Finite Group Databasebeta

← #29653944

Subgroups — #29653944

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456572C211yesmaximal, characteristic, complemented
2#21456567C211yesmaximal, characteristic, complemented
3#21451553C211yesmaximal, characteristic, complemented
4#21444833C214yesmaximal, complemented
5#21444765C214yesmaximal, complemented
6#21444391C211yesmaximal, characteristic, complemented
7#21444387C211yesmaximal, characteristic, complemented
8C2×#16839486C211yesmaximal, characteristic, complemented
9C2×C61×#538C211yesmaximal, characteristic, complemented
10C2×#14519751C2211yescharacteristic
11C2×#14519746—21no—
12#16840090C2212yescomplemented
13#16840080—21no—
14#16840075—21no—
15#16840073C2212yescomplemented
16#16839996—22no—
17#16839993—22no—
18#16839912C2212yescomplemented
19#16839901—21no—
20#16839896—21no—
21#16839895C2212yescomplemented
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839687C2212yescomplemented
25#16839678—22no—
26#16839671C2212yescomplemented
27#16839641—21no—
28#16839622C2211yescharacteristic
29#16839499C2214yescomplemented
30#16839496C2214yescomplemented
31#16839486C2214yescomplemented
32C61×#538—22no—
33C61×#538—22no—
34C61×#538—22no—
35C61×#538C2214yes—
36C2×#126×C61—21no—
37C2×#124×C61—21no—
38C2×#123×C61—21no—
39C2×C22⋊Q8×C61C2211yescharacteristic
40C2×#121×C61—21no—
41C2×#121×C61C2211yescharacteristic
42C2×#120×C61—21no—
43C2×C4×D4×C61—21no—
44C22×C22⋊C4×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C22×C22⋊D61—22no—
48C2×#11758201—22no—
49C2×#11758199—22no—
50C22×#92418—21no—
51C22×#92418—21no—
52#14519751—21no—
53#14519751—21no—
54#14519751—21no—
55#14519751—21no—
56#14519746—44no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—21no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63#14519710—22no—
64#14519708—22no—
65#14519701—22no—
66#14519698—22no—
67#14519697—22no—
68#14519697—22no—
69C22⋊#92418—21no—
70C22⋊#92418—21no—
71C22⋊#92418C2312yes—
72C22⋊#92418C2312yes—
73#14519684—22no—
74#14519684C2312yes—
75C25×C61D411yescharacteristic
76C22×Q8×C61D411yescharacteristic, complemented
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61D411yescharacteristic, complemented
81C23×C4×C61—21no—
82C23×C4×C61D411yescharacteristic
83C23×C4×C61C2311yescharacteristic
84#126×C61—24no—
85#126×C61—24no—
86#124×C61—24no—
87#124×C61—24no—
88#123×C61—24no—
89#123×C61—24no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61D412yescomplemented
93C22⋊Q8×C61C2312yes—
94#121×C61—24no—
95#121×C61—24no—
96#121×C61—21no—
97#121×C61—21no—
98#121×C61D412yescomplemented
99#121×C61C2312yes—
100#120×C61—24no—
101#120×C61—24no—
102C4×D4×C61—24no—
103C4×D4×C61—24no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic, complemented
107C2×C4⋊C4×C61D411yescharacteristic, complemented
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61D412yes—
118C2×C22⋊C4×C61C2312yes—
119C2×C42×C61—21no—
120C24×C61—21no—
121C24×C61—21no—
122C24×C61—21no—
123C24×C61—22no—
124C24×C61—22no—
125C24×C61—81no—
126C24×C61—81no—
127C24×C61C2×D412yes—
128C24×C61C2×D411yescharacteristic
129C23×D61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—81no—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—24no—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61—81no—
141C2×D4×C61—82no—
142C2×D4×C61—82no—
143C2×D4×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—21no—
150C22×C4×C61—81no—
151C22×C4×C61—81no—
152C22×C4×C61—82no—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—44no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—42no—
164C2×C22⋊D61—42no—
165C2×C22⋊D61—82no—
166C2×C22⋊D61—82no—
167C22×Dic61—21no—
168C22×Dic61—22no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—81no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C42×C61—24no—
191C22⋊Dic61—44no—
192C22⋊Dic61—42no—
193#11758201—82no—
194#11758199—82no—
195#11758198—41no—
196#11758198—41no—
197#11758198—82no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—81no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—81no—
204C2×#92418—81no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—81no—
221C23×C61—81no—
222C23×C61—84no—
223C23×C61—84no—
224C23×C61#12011yescharacteristic
225C23×C61#12011yescharacteristic
226C23×C61#13512yes—
227C23×C61AGL(1,ℤ/8)12yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234Q8×C61—42no—
235Q8×C61—81no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—42no—
239D4×C61—44no—
240D4×C61—81no—
241D4×C61—84no—
242D4×C61—84no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—81no—
259C2×C4×C61—84no—
260C2×C4×C61#12011yescharacteristic
261C2×C4×C61#12011yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C22⋊D61—88no—
265C22⋊D61—82no—
266C22⋊D61—82no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—41no—
273D244—42no—
274D244—42no—
275#92418—42no—
276#92418—42no—
277#18296—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—84no—
295C22×C61—84no—
296C22×C61—84no—
297C22×C61#44912yes—
298C22×C61#44912yes—
299C22×C61C2×#12011yescharacteristic
300C22×C61C2×#13511yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C2×D61—82no—
303C2×D61—82no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—42no—
315Dic61—82no—
316Dic61—81no—
317C2×#538D6111yescharacteristic, complemented
318#3187—611no—
319#3186—611no—
320#2617—611no—
321#1612—614no—
322#1600—614no—
323#1520—611no—
324#1519—611no—
325C2×#323—611no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—22no—
329C2×C61—22no—
330C2×C61—82no—
331C2×C61—82no—
332C2×C61C2×#44911yescharacteristic
333C2×C61#301911yescharacteristic
334C2×C61#302611yescharacteristic
335D61—161no—
336#538—22no—
337#538—22no—
338#538—22no—
339#538C2×D6114yes—
340C2×#126—21no—
341C2×#124—21no—
342C2×#123—21no—
343C2×C22⋊Q8C2×D6111yescharacteristic
344C2×#121—21no—
345C2×#121—611no—
346C2×#121C2×D6111yescharacteristic
347C2×#120—21no—
348C2×#120—1221no—
349C2×C4×D4—21no—
350C22×C22⋊C4C2×D6111yescharacteristic
351#483—612no—
352#481—1221no—
353#480—612no—
354#480—1221no—
355#466—1222no—
356#465—1222no—
357#450—612no—
358#448—1221no—
359#447—612no—
360#447—1221no—
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363#397—612no—
364#394—1222no—
365#392—612no—
366#386—1221no—
367#380—611no—
368#328—614no—
369#327—614no—
370#323—614no—
371C61#1829611yescharacteristic, complemented, Sylow
372C25C22⋊D6111yescharacteristic
373C22×Q8C22⋊D6111yescharacteristic, complemented
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4—611no—
378C22×D4—1222no—
379C22×D4—1221no—
380C22×D4C22⋊D6111yescharacteristic, complemented
381C23×C4—21no—
382C23×C4C22⋊D6111yescharacteristic
383C23×C4C22×D6111yescharacteristic
384C2×SD16—1222no—
385C2×D8—1222no—
386C22×C8—1221no—
387C22×C8—1221no—
388#126—24no—
389#126—24no—
390#124—24no—
391#124—24no—
392#123—24no—
393#123—24no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8C22⋊D6112yescomplemented
397C22⋊Q8C22×D6112yes—
398#121—24no—
399#121—24no—
400#121—21no—
401#121—21no—
402#121—1221no—
403#121—1221no—
404#121—1221no—
405#121—1221no—
406#121C22⋊D6112yescomplemented
407#121C22×D6112yes—
408#120—24no—
409#120—24no—
410#120—2444no—
411C4×D4—24no—
412C4×D4—24no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
418C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1222no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4C22⋊D6112yes—
433C2×C22⋊C4C22×D6112yes—
434C2×C42—21no—
435#107—1222no—
436#106—1222no—
437#103—1222no—
438#102—1222no—
439#102—1222no—
440#102—1222no—
441C22⋊C8—612no—
442C22⋊C8—612no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445#95—612no—
446#95—1222no—
447C24—21no—
448C24—21no—
449C24—21no—
450C24—22no—
451C24—22no—
452C24—81no—
453C24—81no—
454C24—1221no—
455C24C2×C22⋊D6112yes—
456C24C2×C22⋊D6111yescharacteristic
457C2×Q8—21no—
458C2×Q8—81no—
459C2×Q8C2×C22⋊D6112yes—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—81no—
468C2×D4—82no—
469C2×D4—82no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—4881no—
473C2×D4—4882no—
474C2×D4—4882no—
475C2×D4—2444no—
476C2×D4—2441no—
477C2×D4—2441no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4C2×C22⋊D6112yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—82no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499SD16—4882no—
500D8—4882no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—4881no—
504C2×C8—4881no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—4881no—
513C4⋊C4C2×C22⋊D6112yes—
514C4⋊C4C2×C22⋊D6112yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—4882no—
527C22⋊C4—2444no—
528C22⋊C4—2442no—
529C22⋊C4—2441no—
530C22⋊C4—2441no—
531C42—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—81no—
548C23—81no—
549C23—84no—
550C23—84no—
551C23—2442no—
552C23—2442no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23#1451974611yescharacteristic
557C23#1451974611yescharacteristic
558C23#1451979712yes—
559C23#1451980212yes—
560C23C22×C22⋊D6111yescharacteristic
561Q8—42no—
562Q8—81no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—81no—
568D4—84no—
569D4—84no—
570D4—4888no—
571D4—4882no—
572D4—4882no—
573D4—2442no—
574D4—2442no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4#1451974611yescharacteristic
599C2×C4#1451974611yescharacteristic
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C8—2442no—
603C8—2442no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—4882no—
624C22—4882no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22#1683990712yes—
629C22#1683990712yes—
630C22C2×#1451974611yescharacteristic
631C22C2×#1451979711yescharacteristic
632C22C2×#1451980211yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—4882no—
642C4—4881no—
643C2—21no—
644C2—22no—
645C2—21no—
646C2—22no—
647C2—82no—
648C2—82no—
649C2—9761no—
650C2C2×#1683990711yescharacteristic
651C2#2145517611yescharacteristic
652C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#538C2
socleC22×C61C2×#120
hypercenterC2×#538D61
hypocenterC61#18296
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. #21456572C2semidirect
2. #21456567C2semidirect
3. #21451553C2semidirect
4. #21444833C2semidirect
5. #21444765C2semidirect
6. #21444391C2semidirect
7. #21444387C2semidirect
8. C2×#16839486C2semidirect
9. C2×C61×#538C2semidirect
12. #16840090C22semidirect
15. #16840073C22semidirect
18. #16839912C22semidirect
21. #16839895C22semidirect
24. #16839687C22semidirect
26. #16839671C22semidirect
29. #16839499C22semidirect
30. #16839496C22semidirect
31. #16839486C22semidirect
76. C22×Q8×C61D4semidirect
80. C22×D4×C61D4semidirect
92. C22⋊Q8×C61D4semidirect
98. #121×C61D4semidirect
106. C2×C4⋊C4×C61D4semidirect
107. C2×C4⋊C4×C61D4semidirect
317. C2×#538D61semidirect
371. C61#18296semidirect
373. C22×Q8C22⋊D61semidirect
380. C22×D4C22⋊D61semidirect
396. C22⋊Q8C22⋊D61semidirect
406. #121C22⋊D61semidirect
417. C2×C4⋊C4C22⋊D61semidirect
418. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214565722
#214565672
#214515532
#214448332
#214447652
#214443912
#214443872
C2×#168394862
C2×C61×#5382
#1829661