Finite Group Databasebeta

← #29714615

Subgroups — #29714615

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455287C211yesmaximal, characteristic, complemented
2C4×#14519807C211yesmaximal, characteristic, complemented
3#21451028C211yesmaximal, characteristic, complemented
4#21447040C212yesmaximal, complemented
5#21446866C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446739C211yesmaximal, characteristic, complemented
8#21446188C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10#21445839C211yesmaximal, characteristic, complemented
11C61×#3511C211yesmaximal, characteristic
12C2×#14519807C2211yescharacteristic
13C2×#14519807C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16839981—22no—
16#16839969—21no—
17#16839959—21no—
18#16839867—21no—
19#16839862—21no—
20C4×#11758202—21no—
21C4×#11758200—21no—
22#16839805C2211yescharacteristic
23#16839786C2211yescharacteristic
24C2×#14519704C2212yes—
25C2×#14519704C2212yes—
26C2×#14519704C2212yes—
27#16839770C2211yescharacteristic
28#16839763C2211yescharacteristic
29#16839763C2212yes—
30C4×#11758194C2211yescharacteristic
31#16839681—22no—
32#16839667C2212yes—
33#16839658—22no—
34#16839631C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839531C2212yes—
37#16839531C2212yes—
38#16839521C2212yes—
39C61×#556—21no—
40C61×#555—21no—
41C61×#554—21no—
42C61×#553—22no—
43C61×#548—22no—
44C61×#547—21no—
45C61×#542—22no—
46C61×#537C2212yes—
47C61×#533—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×Dic122C2311yescharacteristic
54#14519807—22no—
55#14519807—24no—
56#14519807—22no—
57#14519807—42no—
58C2×#11758202—21no—
59C2×#11758202—22no—
60C2×#11758200—21no—
61C2×#11758200—22no—
62C2×#11758194C2311yescharacteristic
63C2×#11758194C2312yes—
64C2×#11758194C2311yescharacteristic
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—24no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61C2312yes—
75#14519710—21no—
76#14519708—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79#14519704—24no—
80#14519704—42no—
81#14519704—42no—
82#14519701—22no—
83#14519701—21no—
84#14519700—21no—
85#14519700—21no—
86#14519698—22no—
87#14519698—21no—
88#14519688—21no—
89C4×#92418—21no—
90#14519684—21no—
91#14519684—24no—
92C2×#41×C61—22no—
93C2×#41×C61C2312yes—
94C2×#41×C61C2311yescharacteristic
95C22×Q8×C61D412yes—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101#127×C61D411yescharacteristic
102#126×C61—24no—
103#125×C61—22no—
104#124×C61—22no—
105#124×C61—24no—
106#124×C61D412yes—
107#123×C61—24no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—24no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112#121×C61—22no—
113#121×C61—22no—
114#121×C61—22no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61—24no—
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C4×Q8×C61D411yescharacteristic
121C4×Q8×C61D411yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C4×D4×C61—21no—
125C4×D4×C61—24no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61D411yescharacteristic
130C4×D4×C61D411yescharacteristic
131#117×C61—22no—
132#117×C61—22no—
133#117×C61—21no—
134#117×C61C2311yescharacteristic
135C2×C4⋊C4×C61—22no—
136C2×C4⋊C4×C61D412yes—
137C2×C4⋊C4×C61C2312yes—
138C2×C42×C61—21no—
139C2×C42×C61C2311yescharacteristic
140#41×C61—22no—
141#41×C61—22no—
142#41×C61—21no—
143#41×C61—21no—
144#41×C61—21no—
145#41×C61—22no—
146#41×C61—22no—
147#41×C61—21no—
148#41×C61—84no—
149C2×Q8×C61—22no—
150C2×Q8×C61—22no—
151C2×Q8×C61—82no—
152C2×Q8×C61C2×D412yes—
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61—22no—
156C2×D4×C61—24no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61C2×D412yes—
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—22no—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D414yes—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—21no—
171C22×Dic61—22no—
172C22×Dic61—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122#4112yescomplemented
179C4⋊C4×C61—24no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C22⋊C4×C61—24no—
192C22⋊C4×C61—24no—
193C22⋊C4×C61—24no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C42×C61—24no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61C2×D411yescharacteristic
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207#11758202—24no—
208#11758202—44no—
209#11758200—24no—
210#11758200—44no—
211C4⋊Dic61—24no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214#11758196—41no—
215#11758196—41no—
216#11758196—84no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—41no—
220C4×Dic61—41no—
221#11758194—21no—
222#11758194—22no—
223#11758194—21no—
224#11758194—22no—
225#11758194—21no—
226#11758194#4112yescomplemented
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—22no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61C22×D411yescharacteristic
233Q8×C61—21no—
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—44no—
238Q8×C61—82no—
239D4×C61—21no—
240D4×C61—21no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—41no—
246D4×C61—42no—
247D4×C61—41no—
248D4×C61—42no—
249D4×C61—84no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—24no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—42no—
263C2×C4×C61—44no—
264C2×C4×C61—84no—
265C2×C4×C61#12011yescharacteristic
266C2×C4×C61#12011yescharacteristic
267C2×C4×C61#12011yescharacteristic
268C2×C4×C61#12011yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×C4×C612−1+411yescharacteristic
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—42no—
278C2×Dic61—41no—
279C2×Dic61—42no—
280C2×Dic61—41no—
281C2×Dic61—42no—
282C2×Dic61—41no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287#92418—22no—
288#92418—41no—
289#92418—42no—
290#24173—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—42no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—42no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61C2×#12011yescharacteristic
300C22×C61#54811yescharacteristic
301C22×C61#55011yescharacteristic
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—44no—
310C4×C61#45612yes—
311C4×C61#52511yescharacteristic
312C4×C61#53711yescharacteristic
313C4×C61#54611yescharacteristic
314C4×C61#54711yescharacteristic
315Dic61—44no—
316Dic61—42no—
317Dic61—81no—
318#3511D6111yescharacteristic
319#3037—611no—
320C4×#137—611no—
321#2567—611no—
322#2039—612no—
323#2007—612no—
324#1997—611no—
325#1980—611no—
326#1882—612no—
327#1859—612no—
328#1806—611no—
329C2×C61—21no—
330C2×C61—21no—
331C2×C61—81no—
332C2×C61—82no—
333C2×C61#240711yescharacteristic
334C2×C61C2×#45611yescharacteristic
335C2×C61#301911yescharacteristic
336C2×#137—611no—
337C2×#137—612no—
338#556—21no—
339#555—21no—
340#554—21no—
341#553—22no—
342#548—22no—
343#547—21no—
344#542—22no—
345#537C2×D6112yes—
346#533—21no—
347#532—21no—
348#532C2×D6111yescharacteristic
349C2×C4⋊Q8C2×D6111yescharacteristic
350#520C2×D6112yes—
351C4×#41C2×D6111yescharacteristic
352C2×C4×Q8—611no—
353#464—1222no—
354#462—1221no—
355#460—1221no—
356#441—1221no—
357#440—1221no—
358C4×Q16—1221no—
359C4×SD16—1221no—
360#428—611no—
361#423—611no—
362C2×#104—612no—
363C2×#104—612no—
364C2×#104—612no—
365#419—611no—
366#417—612no—
367#417—611no—
368C4×M16—611no—
369#395—1222no—
370#391—612no—
371#389—1222no—
372#384—612no—
373C4×C4⋊C4—611no—
374#343—612no—
375#343—612no—
376#339—612no—
377C61#2417311yescharacteristic, complemented, Sylow
378C2×#41—22no—
379C2×#41C22×D6111yescharacteristic
380C2×#41C22×D6112yes—
381C22×Q8—611no—
382C22×Q8C22⋊D6112yes—
383#137—1222no—
384#137—1222no—
385#137—1224no—
386#137—2442no—
387C2×Q16—1221no—
388C2×Q16—1222no—
389C2×SD16—1221no—
390C2×SD16—1222no—
391C2×M16—611no—
392C2×M16—612no—
393C2×M16—611no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398C4⋊Q8C22⋊D6111yescharacteristic
399#127C22⋊D6111yescharacteristic
400#126—24no—
401#125—22no—
402#124—22no—
403#124—24no—
404#124C22⋊D6112yes—
405#123—24no—
406C22⋊Q8—22no—
407C22⋊Q8—24no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410#121—22no—
411#121—22no—
412#121—22no—
413#121—21no—
414#121—21no—
415#121—24no—
416C4×Q8—22no—
417C4×Q8—22no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8C22⋊D6111yescharacteristic
423C4×Q8C22⋊D6111yescharacteristic
424C4×D4—22no—
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—24no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4C22⋊D6111yescharacteristic
432C4×D4C22⋊D6111yescharacteristic
433#117—22no—
434#117—22no—
435#117—21no—
436#117C22×D6111yescharacteristic
437C2×C4⋊C4—22no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1224no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22⋊D6112yes—
443C2×C4⋊C4C22×D6112yes—
444C2×C42—21no—
445C2×C42—612no—
446C2×C42—1221no—
447C2×C42C22×D6111yescharacteristic
448#107—1221no—
449#106—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452#104—1224no—
453#104—2442no—
454#104—2442no—
455#103—1221no—
456#103—1222no—
457#103—1221no—
458#103—1221no—
459#102—1221no—
460#102—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—1221no—
464#95—1224no—
465#41—22no—
466#41—22no—
467#41—21no—
468#41—21no—
469#41—21no—
470#41—22no—
471#41—22no—
472#41—21no—
473#41—84no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—82no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8C2×C22⋊D6112yes—
484C2×Q8C2×C22⋊D6111yescharacteristic
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—24no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4C2×C22⋊D6112yes—
491C2×D4C2×C22⋊D6111yescharacteristic
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6114yes—
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1224no—
505Q16—2444no—
506SD16—1224no—
507SD16—2444no—
508M16—612no—
509M16—1221no—
510M16—1222no—
511M16—1221no—
512M16—1222no—
513M16—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1222no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—1224no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4884no—
531C4⋊C4—2441no—
532C4⋊C4—2441no—
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C4⋊C4C2×C22⋊D6111yescharacteristic
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C42—24no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—1222no—
548C42—1222no—
549C42—2441no—
550C42—2441no—
551C42C2×C22⋊D6111yescharacteristic
552C42C2×C22⋊D6111yescharacteristic
553C42C2×C22⋊D6111yescharacteristic
554C42C2×C22⋊D6111yescharacteristic
555C23—21no—
556C23—22no—
557C23C22×C22⋊D6111yescharacteristic
558Q8—21no—
559Q8—21no—
560Q8—22no—
561Q8—22no—
562Q8—44no—
563Q8—82no—
564Q8—1222no—
565Q8—1222no—
566Q8—2441no—
567Q8—2441no—
568D4—21no—
569D4—21no—
570D4—22no—
571D4—22no—
572D4—42no—
573D4—42no—
574D4—41no—
575D4—42no—
576D4—41no—
577D4—42no—
578D4—84no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—84no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2441no—
602C2×C4#1451974611yescharacteristic
603C2×C4#1451974611yescharacteristic
604C2×C4#1451974611yescharacteristic
605C2×C4#1451974611yescharacteristic
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#1451982311yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—1222no—
613C8—2441no—
614C8—2442no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—41no—
623C22C2×#1451974611yescharacteristic
624C22#1684041911yescharacteristic
625C22#1684043111yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—44no—
632C4—42no—
633C4—41no—
634C4—4881no—
635C4—2444no—
636C4—2442no—
637C4#1683994512yes—
638C4#1684026911yescharacteristic
639C4#1684034011yescharacteristic
640C4#1684040011yescharacteristic
641C4#1684041111yescharacteristic
642C2—21no—
643C2—21no—
644C2—81no—
645C2—82no—
646C2#2144915811yescharacteristic
647C2C2×#1683994511yescharacteristic
648C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#24173
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21455287C2semidirect
2. C4×#14519807C2semidirect
3. #21451028C2semidirect
4. #21447040C2semidirect
5. #21446866C2semidirect
6. #21446809C2semidirect
7. #21446739C2semidirect
8. #21446188C2semidirect
9. #21446045C2semidirect
10. #21445839C2semidirect
178. C2×Dic122#41semidirect
226. #11758194#41semidirect
377. C61#24173semidirect

Maximal subgroups

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

subgroupindex
#214552872
C4×#145198072
#214510282
#214470402
#214468662
#214468092
#214467392
#214461882
#214460452
#214458392
C61×#35112
#2417361