Finite Group Databasebeta

← #29681538

Subgroups — #29681538

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×#14519810C211yesmaximal, characteristic
2#21459020C211yesmaximal, characteristic
3C2×#16839780C211yesmaximal, characteristic
4C2×#16839770C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21446189C218yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519810—22no—
12C2×#14519810—24no—
13C2×#14519810C418yescomplemented
14C2×#14519810C2214yes—
15C22×Dic244—21no—
16C22×#11758190—21no—
17C22×#11758194C2211yescharacteristic
18#16840243—22no—
19#16840242C2214yes—
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#16839770—22no—
30#16839770C2214yescomplemented
31C2×#14519700—22no—
32#16839668C2214yescomplemented
33#16839667—22no—
34#16839623—22no—
35#16839530C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#339C2214yescomplemented
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C2×#11758222D412yes—
47C2×#11758222C2312yes—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122—22no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122C2×C414yescomplemented
57C22×Dic122D411yescharacteristic
58C22×Dic122C2312yes—
59C22×D244D411yescharacteristic
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61—21no—
63#14519810—216no—
64#14519810—28no—
65#14519810C2×C4116yescomplemented
66C2×Dic244—28no—
67C2×Dic244—24no—
68C2×#11758190—28no—
69C2×#11758190—24no—
70C2×#11758194—21no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194C2312yes—
74C22×#92418—21no—
75C4×Dic122—24no—
76#14519742—24no—
77C4×D244—24no—
78#14519739—24no—
79C4×C22⋊D61—28no—
80#14519737—28no—
81#14519736—21no—
82#14519736—21no—
83#14519736—21no—
84#14519736—21no—
85#14519735—24no—
86#14519734—24no—
87#14519733—24no—
88C2×C4⋊Dic61C2312yes—
89C4⋊C4×D61—24no—
90C2×C22⋊Dic61D412yes—
91C22⋊C4×D61—24no—
92C2×C4×Dic61D412yes—
93C42×D61—24no—
94#14519705—28no—
95#14519700—28no—
96#14519684—28no—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—22no—
99C2×M16×C61—21no—
100C2×M16×C61C2×C414yescomplemented
101C2×M16×C61C2312yes—
102C22×C8×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—22no—
106#117×C61—22no—
107#117×C61D412yes—
108#117×C61C2312yes—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D412yes—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2312yes—
115C24×C61C2×D411yescharacteristic
116C23×D61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—22no—
133C22×Dic61—21no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138C22×Dic61—42no—
139C22×Dic61C2×D414yes—
140#11758222—24no—
141#11758222—24no—
142#11758222—24no—
143#11758222—24no—
144#11758222—24no—
145#11758222—22no—
146#11758222—22no—
147#11758222C22⋊C418yes—
148#11758222C22×C414yes—
149#11758222C2×D414yes—
150C2×D244—21no—
151C2×D244—22no—
152C2×D244C22⋊C414yes—
153C2×D244C2×D412yes—
154C2×C4×D61—24no—
155C2×C4×D61—22no—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—28no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—24no—
162C2×Dic122—24no—
163C2×Dic122—24no—
164C2×Dic122—24no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—24no—
170C2×Dic122—22no—
171C2×Dic122—41no—
172C2×Dic122—41no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122C22⋊C414yes—
179C2×Dic122C22×C414yes—
180C2×Dic122C2×D412yes—
181M16×C61—24no—
182M16×C61—22no—
183M16×C61C22×C414yes—
184C2×C8×C61—24no—
185C2×C8×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61C2×D414yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—42no—
192C42×C61—22no—
193C42×C61—22no—
194C42×C61—22no—
195C42×C61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198#11758198—28no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201#11758196—28no—
202C4×Dic61—28no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205#11758194—22no—
206#11758194—22no—
207#11758194—42no—
208C2×#92418—22no—
209C2×#92418—42no—
210Dic244—216no—
211#11758190—216no—
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—
221C2×C4×C61—24no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61—82no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61#12112yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12312yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—24no—
244C22⋊D61—24no—
245C22⋊D61—28no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—44no—
253C2×Dic61—44no—
254C2×Dic61—44no—
255D244—22no—
256D244—24no—
257D244C2×C22⋊C414yes—
258C4×D61—24no—
259C4×D61—24no—
260C4×D61—28no—
261C4×D61—88no—
262Dic122—22no—
263Dic122—24no—
264Dic122—24no—
265Dic122—24no—
266Dic122—42no—
267Dic122—42no—
268Dic122—44no—
269Dic122—44no—
270Dic122C2×C22⋊C414yes—
271C8×C61—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—82no—
289C4×C61#38612yes—
290C4×C61#38612yes—
291C4×C61C22×C22⋊C411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12311yescharacteristic
295Dic61—24no—
296Dic61—44no—
297Dic61—44no—
298C22×#137—611no—
299#3428—611no—
300C2×#421—611no—
301C2×#419—611no—
302#2310—611no—
303#1882—618no—
304#1742—611no—
305#1733D6111yescharacteristic, complemented
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—21no—
309C2×C61—22no—
310C2×C61C2×#38611yescharacteristic
311C2×C61#302212yes—
312D61—44no—
313C22×#41—611no—
314C23×Q8—611no—
315C2×#137—618no—
316C2×#137—614no—
317C2×#137—1222no—
318C2×#137—1224no—
319C22×Q16—1221no—
320C22×SD16—1221no—
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323#520—614no—
324#520—1222no—
325#519—1221no—
326#518—1222no—
327#518—1221no—
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—614no—
340#391—1222no—
341#381—1222no—
342#343—614no—
343#339C2×D6114yescomplemented
344C61#2094411yescharacteristic, complemented, Sylow
345C2×#41—614no—
346C2×#41—612no—
347C2×#41—614no—
348C2×#41—612no—
349C2×#41—1221no—
350C2×#41—1222no—
351C2×#41—1221no—
352C2×#41—1222no—
353C22×Q8—614no—
354C22×Q8—611no—
355C22×Q8—612no—
356C22×Q8—1221no—
357C22×Q8—1222no—
358C22×Q8—1221no—
359C22×Q8—1221no—
360C22×Q8—1222no—
361C22×Q8—1221no—
362C22×Q8—1221no—
363C22×Q8—1222no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—1221no—
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—21no—
377C2×M16—22no—
378C2×M16—612no—
379C2×M16—1221no—
380C2×M16—1222no—
381C2×M16—1222no—
382C2×M16C4×D6114yescomplemented
383C2×M16C22×D6112yes—
384C22×C8—21no—
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—22no—
395#117—22no—
396#117—1224no—
397#117—1224no—
398#117—1221no—
399#117—1221no—
400#117—1221no—
401#117—1221no—
402#117—1224no—
403#117D24412yes—
404#117C22×D6112yes—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4D24412yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—612no—
412C2×C22⋊C4—1224no—
413C2×C42—21no—
414C2×C42—612no—
415C2×C42—1224no—
416C2×C42C22×D6112yes—
417#104—1228no—
418#103—1228no—
419#95—1228no—
420C24—1221no—
421C24D4×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—614no—
433C2×Q8—612no—
434C2×Q8—614no—
435C2×Q8—1222no—
436C2×Q8—1224no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—1224no—
444C2×Q8—1224no—
445C2×Q8—1224no—
446C2×Q8—1224no—
447C2×Q8—2441no—
448C2×Q8—2441no—
449C2×Q8—2442no—
450C2×Q8—2442no—
451C2×Q8—2442no—
452C2×Q8—2442no—
453C2×Q8—2442no—
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—22no—
464C22×C4—614no—
465C22×C4—1228no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2442no—
476C22×C4C2×C4×D6112yes—
477C22×C4C2×D24411yescharacteristic
478C22×C4C2×D24411yescharacteristic
479C22×C4#1175822212yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6112yes—
483C22×C4D4×D6111yescharacteristic
484C22×C4#1175822612yes—
485C22×C4C23×D6111yescharacteristic
486Q16—12216no—
487SD16—12216no—
488M16—22no—
489M16—24no—
490M16—1222no—
491M16—1222no—
492M16—2442no—
493M16C2×C4×D6114yes—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—1222no—
497C2×C8—2442no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—1228no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4C2×D24414yes—
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—
521C23#1451973912yes—
522C23#1451974711yescharacteristic
523C23#1451974711yescharacteristic
524C23#1451975012yes—
525C23C2×D4×D6111yescharacteristic
526Q8—614no—
527Q8—1222no—
528Q8—1224no—
529Q8—1224no—
530Q8—1224no—
531Q8—2442no—
532Q8—2442no—
533Q8—2444no—
534Q8—2444no—
535D4—614no—
536D4—1224no—
537D4—1224no—
538D4—1228no—
539D4—1222no—
540D4—1224no—
541C2×C4—24no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—1222no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1228no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—4888no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4C22⋊C4×D6112yes—
559C2×C4C22⋊C4×D6112yes—
560C2×C4#1451973912yes—
561C2×C4C4×D24412yes—
562C2×C4C4×D24412yes—
563C2×C4#1451974711yescharacteristic
564C2×C4#1451974711yescharacteristic
565C2×C4#1451975012yes—
566C2×C4#1451975212yes—
567C2×C4#1451975212yes—
568C2×C4#1451976212yes—
569C2×C4#1451976212yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C22×D24411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822611yescharacteristic
576C8—24no—
577C8—2442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—1224no—
584C22—4884no—
585C22—2444no—
586C22#1683964012yes—
587C22#1683964012yes—
588C22C2×#1451973911yescharacteristic
589C22C2×#1451974711yescharacteristic
590C22C2×#1451975011yescharacteristic
591C4—44no—
592C4—82no—
593C4—1224no—
594C4—2444no—
595C4—2444no—
596C4#1683964012yes—
597C4#1683964012yes—
598C4C2×C22⋊C4×D6111yescharacteristic
599C4C2×C4×D24411yescharacteristic
600C4C2×#1451975211yescharacteristic
601C4C2×#1451976211yescharacteristic
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—2444no—
607C2C2×#1683964011yescharacteristic
608C2#2145519912yes—
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20944
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. C2×#16839770C2semidirect
5. #21448403C2semidirect
6. #21446189C2semidirect
7. #21445557C2semidirect
8. C61×#1733C2semidirect
13. C2×#14519810C4semidirect
30. #16839770C22semidirect
32. #16839668C22semidirect
35. #16839530C22semidirect
39. C61×#339C22semidirect
56. C22×Dic122C2×C4semidirect
65. #14519810C2×C4semidirect
100. C2×M16×C61C2×C4semidirect
305. #1733D61semidirect
343. #339C2×D61semidirect
344. C61#20944semidirect
382. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
#214590202
C2×#168397802
C2×#168397702
#214484032
#214461892
#214455572
C61×#17332
#2094461