Finite Group Databasebeta

← #29656582

Subgroups — #29656582

Order 15616.

934 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3C2×#16840047C211yesmaximal, characteristic, complemented
4#21445040C212yesmaximal, complemented
5#21445026C212yesmaximal, complemented
6#21445019C212yesmaximal, complemented
7#21445005C212yesmaximal, complemented
8#21444514C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3522C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840157—21no—
14#16840132—21no—
15#16840121—22no—
16#16840112C2212yes—
17#16840108C2212yes—
18#16840047—21no—
19#16840047—21no—
20#16840047C2212yes—
21#16840047C2212yes—
22#16839972C2212yes—
23#16839967C2212yes—
24#16839917—21no—
25#16839900—21no—
26#16839819C2211yescharacteristic
27C2×#14519708—21no—
28C2×C4⋊#92418C2211yescharacteristic
29#16839763—22no—
30C2×#14519701—21no—
31C4×#11758194C2211yescharacteristic
32#16839511C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—21no—
39C61×#559C2211yescharacteristic
40Q82×C61C2211yescharacteristic
41C61×#554—21no—
42C61×#554C2211yescharacteristic
43C61×#550C2211yescharacteristic
44C61×#548—21no—
45C61×#547—22no—
46C61×#547—21no—
47C61×#542—22no—
48C61×#542—21no—
49C61×#532—21no—
50C61×#532C2211yescharacteristic
51C61×#529—21no—
52C61×#529C2211yescharacteristic
53C61×#520—21no—
54C2×C4×Q8×C61C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C2×#11758194—21no—
59C22×#92418—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2312yes—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519701—22no—
73#14519701—24no—
74#14519700—22no—
75#14519700—22no—
76#14519698—24no—
77C22⋊#92418—21no—
78#14519688—21no—
79#14519688—21no—
80C4×#92418C2312yes—
81C2×#41×C61—22no—
82C2×#41×C61—21no—
83C2×#41×C61D412yes—
84C22×Q8×C61D411yescharacteristic
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61—21no—
90#127×C61C2311yescharacteristic
91#126×C61—22no—
92#126×C61—21no—
93#125×C61—21no—
94#125×C61D411yescharacteristic
95#124×C61—22no—
96#124×C61—22no—
97#124×C61—21no—
98#124×C61—21no—
99#124×C61D412yes—
100#124×C61D411yescharacteristic
101#123×C61—22no—
102#123×C61—22no—
103#123×C61—22no—
104#123×C61—22no—
105#123×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61D411yescharacteristic
112#121×C61—22no—
113#121×C61—22no—
114#121×C61—21no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61D411yescharacteristic
118C4×Q8×C61—21no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—22no—
128C4×D4×C61—22no—
129C4×D4×C61—22no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61C2311yescharacteristic
138#117×C61—22no—
139#117×C61—21no—
140C2×C4⋊C4×C61—22no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C42×C61—21no—
145C2×C42×C61C2311yescharacteristic
146#41×C61—44no—
147#41×C61—42no—
148#41×C61—42no—
149#41×C61—84no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—22no—
153C2×Q8×C61—41no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×Q8×C61C2×D411yescharacteristic
156C2×Q8×C61C2×D411yescharacteristic
157C2×Q8×C61C2×D412yes—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—21no—
161C2×D4×C61—22no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61C2×D411yescharacteristic
165C2×D4×C61C2×D412yes—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—22no—
170C22×C4×C61—22no—
171C22×C4×C61—21no—
172C22×C4×C61—22no—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×Dic61—41no—
177C2×Dic122—22no—
178C2×Dic122—41no—
179C2×Dic122—82no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C42×C61—22no—
212C42×C61—21no—
213C42×C61—21no—
214C42×C61—21no—
215C42×C61—21no—
216C42×C61—21no—
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221#11758202—82no—
222#11758200—82no—
223C4⋊Dic61—22no—
224C4⋊Dic61—22no—
225C4⋊Dic61—41no—
226C4⋊Dic61—41no—
227#11758194—42no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—42no—
232C23×C61—21no—
233C23×C61—22no—
234C23×C61#12011yescharacteristic
235Q8×C61—21no—
236Q8×C61—21no—
237Q8×C61—21no—
238Q8×C61—22no—
239Q8×C61—42no—
240Q8×C61—42no—
241Q8×C61—82no—
242Q8×C61#13712yes—
243D4×C61—22no—
244D4×C61—42no—
245D4×C61—44no—
246D4×C61—44no—
247D4×C61—42no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—42no—
251D4×C61—82no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—42no—
265C2×C4×C61—44no—
266C2×C4×C61—42no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—82no—
270C2×C4×C61—82no—
271C2×C4×C61—82no—
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×Dic61—42no—
279C2×Dic61—82no—
280Dic122—42no—
281Dic122—42no—
282Dic122—82no—
283#92418—42no—
284#92418—42no—
285#92418—42no—
286#18573—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—41no—
295C22×C61#45512yes—
296C22×C61C2×#12011yescharacteristic
297C4×C61—21no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61C2×#13711yescharacteristic
310C4×C61#57511yescharacteristic
311C4×C61#57712yes—
312Dic61—82no—
313#3522D6111yescharacteristic
314#3218—611no—
315#3141—611no—
316C2×#475—611no—
317#1645—612no—
318#1643—612no—
319#1642—612no—
320#1640—612no—
321#1551—611no—
322#1491—611no—
323#1483—611no—
324C2×C61—22no—
325C2×C61—81no—
326C2×C61—82no—
327C2×C61#300711yescharacteristic
328C2×C61#301011yescharacteristic
329C2×C61C2×#45511yescharacteristic
330#559—21no—
331#559C2×D6111yescharacteristic
332Q82C2×D6111yescharacteristic
333#554—21no—
334#554C2×D6111yescharacteristic
335#550C2×D6111yescharacteristic
336#548—21no—
337#547—22no—
338#547—21no—
339#542—22no—
340#542—21no—
341#532—21no—
342#532C2×D6111yescharacteristic
343C2×C4⋊Q8—611no—
344#529—21no—
345#529C2×D6111yescharacteristic
346#520—21no—
347C2×C4×Q8C2×D6111yescharacteristic
348#497—1221no—
349#491—1221no—
350#489—1222no—
351#487—612no—
352#486—612no—
353#475—612no—
354#475—612no—
355#475—1221no—
356#475—1221no—
357#462—612no—
358#461—612no—
359#451—1221no—
360#448—1221no—
361#432—611no—
362C2×#106—1221no—
363C2×C4⋊C8—611no—
364#417—1222no—
365C2×#103—1221no—
366C4×M16—611no—
367#333—612no—
368#331—612no—
369Q8⋊C8—612no—
370Q8⋊C8—612no—
371Q8⋊C8—612no—
372D4⋊C8—612no—
373C61#1857311yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—21no—
376C2×#41C22⋊D6112yes—
377C22×Q8—1221no—
378C22×Q8C22⋊D6111yescharacteristic
379C2×Q16—1222no—
380C2×SD16—1222no—
381C2×M16—1221no—
382C22×C8—1221no—
383C4⋊Q8—22no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—612no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8C22×D6111yescharacteristic
391#127—21no—
392#127C22×D6111yescharacteristic
393#126—22no—
394#126—21no—
395#125—21no—
396#125C22⋊D6111yescharacteristic
397#124—22no—
398#124—22no—
399#124—21no—
400#124—21no—
401#124C22⋊D6111yescharacteristic
402#124C22⋊D6112yes—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—21no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—21no—
413C22⋊Q8C22⋊D6111yescharacteristic
414#121—22no—
415#121—22no—
416#121—21no—
417#121—21no—
418#121—21no—
419#121C22⋊D6111yescharacteristic
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8C22⋊D6111yescharacteristic
425C4×Q8C22⋊D6111yescharacteristic
426C4×Q8C22×D6111yescharacteristic
427C4×Q8C22×D6111yescharacteristic
428C4×Q8C22×D6111yescharacteristic
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4C22×D6111yescharacteristic
440#117—22no—
441#117—21no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22⋊D6111yescharacteristic
448C2×C42—21no—
449C2×C42C22×D6111yescharacteristic
450#106—1222no—
451#106—1222no—
452C4⋊C8—612no—
453C4⋊C8—612no—
454C4⋊C8—1221no—
455#103—1222no—
456#103—1222no—
457#103—1224no—
458#103—1222no—
459#103—1222no—
460#102—1224no—
461C22⋊C8—1221no—
462#97—1221no—
463#97—1221no—
464C4×C8—612no—
465#41—44no—
466#41—42no—
467#41—42no—
468#41—84no—
469C2×Q8—22no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—41no—
473C2×Q8—1222no—
474C2×Q8—4882no—
475C2×Q8—2441no—
476C2×Q8C2×C22⋊D6111yescharacteristic
477C2×Q8C2×C22⋊D6111yescharacteristic
478C2×Q8C2×C22⋊D6111yescharacteristic
479C2×Q8C2×C22⋊D6112yes—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C22⋊D6111yescharacteristic
487C2×D4C2×C22⋊D6112yes—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—2441no—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499Q16—4882no—
500SD16—4882no—
501M16—2442no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—2442no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4C2×C22⋊D6111yescharacteristic
526C4⋊C4C2×C22⋊D6111yescharacteristic
527C4⋊C4C2×C22⋊D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42C2×C22⋊D6111yescharacteristic
548C42C2×C22⋊D6111yescharacteristic
549C42C2×C22⋊D6111yescharacteristic
550C42C23×D6111yescharacteristic
551C23—21no—
552C23—22no—
553C23#1451974611yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—22no—
558Q8—42no—
559Q8—42no—
560Q8—82no—
561Q8—4882no—
562Q8—2442no—
563Q8—2442no—
564Q8#1451980712yes—
565D4—22no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—41no—
571D4—41no—
572D4—42no—
573D4—82no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—4882no—
595C2×C4—2442no—
596C2×C4#1451974611yescharacteristic
597C2×C4#1451974611yescharacteristic
598C2×C4#1451974611yescharacteristic
599C2×C4C22×C22⋊D6111yescharacteristic
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C8—2442no—
603C8—2442no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22#1683994012yes—
614C22C2×#1451974611yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—4882no—
628C4C2×#1451980711yescharacteristic
629C4#1684056011yescharacteristic
630C4#1684057012yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2#2145507911yescharacteristic
635C2#2145510511yescharacteristic
636C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3522C2
socleC22×C61C2×#120
hypercenter#3522D61
hypocenterC61#18573
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21456212C2semidirect
3. C2×#16840047C2semidirect
4. #21445040C2semidirect
5. #21445026C2semidirect
6. #21445019C2semidirect
7. #21445005C2semidirect
8. #21444514C2semidirect
9. #21444266C2semidirect
10. #21444232C2semidirect
373. C61#18573semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214562122
C2×#168400472
#214450402
#214450262
#214450192
#214450052
#214445142
#214442662
#214442322
C61×#35222
#1857361