Finite Group Databasebeta

← #29721485

Subgroups — #29721485

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455630C211yesmaximal, characteristic
2C2×#16840055C211yesmaximal, characteristic, complemented
3#137×Dic61C211yesmaximal, characteristic
4#21454481C211yesmaximal, characteristic
5#21447262C211yesmaximal, characteristic
6#21447259C211yesmaximal, characteristic
7#21447115C211yesmaximal, characteristic
8#21446806C211yesmaximal, characteristic, complemented
9#21446605C211yesmaximal, characteristic, complemented
10#21446603C211yesmaximal, characteristic, complemented
11#21446591C211yesmaximal, characteristic, complemented
12#21446476C211yesmaximal, characteristic, complemented
13#21446471C211yesmaximal, characteristic, complemented
14#21445813C211yesmaximal, characteristic, complemented
15C61×#2052C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17#41×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19#16840055—21no—
20#16840055—21no—
21#16840055—21no—
22#16840055—21no—
23#16840055—21no—
24#16840050—21no—
25#16840045—21no—
26#16840040—21no—
27#16839869—21no—
28#16839863—21no—
29Q16×Dic61—21no—
30SD16×Dic61—21no—
31#16839820—21no—
32#16839820—21no—
33#16839816—21no—
34#16839812—21no—
35C2×#14519712C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39C2×#14519711C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41#16839785C2211yescharacteristic
42#16839785C2211yescharacteristic
43C2×#14519706C2211yescharacteristic
44#16839771C2211yescharacteristic
45#16839763C2211yescharacteristic
46#16839763C2211yescharacteristic
47C2×#14519701C2211yescharacteristic
48C2×#14519696C2211yescharacteristic
49#16839729C2211yescharacteristic
50#16839720C2211yescharacteristic
51#16839717C2211yescharacteristic
52M16×Dic61C2211yescharacteristic
53#16839531C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839527C2211yescharacteristic
59#16839526C2211yescharacteristic
60C2×#137×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#417C2211yescharacteristic
63C2×#103×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#408C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519776—21no—
73#14519776—21no—
74#14519776—21no—
75#14519776—21no—
76Q8×Dic61—21no—
77Q8×Dic61—21no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80Q8×Dic61D411yescharacteristic
81Q8×Dic61D411yescharacteristic
82D4×Dic61—21no—
83D4×Dic61—21no—
84D4×Dic61D411yescharacteristic
85D4×Dic61D411yescharacteristic
86#14519736—21no—
87#14519736C2311yescharacteristic
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519706D411yescharacteristic
113#14519706D411yescharacteristic
114#14519701—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519701—21no—
120#14519698—21no—
121#14519698—21no—
122#14519696—22no—
123C22⋊#92418—21no—
124C22⋊#92418—21no—
125#14519687—21no—
126#14519687—21no—
127#14519687—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C2×#41×C61C2311yescharacteristic
132C22×Q8×C61C2311yescharacteristic
133#137×C61—22no—
134#137×C61—22no—
135C2×Q16×C61—21no—
136C2×SD16×C61—21no—
137C2×M16×C61C2311yescharacteristic
138C2×M16×C61C2311yescharacteristic
139C2×M16×C61C2311yescharacteristic
140C22×C8×C61C2311yescharacteristic
141C2×C4⋊C4×C61C2311yescharacteristic
142#108×C61—22no—
143#103×C61—21no—
144#103×C61—21no—
145#103×C61—21no—
146#102×C61—21no—
147#101×C61—22no—
148C22⋊C8×C61—21no—
149C22⋊C8×C61—21no—
150#41×C61—21no—
151#41×C61—21no—
152#41×C61—21no—
153#41×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61C2×D411yescharacteristic
160C2×Q8×C61#4112yes—
161C2×D4×C61—21no—
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61C2411yescharacteristic
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61#4111yescharacteristic
172C22×Dic61#4111yescharacteristic
173C2×Dic122—41no—
174C2×Dic122—41no—
175C2×Dic122—82no—
176Q16×C61—24no—
177SD16×C61—24no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61—21no—
182M16×C61—21no—
183M16×C61#4112yes—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61C2×D411yescharacteristic
190C2×C8×C61C2×D411yescharacteristic
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D411yescharacteristic
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—41no—
212C4×Dic61—41no—
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230C23×C61—21no—
231C23×C61C2×#4111yescharacteristic
232Q8×C61—22no—
233Q8×C61—21no—
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238D4×C61—21no—
239D4×C61—21no—
240D4×C61—41no—
241D4×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—81no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—81no—
264C2×Dic61—81no—
265C2×Dic61—81no—
266C2×Dic61#12111yescharacteristic
267C2×Dic61#12111yescharacteristic
268C2×Dic61#12111yescharacteristic
269C2×Dic61#12111yescharacteristic
270Dic122—82no—
271Dic122—82no—
272C8×C61—22no—
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—21no—
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#24722—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61C2×#12111yescharacteristic
289C22×C61#55211yescharacteristic
290C22×C61#55611yescharacteristic
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61#52511yescharacteristic
295C4×C61#53611yescharacteristic
296C4×C61#54711yescharacteristic
297C4×C61#54811yescharacteristic
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305Dic61#47312yes—
306#3079—611no—
307C2×#477—611no—
308C4×#137—611no—
309#2925—611no—
310#2078—611no—
311#2078—611no—
312#2052—611no—
313#2052D6111yescharacteristic
314#1997—611no—
315#1952—611no—
316#1952—611no—
317#1950—611no—
318#1930—611no—
319#1930—611no—
320#1801—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—81no—
324C2×C61#241511yescharacteristic
325C2×C61C2×#47311yescharacteristic
326C2×C61#306811yescharacteristic
327C2×#137C2×D6111yescharacteristic
328C2×C4⋊Q8—611no—
329C4×#41—611no—
330C2×C4×Q8—611no—
331#477—1221no—
332#477—1221no—
333#477—1221no—
334#477—1221no—
335#477—1221no—
336#476—1221no—
337#475—1221no—
338#474—1221no—
339#441—1221no—
340#440—1221no—
341C4×Q16—1221no—
342C4×SD16—1221no—
343#433—1221no—
344#433—1221no—
345#432—1221no—
346#431—1221no—
347C2×#108—611no—
348C2×#108C2×D6111yescharacteristic
349#428—611no—
350#428—611no—
351#428—611no—
352C2×#107—611no—
353C2×#107—611no—
354#423—611no—
355#423—611no—
356C2×C4⋊C8—611no—
357#419—611no—
358#417—611no—
359#417—611no—
360#417C2×D6111yescharacteristic
361C2×#103—611no—
362C2×#103C2×D6111yescharacteristic
363C2×#101—611no—
364C2×#101C2×D6111yescharacteristic
365#408—611no—
366#408C2×D6111yescharacteristic
367#405—611no—
368#405—611no—
369C4×M16—611no—
370#343—611no—
371#343—611no—
372#343—611no—
373#343—611no—
374#343—611no—
375#343C2×D6111yescharacteristic
376#341—611no—
377#341—611no—
378C61#2472211yescharacteristic, complemented, Sylow
379C2×#41C22×D6111yescharacteristic
380C22×Q8—1221no—
381C22×Q8C22×D6111yescharacteristic
382#137—22no—
383#137—22no—
384C2×Q16—21no—
385C2×SD16—21no—
386C2×M16—611no—
387C2×M16—611no—
388C2×M16—611no—
389C2×M16C22×D6111yescharacteristic
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C22×C8—611no—
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—611no—
405C4×D4—611no—
406C4×D4—1221no—
407C4×D4—1221no—
408#117—611no—
409#117—1221no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418#108—22no—
419#108—1222no—
420#107—1221no—
421#107—1221no—
422#107—1221no—
423#107—1221no—
424#107—1221no—
425#107—1221no—
426#107—1221no—
427#106—1221no—
428#106—1221no—
429#106—1221no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438#103—21no—
439#103—21no—
440#103—21no—
441#103—1221no—
442#103—1221no—
443#103—1221no—
444#103—1221no—
445#103—1221no—
446#103—1221no—
447#102—21no—
448#102—1221no—
449#102—1221no—
450#101—22no—
451#101—1222no—
452C22⋊C8—21no—
453C22⋊C8—21no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456#97—1221no—
457#97—1221no—
458#97—1221no—
459C4×C8—1221no—
460C4×C8—1221no—
461C4×C8—1221no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—4882no—
472C2×Q8—2441no—
473C2×Q8—2441no—
474C2×Q8D4×D6111yescharacteristic
475C2×Q8#1175822412yes—
476C2×D4—21no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4C23×D6111yescharacteristic
488Q16—24no—
489SD16—24no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498M16—1221no—
499M16—1221no—
500M16—1221no—
501M16#1175822412yes—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8D4×D6111yescharacteristic
515C2×C8D4×D6111yescharacteristic
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—611no—
519C4⋊C4—611no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2441no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C4⋊C4—2441no—
530C22⋊C4—1221no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—2441no—
541C42—2441no—
542C23—21no—
543C23#41×D6111yescharacteristic
544Q8—22no—
545Q8—21no—
546Q8—21no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—4882no—
551Q8—4882no—
552D4—21no—
553D4—21no—
554D4—41no—
555D4—41no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4881no—
572C2×C4—4881no—
573C2×C4—4881no—
574C2×C4—2442no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×#1175822411yescharacteristic
581C2×C4C2×#1175822411yescharacteristic
582C2×C4#1451982411yescharacteristic
583C2×C4#1451982411yescharacteristic
584C8—22no—
585C8—21no—
586C8—21no—
587C8—41no—
588C8—41no—
589C8—1221no—
590C8—1221no—
591C8—2441no—
592C8—2441no—
593C8—2441no—
594C22—21no—
595C22—21no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#121×D6111yescharacteristic
600C22#1684044911yescharacteristic
601C22#1684048111yescharacteristic
602C4—41no—
603C4—42no—
604C4—81no—
605C4—612no—
606C4—1221no—
607C4—1221no—
608C4—1221no—
609C4—4881no—
610C4—4881no—
611C4—2441no—
612C4—2441no—
613C4#1684026811yescharacteristic
614C4#1684033411yescharacteristic
615C4#1684041211yescharacteristic
616C4#1684041811yescharacteristic
617C2—21no—
618C2—21no—
619C2—81no—
620C2#2144924111yescharacteristic
621C2#473×D6111yescharacteristic
622C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2052C2
socleC22×C61C2×#121
prime radicalC2×#41×C61C23
hypercenter#2052D61
hypocenterC61#24722
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2052

Complements

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

NH ≅ G/Ntype
2. C2×#16840055C2semidirect
8. #21446806C2semidirect
9. #21446605C2semidirect
10. #21446603C2semidirect
11. #21446591C2semidirect
12. #21446476C2semidirect
13. #21446471C2semidirect
14. #21445813C2semidirect
378. C61#24722semidirect

Maximal subgroups

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

subgroupindex
#214556302
C2×#168400552
#137×Dic612
#214544812
#214472622
#214472592
#214471152
#214468062
#214466052
#214466032
#214465912
#214464762
#214464712
#214458132
C61×#20522
#2472261