Finite Group Databasebeta

← #29682721

Subgroups — #29682721

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839764C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic
3#21448309C211yesmaximal, characteristic
4C2×#16839660C211yesmaximal, characteristic, complemented
5#21446516C212yesmaximal, complemented
6#21446509C212yesmaximal, complemented
7#21446502C212yesmaximal, complemented
8#21446495C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10#21445539C211yesmaximal, characteristic, complemented
11C2×C4×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196—21no—
17#16839764—24no—
18#16839764C418yescomplemented
19C2×#14519702C2212yes—
20C2×#14519702C2212yes—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23#16839668C2211yescharacteristic
24#16839668C2211yescharacteristic
25#16839664C2211yescharacteristic
26#16839664C2211yescharacteristic
27#16839660—21no—
28#16839660—21no—
29#16839660C2212yes—
30#16839660C2212yes—
31#16839641—21no—
32#16839641—21no—
33C4×#11758198—21no—
34C4×#11758198—21no—
35C2×#14519684—21no—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C42×C61C2211yescharacteristic
43C4×M16×C61—24no—
44C2×#97×C61C2212yes—
45C2×C4×C8×C61C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58#14519736—22no—
59#14519736—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C414yes—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—44no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61—21no—
79#14519702—24no—
80#14519702C2×C418yescomplemented
81#14519699—24no—
82#14519699C2×C418yescomplemented
83#14519684—22no—
84#14519684—22no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61C2×C414yes—
97C2×M16×C61D414yes—
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C2×C42×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61D411yescharacteristic
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106#97×C61—24no—
107C4×C8×C61—24no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C22×Dic61—24no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142#11758222—28no—
143#11758222—28no—
144#11758222—44no—
145#11758222—44no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C412yes—
161M16×C61—28no—
162M16×C61—28no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61C22×C414yes—
168C2×C8×C61C2×D414yes—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179#11758198—42no—
180C4⋊Dic61—24no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61C22×C414yes—
190#11758196—44no—
191#11758196—44no—
192C4×Dic61—42no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61C4×D412yes—
198C23×C61#12112yes—
199C23×C61#12412yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61#11714yes—
221C2×C4×C61C4×D414yes—
222C2×C4×C61C4×D412yes—
223C2×C4×C61#12112yes—
224C2×C4×C61#12314yes—
225C2×C4×C61#12412yes—
226C2×C4×C61#12614yes—
227C2×C4×C61AGL(1,ℤ/8)12yes—
228C2×C4×C61#13712yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—48no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—48no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—84no—
253D244—22no—
254D244—22no—
255D244—41no—
256D244—41no—
257C4×D61—48no—
258Dic122—22no—
259Dic122—22no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—24no—
263C8×C61—24no—
264#21093—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61#38814yes—
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×#12411yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57511yescharacteristic
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61#38814yes—
291C4×C61C2×#11711yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×#12311yescharacteristic
294C4×C61C2×#12611yescharacteristic
295Dic61—44no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
299Dic61—82no—
300C2×#417—611no—
301C2×#417—611no—
302C2×C4×M16D6111yescharacteristic, complemented
303#2297—611no—
304C2×#389—611no—
305#1936—612no—
306#1935—612no—
307#1934—612no—
308#1933—612no—
309#1736—611no—
310#1736—611no—
311C2×C61—24no—
312C2×C61C2×#38811yescharacteristic
313C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C61C4×#13711yescharacteristic
315C2×C61#305511yescharacteristic
316C2×C61#305611yescharacteristic
317C2×C61#314111yescharacteristic
318C2×C61#314211yescharacteristic
319D61—82no—
320C22×#41—611no—
321C22×M16C2×D6111yescharacteristic
322C22×M16C2×D6111yescharacteristic
323C2×#117—611no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4—1221no—
327C22×C42C2×D6111yescharacteristic
328#417—618no—
329#417—1224no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#102—612no—
333C2×#102—612no—
334C4×M16—24no—
335C2×#97C2×D6112yes—
336C2×C4×C8C2×D6112yes—
337#391—611no—
338#391—611no—
339#390—611no—
340#390—611no—
341#389—612no—
342#389—612no—
343#389—1221no—
344#389—1221no—
345#386—1221no—
346#386—1221no—
347C4×C22⋊C4—1221no—
348C4×C22⋊C4—1221no—
349C2×#95—1221no—
350#340—612no—
351#340—612no—
352#340—612no—
353#340—612no—
354C61#2109311yescharacteristic, complemented, Sylow
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1222no—
361C22×Q8—611no—
362C22×D4—611no—
363C22×D4—1221no—
364C23×C4—1221no—
365C23×C4—1221no—
366C23×C4—1221no—
367C23×C4D24411yescharacteristic
368C23×C4D24411yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16C4×D6114yes—
375C2×M16C22⋊D6114yes—
376C22×C8C22×D6112yes—
377C22×C8C22×D6112yes—
378#117—1222no—
379#117—1222no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—2444no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—1221no—
401C2×C42D24411yescharacteristic
402C2×C42D24411yescharacteristic
403C2×C42C22×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#103—618no—
406#103—1224no—
407#102—618no—
408#102—1224no—
409#97—24no—
410C4×C8—24no—
411#95—1222no—
412#95—1222no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417C24—1221no—
418C24C2×D24411yescharacteristic
419#41—1228no—
420#41—1228no—
421#41—2444no—
422#41—2444no—
423C2×Q8—612no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×D4—612no—
430C2×D4—1224no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—2442no—
437C2×D4—2442no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4C2×C4×D6112yes—
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4#1175822212yes—
465C22×C4#1175822212yes—
466C22×C4#1175822212yes—
467C22×C4#1175822212yes—
468C22×C4C2×C22⋊D6112yes—
469C22×C4C23×D6111yescharacteristic
470M16—28no—
471M16—28no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C2×C4×D6114yes—
477C2×C8C2×C22⋊D6114yes—
478C4⋊C4—614no—
479C4⋊C4—1222no—
480C4⋊C4—1224no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2444no—
488C4⋊C4—2442no—
489C4⋊C4—2444no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—2442no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1222no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23C4×D24412yes—
514C23#1451975112yes—
515C23#1451976512yes—
516C23C22×D24411yescharacteristic
517Q8—1222no—
518Q8—1222no—
519Q8—2441no—
520Q8—2441no—
521D4—1222no—
522D4—1222no—
523D4—2448no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—1224no—
539C2×C4—1222no—
540C2×C4—4884no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2442no—
550C2×C4—2448no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2448no—
556C2×C4#1451973414yes—
557C2×C4C4×C22⋊D6114yes—
558C2×C4C4×D24412yes—
559C2×C4#1451975112yes—
560C2×C4#1451976014yes—
561C2×C4#1451976512yes—
562C2×C4#1451977114yes—
563C2×C4#1451980512yes—
564C2×C4#1451981012yes—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4C2×#1175822211yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C2×C4C2×#1175822211yescharacteristic
571C8—24no—
572C8—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22#1683965314yes—
587C22C2×C4×D24411yescharacteristic
588C22C2×#1451975111yescharacteristic
589C22C2×#1451976511yescharacteristic
590C22C2×#1451980511yescharacteristic
591C22C2×#1451981011yescharacteristic
592C22#1684056211yescharacteristic
593C22#1684056211yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—4882no—
599C4—4881no—
600C4—4881no—
601C4—2444no—
602C4—2442no—
603C4#1683965314yes—
604C4C2×#1451973411yescharacteristic
605C4C2×C4×C22⋊D6111yescharacteristic
606C4C2×#1451976011yescharacteristic
607C4C2×#1451977111yescharacteristic
608C2—24no—
609C2—4882no—
610C2C2×#1683965311yescharacteristic
611C2C4×#1451980511yescharacteristic
612C2C4×#1451981011yescharacteristic
613C2#2145543811yescharacteristic
614C2#2145544911yescharacteristic
615C2#2145621311yescharacteristic
616C2#2145622011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×M16×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×C4×M16D61
hypocenterC61#21093
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
4. C2×#16839660C2semidirect
5. #21446516C2semidirect
6. #21446509C2semidirect
7. #21446502C2semidirect
8. #21446495C2semidirect
9. #21445539C2semidirect
10. #21445539C2semidirect
11. C2×C4×M16×C61C2semidirect
18. #16839764C4semidirect
80. #14519702C2×C4semidirect
82. #14519699C2×C4semidirect
302. C2×C4×M16D61semidirect
354. C61#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#168397642
#214483092
C2×#168396602
#214465162
#214465092
#214465022
#214464952
#214455392
C2×C4×M16×C612
#2109361