Finite Group Databasebeta

← #29680846

Subgroups — #29680846

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459021C211yesmaximal, characteristic
2#428×D61C212yesmaximal, complemented
3C2×#16839745C211yesmaximal, characteristic
4#21448410C211yesmaximal, characteristic
5#21446036C211yesmaximal, characteristic, complemented
6#21446034C211yesmaximal, characteristic, complemented
7#21446033C211yesmaximal, characteristic, complemented
8#21446030C212yesmaximal, complemented
9#21446027C212yesmaximal, complemented
10#21445527C212yesmaximal, complemented
11C61×#1857C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840244—22no—
15#16840244—22no—
16#16840240C2211yescharacteristic
17#16840236—22no—
18#16840236C2212yes—
19C4×#11758226—22no—
20C4×#11758226—22no—
21C2×#14519734C2211yescharacteristic
22#117×D61—22no—
23#117×D61C2212yes—
24C2×C4⋊C4×D61C2211yescharacteristic
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#107×D61—22no—
31#106×D61—22no—
32#16839745C418yescomplemented
33C2×#14519696C2211yescharacteristic
34C2×#14519693C2212yes—
35#16839669C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839663C2211yescharacteristic
38#16839640—22no—
39#16839621—22no—
40#16839523C2212yes—
41#16839522C2212yes—
42#16839521C2212yes—
43C61×#519C2211yescharacteristic
44C61×#428C2212yes—
45C2×#101×C61C2211yescharacteristic
46C61×#391C2211yescharacteristic
47C61×#339C2212yes—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226—24no—
51C2×#11758226C2×C414yes—
52C2×#11758226Q814yes—
53C22×D244D412yes—
54C22×D244C2311yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61D412yes—
57C22×C4×D61C2311yescharacteristic
58C2×#11758194C2312yes—
59C2×#11758189—22no—
60M16×D61—24no—
61M16×D61C2×C418yescomplemented
62C2×C8×D61—22no—
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65C4×D244—24no—
66C4×D244—24no—
67C4×D244—22no—
68C4×D244—22no—
69#14519739—24no—
70#14519739—24no—
71#14519739—22no—
72#14519739—22no—
73#14519736—22no—
74#14519736C2312yes—
75#14519735—22no—
76#14519735—22no—
77#14519735D412yes—
78#14519735D412yes—
79#14519734—24no—
80#14519734—22no—
81#14519734—24no—
82#14519733—22no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61D412yes—
91C4⋊C4×D61D412yes—
92C2×#11758198—21no—
93C2×#11758198—22no—
94C2×#11758198—22no—
95C22⋊C4×D61—24no—
96C22⋊C4×D61—22no—
97C2×C4×Dic61C2311yescharacteristic
98C42×D61—24no—
99C42×D61—22no—
100#14519711—22no—
101#14519710—22no—
102#14519710—22no—
103#14519709—22no—
104#14519708—22no—
105#14519708—22no—
106#14519696C2×C414yescomplemented
107#14519693C2×C418yescomplemented
108#14519684—22no—
109C22×Q8×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#107×C61—22no—
118#106×C61—22no—
119#101×C61C2×C414yescomplemented
120#95×C61—22no—
121C23×D61C2×D412yes—
122C23×D61C2×D411yescharacteristic
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61C22×C412yes—
126C2×Q8×C61C2×Q812yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134#11758226—28no—
135#11758226—28no—
136#11758226—28no—
137#11758226—28no—
138C2×D244—24no—
139C2×D244—24no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244C4⋊C418yes—
143C2×D244C22×C412yes—
144C2×D244C2×Q812yes—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—42no—
153C2×C4×D61—82no—
154C2×C4×D61C4⋊C418yes—
155C2×C4×D61C22×C412yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×Q812yes—
159C2×C4×D61#4114yes—
160M16×C61—22no—
161M16×C61C22×C414yes—
162C2×C8×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C22⋊C4×C61—22no—
169C42×C61—24no—
170C42×C61—22no—
171C42×C61—22no—
172C22⋊Dic61—22no—
173#11758198—24no—
174#11758198—24no—
175#11758198—22no—
176#11758198—22no—
177#11758198—42no—
178#11758198—82no—
179#11758198—82no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61C2×D412yes—
184C4⋊Dic61C2×D412yes—
185#11758196—22no—
186#11758196—22no—
187#11758196—41no—
188#11758196—41no—
189#11758196—42no—
190C4×Dic61—24no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—22no—
196#11758194C22×C414yes—
197C2×#92418—22no—
198#11758189—28no—
199C8×D61—28no—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61C2×C4⋊C414yes—
208C22×D61C2×C4⋊C412yes—
209C22×D61#12012yes—
210C22×D61#12312yes—
211Q8×C61—24no—
212Q8×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C2×C4⋊C414yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D412yes—
225C2×C4×C61C22×Q811yescharacteristic
226C2×C4×C61C2×#4112yes—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61C2×C4⋊C412yes—
235C2×Dic61#12011yescharacteristic
236C2×Dic61#12011yescharacteristic
237C2×Dic61#12312yes—
238D244—216no—
239D244—24no—
240D244—24no—
241C4×D61—28no—
242C4×D61—28no—
243C4×D61—82no—
244C4×D61—82no—
245C4×D61—84no—
246C4×D61—84no—
247C4×D61—84no—
248C4×D61C4×D414yes—
249C4×D61C4×D414yes—
250C4×D61C22⋊Q814yes—
251C4×D61C22⋊Q814yes—
252C8×C61—24no—
253#92418—24no—
254#20858—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61C22×C4⋊C411yescharacteristic
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×#12311yescharacteristic
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—24no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—44no—
268C2×D61—82no—
269C2×D61#38514yes—
270C4×C61—24no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C2×C4×D412yes—
276C4×C61C2×C22⋊Q812yes—
277Dic61—42no—
278Dic61—42no—
279Dic61—81no—
280Dic61—81no—
281Dic61#38512yes—
282Dic61#38512yes—
283#3428—611no—
284C2×#428—612no—
285C2×#413—611no—
286#2311—611no—
287#1857—611no—
288#1857—611no—
289#1857—611no—
290#1857D6111yescharacteristic, complemented
291#1856—612no—
292#1856—612no—
293#1733—612no—
294C2×C61—21no—
295C2×C61—21no—
296C2×C61C2×#38511yescharacteristic
297C2×C61#302111yescharacteristic
298C2×C61#302211yescharacteristic
299D61—24no—
300D61—44no—
301C22×#41—611no—
302C22×M16—612no—
303#520—1222no—
304#520—1222no—
305#519—611no—
306#519C2×D6111yescharacteristic
307#518—612no—
308#518—1222no—
309C4×#41—1222no—
310C4×#41—1222no—
311C2×#117—611no—
312C2×#117—612no—
313C2×#117—1222no—
314C22×C4⋊C4—611no—
315#428—612no—
316#428—612no—
317#428—612no—
318#428—1222no—
319#428—1222no—
320#428C2×D6112yes—
321C2×#107—1222no—
322C2×#106—1222no—
323#413—618no—
324C2×#101—611no—
325C2×#101C2×D6111yescharacteristic
326C2×#100—612no—
327#391—611no—
328#391C2×D6111yescharacteristic
329#390—611no—
330#390—611no—
331#386—1222no—
332#380—1222no—
333#339—612no—
334#339—612no—
335#339—612no—
336#339C2×D6112yes—
337C61#2085811yescharacteristic, complemented, Sylow
338C2×#41—614no—
339C2×#41—614no—
340C2×#41—1222no—
341C2×#41—1222no—
342C2×#41—1224no—
343C22×Q8C22×D6111yescharacteristic
344C22×D4—612no—
345C22×D4—611no—
346C23×C4—611no—
347C23×C4—612no—
348C23×C4—1221no—
349C2×M16—612no—
350C2×M16—618no—
351C2×M16—1222no—
352C2×M16—1224no—
353C2×M16C22×D6112yes—
354C22×C8—1222no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×D4—1222no—
360C4×D4—1222no—
361C4×D4—1224no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1224no—
365C4×D4—1222no—
366C4×D4—1222no—
367#117—22no—
368#117—612no—
369#117—612no—
370#117—612no—
371#117—1222no—
372#117—1224no—
373#117—1224no—
374#117—1222no—
375#117—1222no—
376#117—1222no—
377#117—1222no—
378#117C22×D6112yes—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1221no—
394C2×C42—611no—
395C2×C42—1222no—
396C2×C42—1224no—
397C2×C42C22×D6111yescharacteristic
398#107—22no—
399#107—1222no—
400#107—1222no—
401#107—1222no—
402#106—22no—
403#106—1222no—
404#106—1222no—
405#106—1222no—
406#101—614no—
407#101C4×D6114yescomplemented
408#100—618no—
409#95—22no—
410#95—1222no—
411C24—611no—
412C24—612no—
413#41—1228no—
414#41—1228no—
415#41—1228no—
416#41—1228no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8C2×C4×D6112yes—
420C2×Q8Q8×D6112yes—
421C2×D4—612no—
422C2×D4—612no—
423C2×D4—618no—
424C2×D4—1224no—
425C2×D4—1224no—
426C2×D4—1222no—
427C2×D4—1222no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—614no—
436C22×C4—618no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—4882no—
447C22×C4—2442no—
448C22×C4D4×D6112yes—
449C22×C4C23×D6111yescharacteristic
450M16—22no—
451M16—614no—
452M16—1228no—
453M16—1222no—
454M16C2×C4×D6114yes—
455C2×C8—22no—
456C2×C8—1228no—
457C2×C8—1222no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—612no—
462C4⋊C4—612no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2442no—
471C4⋊C4D4×D6112yes—
472C4⋊C4D4×D6112yes—
473C22⋊C4—22no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—1222no—
479C22⋊C4—4882no—
480C22⋊C4—4882no—
481C22⋊C4—2442no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—1222no—
486C42—1222no—
487C42—1224no—
488C42—1222no—
489C42—1222no—
490C23—612no—
491C23—612no—
492C23—612no—
493C23—614no—
494C23—1222no—
495C23—1222no—
496C23—2441no—
497C23—2441no—
498C23—2442no—
499C23—2442no—
500C23C2×D4×D6111yescharacteristic
501Q8—24no—
502Q8—24no—
503D4—12216no—
504D4—1224no—
505D4—1224no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—612no—
519C2×C4—614no—
520C2×C4—614no—
521C2×C4—614no—
522C2×C4—614no—
523C2×C4—1228no—
524C2×C4—1228no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—4882no—
528C2×C4—4882no—
529C2×C4—4884no—
530C2×C4—4884no—
531C2×C4—4884no—
532C2×C4—2441no—
533C2×C4—2442no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4C4⋊C4×D6114yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×D4×D6112yes—
540C2×C4C2×Q8×D6111yescharacteristic
541C2×C4#41×D6112yes—
542C8—24no—
543C8—1224no—
544C22—21no—
545C22—21no—
546C22—614no—
547C22—1222no—
548C22—1222no—
549C22—1224no—
550C22—4882no—
551C22—2442no—
552C22—2442no—
553C22—2444no—
554C22—2442no—
555C22—2442no—
556C22C2×C4⋊C4×D6111yescharacteristic
557C22#120×D6111yescharacteristic
558C22#123×D6111yescharacteristic
559C4—24no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—612no—
565C4—612no—
566C4—4881no—
567C4—4881no—
568C4—2442no—
569C4—2442no—
570C4C4×D4×D6112yes—
571C4C22⋊Q8×D6112yes—
572C2—21no—
573C2—21no—
574C2—1224no—
575C2—2444no—
576C2#385×D6111yescharacteristic
577C2#2145518911yescharacteristic
578C2#2145519611yescharacteristic
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1857C2
socleC22×C61C2×#120
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1857D61
hypocenterC61#20858
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
2. #428×D61C2semidirect
5. #21446036C2semidirect
6. #21446034C2semidirect
7. #21446033C2semidirect
8. #21446030C2semidirect
9. #21446027C2semidirect
10. #21445527C2semidirect
11. C61×#1857C2semidirect
32. #16839745C4semidirect
61. M16×D61C2×C4semidirect
106. #14519696C2×C4semidirect
107. #14519693C2×C4semidirect
119. #101×C61C2×C4semidirect
290. #1857D61semidirect
337. C61#20858semidirect
407. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590212
#428×D612
C2×#168397452
#214484102
#214460362
#214460342
#214460332
#214460302
#214460272
#214455272
C61×#18572
#2085861