Finite Group Databasebeta

← #29653953

Subgroups — #29653953

Order 15616.

1084 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
1#21456591C211yesmaximal, characteristic, complemented
2#21456576C211yesmaximal, characteristic, complemented
3#21451624C211yesmaximal, characteristic, complemented
4#21444867C214yesmaximal, complemented
5#21444799C214yesmaximal, complemented
6#21444391C211yesmaximal, characteristic, complemented
7#21444387C211yesmaximal, characteristic, complemented
8C2×#16839486C211yesmaximal, characteristic, complemented
9C2×C61×#538C211yesmaximal, characteristic
10C2×#14519756C2211yescharacteristic
11#16840097—21no—
12#16840095C2212yescomplemented
13#16840092—21no—
14#16840078C2212yescomplemented
15#16840001—22no—
16#16839989—22no—
17#16839918—21no—
18#16839917C2212yescomplemented
19#16839913—21no—
20#16839900C2212yescomplemented
21C2×C22⋊#92418C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839698C2212yescomplemented
24#16839685—22no—
25#16839674C2212yescomplemented
26#16839635—21no—
27#16839622C2211yescharacteristic
28#16839618—21no—
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×Dic122C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758200—22no—
48C22×#92418—21no—
49C22×#92418—21no—
50#14519756—21no—
51#14519756—21no—
52#14519756—21no—
53#14519756—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—22no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62#14519710—22no—
63#14519708—22no—
64#14519701—22no—
65#14519700—22no—
66#14519700—22no—
67#14519698—22no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70C22⋊#92418C2312yes—
71C22⋊#92418C2312yes—
72#14519684—22no—
73#14519684C2312yes—
74C25×C61D411yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#126×C61—24no—
84#126×C61—24no—
85#124×C61—24no—
86#124×C61—24no—
87#123×C61—24no—
88#123×C61—24no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61D412yes—
92C22⋊Q8×C61C2312yes—
93#121×C61—24no—
94#121×C61—24no—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61D412yes—
98#121×C61C2312yes—
99#120×C61—24no—
100#120×C61—24no—
101C4×D4×C61—24no—
102C4×D4×C61—24no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—21no—
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×C61D412yes—
117C2×C22⋊C4×C61C2312yes—
118C2×C42×C61—21no—
119C24×C61—21no—
120C24×C61—21no—
121C24×C61—21no—
122C24×C61—22no—
123C24×C61—22no—
124C24×C61—81no—
125C24×C61—81no—
126C24×C61C2×D412yes—
127C24×C61C2×D411yescharacteristic
128C2×Q8×C61—21no—
129C2×Q8×C61—81no—
130C2×Q8×C61C2×D412yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—81no—
139C2×D4×C61—82no—
140C2×D4×C61—82no—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—81no—
149C22×C4×C61—81no—
150C22×C4×C61—82no—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—81no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61—24no—
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185C22⋊Dic61—42no—
186C22⋊Dic61—82no—
187C22⋊Dic61—82no—
188#11758202—82no—
189#11758200—82no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—81no—
193#11758196—41no—
194#11758196—41no—
195#11758196—82no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—81no—
199C2×#92418—81no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—81no—
216C23×C61—81no—
217C23×C61—84no—
218C23×C61—84no—
219C23×C61#12011yescharacteristic
220C23×C61#12011yescharacteristic
221C23×C61#13512yes—
222C23×C61#13712yes—
223C23×C61C22×D411yescharacteristic
224Q8×C61—42no—
225Q8×C61—81no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—42no—
229D4×C61—44no—
230D4×C61—81no—
231D4×C61—84no—
232D4×C61—84no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—81no—
249C2×C4×C61—84no—
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261Dic122—42no—
262Dic122—42no—
263#92418—42no—
264#92418—42no—
265#18297—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—44no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—82no—
282C22×C61—84no—
283C22×C61—84no—
284C22×C61—84no—
285C22×C61#45212yes—
286C22×C61#45212yes—
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×#13511yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—42no—
298Dic61—82no—
299Dic61—81no—
300Dic61—81no—
301C2×#538D6111yescharacteristic
302#3189—611no—
303#3187—611no—
304#2626—611no—
305#1618—614no—
306#1606—614no—
307#1520—611no—
308#1519—611no—
309C2×#323—611no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61—22no—
314C2×C61—82no—
315C2×C61—82no—
316C2×C61C2×#45211yescharacteristic
317C2×C61#301911yescharacteristic
318C2×C61#302611yescharacteristic
319#538—22no—
320#538—22no—
321#538—22no—
322#538C2×D6114yes—
323C2×#126—21no—
324C2×#124—21no—
325C2×#123—21no—
326C2×C22⋊Q8—611no—
327C2×C22⋊Q8C2×D6111yescharacteristic
328C2×#121—21no—
329C2×#121C2×D6111yescharacteristic
330C2×#120—21no—
331C2×C4×D4—21no—
332C22×C22⋊C4C2×D6111yescharacteristic
333#484—612no—
334#484—1221no—
335#483—1221no—
336#481—612no—
337#467—1222no—
338#465—1222no—
339#451—612no—
340#451—1221no—
341#450—1221no—
342#448—612no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—611no—
345#399—612no—
346#396—1222no—
347#393—612no—
348#385—1221no—
349#380—611no—
350#379—1221no—
351#328—614no—
352#327—614no—
353#323—614no—
354C61#1829711yescharacteristic, complemented, Sylow
355C25C22⋊D6111yescharacteristic
356C22×Q8—611no—
357C22×Q8C22⋊D6111yescharacteristic
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4C22⋊D6111yescharacteristic
362C23×C4—21no—
363C23×C4C22⋊D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C2×Q16—1222no—
366C2×SD16—1222no—
367C22×C8—1221no—
368C22×C8—1221no—
369#126—24no—
370#126—24no—
371#124—24no—
372#124—24no—
373#123—24no—
374#123—24no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8—1221no—
380C22⋊Q8—1221no—
381C22⋊Q8C22⋊D6112yes—
382C22⋊Q8C22×D6112yes—
383#121—24no—
384#121—24no—
385#121—21no—
386#121—21no—
387#121C22⋊D6112yes—
388#121C22×D6112yes—
389#120—24no—
390#120—24no—
391C4×D4—24no—
392C4×D4—24no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22⋊D6111yescharacteristic
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4C22⋊D6112yes—
415C2×C22⋊C4C22×D6112yes—
416C2×C42—21no—
417#107—1222no—
418#106—1222no—
419#103—1222no—
420#103—1222no—
421#103—1222no—
422#102—1222no—
423C22⋊C8—612no—
424C22⋊C8—612no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#95—612no—
428#95—1222no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—81no—
435C24—81no—
436C24C2×C22⋊D6112yes—
437C24C2×C22⋊D6111yescharacteristic
438C2×Q8—21no—
439C2×Q8—81no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8C2×C22⋊D6112yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4C2×C22⋊D6112yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4C2×C22⋊D6112yes—
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474Q16—4882no—
475SD16—4882no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—4881no—
479C2×C8—4881no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4882no—
488C4⋊C4—4881no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4C2×C22⋊D6112yes—
492C4⋊C4C2×C22⋊D6112yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—4882no—
505C22⋊C4—4882no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C22⋊C4—2442no—
509C42—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—81no—
527C23—84no—
528C23—84no—
529C23#1451974611yescharacteristic
530C23#1451974611yescharacteristic
531C23#1451979712yes—
532C23#1451980712yes—
533C23C22×C22⋊D6111yescharacteristic
534Q8—42no—
535Q8—81no—
536Q8—2442no—
537Q8—2442no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—81no—
543D4—84no—
544D4—84no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—81no—
561C2×C4—84no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2442no—
569C2×C4#1451974611yescharacteristic
570C2×C4#1451974611yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×C22⋊D6111yescharacteristic
573C8—2442no—
574C8—2442no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22#1683992412yes—
595C22#1683992412yes—
596C22C2×#1451974611yescharacteristic
597C22C2×#1451979711yescharacteristic
598C22C2×#1451980711yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—4882no—
608C4—4881no—
609C4—4881no—
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—82no—
615C2—82no—
616C2C2×#1683992411yescharacteristic
617C2#2145517611yescharacteristic
618C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#538C2
socleC22×C61C2×#120
hypercenterC2×#538D61
hypocenterC61#18297
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. #21456591C2semidirect
2. #21456576C2semidirect
3. #21451624C2semidirect
4. #21444867C2semidirect
5. #21444799C2semidirect
6. #21444391C2semidirect
7. #21444387C2semidirect
8. C2×#16839486C2semidirect
12. #16840095C22semidirect
14. #16840078C22semidirect
18. #16839917C22semidirect
20. #16839900C22semidirect
23. #16839698C22semidirect
25. #16839674C22semidirect
29. #16839499C22semidirect
30. #16839496C22semidirect
31. #16839486C22semidirect
354. C61#18297semidirect

Maximal subgroups

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

subgroupindex
#214565912
#214565762
#214516242
#214448672
#214447992
#214443912
#214443872
C2×#168394862
C2×C61×#5382
#1829761