Finite Group Databasebeta

← #29743506

Subgroups — #29743506

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#21455659C211yesmaximal, characteristic, complemented
3#21454658C211yesmaximal, characteristic
4#21454412C211yesmaximal, characteristic
5#21447370C211yesmaximal, characteristic, complemented
6#21447333C211yesmaximal, characteristic, complemented
7#21447187C211yesmaximal, characteristic, complemented
8#21447169C211yesmaximal, characteristic, complemented
9#21446601C211yesmaximal, characteristic, complemented
10#21446416C211yesmaximal, characteristic, complemented
11#21446332C211yesmaximal, characteristic
12#21446186C211yesmaximal, characteristic
13#21446092C211yesmaximal, characteristic
14#21446035C211yesmaximal, characteristic, complemented
15C61×#1860C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840393C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840239C2211yescharacteristic
21#16840072—21no—
22#16840068—21no—
23#16840067—21no—
24#16840064—21no—
25#16840056—21no—
26#16840055—21no—
27#16840046—21no—
28#16840040—21no—
29#16839867—21no—
30#16839862—21no—
31C4×#11758202—21no—
32C4×#11758200—21no—
33C2×#14519712C2211yescharacteristic
34#16839805—21no—
35#16839805C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839798—21no—
41#16839798C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519708—21no—
44C2×#14519708C2211yescharacteristic
45#16839790C2211yescharacteristic
46#16839790C2211yescharacteristic
47#16839772C2211yescharacteristic
48#16839758C2211yescharacteristic
49C2×#14519700C2211yescharacteristic
50C2×#14519700C2211yescharacteristic
51C2×#14519695C2211yescharacteristic
52#16839718C2211yescharacteristic
53#16839668C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839526C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839517C2211yescharacteristic
60C61×#520C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#382C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68#14519807—22no—
69#14519807Q814yescomplemented
70C2×#11758202—21no—
71C2×#11758200—21no—
72#14519784—22no—
73#14519784—22no—
74C2×#11758194—21no—
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C22×#92418—21no—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519768—21no—
83#14519768—21no—
84#14519756—41no—
85Q8×Dic61—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88#14519742—21no—
89#14519742—21no—
90D4×Dic61—21no—
91D4×Dic61—41no—
92#14519736—21no—
93#14519736C2311yescharacteristic
94#14519736C2311yescharacteristic
95#14519733—41no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61C2311yescharacteristic
98#14519712—22no—
99#14519711—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710D411yescharacteristic
106#14519710D411yescharacteristic
107#14519709—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708—21no—
113#14519708D411yescharacteristic
114#14519708D411yescharacteristic
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519706—21no—
119#14519702—21no—
120#14519702—21no—
121#14519701D411yescharacteristic
122#14519701D411yescharacteristic
123#14519700—21no—
124#14519700—21no—
125#14519700—21no—
126#14519700—21no—
127#14519698D411yescharacteristic
128#14519698D411yescharacteristic
129#14519695—22no—
130#14519688—21no—
131C4×#92418—21no—
132#14519684—41no—
133C2×#41×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C4×Q8×C61—21no—
137C4×D4×C61—21no—
138C4×D4×C61—41no—
139#117×C61—21no—
140#117×C61C2311yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C42×C61C2311yescharacteristic
145#107×C61—21no—
146#106×C61—21no—
147#104×C61—22no—
148#103×C61—21no—
149#102×C61—21no—
150#95×C61—41no—
151#41×C61—21no—
152#41×C61—21no—
153#41×C61C2×Q812yes—
154#41×C61#4112yes—
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61—21no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—41no—
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—41no—
173C2×Dic122C2×Q812yes—
174M16×C61—21no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×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—
196#11758202—24no—
197#11758200—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203C4⋊Dic61C2×D411yescharacteristic
204C4⋊Dic61C2×D411yescharacteristic
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—41no—
207#11758196—41no—
208#11758196—41no—
209#11758196—41no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—41no—
213C4×Dic61—41no—
214C4×Dic61—41no—
215#11758194—24no—
216#11758194—21no—
217#11758194—21no—
218#11758194C2×Q812yes—
219#11758194#4112yes—
220C2×#92418—24no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231C23×C61—21no—
232C23×C61H(5,𝔽2)11yescharacteristic
233Q8×C61—21no—
234Q8×C61C22⋊Q812yes—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—21no—
238D4×C61C22⋊Q812yes—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
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×C61—81no—
251C2×C4×C61—161no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—81no—
267C2×Dic61—161no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272Dic122—161no—
273C8×C61—41no—
274C8×C61—41no—
275#92418—22no—
276#92418—21no—
277#92418—22no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
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—41no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×C22⋊Q811yescharacteristic
297C4×C61#52511yescharacteristic
298C4×C61#54711yescharacteristic
299C4×C61D4×Q811yescharacteristic
300Dic61—42no—
301Dic61—41no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
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#1908—611no—
317#1882—611no—
318#1866—611no—
319#1860D6111yescharacteristic
320#1857—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—42no—
324C2×C61#244911yescharacteristic
325C2×C61#328911yescharacteristic
326C2×C61#329011yescharacteristic
327C2×#137—611no—
328C2×#131—611no—
329#544—611no—
330#520—611no—
331#520C2×D6111yescharacteristic
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—1221no—
351#428—1221no—
352#428C2×D6111yescharacteristic
353#427—611no—
354#427—1221no—
355C2×#107—611no—
356C2×#106—611no—
357C2×#106—1221no—
358#424—611no—
359#424—611no—
360C2×#104C2×D6111yescharacteristic
361#419—611no—
362#417C2×D6111yescharacteristic
363#416—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#101—611no—
367#405—611no—
368#391—611no—
369#382C2×D6111yescharacteristic
370#343—611no—
371#343—611no—
372#343C2×D6111yescharacteristic
373#341—611no—
374#339—611no—
375#339—611no—
376#339C2×D6111yescharacteristic
377#336—611no—
378C61#2640411yescharacteristic, complemented, Sylow
379C2×#41C22×D6111yescharacteristic
380C22×Q8—611no—
381#137—614no—
382#137—1222no—
383C2×Q16—1221no—
384C2×SD16—1221no—
385#131—1222no—
386#131—1222no—
387C2×M16—611no—
388C2×M16—611no—
389C2×M16—1221no—
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397#125—1221no—
398#125—1221no—
399C22⋊Q8—2441no—
400C4×Q8—21no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—21no—
407C4×D4—41no—
408C4×D4—1221no—
409C4×D4—2441no—
410#117—21no—
411#117—611no—
412#117—611no—
413#117—1221no—
414#117—2441no—
415#117C22×D6111yescharacteristic
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C42C22×D6111yescharacteristic
422#108—1222no—
423#107—21no—
424#107—611no—
425#107—611no—
426#107—1221no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—21no—
433#106—611no—
434#106—611no—
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—22no—
446#103—21no—
447#103—611no—
448#103—611no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#102—21no—
456#102—611no—
457#102—611no—
458#101—1222no—
459#97—1221no—
460C4×C8—1221no—
461#95—41no—
462#95—2441no—
463#41—21no—
464#41—21no—
465#41#1175822212yes—
466#41Q8×D6112yes—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—2441no—
474C2×Q8D4×D6111yescharacteristic
475C2×D4—21no—
476C2×D4D4×D6111yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—2441no—
485C22×C4C23×D6111yescharacteristic
486Q16—1224no—
487SD16—1224no—
488M16—21no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—612no—
493M16—612no—
494M16—1224no—
495M16—1221no—
496M16—1221no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—1224no—
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—611no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—611no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—2441no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4D4×D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—41no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C22⋊C4—2441no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—1221no—
539C42—1221no—
540C42—2441no—
541C42—2441no—
542C42—2441no—
543C23—21no—
544C23#1451982211yescharacteristic
545Q8—21no—
546Q8—1222no—
547Q8—1222no—
548Q8—9761no—
549Q8—2441no—
550Q8—2441no—
551Q8#1451975512yes—
552D4—22no—
553D4—22no—
554D4—21no—
555D4#1451975512yes—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—161no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—9761no—
575C2×C4—4881no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4#1451982411yescharacteristic
585C8—41no—
586C8—41no—
587C8—1222no—
588C8—1221no—
589C8—1221no—
590C8—1222no—
591C8—1221no—
592C8—1221no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22#1684035411yescharacteristic
599C22#1684039411yescharacteristic
600C22#1684046611yescharacteristic
601C4—22no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—4881no—
608C4—2442no—
609C4—2441no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4C2×#1451975511yescharacteristic
614C4#1684027111yescharacteristic
615C4#1684041011yescharacteristic
616C4#1684042511yescharacteristic
617C2—21no—
618C2—21no—
619C2—42no—
620C2#2144963011yescharacteristic
621C2#2145759711yescharacteristic
622C2#2145759811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1860C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1860D61
hypocenterC61#26404
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
2. #21455659C2semidirect
5. #21447370C2semidirect
6. #21447333C2semidirect
7. #21447187C2semidirect
8. #21447169C2semidirect
9. #21446601C2semidirect
10. #21446416C2semidirect
14. #21446035C2semidirect
69. #14519807Q8semidirect
378. C61#26404semidirect

Maximal subgroups

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

subgroupindex
#214556742
#214556592
#214546582
#214544122
#214473702
#214473332
#214471872
#214471692
#214466012
#214464162
#214463322
#214461862
#214460922
#214460352
C61×#18602
#2640461