Finite Group Databasebeta

← #29716011

Subgroups — #29716011

Order 15616.

1290 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840034C211yesmaximal, characteristic, complemented
2#21455477C211yesmaximal, characteristic, complemented
3#21455134C212yesmaximal, complemented
4C2×#16839882C211yesmaximal, characteristic, complemented
5#21447115C212yesmaximal, complemented
6#21446864C212yesmaximal, complemented
7#21446713C211yesmaximal, characteristic, complemented
8#21446691C211yesmaximal, characteristic, complemented
9#21445878C211yesmaximal, characteristic, complemented
10#21445778C212yesmaximal, complemented
11C2×C61×#578C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519797C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840340C2212yes—
16#16840034—24no—
17#16840017—21no—
18#16840012—21no—
19#16839988—22no—
20#16839924—22no—
21#16839918—22no—
22#16839907—22no—
23#16839901—22no—
24#16839882—24no—
25C2×#14519713C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519704C2211yescharacteristic
29C2×#14519704C2212yes—
30C2×#14519703C2211yescharacteristic
31#16839763C2212yes—
32#16839763C2212yes—
33C2×#14519701C2212yes—
34C2×#14519696C2212yes—
35#16839729C2212yes—
36C2×#14519687C2211yescharacteristic
37#16839669C2212yes—
38#16839530C2212yes—
39C2×2−1+4×C61C2212yes—
40C61×#578—24no—
41C61×#578—24no—
42C2×#137×C61—22no—
43C2×#137×C61C2212yes—
44C2×#135×C61—21no—
45C2×#135×C61C2211yescharacteristic
46C22×Q16×C61—21no—
47C22×Q16×C61C2211yescharacteristic
48C2×#131×C61C2211yescharacteristic
49C2×#11758222C2311yescharacteristic
50#14519808—24no—
51#14519808—42no—
52#14519797—22no—
53#14519797—22no—
54C2×#11758202—22no—
55C2×#11758202D411yescharacteristic
56C2×#11758200D411yescharacteristic
57C2×#11758201—22no—
58C2×#11758201D411yescharacteristic
59C2×#11758199D411yescharacteristic
60#14519783—22no—
61#14519783—22no—
62C2×#11758194C2312yes—
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C22×#92418C2311yescharacteristic
66C2×C8×D61—21no—
67#14519756—22no—
68#14519753—22no—
69#14519751—22no—
70#14519749—42no—
71#14519748—22no—
72C2×C4⋊Dic61C2312yes—
73C2×C4×Dic61C2311yescharacteristic
74#14519713—22no—
75#14519712—22no—
76#14519709—21no—
77#14519709—21no—
78#14519704—22no—
79#14519704—42no—
80#14519703—22no—
81#14519701—22no—
82#14519701—22no—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519698—22no—
87#14519696—24no—
88C22⋊#92418—22no—
89C22⋊#92418—22no—
90#14519687—22no—
91#14519684—42no—
922−1+4×C61—24no—
932−1+4×C61—24no—
942−1+4×C61—24no—
952−1+4×C61—42no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61—42no—
99C2×#41×C61C2312yes—
100C2×#41×C61C2311yescharacteristic
101C22×Q8×C61—22no—
102C22×Q8×C61—22no—
103C22×Q8×C61C2312yes—
104#137×C61—216no—
105#137×C61—24no—
106#137×C61—24no—
107#135×C61—28no—
108#135×C61—22no—
109#135×C61—22no—
110C2×Q16×C61—28no—
111C2×Q16×C61—22no—
112C2×Q16×C61—21no—
113C2×Q16×C61—21no—
114C2×Q16×C61—22no—
115C2×Q16×C61D411yescharacteristic
116C2×Q16×C61D414yescomplemented
117C2×SD16×C61—22no—
118C2×SD16×C61D412yes—
119C2×D8×C61D411yescharacteristic
120#131×C61—22no—
121#131×C61D414yescomplemented
122C2×M16×C61—21no—
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61—21no—
125C22×C8×C61C2311yescharacteristic
126#41×C61—21no—
127#41×C61—21no—
128#41×C61—28no—
129#41×C61—22no—
130#41×C61—22no—
131#41×C61—22no—
132#41×C61—22no—
133#41×C61—44no—
134#41×C61—48no—
135#41×C61—44no—
136#41×C61—48no—
137#41×C61—44no—
138#41×C61C2×D412yes—
139#41×C61#4112yes—
140C2×Q8×C61—22no—
141C2×Q8×C61—28no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—42no—
147C2×Q8×C61—44no—
148C2×Q8×C61—44no—
149C2×Q8×C61—44no—
150C2×Q8×C61C2×D414yes—
151C2×Q8×C61C2×D412yes—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61—22no—
155C2×D4×C61—42no—
156C2×D4×C61C2×D412yes—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—42no—
162C22×C4×C61—42no—
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—21no—
165C2×C22⋊D61—42no—
166C22×Dic61—21no—
167C22×Dic61—22no—
168#11758222—21no—
169#11758222—21no—
170#11758222—21no—
171#11758222—21no—
172C2×D244C2×D411yescharacteristic
173C2×C4×D61—21no—
174C2×Dic122C2×D411yescharacteristic
175Q16×C61—28no—
176Q16×C61—22no—
177Q16×C61—24no—
178Q16×C61—24no—
179SD16×C61—28no—
180SD16×C61—24no—
181D8×C61—22no—
182M16×C61—21no—
183M16×C61—24no—
184M16×C61C2×D412yes—
185C2×C8×C61—21no—
186C2×C8×C61—24no—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D412yes—
190C2×C8×C61C2×D412yes—
191C22⋊Dic61—42no—
192#11758202—22no—
193#11758202—44no—
194#11758201—22no—
195#11758201—44no—
196#11758200—22no—
197#11758199—22no—
198#11758198—42no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204#11758194—22no—
205#11758194—21no—
206#11758194—21no—
207#11758194—22no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—22no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214#11758189—42no—
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—22no—
218C23×C61H(5,𝔽2)11yescharacteristic
219C22×D61—21no—
220Q8×C61—21no—
221Q8×C61—24no—
222Q8×C61—22no—
223Q8×C61—24no—
224Q8×C61—22no—
225Q8×C61—48no—
226Q8×C61—42no—
227Q8×C61—42no—
228Q8×C61#12112yes—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—21no—
233D4×C61—22no—
234D4×C61—48no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61#12112yes—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—48no—
244C2×C4×C61—48no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C22⋊D61—41no—
256C22⋊D61—41no—
257C22⋊D61—162no—
258C2×Dic61—21no—
259C2×Dic61—42no—
260C2×Dic61—41no—
261C2×Dic61—42no—
262C2×Dic61—82no—
263D244—21no—
264D244—21no—
265C4×D61—41no—
266C4×D61—41no—
267Dic122—21no—
268Dic122—21no—
269C8×C61—22no—
270C8×C61#12012yes—
271C8×C61#12012yes—
272#92418—21no—
273#92418—21no—
274#92418—41no—
275#92418—42no—
276#24293—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61#53511yescharacteristic
284C22×C61D4211yescharacteristic
285C22×C61#54611yescharacteristic
286C2×D61—41no—
287C2×D61—42no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—44no—
291C4×C61C2×#12011yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61D4211yescharacteristic
294C4×C61#54611yescharacteristic
295Dic61—41no—
296Dic61—81no—
297Dic61—82no—
298C2×#578D6111yescharacteristic, complemented
299C2×#473—611no—
300#3060—611no—
301#3013—612no—
302C2×#444—611no—
303#2052—612no—
304#2007—612no—
305#1975—611no—
306#1971—611no—
307#1815—611no—
308#1793—612no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—42no—
312C2×C61—82no—
313C2×C61#240811yescharacteristic
314C2×C61#328912yes—
315D61—81no—
316C2×2−1+4C2×D6112yes—
317#578—24no—
318#578—24no—
319C2×#137—22no—
320C2×#137—612no—
321C2×#137C2×D6112yes—
322C2×#135—21no—
323C2×#135—611no—
324C2×#135C2×D6111yescharacteristic
325C22×Q16—21no—
326C22×Q16C2×D6111yescharacteristic
327C2×#131—611no—
328C2×#131C2×D6111yescharacteristic
329#537—612no—
330#473—1224no—
331#470—1221no—
332#469—1221no—
333#465—1222no—
334#452—1222no—
335#451—1222no—
336#449—1222no—
337#448—1222no—
338#444—1224no—
339C2×#108—611no—
340C2×#108—611no—
341C2×#106—611no—
342C2×#104—611no—
343C2×#104—611no—
344C2×#104—612no—
345#417—612no—
346#417—612no—
347C2×#103—612no—
348C2×#101—612no—
349#408—612no—
350C2×#97—611no—
351#391—612no—
352#343—612no—
353C61#2429311yescharacteristic, complemented, Sylow
3542−1+4—24no—
3552−1+4—24no—
3562−1+4—24no—
3572−1+4—42no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—42no—
361C2×#41—611no—
362C2×#41C22×D6111yescharacteristic
363C2×#41C22×D6112yes—
364C22×Q8—22no—
365C22×Q8—22no—
366C22×Q8C22×D6112yes—
367#137—216no—
368#137—24no—
369#137—24no—
370#137—1224no—
371#137—2442no—
372#135—28no—
373#135—22no—
374#135—22no—
375#135—1222no—
376#135—1222no—
377C2×Q16—28no—
378C2×Q16—22no—
379C2×Q16—21no—
380C2×Q16—21no—
381C2×Q16—22no—
382C2×Q16—611no—
383C2×Q16—1222no—
384C2×Q16C22⋊D6111yescharacteristic
385C2×Q16C22⋊D6114yescomplemented
386C2×SD16—22no—
387C2×SD16—611no—
388C2×SD16—611no—
389C2×SD16—1222no—
390C2×SD16C22⋊D6112yes—
391C2×D8—611no—
392C2×D8C22⋊D6111yescharacteristic
393#131—22no—
394#131—1222no—
395#131—1222no—
396#131C22⋊D6114yescomplemented
397C2×M16—21no—
398C2×M16—611no—
399C2×M16—612no—
400C2×M16—1221no—
401C2×M16C22×D6111yescharacteristic
402C22×C8—21no—
403C22×C8—611no—
404C22×C8—1221no—
405C22×C8C22×D6111yescharacteristic
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408#121—1222no—
409#121—1222no—
410#121—2442no—
411C2×C4⋊C4—612no—
412C2×C42—611no—
413#108—1222no—
414#108—1222no—
415#106—1221no—
416#106—1221no—
417#104—1222no—
418#104—1222no—
419#104—2442no—
420#103—1222no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#102—1222no—
425#102—1222no—
426#101—1224no—
427C22⋊C8—1222no—
428C22⋊C8—1222no—
429#97—1222no—
430#95—2442no—
431#41—21no—
432#41—21no—
433#41—28no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—44no—
439#41—48no—
440#41—44no—
441#41—48no—
442#41—44no—
443#41—1221no—
444#41—1221no—
445#41—1221no—
446#41—1221no—
447#41#1175822412yes—
448#41C2×C22⋊D6112yes—
449C2×Q8—22no—
450C2×Q8—28no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—611no—
460C2×Q8D4×D6111yescharacteristic
461C2×Q8C2×C22⋊D6114yes—
462C2×Q8C2×C22⋊D6112yes—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—42no—
466C2×D4—611no—
467C2×D4—1221no—
468C2×D4—2442no—
469C2×D4D4×D6111yescharacteristic
470C2×D4C2×C22⋊D6112yes—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—1221no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4C23×D6111yescharacteristic
480Q16—28no—
481Q16—22no—
482Q16—24no—
483Q16—24no—
484Q16—1222no—
485Q16—2444no—
486SD16—28no—
487SD16—24no—
488SD16—1222no—
489SD16—1222no—
490SD16—2444no—
491D8—22no—
492D8—1222no—
493M16—21no—
494M16—24no—
495M16—1221no—
496M16—1222no—
497M16—1221no—
498M16—1222no—
499M16—2442no—
500M16C2×C22⋊D6112yes—
501C2×C8—21no—
502C2×C8—24no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1222no—
508C2×C8—1221no—
509C2×C8—2442no—
510C2×C8C2×C22⋊D6112yes—
511C2×C8C2×C22⋊D6111yescharacteristic
512C2×C8C2×C22⋊D6111yescharacteristic
513C2×C8C2×C22⋊D6112yes—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2442no—
517C22⋊C4—2442no—
518C22⋊C4—2442no—
519C42—1221no—
520C42—1221no—
521C23—21no—
522C23—22no—
523C23—1221no—
524C23#1451982211yescharacteristic
525Q8—21no—
526Q8—24no—
527Q8—22no—
528Q8—24no—
529Q8—22no—
530Q8—48no—
531Q8—42no—
532Q8—42no—
533Q8—1221no—
534Q8—1221no—
535Q8#1451974812yes—
536D4—22no—
537D4—22no—
538D4—22no—
539D4—21no—
540D4—22no—
541D4—48no—
542D4—42no—
543D4—42no—
544D4—1221no—
545D4—1221no—
546D4—9762no—
547D4—2441no—
548D4—2441no—
549D4#1451974812yes—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—1221no—
562C2×C4—4882no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C22×C22⋊D6111yescharacteristic
570C2×C4C22×C22⋊D6111yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C2×#1175822411yescharacteristic
573C2×C4#1451982211yescharacteristic
574C8—22no—
575C8—1221no—
576C8—1221no—
577C8—2441no—
578C8—2442no—
579C8#1451974612yes—
580C8#1451974612yes—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—2442no—
588C22—2441no—
589C22#1684032911yescharacteristic
590C22D4×C22⋊D6111yescharacteristic
591C22#1684040611yescharacteristic
592C4—22no—
593C4—42no—
594C4—44no—
595C4—4881no—
596C4—4882no—
597C4—2441no—
598C4C2×#1451974611yescharacteristic
599C4C2×#1451974811yescharacteristic
600C4D4×C22⋊D6111yescharacteristic
601C4#1684040611yescharacteristic
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2—4881no—
607C2#2144916911yescharacteristic
608C2#2145759012yes—
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#578C2
socleC22×C61D42
hypercenterC2×#578D61
hypocenterC61#24293
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#16840034C2semidirect
2. #21455477C2semidirect
3. #21455134C2semidirect
4. C2×#16839882C2semidirect
5. #21447115C2semidirect
6. #21446864C2semidirect
7. #21446713C2semidirect
8. #21446691C2semidirect
9. #21445878C2semidirect
10. #21445778C2semidirect
11. C2×C61×#578C2semidirect
116. C2×Q16×C61D4semidirect
121. #131×C61D4semidirect
298. C2×#578D61semidirect
353. C61#24293semidirect
385. C2×Q16C22⋊D61semidirect
396. #131C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214554772
#214551342
C2×#168398822
#214471152
#214468642
#214467132
#214466912
#214458782
#214457782
C2×C61×#5782
#2429361