Finite Group Databasebeta

← #29743455

Subgroups — #29743455

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#21455655C211yesmaximal, characteristic, complemented
2#21454665C211yesmaximal, characteristic
3#21454412C211yesmaximal, characteristic
4#21447359C211yesmaximal, characteristic, complemented
5#21447332C211yesmaximal, characteristic, complemented
6#21447185C211yesmaximal, characteristic, complemented
7#21447161C211yesmaximal, characteristic, complemented
8#21446597C211yesmaximal, characteristic, complemented
9#21446415C211yesmaximal, characteristic, complemented
10#21446332C211yesmaximal, characteristic
11#21446188C211yesmaximal, characteristic
12#21446096C211yesmaximal, characteristic
13#21446043C211yesmaximal, characteristic
14#21446032C211yesmaximal, characteristic, complemented
15C61×#3084C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840239C2211yescharacteristic
20#16840066—21no—
21#16840066—21no—
22#16840054—21no—
23#16840044—21no—
24#16839866—21no—
25#16839859—21no—
26#16839854—21no—
27#16839853—21no—
28C2×#14519713C2211yescharacteristic
29#16839805—21no—
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839802—21no—
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35#16839798—21no—
36#16839798C2211yescharacteristic
37C2×#14519708—21no—
38C2×#14519708C2211yescharacteristic
39#16839789C2211yescharacteristic
40C2×#14519704C2211yescharacteristic
41#16839770C2211yescharacteristic
42C2×#14519702C2211yescharacteristic
43C2×#14519700C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45#16839717C2211yescharacteristic
46#16839667C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839526C2211yescharacteristic
52#16839523C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#476—21no—
61C61×#475—21no—
62C61×#428C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C61×#424C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#416C2211yescharacteristic
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
77C2×#11758194C2311yescharacteristic
78C22×#92418—21no—
79C22×#92418C2311yescharacteristic
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519742—21no—
84#14519742—21no—
85D4×Dic61—21no—
86D4×Dic61—41no—
87#14519736—21no—
88#14519736C2311yescharacteristic
89#14519733—41no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519713—22no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710D411yescharacteristic
100#14519710D411yescharacteristic
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708D411yescharacteristic
108#14519708D411yescharacteristic
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111#14519704—22no—
112#14519702—21no—
113#14519702—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700—21no—
117#14519700—21no—
118#14519695—22no—
119#14519687—21no—
120C8×Dic61—21no—
121#14519684—41no—
122C2×#41×C61C2311yescharacteristic
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61—21no—
127#125×C61—21no—
128#125×C61—21no—
129C22⋊Q8×C61—41no—
130C4×Q8×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—41no—
133#117×C61—21no—
134#117×C61C2311yescharacteristic
135#117×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C4⋊C4×C61C2311yescharacteristic
139#107×C61—21no—
140#106×C61—21no—
141#106×C61—21no—
142#106×C61—21no—
143C4⋊C8×C61—21no—
144C4⋊C8×C61—21no—
145#103×C61—21no—
146#103×C61D411yescharacteristic
147#103×C61D411yescharacteristic
148#102×C61—21no—
149#102×C61D411yescharacteristic
150#102×C61D411yescharacteristic
151#41×C61—21no—
152#41×C61—21no—
153#41×C61C2×Q812yes—
154#41×C61#4112yes—
155C2×Q8×C61—41no—
156C2×Q8×C61C2×D411yescharacteristic
157C2×D4×C61—21no—
158C2×D4×C61C2×D411yescharacteristic
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—21no—
167C22×Dic61—41no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C2×Q812yes—
174M16×C61—21no—
175M16×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
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×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—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—41no—
199C22⋊Dic61—21no—
200C22⋊Dic61—41no—
201#11758202—24no—
202#11758200—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61C2×D411yescharacteristic
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—41no—
209#11758196—41no—
210#11758196—82no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—41no—
215C4×Dic61—41no—
216C4×Dic61—41no—
217#11758194—24no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194C2×Q812yes—
223#11758194#4112yes—
224C2×#92418—24no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61H(5,𝔽2)11yescharacteristic
234Q8×C61—21no—
235Q8×C61—161no—
236Q8×C61C22⋊Q812yes—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—21no—
240D4×C61C22⋊Q812yes—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—81no—
251C2×C4×C61—81no—
252C2×C4×C61—161no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—81no—
269C2×Dic61—161no—
270Dic122—22no—
271Dic122—22no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—21no—
275C8×C61—21no—
276C8×C61—41no—
277#92418—22no—
278#92418—21no—
279#92418—22no—
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—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61C2×C22⋊Q811yescharacteristic
299C4×C61#52511yescharacteristic
300C4×C61#54711yescharacteristic
301C4×C61D4×Q811yescharacteristic
302Dic61—42no—
303Dic61—42no—
304Dic61—41no—
305Dic61—41no—
306Dic61—81no—
307#3084D6111yescharacteristic
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#1882—611no—
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×#137—611no—
329C2×#131—611no—
330#544C2×D6111yescharacteristic
331#520—611no—
332#520C2×D6111yescharacteristic
333#519—611no—
334#479—21no—
335#479—1221no—
336#479—1221no—
337#478—21no—
338#477—1221no—
339#476—21no—
340#475—21no—
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—1221no—
352#428—1221no—
353#428C2×D6111yescharacteristic
354#427—611no—
355#427—1221no—
356C2×#106—611no—
357C2×#106—1221no—
358C2×#106C2×D6111yescharacteristic
359#424—611no—
360#424C2×D6111yescharacteristic
361C2×#104—611no—
362#419—611no—
363#417C2×D6111yescharacteristic
364#416C2×D6111yescharacteristic
365C2×#103—611no—
366C2×#103—611no—
367C2×#101—611no—
368#405—611no—
369#391—611no—
370#384—611no—
371#343—611no—
372#343—611no—
373#343—611no—
374#341—611no—
375#339—611no—
376#339—611no—
377#339—611no—
378#336—611no—
379C61#2640111yescharacteristic, complemented, Sylow
380C2×#41C22×D6111yescharacteristic
381C22×Q8—611no—
382#137—614no—
383#137—1222no—
384C2×Q16—1221no—
385C2×SD16—1221no—
386#131—1222no—
387#131—1222no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—611no—
391C2×M16—1221no—
392C2×M16C22×D6111yescharacteristic
393C22×C8—611no—
394C22×C8—1221no—
395C22×C8C22×D6111yescharacteristic
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398#125—21no—
399#125—21no—
400C22⋊Q8—41no—
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—1221no—
414#117—2441no—
415#117C22×D6111yescharacteristic
416#117C22×D6111yescharacteristic
417C2×C4⋊C4—21no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—2441no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C42—611no—
424#108—1222no—
425#107—21no—
426#107—611no—
427#107—611no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—21no—
433#106—21no—
434#106—21no—
435#106—611no—
436#106—611no—
437#106—1221no—
438#106—1221no—
439#106—1221no—
440#106—1221no—
441#106—1221no—
442#106—1221no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447#104—1222no—
448#103—21no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#103C22⋊D6111yescharacteristic
456#103C22⋊D6111yescharacteristic
457#102—21no—
458#102C22⋊D6111yescharacteristic
459#102C22⋊D6111yescharacteristic
460#101—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—2441no—
464#41—21no—
465#41—21no—
466#41Q8×D6112yes—
467#41#1175822612yes—
468C2×Q8—41no—
469C2×Q8—612no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8C2×C22⋊D6111yescharacteristic
476C2×D4—21no—
477C2×D4C2×C22⋊D6111yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2441no—
486C22×C4C23×D6111yescharacteristic
487Q16—1224no—
488SD16—1224no—
489M16—21no—
490M16—21no—
491M16—612no—
492M16—612no—
493M16—1224no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
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×C8C2×C22⋊D6111yescharacteristic
510C2×C8C2×C22⋊D6111yescharacteristic
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—611no—
520C4⋊C4—611no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4882no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4D4×D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C4⋊C4C2×C22⋊D6111yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—41no—
534C22⋊C4—1221no—
535C22⋊C4—2441no—
536C42—21no—
537C42—21no—
538C42—41no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—2441no—
543C42—2441no—
544C42—2441no—
545C23—21no—
546C23#1451982211yescharacteristic
547Q8—21no—
548Q8—161no—
549Q8—1222no—
550Q8—1222no—
551Q8—2441no—
552Q8—2441no—
553Q8#1451975312yes—
554D4—22no—
555D4—22no—
556D4—21no—
557D4#1451975312yes—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—161no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—9761no—
575C2×C4—4881no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C2×Q8×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822611yescharacteristic
586C2×C4#1451982411yescharacteristic
587C8—21no—
588C8—21no—
589C8—41no—
590C8—1222no—
591C8—1221no—
592C8—1221no—
593C8—1222no—
594C8—2441no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22#1684036111yescharacteristic
601C22D4×C22⋊D6111yescharacteristic
602C22#1684046611yescharacteristic
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—4881no—
611C4—2442no—
612C4—2442no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451975311yescharacteristic
616C4#1684027511yescharacteristic
617C4#1684041111yescharacteristic
618C4#1684042511yescharacteristic
619C2—21no—
620C2—21no—
621C2—42no—
622C2#2144962411yescharacteristic
623C2#2145759211yescharacteristic
624C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3084C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#3084D61
hypocenterC61#26401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3084

Complements

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

NH ≅ G/Ntype
1. #21455655C2semidirect
4. #21447359C2semidirect
5. #21447332C2semidirect
6. #21447185C2semidirect
7. #21447161C2semidirect
8. #21446597C2semidirect
9. #21446415C2semidirect
14. #21446032C2semidirect
69. #14519807Q8semidirect
379. C61#26401semidirect

Maximal subgroups

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

subgroupindex
#214556552
#214546652
#214544122
#214473592
#214473322
#214471852
#214471612
#214465972
#214464152
#214463322
#214461882
#214460962
#214460432
#214460322
C61×#30842
#2640161