Finite Group Databasebeta

← #29681585

Subgroups — #29681585

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×#16840311C211yesmaximal, characteristic
2#21459017C211yesmaximal, characteristic, complemented
3#21454337C212yesmaximal, complemented
4C2×#16839743C211yesmaximal, characteristic, complemented
5#21446221C212yesmaximal, complemented
6#21446217C211yesmaximal, characteristic, complemented
7#21446215C211yesmaximal, characteristic, complemented
8#21446209C212yesmaximal, complemented
9#21446208C211yesmaximal, characteristic, complemented
10#339×D61C212yesmaximal, complemented
11C61×#1885C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14C2×M16×D61C2212yes—
15#16840311—24no—
16#16840311C418yescomplemented
17C2×#14519775C2211yescharacteristic
18C2×#14519773C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×#14519750—22no—
21#16840241—24no—
22#16840239C2212yes—
23#16840237C2211yescharacteristic
24#16840235—22no—
25C4×#11758224—24no—
26C2×#14519737—22no—
27C2×#14519733—22no—
28#117×D61C2212yes—
29C2×C42×D61C2211yescharacteristic
30#16839771C2212yes—
31#16839768C2212yes—
32#16839766C2212yes—
33#16839762—24no—
34#16839757—24no—
35#16839743—24no—
36C2×#14519695C2212yes—
37C2×#14519694C2211yescharacteristic
38#16839523C2212yes—
39#16839522C2212yes—
40#16839521C2212yes—
41C2×#124×C61C2211yescharacteristic
42C61×#518C2211yescharacteristic
43C61×#419C2212yes—
44C2×#100×C61C2211yescharacteristic
45C61×#339C2212yes—
46C2×#11758226—22no—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—24no—
50C2×#11758226C2×C414yescomplemented
51C2×#11758224—22no—
52C2×#11758224—22no—
53C2×#11758224—24no—
54C2×#11758224C2×C414yes—
55C2×#11758224D414yes—
56C22×Dic122C2311yescharacteristic
57C22×D244—21no—
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—22no—
60C22×C22⋊D61D412yes—
61C23×Dic61D412yes—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758194C2312yes—
66C2×#11758189—22no—
67M16×D61—24no—
68M16×D61—24no—
69C2×C8×D61—22no—
70#14519775—22no—
71#14519775C2×C414yescomplemented
72#14519773—22no—
73#14519773C2×C414yescomplemented
74#14519765—22no—
75#14519765C2×C414yescomplemented
76#14519750—216no—
77C4×Dic122—24no—
78#14519742—24no—
79D4×Dic61—24no—
80C4×C22⋊D61—28no—
81#14519737—28no—
82#14519737—216no—
83#14519736C2312yes—
84#14519735D414yescomplemented
85#14519734—22no—
86#14519733—24no—
87#14519733—24no—
88#14519733—22no—
89#14519733—24no—
90C2×#11758196—24no—
91C2×#11758196—22no—
92C4⋊C4×D61D414yescomplemented
93C2×C22⋊Dic61—22no—
94C2×#11758198—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#14519701—24no—
103#14519699—24no—
104#14519697—24no—
105#14519695—24no—
106#14519694—22no—
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C2×M16×C61C2312yes—
110#124×C61—22no—
111#124×C61C2×C414yescomplemented
112C4×D4×C61—24no—
113#117×C61C2312yes—
114C2×C22⋊C4×C61—22no—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61C2311yescharacteristic
117#103×C61—24no—
118#100×C61—22no—
119C24×C61C2×D412yes—
120C23×D61—21no—
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61C22×C412yes—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61C22×C412yes—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—28no—
137C2×C22⋊D61—48no—
138C2×C22⋊D61C22⋊C418yes—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61—24no—
144C22×Dic61C22⋊C418yes—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D411yescharacteristic
147#11758226—28no—
148#11758226—28no—
149#11758226—44no—
150#11758226—44no—
151#11758224—28no—
152#11758224—28no—
153#11758224—28no—
154#11758224—28no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—24no—
158C2×D244—21no—
159C2×D244—42no—
160C2×D244—42no—
161C2×D244C22×C412yes—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—24no—
165C2×C4×D61—44no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C4×D61—42no—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×D412yes—
173C2×C4×D61#4114yes—
174C2×Dic122—22no—
175C2×Dic122—22no—
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—28no—
184C22⋊C4×C61—22no—
185C42×C61—21no—
186C42×C61—22no—
187C42×C61C22×C412yes—
188C22⋊Dic61—24no—
189C22⋊Dic61—22no—
190#11758198—24no—
191#11758198—28no—
192C4⋊Dic61C2×D414yes—
193#11758196—28no—
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—44no—
210C23×C61C2×C22⋊C414yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—22no—
213C22×D61—42no—
214C22×D61—44no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61C2×C22⋊C412yes—
218C22×D61#12012yes—
219C22×D61#12312yes—
220Q8×C61—24no—
221Q8×C61—42no—
222D4×C61—24no—
223D4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—42no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—84no—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D412yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4112yes—
235C22⋊D61—216no—
236C22⋊D61—416no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—28no—
242C2×Dic61—28no—
243C2×Dic61—24no—
244C2×Dic61—84no—
245C2×Dic61C2×C22⋊C414yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61#12011yescharacteristic
248C2×Dic61#12011yescharacteristic
249C2×Dic61#12312yes—
250D244—24no—
251D244—48no—
252D244—42no—
253C4×D61—48no—
254C4×D61—44no—
255C4×D61—44no—
256C4×D61—84no—
257C4×D61C4×D414yes—
258C4×D61C4×D414yes—
259C4×D61#12114yes—
260C4×D61#12114yes—
261Dic122—24no—
262Dic122—24no—
263C8×C61—42no—
264#92418—42no—
265#20952—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—82no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C2×#12011yescharacteristic
273C22×C61C2×#12311yescharacteristic
274C2×D61—24no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—44no—
278C2×D61—42no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61#38614yes—
282C4×C61—42no—
283C4×C61—44no—
284C4×C61—42no—
285C4×C61C2×C4×D412yes—
286C4×C61C2×#12112yes—
287Dic61—24no—
288Dic61—24no—
289Dic61—44no—
290Dic61#38612yes—
291Dic61#38612yes—
292C2×#532—611no—
293#3428—611no—
294#2896—612no—
295C2×#413—611no—
296#1888—612no—
297#1887—611no—
298#1887—611no—
299#1886—612no—
300#1885—611no—
301#1885D6111yescharacteristic, complemented
302C2×#339—612no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—44no—
306C2×C61C2×#38611yescharacteristic
307C2×C61#301812yes—
308D61—24no—
309D61—82no—
310C22×#41—611no—
311C22×#41—611no—
312C22×M16—612no—
313#532—618no—
314#532—1224no—
315C2×C4⋊Q8—611no—
316C2×#127—611no—
317C2×#124—611no—
318C2×#124C2×D6111yescharacteristic
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#419C2×D6112yes—
332#418—612no—
333#418—612no—
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×Q8C22×D6111yescharacteristic
357C22×D4—612no—
358C22×D4—611no—
359C22×D4—1221no—
360C22×D4—1222no—
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—614no—
375#127—1222no—
376#124—22no—
377#124—614no—
378#124—1222no—
379#124C4×D6114yescomplemented
380#121—12216no—
381C4×Q8—1224no—
382C4×Q8—1224no—
383C4×D4—24no—
384C4×D4—12216no—
385C4×D4—1228no—
386C4×D4—1224no—
387C4×D4—1228no—
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—22no—
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—24no—
412#103—1224no—
413#102—1224no—
414#102—1224no—
415#101—1224no—
416#100—22no—
417#100—1222no—
418C24—611no—
419C24—1221no—
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—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—612no—
433C2×Q8—612no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8C2×C4×D6112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—618no—
440C2×D4—612no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1228no—
445C2×D4—1222no—
446C2×D4—1221no—
447C2×D4—1222no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2448no—
451C2×D4C2×C4×D6112yes—
452C2×D4D4×D6112yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—614no—
462C22×C4—612no—
463C22×C4—618no—
464C22×C4—1224no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1224no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4—2442no—
475C22×C4—2444no—
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—28no—
493C22⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1222no—
497C42—21no—
498C42—22no—
499C42—612no—
500C42—1224no—
501C42—1224no—
502C42—1228no—
503C42—1221no—
504C42—1222no—
505C42C2×C4×D6112yes—
506C23—22no—
507C23—22no—
508C23—44no—
509C23—612no—
510C23—612no—
511C23—612no—
512C23—1222no—
513C23—2442no—
514C23—2444no—
515C23—2441no—
516C23—2441no—
517C23C22⋊C4×D6114yes—
518C23C2×D4×D6111yescharacteristic
519Q8—24no—
520Q8—42no—
521Q8—1224no—
522Q8—1224no—
523D4—24no—
524D4—24no—
525D4—12216no—
526D4—1224no—
527D4—2448no—
528D4—24416no—
529D4—2442no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—612no—
541C2×C4—614no—
542C2×C4—614no—
543C2×C4—614no—
544C2×C4—614no—
545C2×C4—614no—
546C2×C4—1228no—
547C2×C4—1228no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1224no—
553C2×C4—4884no—
554C2×C4—4884no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2448no—
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—82no—
569C22—614no—
570C22—1224no—
571C22—4882no—
572C22—4882no—
573C22—2442no—
574C22—2442no—
575C22—2444no—
576C22—2442no—
577C22C2×C22⋊C4×D6111yescharacteristic
578C22#120×D6111yescharacteristic
579C22#123×D6111yescharacteristic
580C4—42no—
581C4—42no—
582C4—44no—
583C4—612no—
584C4—612no—
585C4—1224no—
586C4—1224no—
587C4—2444no—
588C4C4×D4×D6112yes—
589C4#121×D6112yes—
590C2—21no—
591C2—21no—
592C2—44no—
593C2—1224no—
594C2—4882no—
595C2#386×D6111yescharacteristic
596C2#2145516912yes—
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1885C2
socleC22×C61C2×#120
prime radicalC2×#16840311C2
cosocleC22×C4×C61C24
hypercenter#1885D61
hypocenterC61#20952
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
2. #21459017C2semidirect
3. #21454337C2semidirect
4. C2×#16839743C2semidirect
5. #21446221C2semidirect
6. #21446217C2semidirect
7. #21446215C2semidirect
8. #21446209C2semidirect
9. #21446208C2semidirect
10. #339×D61C2semidirect
11. C61×#1885C2semidirect
16. #16840311C4semidirect
50. C2×#11758226C2×C4semidirect
71. #14519775C2×C4semidirect
73. #14519773C2×C4semidirect
75. #14519765C2×C4semidirect
84. #14519735D4semidirect
92. C4⋊C4×D61D4semidirect
101. C42×D61C2×C4semidirect
111. #124×C61C2×C4semidirect
301. #1885D61semidirect
344. C61#20952semidirect
379. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403112
#214590172
#214543372
C2×#168397432
#214462212
#214462172
#214462152
#214462092
#214462082
#339×D612
C61×#18852
#2095261