Finite Group Databasebeta

← #29610858

Subgroups — #29610858

Order 15616.

1486 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839679C211yesmaximal, characteristic
2#21444074C212yesmaximal, complemented
3#21444067C212yesmaximal, complemented
4#21444063C212yesmaximal, complemented
5#21444063C212yesmaximal, complemented
6#21444059C212yesmaximal
7#21444055C212yesmaximal, complemented
8#21444051C211yesmaximal, characteristic, complemented
9C61×#1445C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×#11758196C2212yes—
12C22×#11758198C2212yes—
13C22×#11758198C2212yes—
14C22×#11758198C2212yes—
15C22×#11758198C2212yes—
16C22×#11758198C2212yes—
17C22×#11758198C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19#16839679C418yescomplemented
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2211yescharacteristic
31C2×#95×C61C2211yescharacteristic
32C24×D61C2312yes—
33C22×D244—42no—
34C22×D244—41no—
35C22×D244—41no—
36C22×D244—42no—
37C22×D244C2×C414yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C2×#11758196—42no—
42C2×#11758196—42no—
43C2×#11758196—42no—
44C2×#11758198—24no—
45C2×#11758198—24no—
46C2×#11758198—42no—
47C2×#11758198—42no—
48C2×#11758198—42no—
49C2×#11758198—42no—
50C2×#11758198—42no—
51C2×#11758198—42no—
52C2×#11758198—42no—
53C2×#11758198—42no—
54C2×#11758198—42no—
55C2×#11758198—42no—
56C2×#11758198—42no—
57C2×#11758198—42no—
58C2×#11758198—42no—
59C2×#11758198—42no—
60C2×#11758198C2×C418yes—
61C2×#11758198C2×C418yes—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—41no—
64#14519684—24no—
65#14519684—42no—
66#14519684—42no—
67#14519684—42no—
68#14519684—42no—
69#14519684—41no—
70#14519684C2×C414yes—
71C23×C4×C61C2312yes—
72C23×C4×C61C2312yes—
73C23×C4×C61C2311yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C2×C4⋊C4×C61—41no—
77C2×C4⋊C4×C61—41no—
78C2×C4⋊C4×C61C2×C414yes—
79#95×C61—42no—
80#95×C61—41no—
81#95×C61—41no—
82#95×C61—42no—
83C24×C61C2411yescharacteristic
84C23×D61—22no—
85C23×D61—22no—
86C23×D61—22no—
87C23×D61—22no—
88C23×D61—22no—
89C23×D61—22no—
90C23×D61—22no—
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61C22×C414yes—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
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×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
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—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61C22×C414yes—
148C22×Dic61#4114yes—
149C2×D244—44no—
150C2×D244—48no—
151C2×D244—44no—
152C2×D244—48no—
153C2×D244—44no—
154C4⋊C4×C61—44no—
155#11758198—48no—
156#11758198—48no—
157#11758198—48no—
158#11758198—48no—
159#11758198—48no—
160#11758198—48no—
161#11758196—48no—
162C4×Dic61—44no—
163C23×C61C23×C411yescharacteristic
164C23×C61C2×#4112yes—
165C23×C61H(5,𝔽2)11yescharacteristic
166C23×C61H(5,𝔽2)12yes—
167C23×C61H(5,𝔽2)12yes—
168C23×C61H(5,𝔽2)11yescharacteristic
169C23×C61H(5,𝔽2)12yes—
170C23×C61H(5,𝔽2)12yes—
171C23×C612−1+412yes—
172C22×D61—24no—
173C22×D61—24no—
174C22×D61—24no—
175C22×D61—24no—
176C22×D61—24no—
177C22×D61—24no—
178C22×D61—24no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—42no—
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×Dic61—42no—
257D244—416no—
258#14641—611nomaximal, Sylow
259C22×C61C4×#4111yescharacteristic
260C22×C61#51812yes—
261C22×C61#51811yescharacteristic
262C22×C61#51811yescharacteristic
263C22×C61#52012yes—
264C22×C61#53812yes—
265C22×C61#53812yes—
266C22×C61#53912yes—
267C22×C61#54012yes—
268C22×C61#54012yes—
269C22×C61#54212yes—
270C22×C61#56011yescharacteristic
271C22×C61#56012yes—
272C22×C61#56012yes—
273C22×C61#56012yes—
274C22×C61H(3,𝔽4)11yescharacteristic
275C22×C61H(3,𝔽4)11yescharacteristic
276C22×C61#56212yes—
277C22×C61#56212yes—
278C22×C61#56212yes—
279C22×C61#56311yescharacteristic
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C2×D61—82no—
292C2×D61—82no—
293C2×D61—82no—
294C2×D61—82no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—82no—
299C4×C61—82no—
300Dic61—44no—
301Dic61—44no—
302Dic61—82no—
303C2×#394—611no—
304#1445D6111yescharacteristic, complemented
305#1442—612no—
306#1440—612no—
307#1439—612no—
308#1439—612no—
309#1438—612no—
310#1437—612no—
311#1436—611no—
312C2×C61#233311yescharacteristic
313C2×C61#233512yes—
314C2×C61#237511yescharacteristic
315C2×C61#237712yes—
316C2×C61#237911yescharacteristic
317C2×C61#282412yes—
318C2×C61#282812yes—
319C2×C61#282912yes—
320C2×C61#283712yes—
321D61—84no—
322C23×D4—611no—
323C22×C4⋊C4—612no—
324C22×C4⋊C4C2×D6111yescharacteristic
325C22×C22⋊C4—612no—
326C22×C22⋊C4—612no—
327C22×C22⋊C4—612no—
328C22×C22⋊C4—612no—
329C22×C22⋊C4—612no—
330C22×C22⋊C4—612no—
331C22×C42—611no—
332#394—618no—
333C2×#95—611no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—611no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95C2×D6112yes—
341C2×#95C2×D6112yes—
342C2×#95C2×D6111yescharacteristic
343C2×#95C2×D6111yescharacteristic
344C61#1464111yescharacteristic, complemented, Sylow
345C25—612no—
346C22×D4—614no—
347C22×D4—2442no—
348C22×D4—2441no—
349C22×D4—2441no—
350C22×D4—2442no—
351C23×C4—612no—
352C23×C4—612no—
353C23×C4—612no—
354C23×C4C22×D6112yes—
355C23×C4C22×D6112yes—
356C23×C4C22×D6111yescharacteristic
357C23×C4C22×D6111yescharacteristic
358C23×C4C22×D6111yescharacteristic
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—2442no—
362C2×C4⋊C4—2442no—
363C2×C4⋊C4—2442no—
364C2×C4⋊C4C4×D6114yes—
365C2×C22⋊C4—618no—
366C2×C22⋊C4—618no—
367C2×C22⋊C4—1224no—
368C2×C22⋊C4—1224no—
369C2×C22⋊C4—2442no—
370C2×C22⋊C4—2442no—
371C2×C22⋊C4—2442no—
372C2×C22⋊C4—2442no—
373C2×C22⋊C4—2442no—
374C2×C22⋊C4—2442no—
375C2×C22⋊C4—2442no—
376C2×C22⋊C4—2442no—
377C2×C22⋊C4—2442no—
378C2×C22⋊C4—2442no—
379C2×C22⋊C4—2442no—
380C2×C22⋊C4—2442no—
381C2×C22⋊C4—2442no—
382C2×C22⋊C4—2442no—
383C2×C42—2441no—
384C2×C42—2442no—
385#95—42no—
386#95—41no—
387#95—41no—
388#95—42no—
389#95—614no—
390#95—1224no—
391#95—2441no—
392#95—2442no—
393#95—2442no—
394#95—2442no—
395#95—2442no—
396C24—614no—
397C24—1222no—
398C24—1222no—
399C24—1222no—
400C24—1222no—
401C24—1222no—
402C24—1222no—
403C24—1222no—
404C24—1222no—
405C24—1222no—
406C24—1222no—
407C24—1222no—
408C24—1222no—
409C24—1222no—
410C24—1222no—
411C24C23×D6111yescharacteristic
412C2×D4—2448no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4—2448no—
416C2×D4—2444no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
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×C4C2×C4×D6112yes—
465C22×C4C2×C4×D6112yes—
466C22×C4C2×C4×D6112yes—
467C4⋊C4—44no—
468C4⋊C4—2448no—
469C22⋊C4—2448no—
470C22⋊C4—2448no—
471C22⋊C4—2448no—
472C22⋊C4—2448no—
473C22⋊C4—2448no—
474C22⋊C4—2448no—
475C42—2444no—
476C23—1224no—
477C23—1224no—
478C23—1224no—
479C23—1224no—
480C23—1224no—
481C23—1224no—
482C23—1224no—
483C23—2442no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23—2442no—
491C23—2442no—
492C23—2442no—
493C23—2442no—
494C23—2442no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23—2442no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23C22×C4×D6111yescharacteristic
512C23#41×D6112yes—
513C23#1451982212yes—
514C23#1451982211yescharacteristic
515C23#1451982211yescharacteristic
516C23#1451982312yes—
517C23#1451982312yes—
518C23#1451982312yes—
519C23#1451982512yes—
520D4—24416no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1224no—
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×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C22—4882no—
572C22—4882no—
573C22—4882no—
574C22—4882no—
575C22—4882no—
576C22—4882no—
577C22—4882no—
578C22—4882no—
579C22—2444no—
580C22—2444no—
581C22—2444no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22—2444no—
586C22#1684023011yescharacteristic
587C22#1684023412yes—
588C22#1684023511yescharacteristic
589C22#1684023611yescharacteristic
590C22#1684024412yes—
591C22#1684034812yes—
592C22#1684035112yes—
593C22#1684036012yes—
594C22#1684036412yes—
595C22#1684036412yes—
596C22#1684038212yes—
597C22#1684049911yescharacteristic
598C22#1684050012yes—
599C22#1684050212yes—
600C22#1684050312yes—
601C22#1684050411yescharacteristic
602C22#1684050411yescharacteristic
603C22#1684050812yes—
604C22#1684050812yes—
605C22#1684051012yes—
606C22#1684051111yescharacteristic
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4—4882no—
613C4—2444no—
614C4—2444no—
615C2—4884no—
616C2#2144858111yescharacteristic
617C2#2144861512yes—
618C2#2144895611yescharacteristic
619C2#2144897412yes—
620C2#2144899111yescharacteristic
621C2#2145390012yes—
622C2#2145394012yes—
623C2#2145395812yes—
624C2#2145403012yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1445C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenter#1445D61
hypocenterC61#14641
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. #21444074C2semidirect
3. #21444067C2semidirect
4. #21444063C2semidirect
5. #21444063C2semidirect
7. #21444055C2semidirect
8. #21444051C2semidirect
9. C61×#1445C2semidirect
19. #16839679C4semidirect
304. #1445D61semidirect
344. C61#14641semidirect

Maximal subgroups

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

subgroupindex
C2×#168396792
#214440742
#214440672
#214440632
#214440592
#214440552
#214440512
C61×#14452
#1464161