Finite Group Databasebeta

← #29611406

Subgroups — #29611406

Order 15616.

3654 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic, complemented
2C2×#16839674C212yesmaximal, complemented
3C2×#16839618C212yesmaximal
4#21444064C212yesmaximal, complemented
5#21444062C214yesmaximal, complemented
6#21444042C211yesmaximal, characteristic, complemented
7#21444042C212yesmaximal, complemented
8C23×C22⋊C4×C61C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic, complemented
10C2×#14519756—28no—
11C2×#14519756—24no—
12C22×C4⋊Dic61C2211yescharacteristic, complemented
13C22×#11758196C2212yescomplemented
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2214yes—
16C22×C22⋊Dic61C2214yes—
17C22×C22⋊Dic61C2212yes—
18#16839674—28no—
19#16839618—28no—
20C2×#14519684C2214yescomplemented
21C2×#14519684C2212yescomplemented
22C2×#14519684C2212yescomplemented
23C2×#14519684C2214yescomplemented
24C26×C61C2211yescharacteristic
25C24×C4×C61C2211yescharacteristic
26C24×C4×C61C2211yescharacteristic
27C22×C22⋊C4×C61—24no—
28C22×C22⋊C4×C61—28no—
29C22×C22⋊C4×C61C2212yes—
30C22×C22⋊C4×C61C2212yes—
31C22×Dic122—22no—
32C22×Dic122—24no—
33C22×Dic122—24no—
34C22×Dic122—42no—
35C23×Dic61C2314yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38#14519756—232no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—24no—
41C2×#11758196—24no—
42C2×#11758196—24no—
43C2×#11758196—24no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—28no—
48C2×C22⋊Dic61—28no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—28no—
52C2×C22⋊Dic61—28no—
53C2×C22⋊Dic61—44no—
54C2×C22⋊Dic61—44no—
55C2×C22⋊Dic61D418yes—
56#14519684—28no—
57#14519684—24no—
58#14519684—44no—
59#14519684—42no—
60C25×C61—24no—
61C25×C61—22no—
62C25×C61—24no—
63C25×C61—22no—
64C25×C61—24no—
65C25×C61—24no—
66C25×C61—42no—
67C25×C61—42no—
68C25×C61Q814yescomplemented
69C25×C61C2312yes—
70C25×C61C2311yescharacteristic
71C23×C4×C61—24no—
72C23×C4×C61—22no—
73C23×C4×C61—24no—
74C23×C4×C61—24no—
75C23×C4×C61—24no—
76C23×C4×C61—42no—
77C23×C4×C61—42no—
78C23×C4×C61D414yes—
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2312yes—
81C23×C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—232no—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—24no—
85C2×C22⋊C4×C61—24no—
86C2×C22⋊C4×C61—24no—
87C2×C22⋊C4×C61—24no—
88C2×C22⋊C4×C61D418yes—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—28no—
94C24×C61—28no—
95C24×C61—216no—
96C24×C61—28no—
97C24×C61—216no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—24no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—24no—
104C24×C61—24no—
105C24×C61—24no—
106C24×C61—216no—
107C24×C61—48no—
108C24×C61—416no—
109C24×C61—416no—
110C24×C61—416no—
111C24×C61—416no—
112C24×C61—44no—
113C24×C61—44no—
114C24×C61—48no—
115C24×C61—44no—
116C24×C61—48no—
117C24×C61—44no—
118C24×C61C2×D414yes—
119C24×C61C2×D414yes—
120C24×C61C2×D412yes—
121C24×C61C2×Q814yes—
122C24×C61C2×Q812yes—
123C24×C61#4114yes—
124C24×C61#4112yes—
125C24×C61#4114yes—
126C24×C61C2411yescharacteristic
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—28no—
130C22×C4×C61—28no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—24no—
138C22×C4×C61—28no—
139C22×C4×C61—44no—
140C22×C4×C61—416no—
141C22×C4×C61—416no—
142C22×C4×C61—44no—
143C22×C4×C61—44no—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D414yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D414yes—
148C22×Dic61—24no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—24no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—24no—
158C22×Dic61—24no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—24no—
162C22×Dic61—24no—
163C22×Dic61C2×D414yes—
164C22×Dic61C2×D414yes—
165C2×Dic122—28no—
166C2×Dic122—416no—
167C2×Dic122—44no—
168C2×Dic122—44no—
169C22⋊C4×C61—216no—
170C22⋊C4×C61—216no—
171C22⋊Dic61—216no—
172C22⋊Dic61—416no—
173C22⋊Dic61—416no—
174C22⋊Dic61—48no—
175C4⋊Dic61—28no—
176#11758196—216no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—28no—
188C23×C61—28no—
189C23×C61—232no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—28no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61—28no—
198C23×C61—28no—
199C23×C61—28no—
200C23×C61—28no—
201C23×C61—28no—
202C23×C61—464no—
203C23×C61—416no—
204C23×C61—416no—
205C23×C61—416no—
206C23×C61—416no—
207C23×C61—416no—
208C23×C61—464no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—44no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—44no—
215C23×C61—44no—
216C23×C61—44no—
217C23×C61—44no—
218C23×C61—44no—
219C23×C61—416no—
220C23×C61#12014yes—
221C23×C61C22⋊Q818yes—
222C23×C61C22⋊Q814yes—
223C23×C61C22⋊Q818yescomplemented
224C23×C61#12418yes—
225C23×C61#12414yes—
226C23×C61C22×D411yescharacteristic
227C23×C61C22×D412yes—
228C23×C61C22×D412yes—
229C23×C61C22×Q811yescharacteristic
230C23×C61C2×#4111yescharacteristic
231C23×C61C2×#4112yes—
232C23×C61C2×#4112yes—
233C23×C61H(5,𝔽2)12yes—
234C23×C61H(5,𝔽2)12yes—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—28no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—28no—
243C2×C4×C61—416no—
244C2×C4×C61—416no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—44no—
248C2×C4×C61—44no—
249C2×C4×C61#12014yes—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—28no—
253C2×Dic61—28no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259C2×Dic61—44no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—44no—
263C2×Dic61—44no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267Dic122—416no—
268#14717—611nomaximal, Sylow
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—24no—
275C22×C61—216no—
276C22×C61—216no—
277C22×C61—28no—
278C22×C61—216no—
279C22×C61—464no—
280C22×C61—44no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—44no—
286C22×C61—48no—
287C22×C61—48no—
288C22×C61—48no—
289C22×C61#39318yescomplemented
290C22×C61C2×#12011yescharacteristic
291C22×C61C2×#12011yescharacteristic
292C22×C61C2×C22⋊Q812yes—
293C22×C61C2×C22⋊Q811yescharacteristic
294C22×C61C2×C22⋊Q812yes—
295C22×C61#52511yescharacteristic
296C22×C61#52512yes—
297C22×C61C2×#12412yes—
298C22×C61C2×#12411yescharacteristic
299C22×C61#53411yescharacteristic
300C22×C61#53511yescharacteristic
301C22×C61#53812yes—
302C22×C61#54312yes—
303C22×C61D4214yes—
304C22×C61#54614yes—
305C22×C61#54614yes—
306C22×C61#55114yes—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—44no—
310Dic61—44no—
311Dic61—44no—
312Dic61—84no—
313C22×C22⋊Q8—611no—
314C23×C22⋊C4D6111yescharacteristic
315C2×#393—612no—
316C2×#379—612no—
317#1439—614no—
318#1439—612no—
319C22⋊#95—612no—
320C22⋊#95—611no—
321C2×C61—28no—
322C2×C61—48no—
323C2×C61C2×#39311yescharacteristic
324C2×C61#240311yescharacteristic
325C2×C61#240411yescharacteristic
326C2×C61#243511yescharacteristic
327C2×C61#243714yes—
328C2×C61#247214yes—
329C2×C61#256111yescharacteristic
330C2×C61#262611yescharacteristic
331C2×C61#284811yescharacteristic
332C26C2×D6111yescharacteristic
333C23×Q8—611no—
334C24×C4C2×D6111yescharacteristic
335C24×C4C2×D6111yescharacteristic
336C2×C22⋊Q8—1228no—
337C2×C22⋊Q8—1224no—
338C22×C4⋊C4—612no—
339C22×C4⋊C4—611no—
340C22×C22⋊C4—24no—
341C22×C22⋊C4—28no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4—612no—
344C22×C22⋊C4—614no—
345C22×C22⋊C4—614no—
346C22×C22⋊C4C2×D6112yes—
347C22×C22⋊C4C2×D6112yes—
348#393—1228no—
349#379—1228no—
350C2×#95—614no—
351C2×#95—612no—
352C2×#95—614no—
353C2×#95—612no—
354C61#1471711yescharacteristic, complemented, Sylow
355C25—24no—
356C25—22no—
357C25—24no—
358C25—22no—
359C25—24no—
360C25—24no—
361C25—42no—
362C25—42no—
363C25Dic12214yescomplemented
364C25C22×D6112yes—
365C25C22×D6111yescharacteristic
366C22×Q8—1224no—
367C22×Q8—1222no—
368C22×Q8—1224no—
369C22×Q8—2442no—
370C23×C4—24no—
371C23×C4—22no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—42no—
376C23×C4—42no—
377C23×C4—612no—
378C23×C4—612no—
379C23×C4—614no—
380C23×C4C22⋊D6114yes—
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6112yes—
383C23×C4C22×D6111yescharacteristic
384C22⋊Q8—12232no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—1222no—
390C2×C22⋊C4—232no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1228no—
400C2×C22⋊C4—1228no—
401C2×C22⋊C4—1228no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1228no—
406C2×C22⋊C4—2444no—
407C2×C22⋊C4—2444no—
408C2×C22⋊C4C22⋊D6118yes—
409#95—1228no—
410#95—1224no—
411#95—2444no—
412#95—2442no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—28no—
418C24—216no—
419C24—28no—
420C24—24no—
421C24—22no—
422C24—22no—
423C24—24no—
424C24—22no—
425C24—22no—
426C24—24no—
427C24—24no—
428C24—28no—
429C24—216no—
430C24—216no—
431C24—48no—
432C24—416no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—44no—
437C24—44no—
438C24—48no—
439C24—44no—
440C24—48no—
441C24—44no—
442C24C2×Dic12214yes—
443C24C2×Dic12212yes—
444C24#1175822214yes—
445C24#1175822212yes—
446C24D4×D6114yes—
447C24#1175822414yes—
448C24C2×C22⋊D6114yes—
449C24C2×C22⋊D6112yes—
450C24C23×D6111yescharacteristic
451C2×Q8—1228no—
452C2×Q8—24416no—
453C2×Q8—2444no—
454C2×Q8—2444no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—44no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1224no—
488C22×C4—1224no—
489C22×C4C2×C22⋊D6114yes—
490C22×C4C2×C22⋊D6112yes—
491C22×C4C2×C22⋊D6114yes—
492C22×C4C2×C22⋊D6112yes—
493C4⋊C4—12216no—
494C4⋊C4—1228no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—12216no—
498C22⋊C4—24416no—
499C22⋊C4—24416no—
500C22⋊C4—2448no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—232no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—464no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—464no—
531C23—416no—
532C23—416no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—416no—
544C23#1451974514yes—
545C23#1451975418yes—
546C23#1451975614yes—
547C23#1451975618yescomplemented
548C23#1451976318yes—
549C23#1451976414yes—
550C23C2×D4×D6112yes—
551C23C22×C22⋊D6111yescharacteristic
552C23C22×C22⋊D6112yes—
553C23C22×Dic12211yescharacteristic
554C23C2×#1175822412yes—
555C23C2×#1175822211yescharacteristic
556C23C2×#1175822212yes—
557C23#1451982212yes—
558C23#1451982212yes—
559Q8—24416no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1228no—
577C2×C4—1228no—
578C2×C4—2444no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4#1451974614yes—
592C22—216no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—216no—
598C22—24no—
599C22—24no—
600C22—216no—
601C22—28no—
602C22—464no—
603C22—48no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22#1683967418yescomplemented
613C22C2×#1451974511yescharacteristic
614C22C2×#1451974611yescharacteristic
615C22C2×#1451975412yes—
616C22C2×#1451975612yes—
617C22C2×#1451975611yescharacteristic
618C22#1684027311yescharacteristic
619C22#1684027312yes—
620C22C2×#1451976312yes—
621C22C2×#1451976411yescharacteristic
622C22#1684032111yescharacteristic
623C22#1684032911yescharacteristic
624C22#1684035012yes—
625C22#1684038812yes—
626C22D4×C22⋊D6114yes—
627C22#1684040214yes—
628C22#1684040614yes—
629C22#1684043814yes—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—4884no—
634C4—2444no—
635C4—2444no—
636C2—28no—
637C2—48no—
638C2C2×#1683967411yescharacteristic
639C2#2144912911yescharacteristic
640C2#2144913811yescharacteristic
641C2#2144946611yescharacteristic
642C2#2144948014yes—
643C2#2144993914yes—
644C2#2145097711yescharacteristic
645C2#2145162411yescharacteristic
646C2#2145410611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutator, socle, cosocleC24×C61C24
FittingC23×C22⋊C4×C61C2
prime radicalC26×C61C22
hypercenterC23×C22⋊C4D61
hypocenterC61#14717
LCSG ▷ C24×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14519756C2semidirect
2. C2×#16839674C2semidirect
4. #21444064C2semidirect
5. #21444062C2semidirect
6. #21444042C2semidirect
7. #21444042C2semidirect
9. C23×Dic122C22semidirect
12. C22×C4⋊Dic61C22semidirect
13. C22×#11758196C22semidirect
20. C2×#14519684C22semidirect
21. C2×#14519684C22semidirect
22. C2×#14519684C22semidirect
23. C2×#14519684C22semidirect
68. C25×C61Q8semidirect
223. C23×C61C22⋊Q8semidirect
289. C22×C61#393semidirect
354. C61#14717semidirect
363. C25Dic122semidirect
547. C23#14519756semidirect
612. C22#16839674semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168396742
C2×#168396182
#214440642
#214440622
#214440422
C23×C22⋊C4×C612
#1471761