Finite Group Databasebeta

← #29721529

Subgroups — #29721529

Order 15616.

658 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#21455641C211yesmaximal, characteristic
2C2×#16840069C211yesmaximal, characteristic, complemented
3#21454485C211yesmaximal, characteristic
4#21447267C211yesmaximal, characteristic
5#21447262C211yesmaximal, characteristic
6#21447115C211yesmaximal, characteristic
7#21447115C211yesmaximal, characteristic
8#21446812C211yesmaximal, characteristic, complemented
9#21446601C211yesmaximal, characteristic, complemented
10#21446596C211yesmaximal, characteristic, complemented
11#21446594C211yesmaximal, characteristic, complemented
12#21446477C211yesmaximal, characteristic, complemented
13#21446477C211yesmaximal, characteristic, complemented
14#21445815C211yesmaximal, characteristic, complemented
15C4×#137×C61C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17#16840069—21no—
18#16840069—21no—
19#16840069—21no—
20#16840069—21no—
21#16840069—21no—
22#16840069—21no—
23#16840062—21no—
24#16840062—21no—
25#16839823—21no—
26#16839823—21no—
27#16839819—21no—
28#16839815—21no—
29C2×#14519712C2211yescharacteristic
30C2×#14519712C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839805C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35#16839786C2211yescharacteristic
36#16839786C2211yescharacteristic
37C2×C4⋊#92418C2211yescharacteristic
38#16839763C2211yescharacteristic
39#16839763C2211yescharacteristic
40C2×#14519701C2211yescharacteristic
41C2×#14519701C2211yescharacteristic
42C2×#14519696C2211yescharacteristic
43C2×#14519696C2211yescharacteristic
44#16839729C2211yescharacteristic
45#16839729C2211yescharacteristic
46#16839718C2211yescharacteristic
47#16839718C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839530C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839526C2211yescharacteristic
56C2×#137×C61C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C2×C4×Q8×C61C2211yescharacteristic
59C61×#441—21no—
60C61×#440—21no—
61C4×Q16×C61—21no—
62C4×SD16×C61—21no—
63C61×#428C2211yescharacteristic
64C61×#419C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C4×M16×C61C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519769—21no—
74#14519769—21no—
75#14519769—21no—
76#14519769—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81#14519712—22no—
82#14519712—22no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418D411yescharacteristic
96C4⋊#92418D411yescharacteristic
97C4⋊#92418D411yescharacteristic
98C4⋊#92418D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519698—21no—
106#14519698—21no—
107#14519696—22no—
108#14519696—22no—
109C22⋊#92418—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519688—21no—
114#14519688—21no—
115C4×#92418—21no—
116C4×#92418—21no—
117C2×#41×C61C2311yescharacteristic
118C22×Q8×C61C2311yescharacteristic
119#137×C61—22no—
120#137×C61—22no—
121C2×Q16×C61—21no—
122C2×SD16×C61—21no—
123C2×M16×C61C2311yescharacteristic
124C2×M16×C61C2311yescharacteristic
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61D411yescharacteristic
130C4×Q8×C61D411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61D411yescharacteristic
134C4×D4×C61D411yescharacteristic
135#117×C61—21no—
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C42×C61—21no—
140C2×C42×C61—21no—
141C2×C42×C61C2311yescharacteristic
142#107×C61—21no—
143#106×C61—21no—
144#103×C61—21no—
145#103×C61—21no—
146#103×C61—21no—
147#102×C61—21no—
148#97×C61—21no—
149C4×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×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61#4111yescharacteristic
169C22×C4×C61#4111yescharacteristic
170C22×C4×C61C2411yescharacteristic
171C22×Dic61—21no—
172C22×Dic61—21no—
173Q16×C61—24no—
174SD16×C61—24no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61#4112yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—41no—
196C42×C61—41no—
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205#11758196—41no—
206#11758196—41no—
207#11758196—41no—
208#11758196—41no—
209#11758194—21no—
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194—21no—
214#11758194—21no—
215#11758194—21no—
216#11758194—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×D411yescharacteristic
226C2×#92418C2×D411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C23×C61—21no—
230C23×C61C2×#4111yescharacteristic
231Q8×C61—22no—
232Q8×C61—21no—
233Q8×C61—21no—
234Q8×C61—22no—
235Q8×C61—41no—
236Q8×C61—41no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—81no—
254C2×C4×C61—81no—
255C2×C4×C61#12111yescharacteristic
256C2×C4×C61#12111yescharacteristic
257C2×C4×C61#12111yescharacteristic
258C2×C4×C61#12111yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—81no—
268C2×Dic61—81no—
269C8×C61—22no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—21no—
274#92418—21no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#92418—41no—
279#24725—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61C2×#12111yescharacteristic
286C22×C61#55211yescharacteristic
287C22×C61#55611yescharacteristic
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61#52511yescharacteristic
298C4×C61#53611yescharacteristic
299C4×C61#54711yescharacteristic
300C4×C61#54811yescharacteristic
301Dic61—81no—
302Dic61—81no—
303#3080—611no—
304C2×#479—611no—
305C4×#137D6111yescharacteristic
306#2925—611no—
307#2079—611no—
308#2078—611no—
309#2052—611no—
310#2052—611no—
311#1998—611no—
312#1952—611no—
313#1951—611no—
314#1950—611no—
315#1930—611no—
316#1930—611no—
317#1801—611no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—81no—
321C2×C61#241511yescharacteristic
322C2×C61#306711yescharacteristic
323C2×C61#306911yescharacteristic
324C2×#137C2×D6111yescharacteristic
325C2×#125—611no—
326C4×#41C2×D6111yescharacteristic
327C2×C4×Q8C2×D6111yescharacteristic
328#479—1221no—
329#479—1221no—
330#479—1221no—
331#479—1221no—
332#479—1221no—
333#479—1221no—
334#478—1221no—
335#478—1221no—
336#441—21no—
337#440—21no—
338C4×Q16—21no—
339C4×SD16—21no—
340#433—1221no—
341#433—1221no—
342#432—1221no—
343#431—1221no—
344C2×#108—611no—
345C2×#108—611no—
346#428—611no—
347#428—611no—
348#428C2×D6111yescharacteristic
349C2×#107—611no—
350C2×#106—611no—
351#423—611no—
352#423—611no—
353C2×C4⋊C8—611no—
354#419C2×D6111yescharacteristic
355#417—611no—
356#417—611no—
357#417C2×D6111yescharacteristic
358C2×#103—611no—
359C2×#103—611no—
360C2×#101—611no—
361C2×#101—611no—
362#408—611no—
363#408—611no—
364#405—611no—
365#405—611no—
366C4×M16C2×D6111yescharacteristic
367#343—611no—
368#343—611no—
369#343—611no—
370#343—611no—
371#343—611no—
372#343—611no—
373#341—611no—
374#341—611no—
375C61#2472511yescharacteristic, complemented, Sylow
376C2×#41C22×D6111yescharacteristic
377C22×Q8C22×D6111yescharacteristic
378#137—22no—
379#137—22no—
380C2×Q16—21no—
381C2×SD16—21no—
382C2×M16—611no—
383C2×M16—611no—
384C2×M16—611no—
385C2×M16—611no—
386C2×M16C22×D6111yescharacteristic
387C2×M16C22×D6111yescharacteristic
388C22×C8—611no—
389C22×C8—611no—
390#125—1221no—
391#125—1221no—
392#125—1221no—
393#125—1221no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8C22⋊D6111yescharacteristic
399C4×Q8C22⋊D6111yescharacteristic
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C22⋊D6111yescharacteristic
403C4×D4C22⋊D6111yescharacteristic
404#117—21no—
405#117C22×D6111yescharacteristic
406C2×C4⋊C4—21no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C42—21no—
413C2×C42—21no—
414C2×C42C22×D6111yescharacteristic
415#108—1222no—
416#108—1222no—
417#107—21no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#106—21no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426#106—1221no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435#103—21no—
436#103—21no—
437#103—21no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#103—1221no—
443#103—1221no—
444#102—21no—
445#102—1221no—
446#102—1221no—
447#101—1222no—
448#101—1222no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453#97—21no—
454#97—1221no—
455#97—1221no—
456C4×C8—21no—
457C4×C8—1221no—
458C4×C8—1221no—
459#41—21no—
460#41—21no—
461#41—21no—
462#41—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8#1175822212yes—
469C2×Q8C2×C22⋊D6111yescharacteristic
470C2×D4—21no—
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4#1175822411yescharacteristic
480C22×C4#1175822411yescharacteristic
481C22×C4C23×D6111yescharacteristic
482Q16—24no—
483SD16—24no—
484M16—21no—
485M16—21no—
486M16—21no—
487M16—1221no—
488M16—1221no—
489M16—1221no—
490M16—1221no—
491M16—1221no—
492M16—1221no—
493M16—1221no—
494M16—1221no—
495M16#1175822212yes—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2441no—
521C4⋊C4—2441no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4C2×C22⋊D6111yescharacteristic
525C4⋊C4C2×C22⋊D6111yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—41no—
534C42D4×D6111yescharacteristic
535C42D4×D6111yescharacteristic
536C42C2×C22⋊D6111yescharacteristic
537C42C2×C22⋊D6111yescharacteristic
538C23—21no—
539C23C2×#1175822411yescharacteristic
540Q8—22no—
541Q8—21no—
542Q8—21no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—4881no—
565C2×C4—4881no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4#1451974911yescharacteristic
569C2×C4#1451974911yescharacteristic
570C2×C4#1451974911yescharacteristic
571C2×C4#1451974911yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—22no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C22—21no—
589C22—21no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1451974911yescharacteristic
594C22#1684045211yescharacteristic
595C22#1684048211yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—4881no—
606C4—4881no—
607C4#1684027311yescharacteristic
608C4#1684033311yescharacteristic
609C4#1684041711yescharacteristic
610C4#1684042111yescharacteristic
611C2—21no—
612C2—21no—
613C2—81no—
614C2#2144924811yescharacteristic
615C2#2145553611yescharacteristic
616C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#137×C61C2
socleC22×C61C2×#121
prime radicalC2×#41×C61C23
hypercenterC4×#137D61
hypocenterC61#24725
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16840069C2semidirect
8. #21446812C2semidirect
9. #21446601C2semidirect
10. #21446596C2semidirect
11. #21446594C2semidirect
12. #21446477C2semidirect
13. #21446477C2semidirect
14. #21445815C2semidirect
375. C61#24725semidirect

Maximal subgroups

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

subgroupindex
#214556412
C2×#168400692
#214544852
#214472672
#214472622
#214471152
#214468122
#214466012
#214465962
#214465942
#214464772
#214458152
C4×#137×C612
#2472561