Finite Group Databasebeta

← #29602739

Subgroups — #29602739

Order 15616.

1622 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic122C211yesmaximal, characteristic
2C2×#16839629C211yesmaximal, characteristic
3C2×#16839607C211yesmaximal, characteristic
4#21444089C212yesmaximal
5#21444085C212yesmaximal
6#21444053C212yesmaximal
7#21444050C212yesmaximal
8#21444046C211yesmaximal, characteristic
9#21444043C211yesmaximal, characteristic
10#21444043C211yesmaximal, characteristic
11C61×#1434C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×C4×Dic122—24no—
14C2×C4×Dic122—24no—
15C2×C4×Dic122C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×C4×Dic61C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16839629—24no—
23#16839607—24no—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C42×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—22no—
45C22×Dic122—22no—
46C22×Dic122C2×C414yes—
47C23×Dic61C2312yes—
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52C23×Dic61C2311yescharacteristic
53C4×Dic122—232no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61C2×C414yes—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196C2×C414yes—
62C2×#11758196C2×C414yes—
63C2×C4×Dic61—22no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61C2×C414yes—
69C2×C4×Dic61C2×C414yes—
70C2×C4×Dic61D414yes—
71C2×C4×Dic61D414yes—
72#14519684—22no—
73#14519684—22no—
74#14519684—24no—
75#14519684—24no—
76#14519684—24no—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83C23×C4×C61C2312yes—
84C23×C4×C61C2311yescharacteristic
85C23×C4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C2×C42×C61—22no—
89C2×C42×C61—22no—
90C2×C42×C61C2×C414yes—
91#95×C61—22no—
92#95×C61—22no—
93#95×C61—24no—
94#95×C61—24no—
95C24×C61C2411yescharacteristic
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4114yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61C22×C412yes—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61#4114yes—
163C22×Dic61#4112yes—
164C22×Dic61#4112yes—
165C2×Dic122—28no—
166C2×Dic122—28no—
167C2×Dic122—28no—
168C2×Dic122—28no—
169C2×Dic122—28no—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C42×C61—28no—
173C4⋊Dic61—28no—
174#11758196—28no—
175#11758196—28no—
176C4×Dic61—216no—
177C4×Dic61—28no—
178C4×Dic61C22⋊C4116yes—
179C23×C61C23×C411yescharacteristic
180C23×C61C22×D411yescharacteristic
181C23×C61C22×D411yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4111yescharacteristic
185C23×C61C2×#4111yescharacteristic
186C23×C61C2×#4112yes—
187C23×C61C2×#4112yes—
188C23×C612−1+412yes—
189C23×C612−1+412yes—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61C2×C22⋊C414yes—
220C2×C4×C61#11714yes—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C414yes—
257C2×Dic61C2×C22⋊C414yes—
258C2×Dic61#11714yes—
259Dic122—232no—
260#13961—611nomaximal, Sylow
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61C2×#11711yescharacteristic
263C22×C61C2×#11711yescharacteristic
264C22×C61C4×#4111yescharacteristic
265C22×C61C4×#4111yescharacteristic
266C22×C61#51911yescharacteristic
267C22×C61#51911yescharacteristic
268C22×C61#52512yes—
269C22×C61#52511yescharacteristic
270C22×C61#52511yescharacteristic
271C22×C61#52511yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61#53212yes—
274C22×C61#53612yes—
275C22×C61#53612yes—
276C22×C61#55212yes—
277C22×C61#55212yes—
278C22×C61#55212yes—
279C22×C61#55212yes—
280C22×C61#55612yes—
281C22×C61#55612yes—
282C22×C61#55612yes—
283C22×C61#55612yes—
284C4×C61—24no—
285C4×C61—24no—
286C4×C61—44no—
287C4×C61—42no—
288C4×C61—82no—
289Dic61—24no—
290Dic61—24no—
291Dic61—24no—
292Dic61—44no—
293Dic61—42no—
294Dic61—82no—
295C22×C4×Q8—611no—
296C2×#383—611no—
297C2×#376—611no—
298#1446—612no—
299#1445—612no—
300#1436—612no—
301#1436—612no—
302#1434—611no—
303#1434—611no—
304#1434—611no—
305#1434D6111yescharacteristic
306C2×C61#228011yescharacteristic
307C2×C61#229711yescharacteristic
308C2×C61#231011yescharacteristic
309C2×C61#235611yescharacteristic
310C2×C61#235611yescharacteristic
311C2×C61#235611yescharacteristic
312C2×C61#235611yescharacteristic
313C2×C61#259912yes—
314C2×C61#259912yes—
315C2×C61#260412yes—
316C2×C61#260412yes—
317C23×Q8—611no—
318C2×C4×Q8—618no—
319C2×C4×Q8—1224no—
320C2×C4×Q8—1224no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C42—611no—
325C22×C42—611no—
326C22×C42—612no—
327C22×C42C2×D6111yescharacteristic
328#383—1224no—
329#376—1224no—
330C2×#95—611no—
331C2×#95—611no—
332C2×#95—612no—
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95C2×D6111yescharacteristic
342C2×#95C2×D6111yescharacteristic
343C2×#95C2×D6112yes—
344C2×#95C2×D6112yes—
345C61#1396111yescharacteristic, complemented, Sylow
346C22×Q8—614no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8—1222no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—612no—
359C23×C4C22×D6112yes—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6112yes—
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C4×Q8—12232no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—614no—
377C2×C42—614no—
378C2×C42—614no—
379C2×C42—614no—
380C2×C42—1224no—
381C2×C42—1224no—
382C2×C42—1224no—
383C2×C42—1222no—
384C2×C42—1222no—
385C2×C42C4×D6114yes—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—1224no—
391#95—1224no—
392#95—1224no—
393#95—1224no—
394#95—1224no—
395#95—1224no—
396#95—1224no—
397#95—1224no—
398#95—1222no—
399#95—1222no—
400#95—1224no—
401C24C23×D6111yescharacteristic
402C2×Q8—1228no—
403C2×Q8—1228no—
404C2×Q8—1228no—
405C2×Q8—1228no—
406C2×Q8—1228no—
407C2×Q8—1228no—
408C2×Q8—1228no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4C2×C4×D6112yes—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×C4×D6112yes—
473C22×C4#1175822214yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4#1175822412yes—
477C22×C4#1175822612yes—
478C4⋊C4—1228no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C42—28no—
482C42—6116no—
483C42—12216no—
484C42—1228no—
485C23C22×C4×D6111yescharacteristic
486C23C2×D4×D6111yescharacteristic
487C23C2×D4×D6111yescharacteristic
488C23#41×D6111yescharacteristic
489C23#41×D6112yes—
490C23#41×D6111yescharacteristic
491C23C2×#1175822411yescharacteristic
492C23C2×#1175822212yes—
493C23C2×#1175822611yescharacteristic
494C23#1451982412yes—
495C23#1451982412yes—
496Q8—12232no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—614no—
527C2×C4—614no—
528C2×C4—614no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4C22⋊C4×D6114yes—
565C2×C4#1451973514yes—
566C22C2×C22⋊C4×D6111yescharacteristic
567C22#117×D6111yescharacteristic
568C22C2×#1451973511yescharacteristic
569C22#1684023011yescharacteristic
570C22C4×#1175822211yescharacteristic
571C22#1684023911yescharacteristic
572C22#1684023911yescharacteristic
573C22#1684027112yes—
574C22#1684027211yescharacteristic
575C22#1684027211yescharacteristic
576C22#1684027511yescharacteristic
577C22#1684027511yescharacteristic
578C22#1684030912yes—
579C22#1684033412yes—
580C22#1684033412yes—
581C22#1684044512yes—
582C22#1684044812yes—
583C22#1684045012yes—
584C22#1684045212yes—
585C22#1684047912yes—
586C22#1684048012yes—
587C22#1684048112yes—
588C22#1684048512yes—
589C4—24no—
590C4—24no—
591C4—42no—
592C4—44no—
593C4—82no—
594C4—1224no—
595C4—1224no—
596C4—1224no—
597C4—4882no—
598C4—2442no—
599C4—2444no—
600C2#2144820111yescharacteristic
601C2#2144830511yescharacteristic
602C2#2144840411yescharacteristic
603C2#2144877911yescharacteristic
604C2#2144878211yescharacteristic
605C2#2144878511yescharacteristic
606C2#2144878611yescharacteristic
607C2#2145139312yes—
608C2#2145139512yes—
609C2#2145145412yes—
610C2#2145145612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1434C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1434D61
hypocenterC61#13961
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic122C4semidirect
345. C61#13961semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1222
C2×#168396292
C2×#168396072
#214440892
#214440852
#214440532
#214440502
#214440462
#214440432
C61×#14342
#1396161