Finite Group Databasebeta

← #29717292

Subgroups — #29717292

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic
2C2×#16840027C212yesmaximal, complemented
3C2×#16839912C211yesmaximal, characteristic, complemented
4#21454268C211yesmaximal, characteristic, complemented
5#21447109C212yesmaximal, complemented
6#21446930C212yesmaximal, complemented
7#21446325C212yesmaximal, complemented
8#21446288C211yesmaximal, characteristic, complemented
9#21446023C211yesmaximal, characteristic, complemented
10#21445739C211yesmaximal, characteristic, complemented
11C61×#3007C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C2×#14519802—28no—
14C2×#14519802—21no—
15C2×#14519802—21no—
16C2×#14519802C2212yes—
17C2×#14519802C2212yes—
18C22×#11758201—21no—
19C22×#11758201C2212yescomplemented
20C22×#11758199—21no—
21C2×#14519784C2212yescomplemented
22C22×#11758194C2211yescharacteristic
23#16840027—28no—
24#16839912—21no—
25#16839912—21no—
26#16839912—22no—
27C2×#14519712C2212yescomplemented
28C2×#14519697C2212yescomplemented
29C2×#14519696C2212yescomplemented
30C2×#14519693C2212yescomplemented
31C2×#14519693C2212yescomplemented
32#16839729—22no—
33#16839729C2212yescomplemented
34#16839726—21no—
35#16839726C2211yescharacteristic
36C2×C22⋊#92418—21no—
37C2×C22⋊#92418C2211yescharacteristic
38#16839531C2212yescomplemented
39#16839526C2212yescomplemented
40C22×#41×C61C2211yescharacteristic
41C2×#137×C61C2211yescharacteristic, complemented
42C2×#137×C61C2211yescharacteristic, complemented
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#452—21no—
45C61×#452—21no—
46C61×#451—21no—
47C61×#450—21no—
48C61×#417C2211yescharacteristic, complemented
49C61×#417C2211yescharacteristic, complemented
50C61×#407C2211yescharacteristic
51C24×D61—21no—
52C22×D244—21no—
53C22×D244—21no—
54C22×D244—24no—
55C22×D244—21no—
56C22×D244—21no—
57C22×D244—21no—
58C22×D244—42no—
59C22×D244C2312yes—
60#14519802—28no—
61#14519802—28no—
62#14519802—24no—
63#14519802—24no—
64#14519802—24no—
65#14519802D418yes—
66C2×#11758201—28no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—24no—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201—42no—
73C2×#11758199—28no—
74C2×#11758199—22no—
75C2×#11758199—22no—
76#14519784—24no—
77#14519784—42no—
78C2×#11758194—21no—
79C2×#11758194—24no—
80C2×#11758194—22no—
81C2×#11758194C2312yes—
82C2×#11758194C2312yes—
83C22×#92418—21no—
84C22×#92418—22no—
85C22×#92418C2312yes—
86#14519712—24no—
87#14519697—22no—
88#14519697—22no—
89#14519696—24no—
90#14519693—24no—
91#14519693—42no—
92C22⋊#92418—21no—
93C22⋊#92418—21no—
94C22⋊#92418—22no—
95C22⋊#92418—22no—
96C22⋊#92418D411yescharacteristic
97C22⋊#92418D411yescharacteristic
98C22⋊#92418D412yes—
99C2×#41×C61—21no—
100C2×#41×C61—21no—
101C2×#41×C61—24no—
102C2×#41×C61—42no—
103C2×#41×C61D411yescharacteristic
104C2×#41×C61D411yescharacteristic
105C2×#41×C61C2311yescharacteristic
106C2×#41×C61C2311yescharacteristic
107C22×Q8×C61D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61—21no—
110C22×D4×C61D411yescharacteristic
111C23×C4×C61—21no—
112C23×C4×C61C2311yescharacteristic
113#137×C61—24no—
114#137×C61—42no—
115C2×Q16×C61—21no—
116C2×Q16×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×M16×C61C2311yescharacteristic
120C2×M16×C61C2311yescharacteristic
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128#103×C61—21no—
129#103×C61—21no—
130#102×C61—21no—
131#102×C61—21no—
132C22⋊C8×C61—21no—
133C22⋊C8×C61—21no—
134C24×C61—21no—
135C24×C61#4111yescharacteristic
136C23×D61—22no—
137C23×D61—44no—
138C23×D61—41no—
139C23×D61—41no—
140C23×D61—82no—
141C23×D61—82no—
142#41×C61—22no—
143#41×C61—22no—
144#41×C61—22no—
145#41×C61—22no—
146#41×C61—22no—
147#41×C61—44no—
148#41×C61—41no—
149#41×C61—41no—
150#41×C61—44no—
151#41×C61C2×D414yes—
152C2×Q8×C61—21no—
153C2×Q8×C61—22no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—22no—
157C2×Q8×C61—22no—
158C2×Q8×C61—41no—
159C2×Q8×C61C2×D411yescharacteristic
160C2×Q8×C61C2×D411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61—22no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—41no—
168C2×D4×C61—44no—
169C2×D4×C61C2×D411yescharacteristic
170C2×D4×C61C2×D411yescharacteristic
171C22×C4×C61—22no—
172C22×C4×C61—21no—
173C22×C4×C61—22no—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—21no—
179C22×C4×C61—21no—
180C22×C4×C61—44no—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D411yescharacteristic
185C22×C4×C61#4111yescharacteristic
186C22×C4×C61C2411yescharacteristic
187C2×D244—24no—
188C2×D244—24no—
189C2×D244—22no—
190C2×D244—22no—
191C2×D244—22no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—44no—
195C2×D244—44no—
196C2×D244—44no—
197C2×D244—42no—
198C2×D244—42no—
199C2×D244—41no—
200C2×D244—41no—
201C2×D244C2×D414yes—
202Q16×C61—42no—
203Q16×C61—42no—
204SD16×C61—42no—
205SD16×C61—42no—
206M16×C61—22no—
207M16×C61—22no—
208C2×C8×C61—21no—
209C2×C8×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—41no—
213C4⋊C4×C61—41no—
214C22⋊C4×C61—41no—
215C22⋊C4×C61—41no—
216#11758201—28no—
217#11758201—28no—
218#11758201—44no—
219#11758201—44no—
220#11758199—28no—
221#11758199—28no—
222#11758194—24no—
223#11758194—22no—
224#11758194—22no—
225#11758194—22no—
226#11758194—44no—
227#11758194C2×D414yes—
228C2×#92418—24no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—22no—
232C2×#92418—22no—
233C2×#92418—44no—
234C2×#92418C2×D412yes—
235C2×#92418C2×D412yes—
236C23×C61—21no—
237C23×C61—22no—
238C23×C61—22no—
239C23×C61—21no—
240C23×C61—41no—
241C23×C61—42no—
242C23×C61—42no—
243C23×C61—81no—
244C23×C61#12111yescharacteristic
245C23×C61#12111yescharacteristic
246C23×C61C2×#4111yescharacteristic
247C22×D61—24no—
248C22×D61—44no—
249C22×D61—44no—
250C22×D61—42no—
251C22×D61—42no—
252C22×D61—82no—
253C22×D61—82no—
254C22×D61—82no—
255C22×D61—816no—
256C22×D61—82no—
257C22×D61—82no—
258C22×D61—82no—
259Q8×C61—21no—
260Q8×C61—21no—
261Q8×C61—21no—
262Q8×C61—42no—
263Q8×C61—41no—
264Q8×C61—41no—
265Q8×C61—42no—
266Q8×C61#12012yes—
267D4×C61—24no—
268D4×C61—21no—
269D4×C61—21no—
270D4×C61—21no—
271D4×C61—42no—
272D4×C61—42no—
273D4×C61—42no—
274D4×C61—42no—
275D4×C61#12012yes—
276C2×C4×C61—21no—
277C2×C4×C61—22no—
278C2×C4×C61—21no—
279C2×C4×C61—21no—
280C2×C4×C61—21no—
281C2×C4×C61—21no—
282C2×C4×C61—24no—
283C2×C4×C61—22no—
284C2×C4×C61—22no—
285C2×C4×C61—21no—
286C2×C4×C61—42no—
287C2×C4×C61—42no—
288C2×C4×C61—42no—
289C2×C4×C61—42no—
290C2×C4×C61—41no—
291C2×C4×C61—41no—
292C2×C4×C61—42no—
293C2×C4×C61—81no—
294C2×C4×C61#12111yescharacteristic
295C2×C4×C61#12111yescharacteristic
296C2×C4×C61C22×D411yescharacteristic
297C2×C4×C61C22×D412yes—
298C2×C4×C61C22×D411yescharacteristic
299C2×C4×C61H(5,𝔽2)12yes—
300D244—24no—
301D244—24no—
302D244—44no—
303D244—44no—
304D244—42no—
305D244—42no—
306C8×C61—41no—
307C8×C61—41no—
308#92418—24no—
309#92418—22no—
310#92418—22no—
311#92418—42no—
312#24399—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—21no—
318C22×C61—22no—
319C22×C61—21no—
320C22×C61—21no—
321C22×C61—44no—
322C22×C61—41no—
323C22×C61—41no—
324C22×C61—41no—
325C22×C61—41no—
326C22×C61—41no—
327C22×C61—84no—
328C22×C61#47112yescomplemented
329C22×C61C2×#12111yescharacteristic
330C22×C61#54612yes—
331C2×D61—44no—
332C2×D61—88no—
333C2×D61—84no—
334C2×D61—88no—
335C2×D61—82no—
336C2×D61—82no—
337C2×D61—84no—
338C4×C61—21no—
339C4×C61—21no—
340C4×C61—22no—
341C4×C61—41no—
342C4×C61—81no—
343C4×C61—81no—
344C4×C61C2×#12011yescharacteristic
345C4×C61#53411yescharacteristic
346C4×C61D4212yes—
347C22×AGL(1,ℤ/8)—611no—
348C2×#471—612no—
349#3007D6111yescharacteristic, complemented
350C2×#450—611no—
351#2880—611no—
352#2051—612no—
353#2019—612no—
354#1907—612no—
355#1901—611no—
356#1855—611no—
357#1783—611no—
358C2×C61—22no—
359C2×C61—22no—
360C2×C61—41no—
361C2×C61—42no—
362C2×C61—81no—
363C2×C61#241011yescharacteristic
364C2×C61C2×#47111yescharacteristic
365C2×C61#306811yescharacteristic
366D61—84no—
367C22×#41C2×D6111yescharacteristic
368C23×D4—611no—
369C2×#137C2×D6111yescharacteristic, complemented
370C2×#137C2×D6111yescharacteristic, complemented
371C2×AGL(1,ℤ/8)—612no—
372C2×AGL(1,ℤ/8)—612no—
373C2×AGL(1,ℤ/8)—1228no—
374C2×AGL(1,ℤ/8)—1221no—
375C2×AGL(1,ℤ/8)—1221no—
376C22×SD16—612no—
377C22×SD16—1221no—
378C22×D8—1221no—
379C2×#131—612no—
380C22×M16—611no—
381C2×C22⋊Q8C2×D6111yescharacteristic
382#471—1228no—
383#452—21no—
384#452—21no—
385#451—21no—
386#450—21no—
387#450—1221no—
388#450—1221no—
389#450—1222no—
390C2×#108—612no—
391#417C2×D6111yescharacteristic, complemented
392#417C2×D6111yescharacteristic, complemented
393C2×#102—612no—
394C2×#101—612no—
395C2×#100—612no—
396C2×#100—612no—
397#408—612no—
398#408—1222no—
399#407—611no—
400#407—1221no—
401#407C2×D6111yescharacteristic
402C2×C22⋊C8—611no—
403C2×C22⋊C8—1221no—
404#343—612no—
405#341—612no—
406C61#2439911yescharacteristic, complemented, Sylow
407C25—1221no—
408C2×#41—21no—
409C2×#41—21no—
410C2×#41—24no—
411C2×#41—42no—
412C2×#41C22⋊D6111yescharacteristic
413C2×#41C22⋊D6111yescharacteristic
414C2×#41C22×D6111yescharacteristic
415C2×#41C22×D6111yescharacteristic
416C22×Q8C22⋊D6111yescharacteristic
417C22×Q8C22×D6111yescharacteristic
418C22×D4—21no—
419C22×D4—612no—
420C22×D4—1224no—
421C22×D4—1221no—
422C22×D4—1221no—
423C22×D4—1221no—
424C22×D4—1221no—
425C22×D4—1221no—
426C22×D4—2442no—
427C22×D4C22⋊D6111yescharacteristic
428C23×C4—21no—
429C23×C4C22×D6111yescharacteristic
430#137—24no—
431#137—42no—
432AGL(1,ℤ/8)—618no—
433AGL(1,ℤ/8)—1224no—
434AGL(1,ℤ/8)—1224no—
435AGL(1,ℤ/8)—1228no—
436AGL(1,ℤ/8)—1228no—
437AGL(1,ℤ/8)—1224no—
438C2×Q16—21no—
439C2×Q16—21no—
440C2×SD16—21no—
441C2×SD16—21no—
442C2×SD16—1222no—
443C2×SD16—1222no—
444C2×SD16—1228no—
445C2×SD16—1222no—
446C2×SD16—1222no—
447C2×SD16—1224no—
448C2×SD16—2442no—
449C2×D8—1228no—
450C2×D8—1222no—
451C2×D8—1222no—
452#131—1224no—
453#131—2442no—
454C2×M16—612no—
455C2×M16—612no—
456C2×M16—1222no—
457C2×M16—1224no—
458C2×M16—1221no—
459C2×M16C22×D6111yescharacteristic
460C2×M16C22×D6111yescharacteristic
461C22×C8—612no—
462C22×C8—1222no—
463C22×C8—1221no—
464C22⋊Q8—21no—
465C22⋊Q8—21no—
466C22⋊Q8—21no—
467C22⋊Q8—21no—
468C2×C4⋊C4—21no—
469C2×C4⋊C4C22×D6111yescharacteristic
470C2×C22⋊C4—21no—
471#108—1224no—
472#103—21no—
473#103—21no—
474#102—21no—
475#102—21no—
476#102—1222no—
477#102—1222no—
478#101—1224no—
479#100—1224no—
480#100—2442no—
481C22⋊C8—21no—
482C22⋊C8—21no—
483C22⋊C8—611no—
484C22⋊C8—611no—
485C22⋊C8—612no—
486C22⋊C8—1222no—
487C22⋊C8—1222no—
488C22⋊C8—1221no—
489C22⋊C8—1221no—
490C24—21no—
491C24—1222no—
492C24—4882no—
493C24—4882no—
494C24—2444no—
495C24—2441no—
496C24—2441no—
497C24#1175822411yescharacteristic
498#41—22no—
499#41—22no—
500#41—22no—
501#41—22no—
502#41—22no—
503#41—44no—
504#41—41no—
505#41—41no—
506#41—44no—
507#41D4×D6114yes—
508C2×Q8—21no—
509C2×Q8—22no—
510C2×Q8—22no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—22no—
514C2×Q8—41no—
515C2×Q8C2×C22⋊D6111yescharacteristic
516C2×Q8C2×C22⋊D6111yescharacteristic
517C2×D4—21no—
518C2×D4—22no—
519C2×D4—21no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—41no—
524C2×D4—44no—
525C2×D4—614no—
526C2×D4—1224no—
527C2×D4—1224no—
528C2×D4—1222no—
529C2×D4—1222no—
530C2×D4—1222no—
531C2×D4—1222no—
532C2×D4—1222no—
533C2×D4—2444no—
534C2×D4—2444no—
535C2×D4—2444no—
536C2×D4—2442no—
537C2×D4—2442no—
538C2×D4—2441no—
539C2×D4—2441no—
540C2×D4C2×C22⋊D6111yescharacteristic
541C2×D4C2×C22⋊D6111yescharacteristic
542C22×C4—21no—
543C22×C4—22no—
544C22×C4—22no—
545C22×C4—21no—
546C22×C4—21no—
547C22×C4—21no—
548C22×C4—21no—
549C22×C4—21no—
550C22×C4—21no—
551C22×C4—44no—
552C22×C4D4×D6111yescharacteristic
553C22×C4D4×D6111yescharacteristic
554C22×C4#1175822411yescharacteristic
555C22×C4C2×C22⋊D6111yescharacteristic
556C22×C4C2×C22⋊D6111yescharacteristic
557C22×C4C23×D6111yescharacteristic
558Q16—42no—
559Q16—42no—
560SD16—42no—
561SD16—42no—
562SD16—1228no—
563SD16—1228no—
564SD16—2444no—
565SD16—2444no—
566D8—1228no—
567D8—1228no—
568M16—22no—
569M16—22no—
570M16—614no—
571M16—1224no—
572M16—1222no—
573M16—1222no—
574M16—1222no—
575M16—2444no—
576C2×C8—21no—
577C2×C8—21no—
578C2×C8—612no—
579C2×C8—612no—
580C2×C8—1224no—
581C2×C8—1222no—
582C2×C8—1222no—
583C2×C8—1222no—
584C2×C8—1222no—
585C2×C8—2444no—
586C4⋊C4—21no—
587C4⋊C4—21no—
588C4⋊C4—41no—
589C4⋊C4—41no—
590C22⋊C4—41no—
591C22⋊C4—41no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—21no—
596C23—41no—
597C23—42no—
598C23—42no—
599C23—81no—
600C23—1224no—
601C23—4882no—
602C23—4882no—
603C23—4882no—
604C23—4882no—
605C23—4882no—
606C23—48816no—
607C23—4882no—
608C23—2444no—
609C23—2444no—
610C23—2442no—
611C23—2442no—
612C23#1451974911yescharacteristic
613C23#1451974911yescharacteristic
614C23C2×#1175822411yescharacteristic
615Q8—21no—
616Q8—21no—
617Q8—21no—
618Q8—42no—
619Q8—41no—
620Q8—41no—
621Q8—42no—
622Q8#1451974512yes—
623D4—24no—
624D4—21no—
625D4—21no—
626D4—21no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—1224no—
632D4—1224no—
633D4—2444no—
634D4—2444no—
635D4—2442no—
636D4—2442no—
637D4#1451974512yes—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—81no—
656C2×C4#1451974911yescharacteristic
657C2×C4#1451974911yescharacteristic
658C2×C4C2×D4×D6112yes—
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4C22×C22⋊D6111yescharacteristic
661C2×C4#1451982212yes—
662C8—41no—
663C8—41no—
664C8—1224no—
665C8—1222no—
666C8—1222no—
667C8—2442no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—22no—
676C22—44no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—84no—
683C22—4884no—
684C22—4888no—
685C22—4884no—
686C22—4888no—
687C22—4882no—
688C22—4882no—
689C22—2444no—
690C22#1684002712yescomplemented
691C22C2×#1451974911yescharacteristic
692C22#1684039912yes—
693C4—21no—
694C4—21no—
695C4—22no—
696C4—41no—
697C4—81no—
698C4—81no—
699C4C2×#1451974511yescharacteristic
700C4#1684032111yescharacteristic
701C4#1684039412yes—
702C2—22no—
703C2—22no—
704C2—41no—
705C2—42no—
706C2—81no—
707C2—4884no—
708C2#2144919411yescharacteristic
709C2C2×#1684002711yescharacteristic
710C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3007C2
socleC22×C61C2×#121
prime radicalC22×#14519802C2
hypercenter#3007D61
hypocenterC61#24399
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
2. C2×#16840027C2semidirect
3. C2×#16839912C2semidirect
4. #21454268C2semidirect
5. #21447109C2semidirect
6. #21446930C2semidirect
7. #21446325C2semidirect
8. #21446288C2semidirect
9. #21446023C2semidirect
10. #21445739C2semidirect
11. C61×#3007C2semidirect
19. C22×#11758201C22semidirect
21. C2×#14519784C22semidirect
27. C2×#14519712C22semidirect
28. C2×#14519697C22semidirect
29. C2×#14519696C22semidirect
30. C2×#14519693C22semidirect
31. C2×#14519693C22semidirect
33. #16839729C22semidirect
38. #16839531C22semidirect
39. #16839526C22semidirect
41. C2×#137×C61C22semidirect
42. C2×#137×C61C22semidirect
48. C61×#417C22semidirect
49. C61×#417C22semidirect
328. C22×C61#471semidirect
349. #3007D61semidirect
369. C2×#137C2×D61semidirect
370. C2×#137C2×D61semidirect
391. #417C2×D61semidirect
392. #417C2×D61semidirect
406. C61#24399semidirect
690. C22#16840027semidirect

Maximal subgroups

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

subgroupindex
C22×#145198022
C2×#168400272
C2×#168399122
#214542682
#214471092
#214469302
#214463252
#214462882
#214460232
#214457392
C61×#30072
#2439961