q8×C61C2×D411yescharacteristic
173C2×D4×C61C2×D413yes—
174C22×C4×C61—23no—
175C22×C4×C61—23no—
176C22×C4×C61C2×D413yes—
177C22×C4×C61C2×D413yes—
178C22×C4×C61C2413yes—
179C2×C22⋊D61—23no—
180C22×Dic61—23no—
181C22×Dic61—23no—
182Q8×D61—28no—
183#11758224—224no—
184#11758222—26no—
185#11758222C2×Q8112yes—
186C2×D244C2411yescharacteristic
187C2×C4×D61—23no—
188C2×C4×D61—21no—
189C2×C4×D61—23no—
190C2×Dic122—23no—
191C2×Dic122—26no—
192C2×Dic122C2×Q8112yes—
193C2×Dic122C2413yes—
194C4⋊C4×C61—23no—
195C4⋊C4×C61—23no—
196C4⋊C4×C61—26no—
197C4⋊C4×C61—26no—
198C4⋊C4×C61—23no—
199C4⋊C4×C61—23no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—23no—
203C4⋊C4×C61—23no—
204C4⋊C4×C61C2×D413yes—
205C4⋊C4×C61C2×D413yes—
206C4⋊C4×C61C2413yes—
207C4⋊C4×C61C2413yes—
208C4⋊C4×C61C2411yescharacteristic
209C4⋊C4×C61C2411yescharacteristic
210C22⋊C4×C61—23no—
211C22⋊C4×C61—23no—
212C22⋊C4×C61C2×D416yes—
213C42×C61—23no—
214C42×C61—23no—
215C42×C61C2×D413yes—
216C42×C61C2×D413yes—
217C42×C61C2411yescharacteristic
218C42×C61C2413yes—
219C22⋊Dic61—23no—
220#11758198—21no—
221#11758198—23no—
222C4⋊Dic61—23no—
223C4⋊Dic61C2413yes—
224C4⋊Dic61C2411yescharacteristic
225#11758196—26no—
226#11758196—23no—
227#11758196—23no—
228C4×Dic61—23no—
229C4×Dic61—23no—
230C4×Dic61—23no—
231C4×Dic61—26no—
232#11758194—26no—
233#11758194—26no—
234C2×#92418—26no—
235C2×#92418—26no—
236C2×#92418C2411yescharacteristic
237C2×#92418C2413yes—
238C2×#92418C2413yes—
239C2×#92418C2411yescharacteristic
240C23×C61C22×D413yes—
241C22×D61—21no—
242Q8×C61—83no—
243Q8×C61—83no—
244Q8×C61C22⋊Q814yes—
245D4×C61C22⋊Q8112yes—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—23no—
249C2×C4×C61—23no—
250C2×C4×C61—43no—
251C2×C4×C61—43no—
252C2×C4×C61—83no—
253C2×C4×C61—83no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D413yes—
256C2×C4×C61C22×D413yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D413yes—
259C2×C4×C61C22×Q816yes—
260C2×C4×C61C2×#4116yes—
261C2×C4×C61C2511yescharacteristic
262C22⋊D61—212no—
263C2×Dic61—212no—
264C2×Dic61—23no—
265C2×Dic61—21no—
266C2×Dic61—23no—
267C2×Dic61—83no—
268D244—21no—
269D244C22×Q812yes—
270C4×D61—212no—
271C4×D61—42no—
272C4×D61—83no—
273Dic122—212no—
274Dic122—23no—
275Dic122C22×Q816yes—
276#92418—21no—
277#92418—21no—
278#92418—23no—
279#92418—23no—
280#60499—611nomaximal, Sylow
281C22×C61—23no—
282C22×C61—23no—
283C22×C61C2×C22⋊Q816yes—
284C22×C61C23×D411yescharacteristic
285C2×D61—22no—
286C2×D61—41no—
287C2×D61—41no—
288C4×C61—23no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—23no—
292C4×C61—43no—
293C4×C61—41no—
294C4×C61—43no—
295C4×C61—41no—
296C4×C61C2×C22⋊Q816yes—
297C4×C61C23×Q811yescharacteristic
298C4×C61C22×#4111yescharacteristic
299Dic61—26no—
300Dic61—43no—
301Dic61—41no—
302#3504D6111yescharacteristic, complemented
303C4×2−1+4—611no—
304#3084—613no—
305#3083—613no—
306#3082—613no—
307#3082—613no—
308#3080—613no—
309#3078—613no—
310#3076—613no—
311C2×#479—613no—
312#2913—611no—
313#2912—611no—
314#2906—611no—
315#2898—611no—
316#2898—611no—
317C2×C61—26no—
318C2×C61C22×C22⋊Q811yescharacteristic
319C2×C61#358411yescharacteristic
320C2×C61#358611yescharacteristic
321D61—42no—
322C2×2−1+4—611no—
323C2×#131—611no—
324C2×#131—611no—
325#557—21no—
326#557—21no—
327#554—23no—
328#554—23no—
329#544C2×D6113yes—
330#544C2×D6113yes—
331#533C2×D6113yes—
332C2×#125C2×D6113yes—
333#520—611no—
334#520—613no—
335#520—1223no—
336#520C2×D6111yescharacteristic
337#520C2×D6111yescharacteristic
338#519—613no—
339#519—1221no—
340C4×#41—613no—
341C4×#41—1223no—
342C4×#41C2×D6111yescharacteristic
343C2×C4×Q8—613no—
344C2×C4×Q8—1221no—
345#479—613no—
346#479—616no—
347#479—616no—
348#479—616no—
349#479—613no—
350#479—616no—
351#478—611no—
352#478—613no—
353#478—616no—
354#477—613no—
355#477—616no—
356#476—613no—
357#475—613no—
358#475—616no—
359#474—613no—
360#428—613no—
361#428—613no—
362#428—613no—
363#428—613no—
364#427—613no—
365#427—613no—
366C2×#107—613no—
367C2×#106—613no—
368#424—613no—
369#424—613no—
370#423—613no—
371C2×C4⋊C8—613no—
372#419—613no—
373#419—613no—
374#417—613no—
375#417—613no—
376#416—613no—
377#416—613no—
378C2×#103—613no—
379C2×#103—613no—
380C61#6049911yescharacteristic, complemented, Sylow
3812−1+4—618no—
3822−1+4—1224no—
383C2×#41—613no—
384C2×#41—1223no—
385C2×#41C22×D6111yescharacteristic
386C22×Q8—613no—
387C22×Q8—1221no—
388#131—1224no—
389#131—1224no—
390C2×M16—613no—
391C2×M16—613no—
392C22×C8—613no—
393C22×C8—613no—
394C4⋊Q8—23no—
395C4⋊Q8—23no—
396C4⋊Q8C22×D6113yes—
397C4⋊Q8C22×D6113yes—
398#125—23no—
399#125—23no—
400#125—23no—
401#125—23no—
402#125C22×D6113yes—
403#125C22×D6116yes—
404#125C22×D6111yescharacteristic
405C22⋊Q8—23no—
406C22⋊Q8—23no—
407C22⋊Q8—26no—
408C22⋊Q8—26no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—23no—
412C4×Q8—23no—
413C4×Q8—613no—
414C4×Q8—616no—
415C4×Q8—613no—
416C4×Q8—1221no—
417C4×Q8—1223no—
418C4×Q8—1223no—
419C4×Q8—1223no—
420C4×Q8—1223no—
421C4×Q8C22⋊D6111yescharacteristic
422C4×Q8C22⋊D6111yescharacteristic
423C4×D4—23no—
424C4×D4—23no—
425C4×D4—611no—
426C4×D4—613no—
427C4×D4—1223no—
428C4×D4—1223no—
429C4×D4—1223no—
430C4×D4—1226no—
431C4×D4C22⋊D6113yes—
432C4×D4C22⋊D6113yes—
433#117—613no—
434#117—1223no—
435#117—1226no—
436#117—1223no—
437#117C22×D6113yes—
438#117C22×D6113yes—
439#117C22×D6113yes—
440C2×C4⋊C4—23no—
441C2×C4⋊C4—23no—
442C2×C4⋊C4—613no—
443C2×C4⋊C4—1223no—
444C2×C4⋊C4—1223no—
445C2×C4⋊C4C22×D6113yes—
446C2×C4⋊C4C22×D6113yes—
447C2×C42—1223no—
448C2×C42—1223no—
449C2×C42C22×D6113yes—
450#107—613no—
451#107—616no—
452#107—611no—
453#107—613no—
454#107—613no—
455#106—613no—
456#106—616no—
457#106—611no—
458#106—613no—
459#106—613no—
460C4⋊C8—611no—
461C4⋊C8—613no—
462C4⋊C8—613no—
463C4⋊C8—613no—
464C4⋊C8—616no—
465#103—613no—
466#103—613no—
467#103—616no—
468#103—616no—
469#103—613no—
470#103—613no—
471#102—611no—
472#102—611no—
473#102—613no—
474#102—613no—
475#41—6112no—
476#41—12224no—
477#41—1226no—
478#41Q8×D6114yes—
479#41#1175822614yes—
480C2×Q8—23no—
481C2×Q8—23no—
482C2×Q8—6112no—
483C2×Q8—613no—
484C2×Q8—1228no—
485C2×Q8—1223no—
486C2×Q8—1226no—
487C2×Q8C2×C22⋊D6111yescharacteristic
488C2×D4—611no—
489C2×D4—1223no—
490C2×D4C2×C22⋊D6113yes—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—1223no—
494C22×C4—1223no—
495C22×C4—1223no—
496C22×C4—1223no—
497C22×C4—1221no—
498C22×C4C2×C22⋊D6113yes—
499C22×C4C2×C22⋊D6113yes—
500C22×C4C23×D6113yes—
501M16—1226no—
502M16—1226no—
503C2×C8—611no—
504C2×C8—613no—
505C2×C8—613no—
506C2×C8—611no—
507C2×C8—1226no—
508C2×C8—1226no—
509C4⋊C4—23no—
510C4⋊C4—23no—
511C4⋊C4—26no—
512C4⋊C4—26no—
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—613no—
520C4⋊C4—611no—
521C4⋊C4—1223no—
522C4⋊C4—1226no—
523C4⋊C4—1223no—
524C4⋊C4—1223no—
525C4⋊C4C2×C22⋊D6113yes—
526C4⋊C4C2×C22⋊D6113yes—
527C4⋊C4C23×D6113yes—
528C4⋊C4C23×D6111yescharacteristic
529C4⋊C4C23×D6113yes—
530C4⋊C4C23×D6111yescharacteristic
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—1223no—
534C22⋊C4—1221no—
535C22⋊C4—1223no—
536C22⋊C4C2×C22⋊D6116yes—
537C42—23no—
538C42—23no—
539C42—1223no—
540C42—1223no—
541C42—1223no—
542C42—1226no—
543C42C2×C22⋊D6113yes—
544C42C2×C22⋊D6113yes—
545C42C23×D6111yescharacteristic
546C42C23×D6113yes—
547C23—1221no—
548C23C22×C22⋊D6113yes—
549Q8—83no—
550Q8—83no—
551Q8—616no—
552Q8—12212no—
553Q8—1223no—
554Q8#1451975314yes—
555D4—612no—
556D4—12212no—
557D4—1221no—
558D4#14519753112yes—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—23no—
562C2×C4—23no—
563C2×C4—43no—
564C2×C4—43no—
565C2×C4—83no—
566C2×C4—83no—
567C2×C4—12212no—
568C2×C4—1223no—
569C2×C4—12212no—
570C2×C4—1221no—
571C2×C4—1223no—
572C2×C4—4883no—
573C2×C4—4883no—
574C2×C4—2442no—
575C2×C4C22×C22⋊D6111yescharacteristic
576C2×C4C22×C22⋊D6113yes—
577C2×C4C22×C22⋊D6113yes—
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4C22×C22⋊D6113yes—
580C2×C4C2×Q8×D6116yes—
581C2×C4C2×#1175822616yes—
582C2×C4C24×D6111yescharacteristic
583C8—1221no—
584C8—1221no—
585C8—1223no—
586C8—1223no—
587C22—23no—
588C22—23no—
589C22—1222no—
590C22—2441no—
591C22—2441no—
592C22C2×#1451975316yes—
593C22C23×C22⋊D6111yescharacteristic
594C4—21no—
595C4—23no—
596C4—21no—
597C4—23no—
598C4—43no—
599C4—41no—
600C4—43no—
601C4—41no—
602C4—1226no—
603C4—2443no—
604C4—2441no—
605C4C2×#1451975316yes—
606C4C22×Q8×D6111yescharacteristic
607C4C22×#1175822611yescharacteristic
608C2—26no—
609C2—2442no—
610C2C22×#1451975311yescharacteristic
611C2#2146059011yescharacteristic
612C2#2146060511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3504C2
socleC22×C61C23×D4
prime radicalC2×#14519824C22
hypercenter#3504D61
hypocenterC61#60499
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
2. #21455677C2semidirect
3. #21455666C2semidirect
4. #21455655C2semidirect
5. #21455655C2semidirect
6. #21455637C2semidirect
7. #21455620C2semidirect
8. #21455601C2semidirect
9. C2×#16840066C2semidirect
10. #21454417C2semidirect
11. #21454412C2semidirect
12. #21454387C2semidirect
15. C61×#3504C2semidirect
75. #14519824Q8semidirect
302. #3504D61semidirect
380. C61#60499semidirect

Maximal subgroups

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

subgroupindex
C4×#145198242
#214556772
#214556662
#214556552
#214556372
#214556202
#214556012
C2×#168400662
#214544172
#214544122
#214543872
#214543492
C61×#35042
#6049961