Finite Group Databasebeta

← #29712647

Subgroups — #29712647

Order 15616.

1844 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×#14519751C211yesmaximal, characteristic
2C2×#16840076C212yesmaximal, complemented
3C2×#16839622C211yesmaximal, characteristic, complemented
4#21447002C214yesmaximal, complemented
5#21446134C212yesmaximal, complemented
6#21446009C212yesmaximal, complemented
7#21445751C212yesmaximal, complemented
8C61×#1779C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C23×C22⋊D61—21no—
11C2×#14519751—28no—
12C2×#14519751—21no—
13C2×#14519751—21no—
14C2×#14519751C2212yes—
15C2×#14519751C2212yes—
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic, complemented
18C22×C4⋊Dic61C2211yescharacteristic, complemented
19C22×C22⋊Dic61—21no—
20C22×#11758198—21no—
21#16840076—28no—
22C2×#14519711C2214yescomplemented
23C2×#14519699C2214yescomplemented
24C2×#14519699C2214yescomplemented
25#16839622—21no—
26#16839622—21no—
27#16839622C2212yes—
28#16839622C2212yescomplemented
29C2×#14519684—21no—
30#16839525C2214yescomplemented
31C24×C4×C61C2211yescharacteristic
32C2×C22⋊C8×C61C2212yes—
33C2×C22⋊C8×C61C2212yescomplemented
34C2×C22⋊C8×C61C2212yescomplemented
35C24×D61—21no—
36C22×D244—21no—
37C22×D244—24no—
38C22×D244—24no—
39C22×D244—42no—
40C22×D244C2312yes—
41C22×C22⋊D61—28no—
42C22×C22⋊D61—22no—
43C22×C22⋊D61—22no—
44C22×C22⋊D61—44no—
45C22×C22⋊D61—44no—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C23×Dic61—21no—
49#14519751—28no—
50#14519751—28no—
51#14519751—24no—
52#14519751—24no—
53#14519751—24no—
54#14519751D418yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—24no—
57C2×C4⋊Dic61—24no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—42no—
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—44no—
67C2×#11758198—28no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70#14519711—28no—
71#14519699—28no—
72#14519699—44no—
73#14519684—22no—
74#14519684—22no—
75C25×C61D411yescharacteristic
76C23×C4×C61—24no—
77C23×C4×C61—24no—
78C23×C4×C61—42no—
79C23×C4×C61D411yescharacteristic
80C23×C4×C61D411yescharacteristic
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61C2314yes—
86C22⋊C8×C61—24no—
87C22⋊C8×C61—24no—
88C22⋊C8×C61—42no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—41no—
94C24×C61—41no—
95C24×C61D814yes—
96C24×C61C2×D411yescharacteristic
97C24×C61C2×D411yescharacteristic
98C24×C61C2×D411yescharacteristic
99C23×D61—24no—
100C23×D61—22no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
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×C61—22no—
120C22×C4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—416no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4114yes—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—28no—
131C2×C22⋊D61—28no—
132C2×C22⋊D61—432no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C2×D244—24no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—22no—
153C2×D244—416no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244C2×D414yes—
157C2×C8×C61—24no—
158C2×C8×C61—24no—
159C2×C8×C61—24no—
160C22⋊Dic61—44no—
161C22⋊Dic61—44no—
162#11758198—28no—
163#11758198—28no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—24no—
167C4⋊Dic61—24no—
168C4⋊Dic61—42no—
169C4⋊Dic61—42no—
170C4⋊Dic61C2×D414yes—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—44no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61—44no—
191C23×C61—44no—
192C23×C61#12012yes—
193C23×C61#12312yes—
194C23×C61C2×D812yes—
195C23×C61C2×D812yes—
196C23×C61C2×D812yes—
197C23×C61AGL(1,ℤ/8)12yes—
198C23×C61#13712yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—48no—
202C22×D61—48no—
203C22×D61—48no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—84no—
211C22×D61—84no—
212C22×D61—84no—
213C22×D61—84no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—28no—
219C2×C4×C61—28no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61#12012yes—
228C2×C4×C61#12312yes—
229C2×C4×C61C22×D412yes—
230C2×C4×C61C2×#4112yes—
231C2×C4×C61H(5,𝔽2)12yes—
232C22⋊D61—416no—
233C22⋊D61—416no—
234C2×Dic61—24no—
235C2×Dic61—48no—
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—84no—
245C2×Dic61—84no—
246D244—28no—
247D244—48no—
248D244—44no—
249C8×C61—44no—
250#24002—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—416no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#44714yes—
272C22×C61#48014yescomplemented
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C22×C61#53411yescharacteristic
276C22×C61C22×D811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—816no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61#54614yes—
292Dic61—44no—
293Dic61—82no—
294Dic61—82no—
295C22×#121—611no—
296C2×#480—612no—
297C2×#380—611no—
298#2033—614no—
299#1874—612no—
300#1852—612no—
301#1786—612no—
302#1779D6111yescharacteristic, complemented
303C2×C61—24no—
304C2×C61—24no—
305C2×C61—42no—
306C2×C61#240511yescharacteristic
307C2×C61C2×#44711yescharacteristic
308C2×C61#300711yescharacteristic
309C2×C61C2×#48011yescharacteristic
310C2×C61#309411yescharacteristic
311C2×C61#318611yescharacteristic
312C2×C61#319311yescharacteristic
313D61—84no—
314C23×D4—611no—
315C23×D4—1221no—
316C24×C4C2×D6111yescharacteristic
317C2×#121—612no—
318C2×#121—612no—
319C2×#121—1228no—
320C2×#121—1221no—
321C2×#121—1221no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4—611no—
325C22×C22⋊C4—1221no—
326C22×C22⋊C4—1221no—
327#480—1228no—
328C2×#107—614no—
329C2×#102—614no—
330C2×#102—614no—
331C2×C22⋊C8C2×D6112yes—
332C2×C22⋊C8C2×D6112yescomplemented
333C2×C22⋊C8C2×D6112yescomplemented
334#380—612no—
335#380—612no—
336#380—1221no—
337#380—1221no—
338C2×#95—1221no—
339#340—614no—
340C61#2400211yescharacteristic, complemented, Sylow
341C25—1221no—
342C25D24411yescharacteristic
343C22×D4—612no—
344C22×D4—1224no—
345C22×D4—1228no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1221no—
349C22×D4—1224no—
350C22×D4—2442no—
351C22×D4—2444no—
352C22×D4—2444no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—42no—
356C23×C4—1221no—
357C23×C4—1221no—
358C23×C4—1221no—
359C23×C4D24411yescharacteristic
360C23×C4D24411yescharacteristic
361C23×C4D24411yescharacteristic
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C22×C8C22×D6114yes—
366#121—618no—
367#121—1224no—
368#121—1224no—
369#121—1228no—
370#121—1228no—
371#121—1224no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—2442no—
381C2×C22⋊C4—1228no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—2444no—
387#107—1228no—
388#102—1228no—
389#102—2444no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—42no—
393#95—1222no—
394#95—1222no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—1224no—
402C24—1222no—
403C24—2442no—
404C24—2442no—
405C24—2442no—
406C24—2442no—
407C24—2442no—
408C24—2442no—
409C24D48814yes—
410C24C2×D24411yescharacteristic
411C24C2×D24411yescharacteristic
412C24C2×D24411yescharacteristic
413C2×D4—614no—
414C2×D4—1224no—
415C2×D4—1228no—
416C2×D4—1228no—
417C2×D4—1222no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—24416no—
421C2×D4—2442no—
422C2×D4—24432no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4—2444no—
427C2×D4—2442no—
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—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—416no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4C2×D24411yescharacteristic
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4D4×D6114yes—
461C22×C4#1175822414yes—
462C22×C4C23×D6111yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C4⋊C4—614no—
467C4⋊C4—1224no—
468C4⋊C4—1222no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—2442no—
472C4⋊C4—2442no—
473C22⋊C4—1228no—
474C22⋊C4—1228no—
475C22⋊C4—2444no—
476C22⋊C4—2444no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—1224no—
499C23—4884no—
500C23—4884no—
501C23—4884no—
502C23—4884no—
503C23—2448no—
504C23—2448no—
505C23—2448no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23#1451974712yes—
513C23#1451976112yes—
514C23C2×D48812yes—
515C23C2×D48812yes—
516C23C2×D48812yes—
517C23#1451980512yes—
518C23#1451981012yes—
519C23C22×D24411yescharacteristic
520D4—1228no—
521D4—2448no—
522D4—24416no—
523D4—24416no—
524D4—2444no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—1224no—
539C2×C4—4884no—
540C2×C4—4884no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2448no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4#1451974712yes—
551C2×C4#1451976112yes—
552C2×C4C2×D4×D6112yes—
553C2×C4C2×#1175822412yes—
554C2×C4#1451982212yes—
555C8—44no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—48816no—
577C22—4882no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—2444no—
582C22—2444no—
583C22—2444no—
584C22#1683989814yes—
585C22#1684007614yescomplemented
586C22C2×#1451974711yescharacteristic
587C22C2×#1451976111yescharacteristic
588C22#1684032411yescharacteristic
589C22C22×D48811yescharacteristic
590C22C2×#1451980511yescharacteristic
591C22C2×#1451981011yescharacteristic
592C22#1684056211yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—4882no—
597C4—4882no—
598C4—2444no—
599C4#1684039914yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—4884no—
604C2#2144914711yescharacteristic
605C2C2×#1683989811yescharacteristic
606C2#2145508111yescharacteristic
607C2C2×#1684007611yescharacteristic
608C2#2145576011yescharacteristic
609C2#2145657011yescharacteristic
610C2#2145662611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1779C2
socleC23×C61C22×D4
prime radicalC22×#14519751C2
hypercenter#1779D61
hypocenterC61#24002
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16840076C2semidirect
3. C2×#16839622C2semidirect
4. #21447002C2semidirect
5. #21446134C2semidirect
6. #21446009C2semidirect
7. #21445751C2semidirect
8. C61×#1779C2semidirect
17. C22×C4⋊Dic61C22semidirect
18. C22×C4⋊Dic61C22semidirect
22. C2×#14519711C22semidirect
23. C2×#14519699C22semidirect
24. C2×#14519699C22semidirect
28. #16839622C22semidirect
30. #16839525C22semidirect
33. C2×C22⋊C8×C61C22semidirect
34. C2×C22⋊C8×C61C22semidirect
272. C22×C61#480semidirect
302. #1779D61semidirect
332. C2×C22⋊C8C2×D61semidirect
333. C2×C22⋊C8C2×D61semidirect
340. C61#24002semidirect
585. C22#16840076semidirect

Maximal subgroups

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

subgroupindex
C22×#145197512
C2×#168400762
C2×#168396222
#214470022
#214461342
#214460092
#214457512
C61×#17792
#2400261