Finite Group Databasebeta

← #29610383

Subgroups — #29610383

Order 15616.

1506 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839674C211yesmaximal, characteristic
2#21444089C212yesmaximal, complemented
3#21444079C212yesmaximal, complemented
4#21444056C212yesmaximal
5#21444051C212yesmaximal, complemented
6#21444043C212yesmaximal, complemented
7#21444041C212yesmaximal
8#21444041C211yesmaximal, characteristic
9C61×#1439C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C24×Dic61C2211yescharacteristic
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C22×C4×Dic61C2212yes—
15#16839674C418yescomplemented
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C22×C4⋊C4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2212yes—
29C22×C22⋊C4×C61C2211yescharacteristic
30C2×#95×C61C2212yes—
31C2×#95×C61C2211yescharacteristic
32C22×Dic122—41no—
33C22×Dic122—41no—
34C22×Dic122—42no—
35C22×Dic122—42no—
36C22×Dic122C2×C414yescomplemented
37C23×Dic61—22no—
38C23×Dic61—24no—
39C23×Dic61—41no—
40C23×Dic61—42no—
41C23×Dic61—41no—
42C23×Dic61C2312yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46C2×C22⋊Dic61—42no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61—41no—
49C2×C22⋊Dic61—41no—
50C2×C22⋊Dic61C2×C418yes—
51C2×C22⋊Dic61Q814yescomplemented
52C2×C4×Dic61—24no—
53C2×C4×Dic61—42no—
54C2×C4×Dic61—42no—
55#14519684—24no—
56#14519684—24no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—42no—
61#14519684—42no—
62#14519684—42no—
63#14519684—42no—
64#14519684C2×C418yescomplemented
65#14519684C2×C414yescomplemented
66C25×C61C2311yescharacteristic
67C23×C4×C61C2312yes—
68C23×C4×C61C2311yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2312yes—
71C2×C4⋊C4×C61—24no—
72C2×C4⋊C4×C61—41no—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—41no—
76C2×C22⋊C4×C61—41no—
77C2×C22⋊C4×C61—42no—
78C2×C22⋊C4×C61C2×C414yes—
79#95×C61—42no—
80#95×C61—41no—
81C24×C61—22no—
82C24×C61—22no—
83C24×C61—21no—
84C24×C61—22no—
85C24×C61—21no—
86C24×C61—22no—
87C24×C61—21no—
88C24×C61—21no—
89C24×C61C22×C412yes—
90C24×C61C2×Q812yes—
91C24×C61#4112yes—
92C24×C61C2411yescharacteristic
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
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×C61C22×C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61#4114yes—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61—48no—
143C22×Dic61—48no—
144C22×Dic61—44no—
145C22×Dic61—44no—
146C22×Dic61C22×C414yes—
147C22×Dic61C22×C414yes—
148C22×Dic61C2×Q814yes—
149C22×Dic61#4114yes—
150C2×Dic122—48no—
151C2×Dic122—44no—
152C2×Dic122—44no—
153C2×Dic122—48no—
154C2×Dic122—44no—
155C4⋊C4×C61—44no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—48no—
158C22⋊Dic61—48no—
159C22⋊Dic61—44no—
160C4×Dic61—48no—
161C23×C61—24no—
162C23×C61—24no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C4×Q814yes—
184C23×C61C23×C411yescharacteristic
185C23×C61C22×Q811yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4112yes—
188C23×C61C2×#4112yes—
189C23×C61H(5,𝔽2)12yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C61H(5,𝔽2)11yescharacteristic
192C23×C61H(5,𝔽2)11yescharacteristic
193C23×C61H(5,𝔽2)11yescharacteristic
194C23×C612−1+411yescharacteristic
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
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×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—416no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248Dic122—416no—
249#14596—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—28no—
252C22×C61—44no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—44no—
261C22×C61C2×C4×Q811yescharacteristic
262C22×C61C4×#4112yes—
263C22×C61#51812yes—
264C22×C61#51811yescharacteristic
265C22×C61#52011yescharacteristic
266C22×C61#53811yescharacteristic
267C22×C61#53812yes—
268C22×C61#53812yes—
269C22×C61#53911yescharacteristic
270C22×C61#54212yes—
271C22×C61#54212yes—
272C22×C61#54311yescharacteristic
273C22×C61#54411yescharacteristic
274C22×C61#55112yes—
275C22×C61#55111yescharacteristic
276C22×C61#55112yes—
277C22×C61#55111yescharacteristic
278C22×C61#55412yes—
279C22×C61#55412yes—
280C22×C61#55512yes—
281C22×C61#56011yescharacteristic
282C22×C61#56012yes—
283C22×C61H(3,𝔽4)11yescharacteristic
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—44no—
287C4×C61—82no—
288Dic61—44no—
289Dic61—44no—
290Dic61—44no—
291Dic61—44no—
292C2×#393—611no—
293#1446—612no—
294#1443—612no—
295#1439D6111yescharacteristic
296#1437—612no—
297#1436—612no—
298#1434—612no—
299C22⋊#95—612no—
300C22⋊#95—611no—
301C2×C61—44no—
302C2×C61#232911yescharacteristic
303C2×C61#233712yes—
304C2×C61#236812yes—
305C2×C61#237211yescharacteristic
306C2×C61#237511yescharacteristic
307C2×C61#273611yescharacteristic
308C2×C61#273712yes—
309C2×C61#274211yescharacteristic
310C2×C61#276011yescharacteristic
311C2×C61#276311yescharacteristic
312C2×C61#278412yes—
313C23×Q8—611no—
314C24×C4—611no—
315C22×C4⋊C4C2×D6111yescharacteristic
316C22×C22⋊C4—612no—
317C22×C22⋊C4—611no—
318C22×C22⋊C4C2×D6112yes—
319C22×C22⋊C4C2×D6111yescharacteristic
320C22×C42—612no—
321#393—618no—
322C2×#95—611no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—612no—
332C2×#95—612no—
333C2×#95C2×D6112yes—
334C2×#95C2×D6111yescharacteristic
335C61#1459611yescharacteristic, complemented, Sylow
336C25C22×D6111yescharacteristic
337C22×Q8—614no—
338C22×Q8—2442no—
339C22×Q8—2441no—
340C22×Q8—2441no—
341C22×Q8—2442no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—1224no—
347C23×C4—1222no—
348C23×C4—2441no—
349C23×C4—2442no—
350C23×C4—2441no—
351C23×C4C22×D6112yes—
352C23×C4C22×D6111yescharacteristic
353C23×C4C22×D6111yescharacteristic
354C23×C4C22×D6112yes—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—41no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—618no—
363C2×C22⋊C4—614no—
364C2×C22⋊C4—2441no—
365C2×C22⋊C4—2442no—
366C2×C22⋊C4—2442no—
367C2×C22⋊C4—2441no—
368C2×C22⋊C4C4×D6114yes—
369C2×C42—1224no—
370C2×C42—2442no—
371C2×C42—2442no—
372#95—42no—
373#95—41no—
374#95—614no—
375#95—618no—
376#95—1224no—
377#95—1224no—
378#95—2442no—
379#95—2442no—
380#95—2442no—
381#95—2442no—
382#95—2442no—
383#95—2442no—
384#95—2442no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24C2×C4×D6112yes—
394C24#1175822412yes—
395C24Q8×D6112yes—
396C24C23×D6111yescharacteristic
397C2×Q8—2448no—
398C2×Q8—2444no—
399C2×Q8—2444no—
400C2×Q8—2448no—
401C2×Q8—2444no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—614no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—2444no—
452C22×C4—2448no—
453C22×C4—2448no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4C2×C4×D6112yes—
457C22×C4C2×C4×D6112yes—
458C22×C4#1175822414yes—
459C4⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—2448no—
463C22⋊C4—2444no—
464C42—2448no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23#1451974214yes—
488C23C22×C4×D6111yescharacteristic
489C23C2×Q8×D6111yescharacteristic
490C23#41×D6112yes—
491C23C2×#1175822412yes—
492C23C2×#1175822411yescharacteristic
493C23#1451982212yes—
494C23#1451982211yescharacteristic
495C23#1451982211yescharacteristic
496C23#1451982312yes—
497C23#1451982311yescharacteristic
498C23#1451982411yescharacteristic
499Q8—24416no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
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—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—24416no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22C2×#1451974211yescharacteristic
565C22C4×#1175822412yes—
566C22#1684023412yes—
567C22#1684023511yescharacteristic
568C22#1684024211yescharacteristic
569C22#1684034911yescharacteristic
570C22#1684034912yes—
571C22#1684035112yes—
572C22#1684035711yescharacteristic
573C22#1684038012yes—
574C22#1684038112yes—
575C22#1684038711yescharacteristic
576C22#1684039111yescharacteristic
577C22#1684043512yes—
578C22#1684044011yescharacteristic
579C22#1684044312yes—
580C22#1684044311yescharacteristic
581C22#1684046612yes—
582C22#1684047012yes—
583C22#1684047612yes—
584C22#1684049911yescharacteristic
585C22#1684050012yes—
586C22#1684050411yescharacteristic
587C4—44no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—2444no—
592C4—2444no—
593C4—2444no—
594C4—2444no—
595C2—44no—
596C2#2144854911yescharacteristic
597C2#2144862612yes—
598C2#2144890012yes—
599C2#2144892811yescharacteristic
600C2#2144895611yescharacteristic
601C2#2145286611yescharacteristic
602C2#2145287612yes—
603C2#2145294911yescharacteristic
604C2#2145319811yescharacteristic
605C2#2145323611yescharacteristic
606C2#2145352112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1439C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1439D61
hypocenterC61#14596
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. #21444089C2semidirect
3. #21444079C2semidirect
5. #21444051C2semidirect
6. #21444043C2semidirect
15. #16839674C4semidirect
36. C22×Dic122C2×C4semidirect
51. C2×C22⋊Dic61Q8semidirect
64. #14519684C2×C4semidirect
65. #14519684C2×C4semidirect
335. C61#14596semidirect

Maximal subgroups

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

subgroupindex
C2×#168396742
#214440892
#214440792
#214440562
#214440512
#214440432
#214440412
C61×#14392
#1459661