Finite Group Databasebeta

← #29611408

Subgroups — #29611408

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
2C23×#11758198C211yesmaximal, characteristic
3C2×#16839675C212yesmaximal, complemented
4C2×#16839617C212yesmaximal
5#21444065C212yesmaximal, complemented
6#21444063C214yesmaximal, complemented
7#21444041C212yesmaximal, complemented
8C61×C22⋊#95C211yesmaximal, characteristic, complemented
9C25×D61C2211yescharacteristic
10C23×Dic122C2211yescharacteristic, complemented
11C24×Dic61C2211yescharacteristic
12C2×#14519756—28no—
13C2×#14519756—24no—
14C22×C4⋊Dic61C2211yescharacteristic, complemented
15C22×#11758196C2212yescomplemented
16C22×C22⋊Dic61C2212yes—
17C22×#11758198—28no—
18C22×#11758198—24no—
19C22×#11758198C2212yes—
20C22×#11758198C2214yes—
21C22×#11758198C2214yes—
22C22×#11758198C2212yes—
23#16839675—28no—
24#16839617—28no—
25C2×#14519684C2214yescomplemented
26C2×#14519684C2214yescomplemented
27C24×C4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2212yes—
29C2×#95×C61C2212yescomplemented
30C2×#95×C61C2212yescomplemented
31C24×D61—22no—
32C24×D61—22no—
33C24×D61—24no—
34C24×D61—24no—
35C24×D61—24no—
36C24×D61—24no—
37C24×D61—42no—
38C24×D61—42no—
39C24×D61Q814yescomplemented
40C24×D61C2312yes—
41C22×Dic122—22no—
42C22×Dic122—24no—
43C22×Dic122—24no—
44C22×Dic122—42no—
45C23×Dic61—22no—
46C23×Dic61—24no—
47C23×Dic61—24no—
48C23×Dic61—42no—
49C23×Dic61C2312yes—
50C23×Dic61C2312yes—
51C23×Dic61C2312yes—
52#14519756—232no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—24no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×C22⋊Dic61—24no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61D418yes—
61C2×#11758198—232no—
62C2×#11758198—24no—
63C2×#11758198—24no—
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—24no—
67C2×#11758198—28no—
68C2×#11758198—28no—
69C2×#11758198—28no—
70C2×#11758198—28no—
71C2×#11758198—44no—
72C2×#11758198—44no—
73C2×#11758198D418yes—
74#14519684—28no—
75#14519684—44no—
76C25×C61C2311yescharacteristic
77C23×C4×C61—24no—
78C23×C4×C61—24no—
79C23×C4×C61—42no—
80C23×C4×C61D414yes—
81C23×C4×C61C2314yes—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C2×C22⋊C4×C61—24no—
85C2×C22⋊C4×C61—24no—
86C2×C22⋊C4×C61—24no—
87#95×C61—24no—
88#95×C61—42no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61C2×D412yes—
94C24×C61C2×D414yes—
95C24×C61C2×Q812yes—
96C24×C61#4114yes—
97C24×C61#4112yes—
98C24×C61C2411yescharacteristic
99C23×D61—216no—
100C23×D61—28no—
101C23×D61—28no—
102C23×D61—28no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—24no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—24no—
109C23×D61—24no—
110C23×D61—24no—
111C23×D61—216no—
112C23×D61—216no—
113C23×D61—48no—
114C23×D61—416no—
115C23×D61—416no—
116C23×D61—416no—
117C23×D61—416no—
118C23×D61—44no—
119C23×D61—44no—
120C23×D61—48no—
121C23×D61—44no—
122C23×D61—48no—
123C23×D61—44no—
124C23×D61C2×D414yes—
125C23×D61C2×Q814yes—
126C23×D61#4114yes—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—28no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—28no—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—44no—
142C22×C4×C61—416no—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D414yes—
146C22×Dic61—28no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—24no—
158C22×Dic61—24no—
159C22×Dic61—416no—
160C22×Dic61—44no—
161C22×Dic61—44no—
162C22×Dic61C2×D414yes—
163C22×Dic61C2×D414yes—
164C22×Dic61C2×D414yes—
165C2×Dic122—28no—
166C2×Dic122—416no—
167C2×Dic122—44no—
168C2×Dic122—44no—
169C22⋊C4×C61—48no—
170C22⋊Dic61—216no—
171#11758198—216no—
172#11758198—216no—
173#11758198—416no—
174#11758198—416no—
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#12014yes—
190C23×C61C22⋊Q818yes—
191C23×C61C22⋊Q814yes—
192C23×C61#12418yes—
193C23×C61#12414yes—
194C23×C61C22×D411yescharacteristic
195C23×C61C22×D412yes—
196C23×C61C22×D412yes—
197C23×C61C22×Q811yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61C2×#4112yes—
200C23×C61C2×#4112yes—
201C23×C61H(5,𝔽2)12yes—
202C23×C61H(5,𝔽2)12yes—
203C22×D61—232no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—28no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—28no—
212C22×D61—28no—
213C22×D61—28no—
214C22×D61—28no—
215C22×D61—28no—
216C22×D61—464no—
217C22×D61—416no—
218C22×D61—416no—
219C22×D61—416no—
220C22×D61—416no—
221C22×D61—416no—
222C22×D61—464no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—44no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—44no—
229C22×D61—44no—
230C22×D61—44no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61—416no—
234C22×D61C22⋊Q818yescomplemented
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—28no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—416no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61—44no—
247C2×C4×C61—44no—
248C2×C4×C61—44no—
249C2×C4×C61—44no—
250C2×C4×C61#12014yes—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—28no—
255C2×Dic61—28no—
256C2×Dic61—28no—
257C2×Dic61—416no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
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#39318yescomplemented
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×#12011yescharacteristic
280C22×C61C2×C22⋊Q812yes—
281C22×C61C2×C22⋊Q812yes—
282C22×C61C2×C22⋊Q811yescharacteristic
283C22×C61#52512yes—
284C22×C61#52511yescharacteristic
285C22×C61C2×#12412yes—
286C22×C61C2×#12411yescharacteristic
287C22×C61#53411yescharacteristic
288C22×C61#53511yescharacteristic
289C22×C61#53812yes—
290C22×C61#54312yes—
291C22×C61D4214yes—
292C22×C61#54614yes—
293C22×C61#54614yes—
294C22×C61#55114yes—
295C2×D61—28no—
296C2×D61—216no—
297C2×D61—464no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—44no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—48no—
305C2×D61—48no—
306C2×D61—48no—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—84no—
310Dic61—44no—
311Dic61—44no—
312Dic61—44no—
313C22×C22⋊Q8—611no—
314C23×C22⋊C4—611no—
315C2×#393—612no—
316C2×#379—612no—
317#1439—614no—
318#1439—612no—
319C22⋊#95—612no—
320C22⋊#95D6111yescharacteristic, complemented
321C2×C61—28no—
322C2×C61C2×#39311yescharacteristic
323C2×C61#240311yescharacteristic
324C2×C61#240411yescharacteristic
325C2×C61#243511yescharacteristic
326C2×C61#243714yes—
327C2×C61#247214yes—
328C2×C61#256111yescharacteristic
329C2×C61#262611yescharacteristic
330C2×C61#284811yescharacteristic
331D61—48no—
332C26—611no—
333C23×Q8—611no—
334C24×C4—611no—
335C24×C4C2×D6111yescharacteristic
336C2×C22⋊Q8—1228no—
337C2×C22⋊Q8—1224no—
338C22×C4⋊C4—612no—
339C22×C4⋊C4—611no—
340C22×C22⋊C4—612no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4—614no—
344C22×C22⋊C4—614no—
345C22×C22⋊C4—1228no—
346C22×C22⋊C4—1224no—
347C22×C22⋊C4C2×D6112yes—
348#393—1228no—
349#379—1228no—
350C2×#95—614no—
351C2×#95—614no—
352C2×#95C2×D6112yescomplemented
353C2×#95C2×D6112yescomplemented
354C61#1471711yescharacteristic, complemented, Sylow
355C25—612no—
356C25—614no—
357C25—1224no—
358C25—1224no—
359C25—1222no—
360C25—1222no—
361C25—1224no—
362C25—1224no—
363C25—2442no—
364C25—2442no—
365C25C22×D6111yescharacteristic
366C22×Q8—1224no—
367C22×Q8—1222no—
368C22×Q8—1224no—
369C22×Q8—2442no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—42no—
373C23×C4—612no—
374C23×C4—612no—
375C23×C4—612no—
376C23×C4—1224no—
377C23×C4—1222no—
378C23×C4—1224no—
379C23×C4—2442no—
380C23×C4D24414yes—
381C23×C4C22×D6114yes—
382C23×C4C22×D6111yescharacteristic
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—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—618no—
394C2×C22⋊C4—618no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—12232no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1228no—
404C2×C22⋊C4—1228no—
405C2×C22⋊C4—1228no—
406C2×C22⋊C4—1228no—
407C2×C22⋊C4—2444no—
408C2×C22⋊C4—2444no—
409#95—24no—
410#95—42no—
411#95—1228no—
412#95—2444no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—614no—
418C24—614no—
419C24—614no—
420C24—12216no—
421C24—1228no—
422C24—12216no—
423C24—1228no—
424C24—1222no—
425C24—1224no—
426C24—1222no—
427C24—1222no—
428C24—1222no—
429C24—1224no—
430C24—1224no—
431C24—1224no—
432C24—1228no—
433C24—12216no—
434C24—2448no—
435C24—24416no—
436C24—24416no—
437C24—24416no—
438C24—24416no—
439C24—2444no—
440C24—2444no—
441C24—2448no—
442C24—2444no—
443C24—2448no—
444C24—2444no—
445C24C2×D24412yes—
446C24#1175822214yes—
447C24D4×D6114yes—
448C24#1175822412yes—
449C24Q8×D6112yes—
450C24C23×D6111yescharacteristic
451C2×Q8—1228no—
452C2×Q8—24416no—
453C2×Q8—2444no—
454C2×Q8—2444no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—44no—
470C22×C4—416no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—1228no—
475C22×C4—1224no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—24416no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4C2×D24412yes—
491C22×C4C2×D24412yes—
492C22×C4D4×D6114yes—
493C4⋊C4—12216no—
494C4⋊C4—1228no—
495C22⋊C4—48no—
496C22⋊C4—12216no—
497C22⋊C4—12216no—
498C22⋊C4—12216no—
499C22⋊C4—24416no—
500C22⋊C4—24416no—
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—618no—
514C23—12232no—
515C23—1224no—
516C23—1224no—
517C23—1228no—
518C23—1224no—
519C23—1224no—
520C23—1224no—
521C23—1224no—
522C23—1228no—
523C23—1228no—
524C23—1228no—
525C23—1228no—
526C23—1228no—
527C23—24464no—
528C23—24416no—
529C23—24416no—
530C23—24416no—
531C23—24416no—
532C23—24464no—
533C23—24416no—
534C23—2442no—
535C23—2444no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23—2444no—
540C23—2444no—
541C23—2444no—
542C23—2444no—
543C23—2444no—
544C23—24416no—
545C23#1451974714yes—
546C23#1451975518yes—
547C23#1451975714yes—
548C23#1451976318yes—
549C23#1451976514yes—
550C23C2×D4×D6112yes—
551C23C2×D4×D6112yes—
552C23C22×D24411yescharacteristic
553C23C2×Q8×D6111yescharacteristic
554C23#41×D6112yes—
555C23C2×#1175822411yescharacteristic
556C23C2×#1175822212yes—
557C23#1451982212yes—
558C23#1451982312yes—
559Q8—24416no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1228no—
579C2×C4—1228no—
580C2×C4—1228no—
581C2×C4—24416no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4#1451974714yes—
592C22—216no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—216no—
599C22—24no—
600C22—12216no—
601C22—1228no—
602C22—24464no—
603C22—2448no—
604C22—2444no—
605C22—2444no—
606C22—2444no—
607C22—2444no—
608C22—2444no—
609C22—2448no—
610C22—2448no—
611C22—2444no—
612C22#1683967518yescomplemented
613C22C2×#1451974711yescharacteristic
614C22C2×#1451974711yescharacteristic
615C22C22⋊Q8×D6112yes—
616C22C2×#1451975512yes—
617C22C2×#1451975711yescharacteristic
618C22#1684027212yes—
619C22#1684027411yescharacteristic
620C22C2×#1451976312yes—
621C22C2×#1451976511yescharacteristic
622C22#1684032411yescharacteristic
623C22#1684033111yescharacteristic
624C22#1684034912yes—
625C22#1684038712yes—
626C22#1684039414yes—
627C22#1684040414yes—
628C22#1684040814yes—
629C22#1684044114yes—
630C4—24no—
631C4—42no—
632C4—84no—
633C4—2444no—
634C4—2444no—
635C4—2444no—
636C2—28no—
637C2—2448no—
638C2C2×#1683967511yescharacteristic
639C2#2144913211yescharacteristic
640C2#2144914011yescharacteristic
641C2#2144947011yescharacteristic
642C2#2144948414yes—
643C2#2144995014yes—
644C2#2145097211yescharacteristic
645C2#2145162711yescharacteristic
646C2#2145410711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutator, socle, cosocleC24×C61C24
FittingC61×C22⋊#95C2
prime radicalC25×D61C22
hypercenterC22⋊#95D61
hypocenterC61#14717
LCSG ▷ C24×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×#14519756C2semidirect
3. C2×#16839675C2semidirect
5. #21444065C2semidirect
6. #21444063C2semidirect
7. #21444041C2semidirect
8. C61×C22⋊#95C2semidirect
10. C23×Dic122C22semidirect
14. C22×C4⋊Dic61C22semidirect
15. C22×#11758196C22semidirect
25. C2×#14519684C22semidirect
26. C2×#14519684C22semidirect
29. C2×#95×C61C22semidirect
30. C2×#95×C61C22semidirect
39. C24×D61Q8semidirect
234. C22×D61C22⋊Q8semidirect
277. C22×C61#393semidirect
320. C22⋊#95D61semidirect
352. C2×#95C2×D61semidirect
353. C2×#95C2×D61semidirect
354. C61#14717semidirect
612. C22#16839675semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C23×#117581982
C2×#168396752
C2×#168396172
#214440652
#214440632
#214440412
C61×C22⋊#952
#1471761