Finite Group Databasebeta

← #29743463

Subgroups — #29743463

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455681C211yesmaximal, characteristic
2#21455663C211yesmaximal, characteristic, complemented
3#21454663C211yesmaximal, characteristic
4#21454412C211yesmaximal, characteristic
5#21447362C211yesmaximal, characteristic, complemented
6#21447333C211yesmaximal, characteristic, complemented
7#21447186C211yesmaximal, characteristic, complemented
8#21447163C211yesmaximal, characteristic, complemented
9#21446598C211yesmaximal, characteristic, complemented
10#21446421C211yesmaximal, characteristic, complemented
11#21446336C211yesmaximal, characteristic
12#21446092C211yesmaximal, characteristic
13#21446047C211yesmaximal, characteristic
14#21446033C211yesmaximal, characteristic, complemented
15C61×#1882C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840071—21no—
21#16840069—21no—
22#16840067—21no—
23#16840062—21no—
24#16840055—21no—
25#16840052—21no—
26#16840047—21no—
27#16840045—21no—
28#16839869—21no—
29#16839863—21no—
30Q16×Dic61—21no—
31SD16×Dic61—21no—
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839798—21no—
41#16839798C2211yescharacteristic
42C2×#14519708—21no—
43C2×#14519708C2211yescharacteristic
44C2×#14519708C2211yescharacteristic
45#16839790C2211yescharacteristic
46#16839789C2211yescharacteristic
47#16839763C2211yescharacteristic
48#16839758C2211yescharacteristic
49C2×#14519701C2211yescharacteristic
50C2×#14519701C2211yescharacteristic
51C2×#14519696C2211yescharacteristic
52#16839720C2211yescharacteristic
53#16839632C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839526C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839517C2211yescharacteristic
60C2×#137×C61C2211yescharacteristic
61C61×#520C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#419C2211yescharacteristic
64C61×#391C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519769—21no—
78#14519768—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81Q8×Dic61—21no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84Q8×Dic61—21no—
85D4×Dic61—21no—
86D4×Dic61—41no—
87#14519736—21no—
88#14519736—41no—
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—41no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710D411yescharacteristic
102#14519710D411yescharacteristic
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708D411yescharacteristic
113#14519708D411yescharacteristic
114C4⋊#92418—21no—
115C4⋊#92418—21no—
116#14519706—21no—
117#14519706—21no—
118#14519701—21no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519701D411yescharacteristic
124#14519701D411yescharacteristic
125#14519698—21no—
126#14519698D411yescharacteristic
127#14519698D411yescharacteristic
128#14519696—22no—
129#14519687—21no—
130C8×Dic61—21no—
131C2×#41×C61C2311yescharacteristic
132C22×Q8×C61C2311yescharacteristic
133#137×C61—22no—
134#137×C61Q814yescomplemented
135C2×Q16×C61—21no—
136C2×SD16×C61—21no—
137C2×M16×C61C2311yescharacteristic
138C2×M16×C61C2311yescharacteristic
139C4×Q8×C61—21no—
140C4×D4×C61—21no—
141C4×D4×C61—41no—
142#117×C61—21no—
143#117×C61C2311yescharacteristic
144C2×C4⋊C4×C61—21no—
145C2×C4⋊C4×C61C2311yescharacteristic
146C2×C42×C61C2311yescharacteristic
147#104×C61—22no—
148#103×C61—21no—
149#103×C61—21no—
150#95×C61—41no—
151#41×C61—21no—
152#41×C61—21no—
153#41×C61C2×Q812yes—
154#41×C61#4112yes—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61C2×D411yescharacteristic
161C2×Q8×C61C2×Q812yes—
162C2×D4×C61—21no—
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—41no—
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—41no—
173C2×Dic122—41no—
174Q16×C61—24no—
175SD16×C61—24no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61C2×Q812yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—41no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—82no—
205C4⋊Dic61C2×D411yescharacteristic
206C4⋊Dic61C2×D411yescharacteristic
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210#11758196—41no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—41no—
214C4×Dic61—41no—
215C4×Dic61—41no—
216#11758194—24no—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194#4112yes—
221C2×#92418—24no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61H(5,𝔽2)11yescharacteristic
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61C22⋊Q812yes—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—21no—
243D4×C61C22⋊Q812yes—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—81no—
257C2×C4×C61—161no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×Q811yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—81no—
271C2×Dic61—81no—
272C2×Dic61—161no—
273Dic122—161no—
274C8×C61—22no—
275C8×C61—41no—
276#92418—21no—
277#92418—22no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—41no—
282#26401—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61#53911yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61#55411yescharacteristic
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61#52511yescharacteristic
299C4×C61#54711yescharacteristic
300C4×C61D4×Q811yescharacteristic
301Dic61—41no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307#3084—611no—
308#3082—611no—
309#2947—611no—
310#2912—611no—
311#2096—611no—
312#2090—611no—
313#2064—611no—
314#2060—611no—
315#1951—611no—
316#1922—611no—
317#1908—611no—
318#1882D6111yescharacteristic
319#1866—611no—
320#1859—611no—
321#1857—611no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—42no—
325C2×C61#244911yescharacteristic
326C2×C61#328911yescharacteristic
327C2×C61#329111yescharacteristic
328C2×#137C2×D6111yescharacteristic
329C2×#131—611no—
330#544—611no—
331#520—611no—
332#520C2×D6111yescharacteristic
333#519—611no—
334#479—1221no—
335#479—1221no—
336#479—1221no—
337#478—1221no—
338#477—1221no—
339#476—1221no—
340#475—1221no—
341#475—1221no—
342#441—1221no—
343#440—1221no—
344C4×Q16—1221no—
345C4×SD16—1221no—
346C2×#108—611no—
347#428—611no—
348#428—611no—
349#428—611no—
350#428—611no—
351#428—611no—
352#428—1221no—
353#428—1221no—
354#427—611no—
355#427—1221no—
356C2×#106—611no—
357C2×#106—611no—
358C2×#106—1221no—
359#424—611no—
360#424—611no—
361C2×#104C2×D6111yescharacteristic
362#419C2×D6111yescharacteristic
363#417—611no—
364#416—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#101—611no—
368#405—611no—
369#391C2×D6111yescharacteristic
370#384—611no—
371#343—611no—
372#343—611no—
373#343C2×D6111yescharacteristic
374#341—611no—
375#339—611no—
376#339—611no—
377#339C2×D6111yescharacteristic
378#336—611no—
379C61#2640111yescharacteristic, complemented, Sylow
380C2×#41C22×D6111yescharacteristic
381C22×Q8C22×D6111yescharacteristic
382#137—22no—
383#137Dic12214yescomplemented
384C2×Q16—21no—
385C2×SD16—21no—
386#131—1222no—
387#131—1222no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—1221no—
391C2×M16C22×D6111yescharacteristic
392C2×M16C22×D6111yescharacteristic
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398#125—1221no—
399#125—1221no—
400C22⋊Q8—2441no—
401C4×Q8—21no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×D4—21no—
408C4×D4—41no—
409C4×D4—1221no—
410C4×D4—2441no—
411#117—21no—
412#117—611no—
413#117—611no—
414#117—1221no—
415#117—2441no—
416#117C22×D6111yescharacteristic
417C2×C4⋊C4—21no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2441no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#108—1222no—
425#107—611no—
426#107—611no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—611no—
433#106—611no—
434#106—1221no—
435#106—1221no—
436#106—1221no—
437#106—1221no—
438#106—1221no—
439#106—1221no—
440#106—1221no—
441#106—1221no—
442#106—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447#104—22no—
448#103—21no—
449#103—21no—
450#103—611no—
451#103—611no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#103—1221no—
456#103—1221no—
457#102—611no—
458#102—611no—
459#102—1221no—
460#101—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—41no—
464#41—21no—
465#41—21no—
466#41C2×Dic12212yes—
467#41#1175822412yes—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—2441no—
474C2×Q8C2×Dic12212yes—
475C2×Q8D4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2441no—
486C22×C4C23×D6111yescharacteristic
487Q16—24no—
488SD16—24no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—612no—
493M16—1224no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16C2×Dic12212yes—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—1224no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4882no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C4⋊C4D4×D6111yescharacteristic
530C4⋊C4D4×D6111yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—41no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—2441no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42—1221no—
541C42—1221no—
542C42—2441no—
543C42—2441no—
544C42—2441no—
545C23—21no—
546C23#1451982211yescharacteristic
547Q8—21no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—9761no—
553Q8#1451975412yes—
554D4—22no—
555D4—22no—
556D4—21no—
557D4#1451975412yes—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—81no—
571C2×C4—161no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—9761no—
577C2×C4—4881no—
578C2×C4—4881no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×Dic12211yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4#1451982411yescharacteristic
587C8—22no—
588C8—41no—
589C8—1221no—
590C8—1221no—
591C8—1222no—
592C8—1221no—
593C8—1221no—
594C8—2441no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22#1684035711yescharacteristic
601C22#1684039411yescharacteristic
602C22#1684047111yescharacteristic
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—4881no—
610C4—4881no—
611C4—2441no—
612C4—2441no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451975411yescharacteristic
616C4#1684027211yescharacteristic
617C4#1684041011yescharacteristic
618C4D4×Dic12211yescharacteristic
619C2—21no—
620C2—21no—
621C2—42no—
622C2#2144963211yescharacteristic
623C2#2145759711yescharacteristic
624C2#2145760911yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1882C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1882D61
hypocenterC61#26401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1882

Complements

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

NH ≅ G/Ntype
2. #21455663C2semidirect
5. #21447362C2semidirect
6. #21447333C2semidirect
7. #21447186C2semidirect
8. #21447163C2semidirect
9. #21446598C2semidirect
10. #21446421C2semidirect
14. #21446033C2semidirect
134. #137×C61Q8semidirect
379. C61#26401semidirect
383. #137Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556812
#214556632
#214546632
#214544122
#214473622
#214473332
#214471862
#214471632
#214465982
#214464212
#214463362
#214460922
#214460472
#214460332
C61×#18822
#2640161