Finite Group Databasebeta

← #29743504

Subgroups — #29743504

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#21455670C211yesmaximal, characteristic
2#21455664C211yesmaximal, characteristic, complemented
3#21454658C211yesmaximal, characteristic
4#21447368C211yesmaximal, characteristic, complemented
5#21447330C211yesmaximal, characteristic, complemented
6#21447183C211yesmaximal, characteristic, complemented
7#21447167C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446420C211yesmaximal, characteristic, complemented
10#21446334C211yesmaximal, characteristic
11#21446188C211yesmaximal, characteristic
12#21446094C211yesmaximal, characteristic
13#21446052C211yesmaximal, characteristic
14#21446032C211yesmaximal, characteristic, complemented
15C61×#2912C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840239C2211yescharacteristic
19#16840072—21no—
20#16840069—21no—
21#16840068—21no—
22#16840062—21no—
23#16840057—21no—
24#16840056—21no—
25#16840046—21no—
26#16840042—21no—
27#16839867—21no—
28#16839862—21no—
29C4×#11758202—21no—
30C4×#11758200—21no—
31C2×#14519712C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35C2×#14519710C2211yescharacteristic
36#16839789C2211yescharacteristic
37#16839788C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39#16839770C2211yescharacteristic
40#16839763C2211yescharacteristic
41#16839758C2211yescharacteristic
42C2×#14519702C2211yescharacteristic
43C2×#14519700C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839667C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839523C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#131×C61C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#428—21no—
60C61×#428—21no—
61C61×#428C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C61×#427—21no—
64C61×#427C2211yescharacteristic
65C2×#106×C61—21no—
66C2×#106×C61C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68#14519807—22no—
69#14519807Q814yescomplemented
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519777—21no—
77#14519776—21no—
78#14519769—21no—
79#14519768—21no—
80#14519754—41no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519742—21no—
84#14519742—21no—
85#14519736C2311yescharacteristic
86#14519733—41no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519712—22no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—21no—
98C4⋊#92418—21no—
99C4⋊#92418—21no—
100#14519706—21no—
101#14519706—21no—
102#14519704—22no—
103#14519702—21no—
104#14519702—21no—
105#14519701—21no—
106#14519701D411yescharacteristic
107#14519701D411yescharacteristic
108#14519700—21no—
109#14519700—21no—
110#14519700—21no—
111#14519700—21no—
112#14519698—21no—
113#14519698D411yescharacteristic
114#14519698D411yescharacteristic
115#14519695—22no—
116#14519688—21no—
117C4×#92418—21no—
118#14519684—41no—
119#14519684—41no—
120C2×#41×C61C2311yescharacteristic
121#131×C61—22no—
122#131×C61—22no—
123C2×M16×C61—21no—
124C2×M16×C61C2311yescharacteristic
125C22×C8×C61—21no—
126C22×C8×C61C2311yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—41no—
132C4×D4×C61—41no—
133#117×C61—21no—
134#117×C61—21no—
135#117×C61C2311yescharacteristic
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138C2×C4⋊C4×C61—21no—
139C2×C4⋊C4×C61C2311yescharacteristic
140C2×C4⋊C4×C61C2311yescharacteristic
141#107×C61—21no—
142#107×C61—21no—
143#107×C61D411yescharacteristic
144#107×C61D411yescharacteristic
145#106×C61—21no—
146#106×C61—21no—
147#106×C61—21no—
148#106×C61—21no—
149#106×C61D411yescharacteristic
150#106×C61D411yescharacteristic
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—41no—
162C22×C4×C61—41no—
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—41no—
173C2×Dic122C2×Q812yes—
174M16×C61—24no—
175M16×C61—21no—
176M16×C61#4112yes—
177C2×C8×C61—24no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—41no—
200C42×C61—41no—
201C22⋊Dic61—21no—
202#11758202—24no—
203#11758200—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217C4×Dic61—41no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194C2×Q812yes—
224C2×#92418—21no—
225C2×#92418—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×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—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—81no—
249C2×C4×C61—81no—
250C2×C4×C61—161no—
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—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—81no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272Dic122—161no—
273C8×C61—21no—
274C8×C61—22no—
275C8×C61—21no—
276#92418—22no—
277#92418—21no—
278#92418—21no—
279#92418—41no—
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—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61#52511yescharacteristic
299C4×C61#54711yescharacteristic
300C4×C61D4×Q811yescharacteristic
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306#3083—611no—
307#3082—611no—
308#2947—611no—
309#2912D6111yescharacteristic
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#1860—611no—
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×#131C2×D6111yescharacteristic
329#544—611no—
330#520C2×D6111yescharacteristic
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—21no—
347#428—21no—
348#428—611no—
349#428—611no—
350#428—611no—
351#428C2×D6111yescharacteristic
352#428C2×D6111yescharacteristic
353#427—21no—
354#427C2×D6111yescharacteristic
355C2×#107—611no—
356C2×#106—21no—
357C2×#106C2×D6111yescharacteristic
358#424—611no—
359#424—611no—
360C2×#104—611no—
361#419—611no—
362#417—611no—
363#416—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#101—611no—
367#405—611no—
368#391—611no—
369#382—611no—
370#343—611no—
371#343—611no—
372#343—611no—
373#341—611no—
374#339—611no—
375#339—611no—
376#339—611no—
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—22no—
386#131—22no—
387C2×M16—21no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—611no—
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—21no—
401C4×Q8—21no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—41no—
409C4×D4—41no—
410#117—21no—
411#117—21no—
412#117—611no—
413#117—2441no—
414#117C22×D6111yescharacteristic
415#117C22×D6111yescharacteristic
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C42—611no—
422#108—1222no—
423#107—21no—
424#107—21no—
425#107—1221no—
426#107—1221no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107C22⋊D6111yescharacteristic
431#107C22⋊D6111yescharacteristic
432#106—21no—
433#106—21no—
434#106—21no—
435#106—21no—
436#106—1221no—
437#106—1221no—
438#106—1221no—
439#106C22⋊D6111yescharacteristic
440#106C22⋊D6111yescharacteristic
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445#104—1222no—
446#103—611no—
447#103—611no—
448#103—1221no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#102—611no—
456#102—611no—
457#102—1221no—
458#101—1222no—
459#97—1221no—
460C4×C8—1221no—
461#95—2441no—
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—41no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4C23×D6111yescharacteristic
486Q16—1224no—
487SD16—1224no—
488M16—24no—
489M16—21no—
490M16—612no—
491M16—1221no—
492M16—1221no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496M16#1175822212yes—
497C2×C8—24no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8C2×C22⋊D6111yescharacteristic
509C2×C8C2×C22⋊D6111yescharacteristic
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2441no—
521C4⋊C4—2441no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4C2×C22⋊D6111yescharacteristic
526C4⋊C4C2×C22⋊D6111yescharacteristic
527C4⋊C4C2×C22⋊D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—1221no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—41no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
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—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—161no—
568C2×C4—161no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—4881no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4C2×#1175822211yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4#1451982511yescharacteristic
585C8—21no—
586C8—22no—
587C8—21no—
588C8—1222no—
589C8—1221no—
590C8—1221no—
591C8—2441no—
592C8—2441no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22#1684035911yescharacteristic
599C22D4×C22⋊D6111yescharacteristic
600C22#1684046711yescharacteristic
601C4—22no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—4881no—
609C4—4881no—
610C4—2442no—
611C4—2441no—
612C4—2441no—
613C4C2×#1451975511yescharacteristic
614C4#1684027311yescharacteristic
615C4#1684041311yescharacteristic
616C4Q8×C22⋊D6111yescharacteristic
617C2—21no—
618C2—21no—
619C2—42no—
620C2#2144962811yescharacteristic
621C2#2145759211yescharacteristic
622C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2912C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#2912D61
hypocenterC61#26404
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #21455664C2semidirect
4. #21447368C2semidirect
5. #21447330C2semidirect
6. #21447183C2semidirect
7. #21447167C2semidirect
8. #21446601C2semidirect
9. #21446420C2semidirect
14. #21446032C2semidirect
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
#214556702
#214556642
#214546582
#214473682
#214473302
#214471832
#214471672
#214466012
#214464202
#214463342
#214461882
#214460942
#214460522
#214460322
C61×#29122
#2640461