Finite Group Databasebeta

← #29712204

Subgroups — #29712204

Order 15616.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840458C211yesmaximal, characteristic, complemented
2C2×#16839896C211yesmaximal, characteristic, complemented
3C22×#14519698C211yesmaximal, characteristic, complemented
4#21446965C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446245C211yesmaximal, characteristic, complemented
7#21446170C212yesmaximal, complemented
8#21446142C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445907C211yesmaximal, characteristic, complemented
11C61×#2408C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C23×#92418C2211yescharacteristic
14#16840458—24no—
15#16840458—24no—
16C2×#14519772—21no—
17C2×#14519769C2211yescharacteristic, complemented
18C2×#14519759—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519751C2211yescharacteristic, complemented
21C2×#14519749—21no—
22C2×C4×D244C2211yescharacteristic, complemented
23C2×#14519737—21no—
24C2×#14519736C2211yescharacteristic, complemented
25C22×C4⋊Dic61C2211yescharacteristic
26#16839896—24no—
27C2×#14519698—24no—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yescomplemented
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×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61—21no—
52C2×#11758199—24no—
53C2×#11758199—24no—
54C2×#11758199—42no—
55C22×#92418—22no—
56C22×#92418—42no—
57C22×#92418C2312yes—
58C22×#92418C2312yes—
59#14519772—28no—
60#14519769—22no—
61#14519769—22no—
62#14519759—28no—
63#14519751—22no—
64#14519751—22no—
65#14519751—22no—
66#14519751D414yescomplemented
67#14519749—28no—
68C4×D244—22no—
69C4×D244—22no—
70#14519737—28no—
71#14519736—22no—
72#14519736—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D414yescomplemented
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86#14519698—24no—
87#14519698—24no—
88#14519698—24no—
89#14519697—42no—
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
126C23×D61—21no—
127C2×D4×C61—28no—
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—22no—
135C2×D4×C61—22no—
136C2×D4×C61—48no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61—44no—
141C2×D4×C61—42no—
142C2×D4×C61—82no—
143C2×D4×C61#4114yes—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—24no—
159C2×C22⋊D61—24no—
160C2×C22⋊D61—42no—
161C2×C22⋊D61—42no—
162C2×C22⋊D61—82no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—24no—
167C22×Dic61—82no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—21no—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—82no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—44no—
180C22⋊C4×C61—44no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—82no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—24no—
186#11758199—48no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—24no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61C2×D412yes—
196C4⋊Dic61C2×D412yes—
197#11758196—24no—
198#11758196—24no—
199C4×Dic61—24no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—24no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—48no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—88no—
223C23×C61—88no—
224C23×C61—84no—
225C23×C61—84no—
226C23×C61#12012yes—
227C23×C61#13512yes—
228C23×C61AGL(1,ℤ/8)12yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—42no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—42no—
235D4×C61—24no—
236D4×C61—24no—
237D4×C61—48no—
238D4×C61—44no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—88no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61#12012yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4112yes—
263C2×C4×C61H(5,𝔽2)12yes—
264C22⋊D61—48no—
265C22⋊D61—84no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—84no—
274D244—24no—
275D244—41no—
276D244—41no—
277C4×D61—84no—
278#92418—42no—
279#92418—42no—
280#23963—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—41no—
286C22×C61—44no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—42no—
290C22×C61—44no—
291C22×C61—42no—
292C22×C61—44no—
293C22×C61—816no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61—82no—
297C22×C61—82no—
298C22×C61#44914yes—
299C22×C61C2×#12011yescharacteristic
300C22×C61#54612yes—
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C61#57612yes—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—82no—
308C2×D61—82no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—82no—
314C4×C61#52511yescharacteristic
315C4×C61#53511yescharacteristic
316C4×C61#54612yes—
317Dic61—42no—
318Dic61—42no—
319Dic61—42no—
320C2×#553—611no—
321C2×#447—611no—
322C22×#102—611no—
323#2408D6111yescharacteristic, complemented
324#2025—612no—
325#1999—612no—
326#1893—611no—
327#1879—612no—
328#1875—612no—
329#1869—611no—
330#1824—611no—
331C2×C61—24no—
332C2×C61—81no—
333C2×C61—84no—
334C2×C61#240411yescharacteristic
335C2×C61C2×#44911yescharacteristic
336C2×C61#300811yescharacteristic
337C2×C61#329712yes—
338C2×C61#331812yes—
339D61—82no—
340C23×D4C2×D6111yescharacteristic
341C22×D8—612no—
342C23×C8—611no—
343#553—1224no—
344#553—1224no—
345C2×#126—1221no—
346C2×#125—611no—
347C2×#123—1221no—
348C2×#121—611no—
349C2×#121—611no—
350C2×#121—1221no—
351C2×#121C2×D6112yes—
352C2×#120—21no—
353C2×C4×D4—611no—
354C2×C4×D4—1221no—
355C2×C4×D4C2×D6111yescharacteristic
356C2×#117—611no—
357C22×C4⋊C4—611no—
358#447—1224no—
359C2×#102—612no—
360C2×#102—612no—
361C2×#102—612no—
362C2×#102—612no—
363C2×#102—1224no—
364C2×C22⋊C8—611no—
365C2×C22⋊C8—612no—
366#394—21no—
367#392—21no—
368#390C2×D6112yes—
369#380C2×D6111yescharacteristic
370#379—21no—
371#340—612no—
372#340—612no—
373#340—612no—
374#336—612no—
375C61#2396311yescharacteristic, complemented, Sylow
376C25—21no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—21no—
382C22×D4—42no—
383C22×D4—611no—
384C22×D4—1221no—
385C22×D4—1221no—
386C22×D4C22⋊D6111yescharacteristic
387C22×D4C22×D6112yes—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4C22×D6111yescharacteristic
391C23×C4C22×D6111yescharacteristic
392C2×D8—1224no—
393C2×D8—1224no—
394C2×D8—2442no—
395C22×C8—612no—
396C22×C8—612no—
397C22×C8—1222no—
398C22×C8—2442no—
399#126—1228no—
400#125—1222no—
401#125—1222no—
402#123—1228no—
403#121—24no—
404#121—42no—
405#121—614no—
406#121—1222no—
407#121—1222no—
408#121—1228no—
409#121—1222no—
410#120—44no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1228no—
416#117—1222no—
417#117—1222no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4C22⋊D6111yescharacteristic
427C2×C4⋊C4C22×D6112yes—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4C22⋊D6111yescharacteristic
437C2×C22⋊C4C22⋊D6111yescharacteristic
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440#102—1224no—
441#102—1224no—
442#102—1224no—
443#102—2442no—
444C22⋊C8—1222no—
445C22⋊C8—2442no—
446#95—21no—
447C24—22no—
448C24—44no—
449C24—42no—
450C24—42no—
451C24—42no—
452C24—82no—
453C24—1221no—
454C24C2×C22⋊D6111yescharacteristic
455C24C2×C22⋊D6111yescharacteristic
456C2×D4—28no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—48no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—82no—
472C2×D4—612no—
473C2×D4—1224no—
474C2×D4—1224no—
475C2×D4—1221no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—4882no—
479C2×D4—2442no—
480C2×D4—2442no—
481C2×D4#1175822414yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—1224no—
490C22×C4—1224no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4882no—
495C22×C4—4882no—
496C22×C4D4×D6112yes—
497C22×C4D4×D6112yes—
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503D8—2448no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1224no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—82no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C22⋊C4—1224no—
532C22⋊C4—1224no—
533C42—21no—
534C42—21no—
535C42—1224no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—88no—
552C23—88no—
553C23—84no—
554C23—84no—
555C23—1222no—
556C23—2441no—
557C23—2441no—
558C23—2442no—
559C23—2442no—
560C23#1451974512yes—
561C23#1451979712yes—
562C23#1451980312yes—
563C23C22×C22⋊D6111yescharacteristic
564D4—24no—
565D4—24no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—42no—
570D4—88no—
571D4—1224no—
572D4—4884no—
573D4—2448no—
574D4—2441no—
575D4—2441no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—4884no—
597C2×C4—4884no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4#1451974512yes—
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4C2×#1175822412yes—
606C2×C4#1451982212yes—
607C8—2442no—
608C8—2442no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—41no—
618C22—41no—
619C22—44no—
620C22—41no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—4882no—
627C22—4882no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22#1683990514yes—
632C22C2×#1451974511yescharacteristic
633C22#1684040612yes—
634C22C2×#1451979711yescharacteristic
635C22C2×#1451980311yescharacteristic
636C22#1684056512yes—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—82no—
642C4—2442no—
643C4—2442no—
644C4—2442no—
645C4#1684026811yescharacteristic
646C4#1684032711yescharacteristic
647C4#1684039912yes—
648C2—24no—
649C2—81no—
650C2—84no—
651C2—4882no—
652C2#2144913511yescharacteristic
653C2C2×#1683990511yescharacteristic
654C2#2145508511yescharacteristic
655C2#2145766912yes—
656C2#2145796012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2408C2
socleC23×C61C22×D4
hypercenter#2408D61
hypocenterC61#23963
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×#16840458C2semidirect
2. C2×#16839896C2semidirect
3. C22×#14519698C2semidirect
4. #21446965C2semidirect
5. #21446815C2semidirect
6. #21446245C2semidirect
7. #21446170C2semidirect
8. #21446142C2semidirect
9. #21446112C2semidirect
10. #21445907C2semidirect
11. C61×#2408C2semidirect
17. C2×#14519769C22semidirect
20. C2×#14519751C22semidirect
22. C2×C4×D244C22semidirect
24. C2×#14519736C22semidirect
31. C2×#14519697C22semidirect
33. C2×C22⋊#92418C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
66. #14519751D4semidirect
75. C2×C4⋊Dic61D4semidirect
323. #2408D61semidirect
375. C61#23963semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
C2×#168398962
C22×#145196982
#214469652
#214468152
#214462452
#214461702
#214461422
#214461122
#214459072
C61×#24082
#2396361