Finite Group Databasebeta

← #29712441

Subgroups — #29712441

Order 15616.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic, complemented
2C2×#16839913C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4#21446965C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446177C212yesmaximal, complemented
8#21446137C212yesmaximal, complemented
9#21446117C211yesmaximal, characteristic, complemented
10#21445911C211yesmaximal, characteristic, complemented
11C61×#2408C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C22×#11758194C2211yescharacteristic
14#16840452—24no—
15#16840452—24no—
16C2×#14519770—21no—
17C2×#14519769C2211yescharacteristic, complemented
18C2×#14519767—21no—
19C2×#14519758—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519756C2211yescharacteristic, complemented
22C2×C4×Dic122C2211yescharacteristic, complemented
23C2×#14519736C2211yescharacteristic
24C2×#14519733—21no—
25C22×C4⋊Dic61C2211yescharacteristic, complemented
26#16839913—24no—
27#16839767—24no—
28C2×#14519700C2212yescomplemented
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519698C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2212yescomplemented
34#16839524C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#121×C61C2212yes—
40C2×#120×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C61×#394—21no—
43C61×#392—21no—
44C61×#390C2212yes—
45C61×#380C2211yescharacteristic
46C61×#379—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C2×#11758200—24no—
50C2×#11758200—24no—
51C2×#11758200—42no—
52C2×#11758194—22no—
53C2×#11758194—42no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56#14519770—28no—
57#14519769—22no—
58#14519769—22no—
59#14519767—28no—
60#14519758—28no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64#14519756D414yescomplemented
65C4×Dic122—22no—
66C4×Dic122—22no—
67#14519736—22no—
68#14519736D414yescomplemented
69#14519733—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86#14519700—42no—
87#14519698—24no—
88#14519698—24no—
89#14519698—24no—
90C22⋊#92418—22no—
91C22⋊#92418—42no—
92C25×C61—21no—
93C22×D4×C61—24no—
94C22×D4×C61—22no—
95C22×D4×C61—22no—
96C22×D4×C61—22no—
97C22×D4×C61—21no—
98C22×D4×C61—42no—
99C22×D4×C61D411yescharacteristic
100C22×D4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102C23×C4×C61C2311yescharacteristic
103#121×C61—24no—
104#121×C61—42no—
105#120×C61—44no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61—22no—
113C2×C22⋊C4×C61—22no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—42no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—82no—
124C24×C61C2×D411yescharacteristic
125C24×C61C2×D411yescharacteristic
126C2×D4×C61—28no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—48no—
135C2×D4×C61—41no—
136C2×D4×C61—44no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61—42no—
141C2×D4×C61—82no—
142C2×D4×C61#4114yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—21no—
167C2×Dic122C2×D412yes—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—44no—
174C22⋊C4×C61—44no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—82no—
177C42×C61—21no—
178C42×C61—21no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—24no—
182#11758200—48no—
183C4⋊Dic61—24no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C2×D412yes—
188C4⋊Dic61C2×D412yes—
189C4⋊Dic61C2×D412yes—
190#11758196—24no—
191#11758196—24no—
192#11758196—24no—
193#11758196—24no—
194#11758196—24no—
195C4×Dic61—24no—
196C4×Dic61—24no—
197#11758194—44no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—24no—
213C23×C61—48no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—84no—
221C23×C61—84no—
222C23×C61—88no—
223C23×C61—88no—
224C23×C61#12012yes—
225C23×C61AGL(1,ℤ/8)12yes—
226C23×C61C22×D411yescharacteristic
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—48no—
230D4×C61—44no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—88no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61#12012yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4112yes—
255C2×C4×C61H(5,𝔽2)12yes—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—48no—
261C2×Dic61—42no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267Dic122—24no—
268Dic122—41no—
269Dic122—41no—
270#92418—42no—
271#92418—42no—
272#23982—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—24no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—44no—
286C22×C61—816no—
287C22×C61—82no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61C2×#12011yescharacteristic
292C22×C61#54612yes—
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57712yes—
296C4×C61—22no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—82no—
301C4×C61#52511yescharacteristic
302C4×C61#53511yescharacteristic
303C4×C61#54612yes—
304Dic61—42no—
305Dic61—42no—
306Dic61—42no—
307Dic61—42no—
308C2×#552—611no—
309C2×#450—611no—
310C2×#418—611no—
311#2408D6111yescharacteristic
312#2025—612no—
313#2000—612no—
314#1894—611no—
315#1880—612no—
316#1874—612no—
317#1870—611no—
318#1825—611no—
319C2×C61—42no—
320C2×C61—42no—
321C2×C61—84no—
322C2×C61#240411yescharacteristic
323C2×C61#300611yescharacteristic
324C2×C61#301011yescharacteristic
325C2×C61#331012yes—
326C2×C61#332312yes—
327C23×D4C2×D6111yescharacteristic
328C22×SD16—612no—
329C22×M16—611no—
330#552—1224no—
331#552—1224no—
332C2×#126—1221no—
333C2×#125—611no—
334C2×#125—1221no—
335C2×#123—1221no—
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8—611no—
338C2×#121C2×D6112yes—
339C2×#120—21no—
340C2×C4×Q8—611no—
341C2×C4×D4C2×D6111yescharacteristic
342C2×#117—611no—
343C2×#117—1221no—
344C22×C4⋊C4—611no—
345#450—1224no—
346#418—1224no—
347C2×#103—612no—
348C2×#102—612no—
349C2×#102—612no—
350C2×#102—612no—
351C2×C22⋊C8—611no—
352C2×C22⋊C8—612no—
353#394—21no—
354#392—21no—
355#390C2×D6112yes—
356#380C2×D6111yescharacteristic
357#379—21no—
358#340—612no—
359#340—612no—
360#340—612no—
361#336—612no—
362C61#2398211yescharacteristic, complemented, Sylow
363C25—21no—
364C22×Q8—611no—
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—42no—
371C22×D4C22⋊D6111yescharacteristic
372C22×D4C22×D6112yes—
373C23×C4—1221no—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C2×SD16—1224no—
377C2×SD16—1224no—
378C2×SD16—2442no—
379C2×M16—1222no—
380C2×M16—2442no—
381C22×C8—612no—
382C22×C8—612no—
383#126—1228no—
384#125—1222no—
385#125—1222no—
386#125—1228no—
387#123—1228no—
388C22⋊Q8—614no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392#121—24no—
393#121—42no—
394#120—44no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×D4—22no—
398C4×D4—22no—
399#117—614no—
400#117—1228no—
401#117—1222no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C4⋊C4C22×D6112yes—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427#103—2442no—
428#102—1224no—
429#102—1224no—
430#102—1224no—
431C22⋊C8—1222no—
432C22⋊C8—2442no—
433#95—21no—
434C24—22no—
435C24—44no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24C2×C22⋊D6111yescharacteristic
441C24C2×C22⋊D6111yescharacteristic
442C2×Q8—612no—
443C2×Q8—1221no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—48no—
455C2×D4—41no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4#1175822414yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—1224no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6112yes—
478C22×C4D4×D6112yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—2448no—
485M16—2444no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1222no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1224no—
501C4⋊C4—1224no—
502C4⋊C4—1224no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1224no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—82no—
513C22⋊C4—1224no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C42—21no—
517C42—21no—
518C42—1224no—
519C42—1224no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23—88no—
539C23—88no—
540C23#1451974512yes—
541C23#1451980312yes—
542C23C22×C22⋊D6111yescharacteristic
543Q8—1224no—
544Q8—2441no—
545Q8—2441no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—44no—
550D4—42no—
551D4—42no—
552D4—88no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—2448no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4#1451974512yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×#1175822412yes—
585C2×C4#1451982212yes—
586C8—2442no—
587C8—2442no—
588C22—22no—
589C22—24no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C2×#1451974511yescharacteristic
607C22#1684040612yes—
608C22C2×#1451980311yescharacteristic
609C22#1684056011yescharacteristic
610C22#1684057312yes—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—82no—
616C4—2442no—
617C4—2442no—
618C4—2442no—
619C4—2442no—
620C4#1684026811yescharacteristic
621C4#1684032711yescharacteristic
622C4#1684039912yes—
623C2—42no—
624C2—42no—
625C2—84no—
626C2#2144913511yescharacteristic
627C2#2145506911yescharacteristic
628C2#2145510311yescharacteristic
629C2#2145784012yes—
630C2#2145803512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2408C2
socleC23×C61C22×D4
hypercenter#2408D61
hypocenterC61#23982
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#16840452C2semidirect
2. C2×#16839913C2semidirect
3. C2×#16839767C2semidirect
4. #21446965C2semidirect
5. #21446822C2semidirect
6. #21446250C2semidirect
7. #21446177C2semidirect
8. #21446137C2semidirect
9. #21446117C2semidirect
10. #21445911C2semidirect
17. C2×#14519769C22semidirect
21. C2×#14519756C22semidirect
22. C2×C4×Dic122C22semidirect
25. C22×C4⋊Dic61C22semidirect
28. C2×#14519700C22semidirect
33. C2×C22⋊#92418C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
64. #14519756D4semidirect
68. #14519736D4semidirect
362. C61#23982semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168399132
C2×#168397672
#214469652
#214468222
#214462502
#214461772
#214461372
#214461172
#214459112
C61×#24082
#2398261