Finite Group Databasebeta

← #29681540

Subgroups — #29681540

Order 15616.

1702 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
1C22×#14519808C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic
3C2×#16839780C211yesmaximal, characteristic
4C2×#16839771C211yesmaximal, characteristic, complemented
5#21446191C218yesmaximal, complemented
6#21445558C211yesmaximal, characteristic, complemented
7#21445529C211yesmaximal, characteristic, complemented
8C61×#2310C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519808—22no—
11C2×#14519808—24no—
12C2×#14519808C418yescomplemented
13C2×#14519808C2214yes—
14C22×#11758202—21no—
15C22×#11758201—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18#16840243C2214yes—
19#16840242—22no—
20#16840239—21no—
21#16840236—21no—
22#16840235—22no—
23C4×#11758222—22no—
24#16840230—22no—
25C2×#14519736C2211yescharacteristic
26#117×D61—22no—
27#16839780—24no—
28C2×#14519705C2214yes—
29#16839771—22no—
30#16839771C2214yescomplemented
31C2×#14519701—22no—
32#16839531C2214yescomplemented
33#16839522C2214yescomplemented
34C23×Q8×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#391—22no—
38C61×#391C2214yescomplemented
39C61×#381—22no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C2×#11758222D412yes—
47C2×#11758222C2312yes—
48C22×Dic122D411yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52#14519808—216no—
53#14519808—28no—
54#14519808C2×C4116yescomplemented
55C2×#11758202—28no—
56C2×#11758202—24no—
57C2×#11758201—28no—
58C2×#11758201—24no—
59C2×#11758194—22no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194C2×C414yescomplemented
65C2×#11758194C2312yes—
66C2×#11758194C2312yes—
67C22×#92418—21no—
68C22×#92418—21no—
69C4×Dic122—24no—
70#14519742—24no—
71C4×D244—24no—
72#14519739—24no—
73C4×C22⋊D61—28no—
74#14519737—28no—
75#14519736—21no—
76#14519736—21no—
77#14519736D412yes—
78#14519736C2312yes—
79#14519735—24no—
80#14519734—24no—
81#14519733—24no—
82C2×C4⋊Dic61D412yes—
83C4⋊C4×D61—24no—
84C2×C22⋊Dic61—21no—
85C22⋊C4×D61—24no—
86C2×C4×Dic61—21no—
87C42×D61—24no—
88#14519705—28no—
89#14519701—28no—
90C22×Q8×C61—21no—
91C22×Q8×C61—22no—
92C22×Q8×C61—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61—21no—
95C22×Q8×C61—21no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61C2×C414yescomplemented
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102#117×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D412yes—
112C2×C42×C61D412yes—
113C2×C42×C61C2312yes—
114#95×C61—28no—
115C24×C61C2×D411yescharacteristic
116C23×D61—21no—
117C2×Q8×C61—24no—
118C2×Q8×C61—22no—
119C2×Q8×C61—24no—
120C2×Q8×C61—24no—
121C2×Q8×C61—24no—
122C2×Q8×C61—24no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—41no—
128C2×Q8×C61—41no—
129C2×Q8×C61—42no—
130C2×Q8×C61—42no—
131C2×Q8×C61—42no—
132C2×Q8×C61—42no—
133C2×Q8×C61—42no—
134C2×Q8×C61C22×C414yes—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—41no—
139C22×C4×C61—41no—
140C22×C4×C61—42no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61#4112yes—
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—24no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—22no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158#11758222—24no—
159#11758222—24no—
160#11758222—24no—
161#11758222—24no—
162#11758222—24no—
163#11758222—22no—
164#11758222—22no—
165#11758222C22⋊C418yes—
166#11758222C22×C414yes—
167#11758222C2×D414yes—
168C2×D244—21no—
169C2×D244—22no—
170C2×D244C22⋊C414yes—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—22no—
174C2×C4×D61—24no—
175C2×C4×D61—22no—
176C2×C4×D61—28no—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122C22⋊C414yes—
180C2×Dic122C2×D412yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—42no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—42no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—22no—
190C42×C61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758202—216no—
194#11758201—216no—
195#11758198—28no—
196C4⋊Dic61—22no—
197C4⋊Dic61C2×D414yes—
198#11758196—28no—
199C4×Dic61—28no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202#11758194—24no—
203#11758194—22no—
204#11758194—22no—
205#11758194—22no—
206#11758194—42no—
207#11758194C22×C414yes—
208C2×#92418—24no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—42no—
212C23×C61—42no—
213C23×C61C4×D412yes—
214C23×C61#12011yescharacteristic
215C23×C61#12011yescharacteristic
216C23×C61#12112yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—24no—
220C22×D61—82no—
221Q8×C61—24no—
222Q8×C61—24no—
223Q8×C61—42no—
224Q8×C61—42no—
225Q8×C61—44no—
226Q8×C61—44no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12312yes—
247C2×C4×C61#12312yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—24no—
255C22⋊D61—24no—
256C22⋊D61—28no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—82no—
261D244—22no—
262D244—24no—
263D244C2×C22⋊C414yes—
264C4×D61—24no—
265C4×D61—24no—
266C4×D61—28no—
267C4×D61—88no—
268Dic122—22no—
269Dic122—24no—
270Dic122C2×C22⋊C414yes—
271#92418—24no—
272#92418—42no—
273#20944—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61#38612yes—
280C22×C61#38612yes—
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C2×#12011yescharacteristic
283C22×C61C2×#12111yescharacteristic
284C2×D61—24no—
285C2×D61—44no—
286C2×D61—84no—
287C4×C61—44no—
288C4×C61—44no—
289C4×C61—82no—
290C4×C61#38612yes—
291C4×C61#38612yes—
292C4×C61C22×C22⋊C411yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×#12111yescharacteristic
295C4×C61C2×#12311yescharacteristic
296Dic61—24no—
297Dic61—44no—
298C22×#137—611no—
299#3428—611no—
300C2×#421—611no—
301C2×#419—611no—
302#2310D6111yescharacteristic, complemented
303#1882—618no—
304#1742—611no—
305#1733—611no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—21no—
309C2×C61—22no—
310C2×C61C2×#38611yescharacteristic
311C2×C61#302212yes—
312D61—44no—
313C22×#41—611no—
314C23×Q8C2×D6111yescharacteristic
315C2×#137—618no—
316C2×#137—614no—
317C2×#137—1222no—
318C2×#137—1224no—
319C22×Q16—1221no—
320C22×SD16—1221no—
321C22×M16—611no—
322C22×M16—611no—
323#520—614no—
324#520—1222no—
325#519—1221no—
326#518—1221no—
327#518—1222no—
328C4×#41—1222no—
329C4×#41—1222no—
330C2×#117—611no—
331C2×#117—1222no—
332C2×#117C2×D6111yescharacteristic
333C2×#117C2×D6111yescharacteristic
334#421—1224no—
335C2×#104—614no—
336#419—614no—
337#419—1222no—
338C2×#103—1222no—
339#391—22no—
340#391C2×D6114yescomplemented
341#381—22no—
342#343—614no—
343#339—614no—
344C61#2094411yescharacteristic, complemented, Sylow
345C2×#41—614no—
346C2×#41—612no—
347C2×#41—614no—
348C2×#41—612no—
349C2×#41—1222no—
350C2×#41—1221no—
351C2×#41—1222no—
352C2×#41—1221no—
353C22×Q8—22no—
354C22×Q8—21no—
355C22×Q8—21no—
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—21no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8—611no—
362C22×Q8C4×D6114yescomplemented
363C22×Q8C22×D6112yes—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—21no—
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369#137—6116no—
370#137—1228no—
371#137—12216no—
372C2×Q16—1224no—
373C2×Q16—1228no—
374C2×SD16—1224no—
375C2×SD16—1228no—
376C2×M16—614no—
377C2×M16—612no—
378C2×M16—612no—
379C2×M16—1221no—
380C2×M16—1222no—
381C2×M16—1221no—
382C2×M16—1222no—
383C2×M16—1222no—
384C22×C8—1221no—
385C22×C8—1221no—
386C4×Q8—1224no—
387C4×Q8—1224no—
388C4×D4—1224no—
389C4×D4—1228no—
390C4×D4—1224no—
391C4×D4—1228no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—22no—
397#117—22no—
398#117—612no—
399#117—612no—
400#117—1224no—
401#117—1224no—
402#117—1224no—
403#117—1221no—
404#117—1221no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4C22×D6112yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1224no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4C22⋊D6112yes—
413C2×C42—1224no—
414C2×C42—1221no—
415C2×C42C22⋊D6112yes—
416C2×C42C22×D6112yes—
417#104—1228no—
418#103—1228no—
419#95—28no—
420C24—1221no—
421C24C2×C22⋊D6111yescharacteristic
422#41—614no—
423#41—614no—
424#41—618no—
425#41—1222no—
426#41—1222no—
427#41—1224no—
428#41—1224no—
429#41—1224no—
430#41—1224no—
431#41—1224no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—612no—
450C2×Q8—614no—
451C2×Q8—1221no—
452C2×Q8—1222no—
453C2×Q8C2×C4×D6114yes—
454C2×D4—612no—
455C2×D4—614no—
456C2×D4—1222no—
457C2×D4—1224no—
458C2×D4—1222no—
459C2×D4—1224no—
460C2×D4—1221no—
461C2×D4—1222no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—1228no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4—1224no—
474C22×C4—1222no—
475C22×C4C2×C4×D6112yes—
476C22×C4#1175822212yes—
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6112yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4#1175822412yes—
483C22×C4C2×C22⋊D6114yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486Q16—12216no—
487SD16—12216no—
488M16—614no—
489M16—1222no—
490M16—1224no—
491M16—1222no—
492M16—1222no—
493M16—2442no—
494C2×C8—1222no—
495C2×C8—1224no—
496C2×C8—1222no—
497C2×C8—2442no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—614no—
502C4⋊C4—1228no—
503C4⋊C4—1222no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—1228no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—1228no—
515C42—1222no—
516C42—1222no—
517C23—42no—
518C23—1224no—
519C23—1222no—
520C23—4882no—
521C23C4×C22⋊D6112yes—
522C23#1451974511yescharacteristic
523C23#1451974511yescharacteristic
524C23#1451974812yes—
525C23C22×C22⋊D6111yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—42no—
529Q8—42no—
530Q8—44no—
531Q8—44no—
532Q8—614no—
533Q8—1222no—
534Q8—1224no—
535D4—614no—
536D4—1224no—
537D4—1224no—
538D4—1228no—
539D4—1222no—
540D4—1224no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—1222no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1228no—
555C2×C4—1224no—
556C2×C4—4888no—
557C2×C4—4882no—
558C2×C4C22⋊C4×D6112yes—
559C2×C4C22⋊C4×D6112yes—
560C2×C4#1451973712yes—
561C2×C4#1451973712yes—
562C2×C4C4×C22⋊D6112yes—
563C2×C4#1451974511yescharacteristic
564C2×C4#1451974511yescharacteristic
565C2×C4#1451974812yes—
566C2×C4#1451975012yes—
567C2×C4#1451975012yes—
568C2×C4#1451975912yes—
569C2×C4#1451975912yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C8—1224no—
577C8—2442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—1224no—
584C22—4884no—
585C22—2444no—
586C22#1683963812yes—
587C22#1683963812yes—
588C22C2×C4×C22⋊D6111yescharacteristic
589C22C2×#1451974511yescharacteristic
590C22C2×#1451974811yescharacteristic
591C4—44no—
592C4—44no—
593C4—82no—
594C4—1224no—
595C4—2444no—
596C4#1683963812yes—
597C4#1683963812yes—
598C4C2×C22⋊C4×D6111yescharacteristic
599C4C2×#1451973711yescharacteristic
600C4C2×#1451975011yescharacteristic
601C4C2×#1451975911yescharacteristic
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—2444no—
607C2C2×#1683963811yescharacteristic
608C2#2145519712yes—
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2310D61
hypocenterC61#20944
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C2×#16839771C2semidirect
5. #21446191C2semidirect
6. #21445558C2semidirect
7. #21445529C2semidirect
8. C61×#2310C2semidirect
12. C2×#14519808C4semidirect
30. #16839771C22semidirect
32. #16839531C22semidirect
33. #16839522C22semidirect
38. C61×#391C22semidirect
54. #14519808C2×C4semidirect
64. C2×#11758194C2×C4semidirect
98. C22×Q8×C61C2×C4semidirect
302. #2310D61semidirect
340. #391C2×D61semidirect
344. C61#20944semidirect
362. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
#214590202
C2×#168397802
C2×#168397712
#214461912
#214455582
#214455292
C61×#23102
#2094461