Finite Group Databasebeta

← #29714689

Subgroups — #29714689

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455620C211yesmaximal, characteristic, complemented
2#21455287C211yesmaximal, characteristic
3C4×#14519775C211yesmaximal, characteristic, complemented
4#21447033C212yesmaximal, complemented
5#21446853C212yesmaximal, complemented
6#21446797C211yesmaximal, characteristic, complemented
7#21446648C211yesmaximal, characteristic
8#21446188C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10#21445815C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519775C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16840066—22no—
16#16840054—21no—
17#16840044—21no—
18#16839981—22no—
19#16839969—21no—
20#16839959—21no—
21#16839805C2212yes—
22#16839786C2211yescharacteristic
23#16839786C2211yescharacteristic
24C2×#14519704C2212yes—
25C2×#14519704C2212yes—
26#16839770C2212yes—
27#16839763C2212yes—
28C2×#14519696C2212yes—
29#16839669C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839657—22no—
32#16839631C2211yescharacteristic
33C4×#11758196—22no—
34C42×Dic61C2211yescharacteristic
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38C61×#559—21no—
39C61×#557—21no—
40C61×#556—21no—
41C61×#556—22no—
42C61×#554—21no—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536C2212yes—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520C2212yes—
50C2×C4×Q8×C61C2211yescharacteristic
51C22×Dic122C2311yescharacteristic
52#14519807—24no—
53#14519807—42no—
54C2×#11758202—22no—
55C2×#11758200—22no—
56C2×#11758194C2312yes—
57C2×#11758194C2312yes—
58#14519775—22no—
59#14519775—22no—
60C4×Dic122—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—22no—
66C2×#11758196—21no—
67C2×#11758196—22no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73#14519710—22no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79#14519704—24no—
80#14519704—42no—
81#14519701—22no—
82#14519700—22no—
83#14519700—22no—
84#14519698—22no—
85#14519696—42no—
86#14519684—22no—
87#14519684—22no—
88C2×#41×C61C2312yes—
89C22×Q8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61D411yescharacteristic
94#127×C61D411yescharacteristic
95#126×C61—24no—
96#126×C61—24no—
97#125×C61—24no—
98#125×C61—22no—
99#125×C61D412yes—
100#124×C61—22no—
101#124×C61—22no—
102#124×C61D412yes—
103#123×C61—24no—
104#123×C61—24no—
105C22⋊Q8×C61—24no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110#121×C61—21no—
111#121×C61—21no—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61—21no—
119C4×Q8×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—24no—
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125#117×C61—24no—
126#117×C61—22no—
127#117×C61C2312yes—
128C2×C4⋊C4×C61—22no—
129C2×C4⋊C4×C61D412yes—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61D412yes—
132C2×C42×C61C2311yescharacteristic
133#41×C61—22no—
134#41×C61—22no—
135#41×C61—22no—
136#41×C61—22no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D412yes—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C2×D412yes—
150C22×C4×C61—24no—
151C22×C4×C61—22no—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122#4112yes—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—24no—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196#11758202—44no—
197#11758200—44no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—28no—
201#11758196—44no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—24no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C4×Dic61—41no—
210C4×Dic61—41no—
211C4×Dic61#4112yes—
212#11758194—22no—
213#11758194—22no—
214#11758194—22no—
215#11758194—22no—
216C2×#92418—22no—
217C2×#92418—22no—
218C23×C61—22no—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—42no—
227Q8×C61—81no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—81no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—84no—
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272#92418—42no—
273#92418—42no—
274#24179—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61C2×#12011yescharacteristic
281C22×C61#52511yescharacteristic
282C22×C61#53711yescharacteristic
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61#45612yes—
292C4×C61#54611yescharacteristic
293C4×C61#54711yescharacteristic
294C4×C61#54811yescharacteristic
295C4×C61#55011yescharacteristic
296Dic61—24no—
297Dic61—42no—
298Dic61—42no—
299Dic61—81no—
300#3507D6111yescharacteristic
301#3078—611no—
302#3037—611no—
303C4×C4⋊Q8—611no—
304#2038—612no—
305#2005—612no—
306#1994—611no—
307#1962—611no—
308#1882—612no—
309#1859—612no—
310#1801—611no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61—82no—
314C2×C61#240711yescharacteristic
315C2×C61C2×#45611yescharacteristic
316C2×C61#301811yescharacteristic
317C2×#137—612no—
318#559—21no—
319#557—21no—
320#556—21no—
321#556—22no—
322#554—21no—
323#552—22no—
324#539—22no—
325#536C2×D6112yes—
326#533C2×D6111yescharacteristic
327#532C2×D6111yescharacteristic
328C2×C4⋊Q8—611no—
329#529—22no—
330#520C2×D6112yes—
331C2×C4×Q8—611no—
332C2×C4×Q8C2×D6111yescharacteristic
333#479—1222no—
334#477—1221no—
335#475—1221no—
336#464—1222no—
337#462—1221no—
338#460—1221no—
339#428—612no—
340#423—611no—
341#423—611no—
342C2×#104—612no—
343C2×#104—612no—
344#419—612no—
345#417—612no—
346C2×#101—612no—
347#391—611no—
348#391—611no—
349#389—1222no—
350#384—611no—
351C4×C4⋊C4—1222no—
352C43—611no—
353#343—612no—
354#339—612no—
355#339—612no—
356C61#2417911yescharacteristic, complemented, Sylow
357C2×#41C22×D6112yes—
358C22×Q8—611no—
359C22×Q8C22×D6111yescharacteristic
360#137—1224no—
361#137—2442no—
362C2×Q16—1222no—
363C2×SD16—1222no—
364C2×M16—612no—
365C2×M16—612no—
366C4⋊Q8—22no—
367C4⋊Q8—1222no—
368C4⋊Q8—1222no—
369C4⋊Q8C22⋊D6111yescharacteristic
370C4⋊Q8C22⋊D6111yescharacteristic
371C4⋊Q8C22⋊D6111yescharacteristic
372#127C22⋊D6111yescharacteristic
373#126—24no—
374#126—24no—
375#125—22no—
376#125—24no—
377#125C22⋊D6112yes—
378#124—22no—
379#124—22no—
380#124C22⋊D6112yes—
381#123—24no—
382#123—24no—
383C22⋊Q8—24no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388#121—21no—
389#121—21no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—24no—
405C4×D4—21no—
406C4×D4—21no—
407#117—24no—
408#117—22no—
409#117C22×D6112yes—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4C22⋊D6112yes—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22⋊D6112yes—
423C2×C42C22×D6111yescharacteristic
424#107—1222no—
425#106—1222no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430#104—1224no—
431#104—2442no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#102—1222no—
436#101—2442no—
437#95—1222no—
438#95—1222no—
439#41—22no—
440#41—22no—
441#41—22no—
442#41—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8C2×C22⋊D6112yes—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C22⋊D6112yes—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4C2×C22⋊D6112yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6112yes—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473Q16—2444no—
474SD16—2444no—
475M16—1222no—
476M16—1222no—
477M16—1222no—
478M16—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—1228no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2444no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4C2×C22⋊D6112yes—
501C4⋊C4C2×C22⋊D6112yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—612no—
513C42—1224no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—2442no—
518C42—2441no—
519C42—2441no—
520C42C2×C22⋊D6111yescharacteristic
521C42C2×C22⋊D6111yescharacteristic
522C42C2×C22⋊D6111yescharacteristic
523C42C2×C22⋊D6111yescharacteristic
524C23—22no—
525C23C22×C22⋊D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—81no—
534Q8—1222no—
535Q8—1222no—
536Q8—2441no—
537Q8—2441no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—81no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4#1451974611yescharacteristic
569C2×C4#1451974611yescharacteristic
570C2×C4#1451974611yescharacteristic
571C2×C4#1451974611yescharacteristic
572C2×C4C22×C22⋊D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#1451982311yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—2442no—
579C8—2442no—
580C22—21no—
581C22—21no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C2×#1451974611yescharacteristic
586C22#1684026911yescharacteristic
587C22#1684034011yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—1224no—
597C4—4881no—
598C4—2442no—
599C4—2442no—
600C4#1683994512yes—
601C4#1684040011yescharacteristic
602C4#1684041111yescharacteristic
603C4#1684041911yescharacteristic
604C4#1684043111yescharacteristic
605C2—21no—
606C2—21no—
607C2—82no—
608C2#2144915811yescharacteristic
609C2C2×#1683994511yescharacteristic
610C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#24179
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21455620C2semidirect
3. C4×#14519775C2semidirect
4. #21447033C2semidirect
5. #21446853C2semidirect
6. #21446797C2semidirect
8. #21446188C2semidirect
9. #21446045C2semidirect
10. #21445815C2semidirect
356. C61#24179semidirect

Maximal subgroups

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

subgroupindex
#214556202
#214552872
C4×#145197752
#214470332
#214468532
#214467972
#214466482
#214461882
#214460452
#214458152
C61×#35072
#2417961