Finite Group Databasebeta

← #29707621

Subgroups — #29707621

Order 15616.

1586 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840326C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C2×#16839779C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21447027C218yesmaximal, complemented
7#21445530C211yesmaximal, characteristic, complemented
8C61×#2896C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840326—22no—
12#16840326—22no—
13#16840326—22no—
14#16840326C2214yes—
15C2×#14519773—21no—
16C2×#14519766—21no—
17C2×#14519751—22no—
18C2×#14519745—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839805—22no—
22#16839802C2214yescomplemented
23#16839798—22no—
24#16839779—24no—
25C2×#14519704C2214yescomplemented
26C2×#14519693—24no—
27C2×#14519693C2214yescomplemented
28#16839664—22no—
29#16839663C2214yescomplemented
30#16839617—22no—
31#16839523C2214yescomplemented
32C22×#41×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C61×#419—21no—
36C61×#418—21no—
37C61×#417C2214yescomplemented
38C61×#416—22no—
39C24×D61D412yes—
40C22×D244—22no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244D412yes—
49C22×D244C2312yes—
50C22×C22⋊D61—24no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C2×#11758194D414yescomplemented
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519773—24no—
57#14519773—24no—
58#14519766—24no—
59#14519766—24no—
60#14519751—28no—
61#14519751—28no—
62#14519745—28no—
63#14519745—28no—
64#14519736—21no—
65#14519736—21no—
66#14519736—22no—
67#14519736—22no—
68#14519736D412yes—
69#14519736C2312yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D412yes—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—24no—
75C2×#11758198—28no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2312yes—
78#14519710—24no—
79#14519708—24no—
80#14519704—28no—
81#14519693—28no—
82C2×#41×C61—21no—
83C2×#41×C61—21no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61D412yes—
89C2×#41×C61C2312yes—
90C22×Q8×C61D411yescharacteristic
91C22×D4×C61—21no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61C2311yescharacteristic
95C2×M16×C61—21no—
96C2×M16×C61—22no—
97C2×M16×C61—22no—
98C2×M16×C61C2312yes—
99C22×C8×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107#103×C61—24no—
108#102×C61—24no—
109C24×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—24no—
114C23×D61—24no—
115C23×D61—84no—
116C23×D61C2×D414yes—
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C2×D412yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—216no—
156C2×C22⋊D61—88no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—24no—
162C2×D244—24no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—42no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244—42no—
173C2×D244—42no—
174C2×D244—42no—
175C2×D244—42no—
176C2×D244—42no—
177M16×C61—22no—
178M16×C61—22no—
179M16×C61—42no—
180C2×C8×C61—22no—
181C2×C8×C61—42no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—41no—
187C42×C61—41no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—42no—
191#11758198—216no—
192#11758198—88no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4⋊Dic61C2×D414yes—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200#11758194—24no—
201#11758194—22no—
202#11758194C2×D414yescomplemented
203C2×#92418—24no—
204C2×#92418—22no—
205C23×C61—22no—
206C23×C61—42no—
207C23×C61—44no—
208C23×C61—42no—
209C23×C61#12011yescharacteristic
210C23×C61#12011yescharacteristic
211C23×C61#12112yes—
212C23×C61#12312yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61—84no—
222C22×D61—88no—
223C22×D61—88no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—42no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—42no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—42no—
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12112yes—
247C2×C4×C61#12312yes—
248C2×C4×C61#12312yes—
249C2×C4×C61#12312yes—
250C2×C4×C61#12412yes—
251C2×C4×C61#12412yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—816no—
259C2×Dic61—24no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263D244—42no—
264D244—42no—
265D244—42no—
266D244—42no—
267D244—44no—
268D244—44no—
269C8×C61—42no—
270#92418—24no—
271#23493—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—84no—
280C22×C61#39412yes—
281C22×C61#39412yes—
282C22×C61C2×#12011yescharacteristic
283C22×C61C2×#12111yescharacteristic
284C22×C61C2×#12311yescharacteristic
285C2×D61—48no—
286C2×D61—44no—
287C2×D61—48no—
288C2×D61—88no—
289C2×D61—88no—
290C2×D61—84no—
291C4×C61—42no—
292C4×C61—82no—
293C4×C61#39412yes—
294C4×C61#39412yes—
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×#12311yescharacteristic
298C4×C61C2×#12411yescharacteristic
299Dic61—44no—
300Dic61—82no—
301C2×#535—611no—
302#2911—611no—
303C2×#421—611no—
304#2896D6111yescharacteristic, complemented
305C22×#100—611no—
306#2309—611no—
307#2037—618no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—22no—
312C2×C61—22no—
313C2×C61—82no—
314C2×C61C2×#39411yescharacteristic
315C2×C61#301912yes—
316D61—84no—
317C22×#41C2×D6111yescharacteristic
318C23×D4—611no—
319C22×M16—611no—
320C22×M16C2×D6111yescharacteristic
321#535—614no—
322#535—1222no—
323#535—1222no—
324#535—1222no—
325C2×#127—1221no—
326C2×#124—1221no—
327C2×#121—1222no—
328C2×#120—1222no—
329C2×#117—611no—
330C2×#117—611no—
331C2×#117C2×D6111yescharacteristic
332#428—614no—
333#428—1222no—
334#427—1222no—
335#421—1224no—
336C2×#104—614no—
337#419—21no—
338#418—21no—
339#417C2×D6114yescomplemented
340#416—22no—
341C2×#100—614no—
342C2×#100—1224no—
343#390—614no—
344#390—1222no—
345#379—1222no—
346#339—614no—
347C61#2349311yescharacteristic, complemented, Sylow
348C25—612no—
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C22⋊D6112yes—
356C2×#41C22×D6112yes—
357C22×Q8C22⋊D6111yescharacteristic
358C22×D4—21no—
359C22×D4—612no—
360C22×D4—612no—
361C22×D4—1224no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—1221no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4C22⋊D6111yescharacteristic
371C23×C4—21no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—21no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—614no—
377C2×M16—612no—
378C2×M16—1222no—
379C2×M16—1221no—
380C2×M16C22×D6112yes—
381C22×C8—21no—
382C22×C8—1221no—
383#127—1224no—
384#127—1224no—
385#124—1224no—
386#124—1224no—
387#121—1228no—
388#121—1228no—
389#120—1228no—
390#120—1228no—
391#117—21no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—612no—
396#117—612no—
397#117—1221no—
398#117—1221no—
399#117—1222no—
400#117—1222no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6112yes—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1228no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—612no—
411C2×C42—1221no—
412#107—1224no—
413#106—1224no—
414#104—1228no—
415#103—24no—
416#102—24no—
417#100—1228no—
418C24—21no—
419C24—614no—
420C24—1222no—
421C24—1224no—
422C24—1224no—
423C24—1222no—
424C24—4884no—
425C24D4×D6111yescharacteristic
426#41—22no—
427#41—22no—
428#41—22no—
429#41—22no—
430#41—24no—
431#41—24no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—42no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C2×C22⋊D6112yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—12216no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—4888no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4—2442no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4#1175822212yes—
477C22×C4#1175822212yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6112yes—
481C22×C4D4×D6111yescharacteristic
482C22×C4#1175822412yes—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—22no—
487M16—22no—
488M16—42no—
489M16—614no—
490M16—1222no—
491M16—1224no—
492C2×C8—22no—
493C2×C8—42no—
494C2×C8—1222no—
495C2×C8—1224no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—614no—
499C4⋊C4—1222no—
500C4⋊C4—2442no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—12216no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—4888no—
507C22⋊C4—2442no—
508C42—41no—
509C42—41no—
510C42—1222no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C23—22no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—1224no—
519C23—1224no—
520C23—1224no—
521C23—4884no—
522C23—4888no—
523C23—4888no—
524C23—2448no—
525C23—2448no—
526C23—2444no—
527C23—2444no—
528C23#1451974511yescharacteristic
529C23#1451974511yescharacteristic
530C23#1451975012yes—
531C23#1451975912yes—
532C23C2×D4×D6111yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—42no—
542D4—48816no—
543D4—2442no—
544D4—2442no—
545D4—2442no—
546D4—2442no—
547D4—2444no—
548D4—2444no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—1224no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4#1451974511yescharacteristic
561C2×C4#1451974511yescharacteristic
562C2×C4#1451974912yes—
563C2×C4#1451974912yes—
564C2×C4#1451975012yes—
565C2×C4#1451975012yes—
566C2×C4#1451975012yes—
567C2×C4#1451975912yes—
568C2×C4#1451976012yes—
569C2×C4#1451976012yes—
570C2×C4#1451976312yes—
571C2×C4#1451976312yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822211yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—42no—
579C8—1224no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—84no—
588C22—4884no—
589C22—4888no—
590C22—4888no—
591C22—2448no—
592C22—2444no—
593C22—2448no—
594C22#1683967712yes—
595C22#1683967712yes—
596C22C2×#1451974511yescharacteristic
597C22C2×#1451975011yescharacteristic
598C22C2×#1451975911yescharacteristic
599C4—42no—
600C4—82no—
601C4—4882no—
602C4—2444no—
603C4#1683967712yes—
604C4#1683967712yes—
605C4C2×#1451974911yescharacteristic
606C4C2×#1451975011yescharacteristic
607C4C2×#1451976011yescharacteristic
608C4C2×#1451976311yescharacteristic
609C2—21no—
610C2—22no—
611C2—22no—
612C2—21no—
613C2—82no—
614C2—4884no—
615C2C2×#1683967711yescharacteristic
616C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radicalC2×#16840326C2
hypercenter#2896D61
hypocenterC61#23493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
3. C2×#16839779C2semidirect
4. C22×#14519693C2semidirect
5. #21448399C2semidirect
6. #21447027C2semidirect
7. #21445530C2semidirect
8. C61×#2896C2semidirect
22. #16839802C22semidirect
25. C2×#14519704C22semidirect
27. C2×#14519693C22semidirect
29. #16839663C22semidirect
31. #16839523C22semidirect
37. C61×#417C22semidirect
53. C2×#11758194D4semidirect
202. #11758194C2×D4semidirect
304. #2896D61semidirect
339. #417C2×D61semidirect
347. C61#23493semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
#214544082
C2×#168397792
C22×#145196932
#214483992
#214470272
#214455302
C61×#28962
#2349361