Finite Group Databasebeta

← #29712985

Subgroups — #29712985

Order 15616.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840073C211yesmaximal, characteristic, complemented
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21450127C211yesmaximal, characteristic, complemented
4#21447055C212yesmaximal, complemented
5#21447019C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446112C211yesmaximal, characteristic, complemented
8#21446073C212yesmaximal, complemented
9#21445759C212yesmaximal, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C2×C61×#548C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519758—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15#16840073—24no—
16C2×#14519710C2212yes—
17C2×#14519701—24no—
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×C22⋊#92418C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26#16839692—21no—
27#16839687—21no—
28#16839660C2212yes—
29#16839623—21no—
30#16839622C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C22×#41×C61C2211yescharacteristic
35C23×Q8×C61C2211yescharacteristic
36C61×#548—24no—
37C61×#548—24no—
38C2×#124×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#121×C61—21no—
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61—21no—
46C2×C4×D4×C61C2211yescharacteristic
47C23×Dic61—21no—
48C22×#92418—22no—
49C22×#92418—42no—
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52#14519758—44no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×#11758196—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—21no—
61#14519710—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—24no—
65#14519701—42no—
66#14519698—24no—
67#14519698—42no—
68C22⋊#92418—22no—
69C22⋊#92418—42no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—24no—
76C2×#41×C61—42no—
77C22×Q8×C61—22no—
78C22×Q8×C61—21no—
79C22×Q8×C61—24no—
80C22×Q8×C61—42no—
81C22×Q8×C61D414yes—
82C22×Q8×C61C2311yescharacteristic
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61D411yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92#124×C61—28no—
93#124×C61—22no—
94#124×C61—22no—
95C22⋊Q8×C61—28no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98#121×C61—28no—
99#121×C61—22no—
100#121×C61D414yes—
101C4×Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×D4×C61—28no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121#41×C61—48no—
122#41×C61—44no—
123#41×C61—44no—
124C2×Q8×C61—24no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—28no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—44no—
132C2×Q8×C61—44no—
133C2×Q8×C61—48no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61#4114yes—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—44no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—24no—
153C22×C4×C61—24no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—44no—
161C22×C4×C61—44no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—42no—
172C22×Dic61—42no—
173C22×Dic61—42no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—22no—
185C42×C61—24no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—44no—
189C22⋊Dic61—42no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195#11758196—44no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—44no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—42no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—42no—
215C23×C61#12012yes—
216C23×C61#13512yes—
217C23×C61AGL(1,ℤ/8)12yes—
218C23×C61C22×D411yescharacteristic
219Q8×C61—22no—
220Q8×C61—24no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—48no—
224Q8×C61—42no—
225Q8×C61#12314yescomplemented
226D4×C61—24no—
227D4×C61—48no—
228D4×C61—44no—
229D4×C61—48no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—48no—
248C2×C4×C61—48no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61#12012yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4112yes—
259C2×C4×C61H(5,𝔽2)12yes—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—84no—
267#92418—42no—
268#92418—42no—
269#24035—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—82no—
283C22×C61—82no—
284C22×C61#44914yes—
285C22×C61C2×#12011yescharacteristic
286C22×C61#54612yes—
287C22×C61C2×#13511yescharacteristic
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61#57812yes—
290C4×C61—22no—
291C4×C61—24no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61C2×#12311yescharacteristic
298C4×C61#53411yescharacteristic
299C4×C61#54612yes—
300Dic61—82no—
301Dic61—82no—
302C2×#548D6111yescharacteristic
303C2×#480—611no—
304C22×#103—611no—
305#2486—611no—
306#2042—612no—
307#2036—612no—
308#1893—611no—
309#1869—611no—
310#1863—612no—
311#1788—612no—
312#1787—611no—
313C2×C61—24no—
314C2×C61—81no—
315C2×C61—82no—
316C2×C61#240511yescharacteristic
317C2×C61C2×#44911yescharacteristic
318C2×C61#301111yescharacteristic
319C2×C61#330012yes—
320C2×C61#330812yes—
321C22×#41C2×D6111yescharacteristic
322C23×Q8C2×D6111yescharacteristic
323C23×C8—611no—
324#548—24no—
325#548—24no—
326C2×#124—21no—
327C2×#124C2×D6111yescharacteristic
328C2×#123—1221no—
329C2×C22⋊Q8—21no—
330C2×C22⋊Q8C2×D6111yescharacteristic
331C2×#121—21no—
332C2×#121C2×D6111yescharacteristic
333C2×C4×Q8C2×D6111yescharacteristic
334C2×C4×D4—21no—
335C2×C4×D4C2×D6111yescharacteristic
336C22×C4⋊C4—611no—
337#480—1224no—
338C2×#107—612no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103—612no—
343C2×#103—1224no—
344C2×#102—612no—
345C2×#102—612no—
346C2×C22⋊C8—611no—
347C2×C22⋊C8—612no—
348#398—1221no—
349#397—1221no—
350#389—612no—
351#381—1221no—
352#380—611no—
353#380—612no—
354#340—612no—
355#336—612no—
356C61#2403511yescharacteristic, complemented, Sylow
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—24no—
360C2×#41—42no—
361C22×Q8—21no—
362C22×Q8—22no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C22⋊D6114yes—
366C22×Q8C22×D6111yescharacteristic
367C22×Q8C22×D6111yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22⋊D6111yescharacteristic
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—1221no—
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C22×C8—612no—
378C22×C8—612no—
379C22×C8—1222no—
380C22×C8—2442no—
381#124—28no—
382#124—22no—
383#124—22no—
384#123—2444no—
385C22⋊Q8—28no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388#121—28no—
389#121—22no—
390#121C22⋊D6114yes—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×D4—28no—
394C4×D4—22no—
395C4×D4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4C22⋊D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4C22⋊D6111yescharacteristic
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C42—21no—
415C2×C42C22×D6111yescharacteristic
416#107—1224no—
417#103—1224no—
418#103—1224no—
419#103—1224no—
420#103—2442no—
421#102—1224no—
422#102—2442no—
423C22⋊C8—1222no—
424C22⋊C8—2442no—
425#95—1222no—
426#95—1221no—
427#95—1222no—
428C24—21no—
429C24C2×C22⋊D6111yescharacteristic
430C24C2×C22⋊D6111yescharacteristic
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8#1175822414yes—
445C2×Q8C2×C22⋊D6112yes—
446C2×Q8C2×C22⋊D6112yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C22⋊D6112yes—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6112yes—
481C22×C4C2×C22⋊D6112yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—2444no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—2444no—
500C4⋊C4—2442no—
501C4⋊C4C2×C22⋊D6112yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—2444no—
509C22⋊C4—2442no—
510C42—24no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23#1451974612yes—
526C23#1451979712yes—
527C23#1451980212yes—
528C23C22×C22⋊D6111yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1451975814yescomplemented
536D4—24no—
537D4—48no—
538D4—44no—
539D4—48no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—4884no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4#1451974612yes—
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822412yes—
576C2×C4#1451982212yes—
577C8—2442no—
578C8—2442no—
579C22—22no—
580C22—24no—
581C22—24no—
582C22—22no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—82no—
592C22—82no—
593C22#1683990714yes—
594C22C2×#1451974611yescharacteristic
595C22#1684040612yes—
596C22C2×#1451979711yescharacteristic
597C22C2×#1451980211yescharacteristic
598C22#1684057712yes—
599C4—22no—
600C4—24no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—42no—
606C4—4882no—
607C4—4882no—
608C4C2×#1451975811yescharacteristic
609C4#1684032111yescharacteristic
610C4#1684040612yes—
611C2—24no—
612C2—81no—
613C2—82no—
614C2#2144914211yescharacteristic
615C2C2×#1683990711yescharacteristic
616C2#2145511411yescharacteristic
617C2#2145770612yes—
618C2#2145781012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#548C2
socleC23×C61C22×D4
hypercenterC2×#548D61
hypocenterC61#24035
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#16840073C2semidirect
2. C22×#14519701C2semidirect
3. #21450127C2semidirect
4. #21447055C2semidirect
5. #21447019C2semidirect
6. #21446248C2semidirect
7. #21446112C2semidirect
8. #21446073C2semidirect
9. #21445759C2semidirect
10. #21445754C2semidirect
225. Q8×C61#123semidirect
356. C61#24035semidirect
535. Q8#14519758semidirect

Maximal subgroups

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

subgroupindex
C2×#168400732
C22×#145197012
#214501272
#214470552
#214470192
#214462482
#214461122
#214460732
#214457592
#214457542
C2×C61×#5482
#2403561