Finite Group Databasebeta

← #29681589

Subgroups — #29681589

Order 15616.

1818 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2#21459017C211yesmaximal, characteristic, complemented
3#21454333C212yesmaximal, complemented
4C2×#16839743C211yesmaximal, characteristic, complemented
5#21446219C212yesmaximal, complemented
6#21446218C211yesmaximal, characteristic, complemented
7#21446211C212yesmaximal, complemented
8#21446207C211yesmaximal, characteristic, complemented
9#21446205C211yesmaximal, characteristic, complemented
10#339×D61C212yesmaximal, complemented
11C61×#1887C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14C2×M16×D61C2212yes—
15#16840310—24no—
16#16840310C418yescomplemented
17C2×#14519777C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519748—22no—
21#16840241—24no—
22#16840239C2212yes—
23#16840237C2211yescharacteristic
24#16840235—22no—
25C4×#11758224—24no—
26C2×D4×Dic61—22no—
27C2×#14519733—22no—
28#117×D61C2212yes—
29C2×C42×D61C2211yescharacteristic
30#16839772C2212yes—
31#16839770C2212yes—
32#16839767C2212yes—
33#16839760—24no—
34#16839755—24no—
35#16839743—24no—
36C2×#14519695C2212yes—
37C2×#14519694C2211yescharacteristic
38#16839523C2212yes—
39#16839522C2212yes—
40#16839521C2212yes—
41C2×#127×C61C2211yescharacteristic
42C61×#518C2211yescharacteristic
43C61×#418C2212yes—
44C2×#100×C61C2211yescharacteristic
45C61×#339C2212yes—
46C2×#11758224—22no—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—24no—
52C2×#11758224—24no—
53C2×#11758224C2×C414yes—
54C2×#11758224C2×C414yescomplemented
55C2×#11758224D414yes—
56C22×Dic122C2311yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C22⋊D61D412yes—
60C23×Dic61—21no—
61C23×Dic61D412yes—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C2×#11758194C2312yes—
65C2×#11758189—22no—
66M16×D61—24no—
67M16×D61—24no—
68C2×C8×D61—22no—
69#14519777—22no—
70#14519777C2×C414yescomplemented
71#14519764—22no—
72#14519764—22no—
73#14519764C2×C414yescomplemented
74#14519764C2×C414yescomplemented
75#14519748—216no—
76C4×Dic122—24no—
77#14519742—24no—
78D4×Dic61—216no—
79D4×Dic61—24no—
80C4×C22⋊D61—28no—
81#14519737—28no—
82#14519736C2312yes—
83#14519735D414yescomplemented
84#14519734—22no—
85#14519733—24no—
86#14519733—24no—
87#14519733—22no—
88#14519733—24no—
89C2×C4⋊Dic61—22no—
90C2×#11758196—24no—
91C4⋊C4×D61D414yescomplemented
92C2×C22⋊Dic61—22no—
93C2×C22⋊Dic61—22no—
94C2×C22⋊Dic61—22no—
95C22⋊C4×D61—22no—
96C2×C4×Dic61—24no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C42×D61—22no—
100C42×D61—22no—
101C42×D61C2×C414yescomplemented
102#14519702—24no—
103#14519700—24no—
104#14519698—24no—
105#14519695—24no—
106#14519694—22no—
107C22×D4×C61—22no—
108C22×D4×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111#127×C61—22no—
112#127×C61C2×C414yescomplemented
113C4×D4×C61—24no—
114#117×C61C2312yes—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61C2311yescharacteristic
117#102×C61—24no—
118#100×C61—22no—
119C24×C61—21no—
120C24×C61C2×D412yes—
121C23×D61C2×D411yescharacteristic
122C2×D4×C61—28no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—48no—
129C2×D4×C61C22×C412yes—
130C2×D4×C61C22×C412yes—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—24no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61C22⋊C418yes—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—24no—
148C22×Dic61—41no—
149C22×Dic61—41no—
150C22×Dic61—42no—
151C22×Dic61C22⋊C418yes—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D411yescharacteristic
154#11758224—28no—
155#11758224—28no—
156#11758224—28no—
157#11758224—28no—
158#11758224—28no—
159#11758224—28no—
160#11758224—44no—
161#11758224—44no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—44no—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61#4114yes—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×Dic122C22×C412yes—
177C2×Dic122C2×D412yes—
178M16×C61—22no—
179M16×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61C2×D414yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—22no—
184C42×C61—21no—
185C42×C61—22no—
186C42×C61C22×C412yes—
187C22⋊Dic61—24no—
188C22⋊Dic61—28no—
189C22⋊Dic61—22no—
190C22⋊Dic61—28no—
191#11758198—24no—
192C4⋊Dic61—28no—
193C4⋊Dic61C2×D414yes—
194#11758196—28no—
195#11758196—24no—
196C4×Dic61—28no—
197C4×Dic61—21no—
198C4×Dic61—22no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C4×Dic61C22×C412yes—
202#11758194—22no—
203#11758194—22no—
204C2×#92418—22no—
205#11758189—44no—
206C8×D61—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—44no—
211C23×C61—44no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61C2×C22⋊C414yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—42no—
217C22×D61C2×C22⋊C412yes—
218C22×D61#12012yes—
219C22×D61#12312yes—
220D4×C61—24no—
221D4×C61—24no—
222D4×C61—24no—
223D4×C61—416no—
224D4×C61—42no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—84no—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4112yes—
233C22⋊D61—216no—
234C22⋊D61—48no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—28no—
238C2×Dic61—28no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—44no—
246C2×Dic61—44no—
247C2×Dic61—84no—
248C2×Dic61C2×C22⋊C414yes—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61#12011yescharacteristic
251C2×Dic61#12011yescharacteristic
252C2×Dic61#12312yes—
253C4×D61—48no—
254C4×D61—84no—
255C4×D61C4×D414yes—
256C4×D61C4×D414yes—
257C4×D61#12114yes—
258C4×D61#12114yes—
259Dic122—24no—
260Dic122—24no—
261Dic122—24no—
262Dic122—42no—
263C8×C61—42no—
264#92418—42no—
265#20952—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—42no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C2×#12011yescharacteristic
277C22×C61C2×#12311yescharacteristic
278C2×D61—24no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61#38614yes—
282C4×C61—24no—
283C4×C61—44no—
284C4×C61C2×C4×D412yes—
285C4×C61C2×#12112yes—
286Dic61—24no—
287Dic61—42no—
288Dic61—44no—
289Dic61—42no—
290Dic61#38612yes—
291Dic61#38612yes—
292C2×#532—611no—
293#3428—611no—
294#2896—612no—
295C2×#413—611no—
296#1888—612no—
297#1887—611no—
298#1887D6111yescharacteristic, complemented
299#1886—612no—
300#1885—611no—
301#1885—611no—
302C2×#339—612no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—44no—
306C2×C61—82no—
307C2×C61C2×#38611yescharacteristic
308C2×C61#301812yes—
309D61—24no—
310C22×#41—611no—
311C22×#41—611no—
312C22×M16—612no—
313#532—618no—
314#532—1224no—
315C2×C4⋊Q8—611no—
316C2×#127C2×D6111yescharacteristic
317C2×#124—611no—
318C2×#124—611no—
319C2×#121—1222no—
320#520—1224no—
321#519—612no—
322#518—611no—
323#518—1222no—
324#518C2×D6111yescharacteristic
325C4×#41—1224no—
326C2×C4×D4—1222no—
327C2×#117—612no—
328C2×#117—1222no—
329C22×C42—611no—
330#419—612no—
331#419—612no—
332#418—612no—
333#418C2×D6112yes—
334#417—1224no—
335#416—1224no—
336#413—1224no—
337C2×#101—612no—
338C2×#100—611no—
339C2×#100C2×D6111yescharacteristic
340#339—612no—
341#339—612no—
342#339—612no—
343#339C2×D6112yes—
344C61#2095211yescharacteristic, complemented, Sylow
345C2×#41—614no—
346C2×#41—614no—
347C2×#41—614no—
348C2×#41—1222no—
349C2×#41—1222no—
350C2×#41—1224no—
351C2×#41—1222no—
352C2×#41—1224no—
353C2×#41—1222no—
354C2×#41—1222no—
355C22×Q8—611no—
356C22×Q8—611no—
357C22×D4—22no—
358C22×D4—612no—
359C22×D4—1221no—
360C22×D4C22×D6111yescharacteristic
361C22×D4C22×D6111yescharacteristic
362C23×C4—611no—
363C23×C4—612no—
364C23×C4—1221no—
365C23×C4—1221no—
366C2×M16—612no—
367C2×M16—1222no—
368C2×M16—1224no—
369C2×M16—1224no—
370C2×M16C22×D6112yes—
371C22×C8—1222no—
372C4⋊Q8—614no—
373C4⋊Q8—1222no—
374#127—22no—
375#127C4×D6114yescomplemented
376#124—614no—
377#124—614no—
378#124—1222no—
379#124—1222no—
380#121—12216no—
381C4×Q8—1224no—
382C4×Q8—1224no—
383C4×D4—24no—
384C4×D4—1228no—
385C4×D4—1224no—
386C4×D4—1228no—
387C4×D4—12216no—
388#117—612no—
389#117—614no—
390#117—1222no—
391#117—1222no—
392#117—1224no—
393#117—1224no—
394#117—1224no—
395#117C22×D6112yes—
396C2×C4⋊C4—614no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1224no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1222no—
404C2×C42—611no—
405C2×C42—614no—
406C2×C42—1222no—
407C2×C42—1222no—
408C2×C42—1221no—
409C2×C42—1224no—
410C2×C42C22×D6111yescharacteristic
411#103—1224no—
412#103—1224no—
413#102—24no—
414#102—1224no—
415#101—1224no—
416#100—22no—
417#100—1222no—
418C24—21no—
419C24—611no—
420C24D4×D6112yes—
421#41—1228no—
422#41—1228no—
423#41—1228no—
424#41—1228no—
425#41—1228no—
426#41—1228no—
427#41—2444no—
428#41—2444no—
429C2×Q8—612no—
430C2×Q8—612no—
431C2×Q8—612no—
432C2×Q8—1222no—
433C2×Q8—1221no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—48no—
444C2×D4—618no—
445C2×D4—1224no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4C2×C4×D6112yes—
451C2×D4C2×C4×D6112yes—
452C2×D4D4×D6112yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—611no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—612no—
462C22×C4—618no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1224no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—2444no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2442no—
476C22×C4C23×D6111yescharacteristic
477M16—22no—
478M16—22no—
479M16—1222no—
480M16—1222no—
481M16—2444no—
482C2×C8—22no—
483C2×C8—1222no—
484C2×C8—2444no—
485C4⋊C4—614no—
486C4⋊C4—1224no—
487C4⋊C4—1228no—
488C4⋊C4—1228no—
489C4⋊C4D4×D6114yes—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—1224no—
493C22⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—1222no—
496C22⋊C4—1228no—
497C42—21no—
498C42—22no—
499C42—612no—
500C42—1224no—
501C42—1224no—
502C42—1228no—
503C42—1221no—
504C42—1222no—
505C42C2×C4×D6112yes—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—44no—
510C23—44no—
511C23—41no—
512C23—41no—
513C23—612no—
514C23—612no—
515C23—612no—
516C23—2442no—
517C23C22⋊C4×D6114yes—
518C23C2×D4×D6111yescharacteristic
519Q8—1224no—
520Q8—1224no—
521Q8—1224no—
522Q8—2442no—
523D4—24no—
524D4—24no—
525D4—24no—
526D4—416no—
527D4—42no—
528D4—12216no—
529D4—2448no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—84no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—612no—
538C2×C4—614no—
539C2×C4—614no—
540C2×C4—614no—
541C2×C4—614no—
542C2×C4—614no—
543C2×C4—1228no—
544C2×C4—1228no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1224no—
550C2×C4—4884no—
551C2×C4—4884no—
552C2×C4—2448no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C2×D4×D6112yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4#41×D6112yes—
562C8—42no—
563C8—2442no—
564C22—22no—
565C22—24no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—614no—
574C22—1224no—
575C22—2442no—
576C22—2442no—
577C22C2×C22⋊C4×D6111yescharacteristic
578C22#120×D6111yescharacteristic
579C22#123×D6111yescharacteristic
580C4—24no—
581C4—44no—
582C4—612no—
583C4—612no—
584C4—1224no—
585C4—2442no—
586C4—2442no—
587C4—2444no—
588C4C4×D4×D6112yes—
589C4#121×D6112yes—
590C2—21no—
591C2—21no—
592C2—44no—
593C2—82no—
594C2—1224no—
595C2#386×D6111yescharacteristic
596C2#2145516412yes—
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1887C2
socleC22×C61C2×#120
prime radicalC2×#16840310C2
cosocleC22×C4×C61C24
hypercenter#1887D61
hypocenterC61#20952
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1887

Complements

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

NH ≅ G/Ntype
2. #21459017C2semidirect
3. #21454333C2semidirect
4. C2×#16839743C2semidirect
5. #21446219C2semidirect
6. #21446218C2semidirect
7. #21446211C2semidirect
8. #21446207C2semidirect
9. #21446205C2semidirect
10. #339×D61C2semidirect
11. C61×#1887C2semidirect
16. #16840310C4semidirect
54. C2×#11758224C2×C4semidirect
70. #14519777C2×C4semidirect
73. #14519764C2×C4semidirect
74. #14519764C2×C4semidirect
83. #14519735D4semidirect
91. C4⋊C4×D61D4semidirect
101. C42×D61C2×C4semidirect
112. #127×C61C2×C4semidirect
298. #1887D61semidirect
344. C61#20952semidirect
375. #127C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
#214590172
#214543332
C2×#168397432
#214462192
#214462182
#214462112
#214462072
#214462052
#339×D612
C61×#18872
#2095261