Finite Group Databasebeta

← #29681437

Subgroups — #29681437

Order 15616.

1926 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
1C22×#14519805C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic, complemented
3C2×#16839756C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446175C214yesmaximal, complemented
7#21446168C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519805—24no—
13C2×#14519805—24no—
14C2×#14519805C418yescomplemented
15C22×#11758190C2212yes—
16C22×D488C2212yes—
17C23×#92418C2211yescharacteristic
18#16840243—24no—
19#16840243—24no—
20C2×#14519742C2212yescomplemented
21C2×#14519739C2212yescomplemented
22C2×C4×C22⋊D61—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519735—21no—
25C2×C4⋊C4×D61—21no—
26#16839756—24no—
27C2×#14519700C2212yescomplemented
28C2×#14519697—24no—
29C2×#14519697C2214yes—
30C2×#14519697C2212yescomplemented
31#16839664—21no—
32#16839664—21no—
33#16839663C2214yes—
34#16839617—22no—
35#16839516C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#340C2214yescomplemented
40C24×D61—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C2×#11758222D414yes—
47C22×Dic122C2311yescharacteristic
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—22no—
54C22×D244C2×C414yescomplemented
55C22×D244C2312yes—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C22×C4×D61—21no—
59C22×C4×D61—22no—
60#14519805—232no—
61C2×#11758190—24no—
62C2×#11758190—24no—
63C2×#11758190C2×C418yescomplemented
64C2×D488—24no—
65C2×D488—24no—
66C2×D488C2×C418yescomplemented
67C22×#92418—22no—
68C22×#92418—22no—
69C22×#92418—22no—
70C22×#92418C2312yes—
71#14519742—24no—
72#14519742—24no—
73#14519739—24no—
74#14519739—24no—
75C4×C22⋊D61—216no—
76#14519736—22no—
77#14519736—22no—
78#14519735—28no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61D411yescharacteristic
81C2×#11758196—22no—
82C4⋊C4×D61—28no—
83C2×C22⋊Dic61D412yes—
84C2×#11758198—22no—
85C2×#11758198—24no—
86C2×#11758198—24no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61C2312yes—
89#14519700—24no—
90#14519697—24no—
91#14519697—28no—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61C2×C414yescomplemented
97C22×C8×C61C2312yes—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61D414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—21no—
109C23×D61—22no—
110C23×D61—24no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61—82no—
114C23×D61—82no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—82no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C22×Dic61—21no—
131C22×Dic61C2×D412yes—
132C22×Dic61C2×D412yes—
133#11758222—28no—
134#11758222—28no—
135#11758222—28no—
136#11758222C22⋊C4116yescomplemented
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—28no—
141C2×D244—28no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—22no—
148C2×D244—42no—
149C2×D244—44no—
150C2×D244—42no—
151C2×D244—44no—
152C2×D244C22×C414yes—
153C2×D244C22×C412yes—
154C2×D244C2×D412yes—
155C2×D244C2×D412yes—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—28no—
160C2×C4×D61—84no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C22×C412yes—
165C2×Dic122C2×D412yes—
166C2×Dic122C2×D412yes—
167M16×C61—28no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61C22×C414yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2×D414yes—
176C22⋊C4×C61—42no—
177C42×C61—42no—
178C22⋊Dic61—24no—
179#11758198—28no—
180#11758198—84no—
181#11758198—84no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184#11758196—28no—
185C4×Dic61—28no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—44no—
192D488—216no—
193#11758190—216no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61C4×D412yes—
197C23×C61#12012yes—
198C23×C61#12112yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—24no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—48no—
211C22×D61—88no—
212C22×D61—84no—
213C22×D61—84no—
214C22×D61—88no—
215C2×C4×C61—24no—
216C2×C4×C61—44no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—84no—
221C2×C4×C61C2×C22⋊C414yes—
222C2×C4×C61C4×D414yes—
223C2×C4×C61C4×D412yes—
224C2×C4×C61#12012yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12314yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—216no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242D244—24no—
243D244—22no—
244D244—24no—
245D244—22no—
246D244—22no—
247D244—42no—
248D244—42no—
249D244—48no—
250D244C2×C22⋊C414yes—
251C4×D61—216no—
252C4×D61—88no—
253Dic122—22no—
254Dic122—22no—
255Dic122—22no—
256Dic122C2×C22⋊C414yes—
257C8×C61—24no—
258#92418—42no—
259#20932—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61#38614yes—
265C22×C61C2×C4×D411yescharacteristic
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#12111yescharacteristic
268C22×C61#57612yes—
269C22×C61#57712yes—
270C2×D61—24no—
271C2×D61—44no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—44no—
275C2×D61—48no—
276C2×D61—44no—
277C2×D61—816no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—84no—
281C4×C61—44no—
282C4×C61—82no—
283C4×C61#38614yes—
284C4×C61C22×C22⋊C411yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286C4×C61C2×#12111yescharacteristic
287C4×C61C2×#12311yescharacteristic
288Dic61—24no—
289Dic61—42no—
290Dic61—42no—
291C22×AGL(1,ℤ/8)—611no—
292C2×#520—611no—
293C2×#416—611no—
294C22×#102—611no—
295#2309—611no—
296#1880—614no—
297#1879—614no—
298#1736D6111yescharacteristic, complemented
299#1729—611no—
300C2×C61—24no—
301C2×C61C2×#38611yescharacteristic
302C2×C61#294612yes—
303C2×C61#300811yescharacteristic
304C2×C61#301011yescharacteristic
305C2×C61#304512yes—
306D61—44no—
307D61—84no—
308C22×#41—611no—
309C23×D4—611no—
310C2×AGL(1,ℤ/8)—618no—
311C2×AGL(1,ℤ/8)—1224no—
312C2×AGL(1,ℤ/8)—1224no—
313C22×SD16—612no—
314C22×D8—612no—
315C22×M16C2×D6111yescharacteristic
316C23×C8—611no—
317#520—1224no—
318#520—1224no—
319C2×C4×Q8—612no—
320C2×C4×D4—612no—
321C2×C4×D4—1222no—
322C2×#117—611no—
323C2×#117—1221no—
324C2×#117C2×D6111yescharacteristic
325C22×C4⋊C4—1221no—
326C22×C4⋊C4C2×D6111yescharacteristic
327#416—1224no—
328C2×#103—612no—
329C2×#102—614no—
330C2×#102—612no—
331C2×#102—1224no—
332#390—614no—
333#390—1221no—
334#390—1221no—
335#379—1222no—
336#340C2×D6114yescomplemented
337#336—614no—
338C61#2093211yescharacteristic, complemented, Sylow
339C25—1221no—
340C2×#41—614no—
341C2×#41—614no—
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C22×Q8—611no—
347C22×D4—614no—
348C22×D4—611no—
349C22×D4—612no—
350C22×D4—1221no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1221no—
356C22×D4—1222no—
357C23×C4—21no—
358C23×C4—1222no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361AGL(1,ℤ/8)—12232no—
362C2×SD16—618no—
363C2×SD16—1224no—
364C2×SD16—1224no—
365C2×D8—618no—
366C2×D8—1224no—
367C2×D8—1224no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16C4×D6114yescomplemented
371C22×C8—612no—
372C22×C8—1222no—
373C22×C8—1222no—
374C22×C8—1222no—
375C22×C8C22×D6112yes—
376C4×Q8—1224no—
377C4×Q8—1224no—
378C4×D4—1224no—
379C4×D4—12216no—
380C4×D4—1224no—
381#117—22no—
382#117—22no—
383#117—1228no—
384#117—1222no—
385#117—1222no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1228no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4D24414yes—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—612no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C42—21no—
401C2×C42—612no—
402C2×C42—1222no—
403#103—1224no—
404#102—1224no—
405#102—1228no—
406C24—1221no—
407C24—1224no—
408C24—1222no—
409C24—4882no—
410C24—4882no—
411C24—2442no—
412C24—2442no—
413C24D4×D6111yescharacteristic
414#41—6116no—
415#41—1228no—
416#41—1228no—
417#41—1228no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×D4—612no—
425C2×D4—614no—
426C2×D4—612no—
427C2×D4—612no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1221no—
434C2×D4—1222no—
435C2×D4—1221no—
436C2×D4—1222no—
437C2×D4—1221no—
438C2×D4—1222no—
439C2×D4—1228no—
440C2×D4—1224no—
441C2×D4—1228no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4—2444no—
445C2×D4—2444no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—82no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—1228no—
453C22×C4—1221no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—4884no—
458C22×C4C2×C4×D6112yes—
459C22×C4C2×D24412yes—
460C22×C4#1175822212yes—
461C22×C4D4×D6112yes—
462C22×C4D4×D6112yes—
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822612yes—
465C22×C4C23×D6111yescharacteristic
466SD16—12216no—
467D8—12216no—
468M16—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—2444no—
475C2×C8C2×C4×D6114yes—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—1228no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4C2×D24414yes—
484C22⋊C4—42no—
485C22⋊C4—1228no—
486C22⋊C4—1224no—
487C22⋊C4—4884no—
488C22⋊C4—4884no—
489C42—42no—
490C42—1228no—
491C42—1222no—
492C42—1222no—
493C23—22no—
494C23—22no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—1224no—
499C23—4884no—
500C23—4884no—
501C23—4888no—
502C23—4888no—
503C23—2442no—
504C23—2442no—
505C23—2444no—
506C23—2444no—
507C23—2448no—
508C23—2442no—
509C23—2442no—
510C23#1451973912yes—
511C23#1451974712yes—
512C23#1451975012yes—
513C23C2×D4×D6111yescharacteristic
514Q8—614no—
515Q8—1222no—
516Q8—1222no—
517Q8—1222no—
518D4—614no—
519D4—12216no—
520D4—1224no—
521D4—1222no—
522D4—1224no—
523D4—1222no—
524D4—1222no—
525D4—2442no—
526D4—2442no—
527D4—2448no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—12216no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—4888no—
543C2×C4C22⋊C4×D6114yes—
544C2×C4#1451973912yes—
545C2×C4C4×D24414yes—
546C2×C4#1451974712yes—
547C2×C4#1451975012yes—
548C2×C4#1451975214yes—
549C2×C4#1451976214yes—
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C8—24no—
557C8—2442no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—1224no—
563C22—48816no—
564C22—4882no—
565C22—4882no—
566C22—4884no—
567C22—2444no—
568C22—2448no—
569C22—2442no—
570C22—2444no—
571C22—2442no—
572C22—2444no—
573C22#1683964014yes—
574C22C2×#1451973911yescharacteristic
575C22C2×#1451974711yescharacteristic
576C22C2×#1451975011yescharacteristic
577C22#1684056612yes—
578C22#1684057112yes—
579C4—44no—
580C4—82no—
581C4—1224no—
582C4—2442no—
583C4—2442no—
584C4#1683964014yes—
585C4C2×C22⋊C4×D6111yescharacteristic
586C4C2×C4×D24411yescharacteristic
587C4C2×#1451975211yescharacteristic
588C4C2×#1451976211yescharacteristic
589C2—24no—
590C2—4884no—
591C2—2444no—
592C2C2×#1683964011yescharacteristic
593C2#2145464912yes—
594C2#2145508811yescharacteristic
595C2#2145510611yescharacteristic
596C2#2145535712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#14519805C2
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20932
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#16840243C2semidirect
3. C2×#16839756C2semidirect
4. C22×#14519697C2semidirect
5. #21448399C2semidirect
6. #21446175C2semidirect
7. #21446168C2semidirect
8. #21445517C2semidirect
9. C61×#1736C2semidirect
14. C2×#14519805C4semidirect
20. C2×#14519742C22semidirect
21. C2×#14519739C22semidirect
27. C2×#14519700C22semidirect
30. C2×#14519697C22semidirect
35. #16839516C22semidirect
39. C61×#340C22semidirect
54. C22×D244C2×C4semidirect
63. C2×#11758190C2×C4semidirect
66. C2×D488C2×C4semidirect
96. C2×M16×C61C2×C4semidirect
136. #11758222C22⋊C4semidirect
298. #1736D61semidirect
336. #340C2×D61semidirect
338. C61#20932semidirect
370. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
C2×#168402432
C2×#168397562
C22×#145196972
#214483992
#214461752
#214461682
#214455172
C61×#17362
#2093261