Finite Group Databasebeta

← #29743510

Subgroups — #29743510

Order 15616.

706 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#21455674C211yesmaximal, characteristic
2#21455657C211yesmaximal, characteristic, complemented
3#21454663C211yesmaximal, characteristic
4#21454414C211yesmaximal, characteristic
5#21447369C211yesmaximal, characteristic, complemented
6#21447331C211yesmaximal, characteristic, complemented
7#21447182C211yesmaximal, characteristic, complemented
8#21447170C211yesmaximal, characteristic, complemented
9#21446603C211yesmaximal, characteristic, complemented
10#21446417C211yesmaximal, characteristic, complemented
11#21446190C211yesmaximal, characteristic
12#21446094C211yesmaximal, characteristic
13#21446050C211yesmaximal, characteristic
14#21446036C211yesmaximal, characteristic, complemented
15C61×#1908C211yesmaximal, characteristic
16#16840393C2211yescharacteristic
17#16840245C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840071—21no—
21#16840067—21no—
22#16840067—21no—
23#16840064—21no—
24#16840055—21no—
25#16840055—21no—
26#16840045—21no—
27#16840040—21no—
28#16839869—21no—
29#16839863—21no—
30Q16×Dic61—21no—
31SD16×Dic61—21no—
32C2×#14519712C2211yescharacteristic
33#16839804—21no—
34#16839804—21no—
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839800—21no—
41#16839800C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519709—21no—
44C2×#14519709C2211yescharacteristic
45#16839790C2211yescharacteristic
46#16839788C2211yescharacteristic
47C2×#14519704C2211yescharacteristic
48#16839771C2211yescharacteristic
49#16839758C2211yescharacteristic
50C2×#14519701C2211yescharacteristic
51#16839720C2211yescharacteristic
52#16839669C2211yescharacteristic
53#16839627C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839523C2211yescharacteristic
57#16839523C2211yescharacteristic
58#16839523C2211yescharacteristic
59#16839516C2211yescharacteristic
60C2×#137×C61C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C61×#417C2211yescharacteristic
63C2×#103×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#341C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—21no—
70#14519776—21no—
71#14519768—21no—
72#14519768—21no—
73#14519756—41no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76Q8×Dic61—21no—
77Q8×Dic61—21no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80D4×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—41no—
83D4×Dic61—41no—
84#14519736—21no—
85#14519736—21no—
86#14519736—41no—
87#14519736C2311yescharacteristic
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711D411yescharacteristic
101#14519711D411yescharacteristic
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519709—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709—21no—
111#14519709D411yescharacteristic
112#14519709D411yescharacteristic
113#14519708—21no—
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519704—22no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519701D411yescharacteristic
124#14519701D411yescharacteristic
125#14519698D411yescharacteristic
126#14519698D411yescharacteristic
127#14519687—21no—
128C8×Dic61—21no—
129#14519684—41no—
130#14519684—41no—
131C2×#41×C61C2311yescharacteristic
132C22×Q8×C61C2311yescharacteristic
133#137×C61—22no—
134#137×C61Q814yescomplemented
135C2×Q16×C61—21no—
136C2×SD16×C61—21no—
137#131×C61—22no—
138#131×C61—22no—
139C2×M16×C61—21no—
140C2×M16×C61C2311yescharacteristic
141C2×M16×C61C2311yescharacteristic
142C2×M16×C61C2311yescharacteristic
143C22×C8×C61—21no—
144C22×C8×C61C2311yescharacteristic
145C2×C4⋊C4×C61C2311yescharacteristic
146#103×C61—21no—
147#103×C61—21no—
148#103×C61—21no—
149#102×C61—21no—
150#101×C61—22no—
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×C61C2411yescharacteristic
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—41no—
172C22×Dic61—41no—
173C2×Dic122—41no—
174Q16×C61—24no—
175SD16×C61—24no—
176M16×C61—24no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61C2×Q812yes—
182M16×C61#4112yes—
183C2×C8×C61—24no—
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—
196C22⋊Dic61—41no—
197C22⋊Dic61—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211C4⋊Dic61C2×D411yescharacteristic
212C4⋊Dic61C2×D411yescharacteristic
213#11758196—41no—
214#11758196—41no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—41no—
221C4×Dic61—41no—
222C4×Dic61—41no—
223C4×Dic61—41no—
224#11758194—21no—
225#11758194—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231C23×C61—21no—
232C23×C61H(5,𝔽2)11yescharacteristic
233Q8×C61—21no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61C22⋊Q812yes—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—21no—
242D4×C61C22⋊Q812yes—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—81no—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—81no—
269C2×Dic61—81no—
270C2×Dic61—161no—
271C2×Dic61—161no—
272Dic122—161no—
273C8×C61—22no—
274C8×C61—21no—
275C8×C61—22no—
276C8×C61—21no—
277C8×C61—41no—
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#26404—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61#53911yescharacteristic
288C22×C61D4211yescharacteristic
289C22×C61#55411yescharacteristic
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61C2×C22⋊Q811yescharacteristic
294C4×C61#52511yescharacteristic
295C4×C61#54711yescharacteristic
296C4×C61D4×Q811yescharacteristic
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305Dic61—81no—
306#3083—611no—
307#3082—611no—
308#2947—611no—
309#2912—611no—
310#2097—611no—
311#2090—611no—
312#2064—611no—
313#2061—611no—
314#1952—611no—
315#1922—611no—
316#1908D6111yescharacteristic
317#1882—611no—
318#1866—611no—
319#1860—611no—
320#1857—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—42no—
324C2×C61#244911yescharacteristic
325C2×C61#328911yescharacteristic
326C2×C61#329011yescharacteristic
327C2×#137C2×D6111yescharacteristic
328C2×#131C2×D6111yescharacteristic
329#544—611no—
330#520—611no—
331#520—611no—
332#519—611no—
333#479—1221no—
334#479—1221no—
335#479—1221no—
336#478—1221no—
337#477—1221no—
338#477—1221no—
339#475—1221no—
340#474—1221no—
341#441—1221no—
342#440—1221no—
343C4×Q16—1221no—
344C4×SD16—1221no—
345C2×#108—611no—
346#428—611no—
347#428—611no—
348#428—611no—
349#428—611no—
350#428—611no—
351#428—1221no—
352#428—1221no—
353#427—611no—
354#427—1221no—
355C2×#107—611no—
356C2×#106—611no—
357C2×#106—1221no—
358#424—611no—
359#424—611no—
360C2×#104—611no—
361#419—611no—
362#417C2×D6111yescharacteristic
363#416—611no—
364C2×#103—611no—
365C2×#103C2×D6111yescharacteristic
366C2×#101C2×D6111yescharacteristic
367#405—611no—
368#391—611no—
369#382—611no—
370#343—611no—
371#343—611no—
372#343C2×D6111yescharacteristic
373#341C2×D6111yescharacteristic
374#339—611no—
375#339—611no—
376#339—611no—
377#336—611no—
378C61#2640411yescharacteristic, complemented, Sylow
379C2×#41C22×D6111yescharacteristic
380C22×Q8C22×D6111yescharacteristic
381#137—22no—
382#137Dic12214yescomplemented
383C2×Q16—21no—
384C2×SD16—21no—
385#131—22no—
386#131—22no—
387C2×M16—21no—
388C2×M16—611no—
389C2×M16C22×D6111yescharacteristic
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C22×C8—21no—
393C22×C8—611no—
394C22×C8C22×D6111yescharacteristic
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397#125—1221no—
398#125—1221no—
399C22⋊Q8—2441no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4—2441no—
409C4×D4—2441no—
410#117—611no—
411#117—611no—
412#117—611no—
413#117—1221no—
414#117—1221no—
415#117—2441no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C42—611no—
422#108—1222no—
423#107—611no—
424#107—611no—
425#107—1221no—
426#107—1221no—
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—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445#104—1222no—
446#103—21no—
447#103—21no—
448#103—21no—
449#103—611no—
450#103—611no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#102—21no—
456#102—611no—
457#102—611no—
458#101—22no—
459#97—1221no—
460C4×C8—1221no—
461#95—2441no—
462#95—2441no—
463#41—21no—
464#41—21no—
465#41C2×Dic12212yes—
466#41#1175822412yes—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—2441no—
473C2×Q8C2×Dic12212yes—
474C2×Q8D4×D6111yescharacteristic
475C2×D4—21no—
476C2×D4D4×D6111yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4C23×D6111yescharacteristic
486Q16—24no—
487SD16—24no—
488M16—24no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—1221no—
494M16—1221no—
495M16C2×Dic12212yes—
496M16#1175822412yes—
497C2×C8—24no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8D4×D6111yescharacteristic
509C2×C8D4×D6111yescharacteristic
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—611no—
513C4⋊C4—611no—
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—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C22⋊C4—1221no—
530C22⋊C4—1221no—
531C22⋊C4—1221no—
532C22⋊C4—2441no—
533C22⋊C4—2441no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C42—2441no—
540C42—2441no—
541C42—2441no—
542C42—2441no—
543C23—21no—
544C23#1451982311yescharacteristic
545Q8—21no—
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—9761no—
551Q8#1451975412yes—
552D4—22no—
553D4—22no—
554D4—21no—
555D4#1451975412yes—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—9761no—
571C2×C4—9761no—
572C2×C4—4881no—
573C2×C4—4881no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4C22×Dic12211yescharacteristic
582C2×C4C2×#1175822411yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4#1451982411yescharacteristic
585C8—22no—
586C8—21no—
587C8—22no—
588C8—21no—
589C8—41no—
590C8—1221no—
591C8—1221no—
592C8—2441no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22#1684035611yescharacteristic
599C22#1684039611yescharacteristic
600C22#1684046911yescharacteristic
601C4—22no—
602C4—42no—
603C4—81no—
604C4—4881no—
605C4—4881no—
606C4—4881no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4C2×#1451975411yescharacteristic
614C4#1684026811yescharacteristic
615C4#1684041211yescharacteristic
616C4D4×Dic12211yescharacteristic
617C2—21no—
618C2—21no—
619C2—42no—
620C2#2144963411yescharacteristic
621C2#2145759311yescharacteristic
622C2#2145760211yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1908C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1908D61
hypocenterC61#26404
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1908

Complements

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

NH ≅ G/Ntype
2. #21455657C2semidirect
5. #21447369C2semidirect
6. #21447331C2semidirect
7. #21447182C2semidirect
8. #21447170C2semidirect
9. #21446603C2semidirect
10. #21446417C2semidirect
14. #21446036C2semidirect
134. #137×C61Q8semidirect
378. C61#26404semidirect
382. #137Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556742
#214556572
#214546632
#214544142
#214473692
#214473312
#214471822
#214471702
#214466032
#214464172
#214461902
#214460942
#214460502
#214460362
C61×#19082
#2640461