Finite Group Databasebeta

← #29680896

Subgroups — #29680896

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840243C211yesmaximal, characteristic
2#21454409C211yesmaximal, characteristic, complemented
3C2×#16839778C211yesmaximal, characteristic
4#21454339C211yesmaximal, characteristic
5#21448443C211yesmaximal, characteristic, complemented
6#21446055C214yesmaximal, complemented
7#21446048C214yesmaximal, complemented
8#21445530C211yesmaximal, characteristic, complemented
9C2×C61×#343C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840243—22no—
12#16840243—22no—
13#16840243—22no—
14#16840243C2214yes—
15C2×#14519742—21no—
16C2×#14519739—21no—
17C2×C4×C22⋊D61—22no—
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C2×#14519735—21no—
22C2×C4⋊C4×D61—21no—
23#16839804—21no—
24#16839804—21no—
25#16839804C2212yescomplemented
26#16839804C2212yescomplemented
27#16839800—22no—
28#16839778C418yescomplemented
29C2×#14519703C2214yes—
30#16839772—21no—
31#16839768—21no—
32#16839764C2212yes—
33#16839764C2212yes—
34#16839759—22no—
35#16839658—22no—
36#16839635—22no—
37#16839631C2212yescomplemented
38#16839627C2212yescomplemented
39#16839523C2214yescomplemented
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#343C2214yescomplemented
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222D412yes—
50C2×#11758222Q814yes—
51C2×#11758222C2312yes—
52C22×Dic122D411yescharacteristic
53C22×D244D411yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57#14519742—24no—
58#14519742—24no—
59#14519739—24no—
60#14519739—24no—
61C4×C22⋊D61—28no—
62C4×C22⋊D61—28no—
63#14519736—21no—
64#14519736—21no—
65#14519736—22no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736C2×C414yescomplemented
71#14519736D412yes—
72#14519736C2312yes—
73#14519735—24no—
74#14519735—24no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—22no—
81C2×#11758196—24no—
82C4⋊C4×D61—24no—
83C4⋊C4×D61—24no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—22no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2312yes—
92#14519711—22no—
93#14519711—22no—
94#14519709—22no—
95#14519709—22no—
96#14519703C2×C4116yescomplemented
97#14519702—22no—
98#14519702—22no—
99#14519699—22no—
100#14519699—22no—
101#14519684—24no—
102C23×C4×C61D412yes—
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—21no—
105C2×M16×C61—22no—
106C2×M16×C61—21no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61C2×C414yescomplemented
110C2×M16×C61C2312yes—
111C2×M16×C61C2312yes—
112C22×C8×C61—21no—
113C22×C8×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61D412yes—
119C2×C4⋊C4×C61C2312yes—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—82no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×Q812yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—22no—
152#11758222—22no—
153#11758222C4⋊C418yescomplemented
154#11758222C22×C414yes—
155#11758222C2×Q814yes—
156C2×D244—21no—
157C2×D244—22no—
158C2×D244C4⋊C414yescomplemented
159C2×D244C2×D412yes—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—28no—
165C2×C4×D61—84no—
166C2×Dic122—21no—
167C2×Dic122—22no—
168C2×Dic122C4⋊C414yescomplemented
169C2×Dic122C2×D412yes—
170M16×C61—24no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—22no—
174M16×C61—42no—
175M16×C61C22×C414yescomplemented
176C2×C8×C61—24no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—42no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C22⋊Dic61—24no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—41no—
188C22⋊Dic61—41no—
189#11758198—28no—
190C4⋊Dic61—24no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D412yes—
198C4⋊Dic61C2×D412yes—
199#11758196—28no—
200#11758196—84no—
201C4×Dic61—28no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—41no—
206C4×Dic61—41no—
207C4×Dic61C22×C414yescomplemented
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C4×D412yes—
211C23×C61#12011yescharacteristic
212C23×C61#12011yescharacteristic
213C23×C61C22⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—42no—
218C22×D61—42no—
219C2×C4×C61—24no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—82no—
223C2×C4×C61C2×C4⋊C412yes—
224C2×C4×C61C2×C4⋊C412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61C22⋊Q812yes—
231C2×C4×C61C22⋊Q812yes—
232C2×C4×C61C22⋊Q812yes—
233C2×C4×C61#12312yes—
234C2×C4×C61#12312yes—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×Q811yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C22⋊D61—24no—
242C22⋊D61—24no—
243C22⋊D61—28no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257D244—22no—
258D244—24no—
259D244C2×C4⋊C414yes—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—28no—
263C4×D61—88no—
264Dic122—22no—
265Dic122—24no—
266Dic122C2×C4⋊C414yes—
267C8×C61—24no—
268C8×C61—42no—
269#20864—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61#38512yes—
279C22×C61#38512yes—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C2×#12011yescharacteristic
282C22×C61C2×C22⋊Q811yescharacteristic
283C2×D61—24no—
284C2×D61—48no—
285C2×D61—42no—
286C2×D61—42no—
287C4×C61—44no—
288C4×C61#38512yes—
289C4×C61#38512yes—
290C4×C61C22×C4⋊C411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×C22⋊Q811yescharacteristic
293C4×C61C2×#12311yescharacteristic
294Dic61—24no—
295Dic61—44no—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C2×#520—611no—
301#2911—611no—
302C2×#421—611no—
303#2896—611no—
304#2317—611no—
305#1860—614no—
306#1859—614no—
307C2×#343D6111yescharacteristic, complemented
308#1733—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61C2×#38511yescharacteristic
314C2×C61#301911yescharacteristic
315C2×C61#302011yescharacteristic
316D61—44no—
317C22×#41—611no—
318C22×M16C2×D6111yescharacteristic
319C22×M16C2×D6111yescharacteristic
320#520—614no—
321#520—1222no—
322#520—1222no—
323#520—1222no—
324C2×C4×Q8—1221no—
325C2×C4×D4—1221no—
326C2×C4×D4—1222no—
327C2×#117—611no—
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—1221no—
331C22×C4⋊C4—1221no—
332C22×C4⋊C4C2×D6111yescharacteristic
333#428—612no—
334#428—612no—
335#428—1221no—
336#428—1221no—
337#427—1222no—
338#421—618no—
339C2×#104—614no—
340#419—1221no—
341#418—1221no—
342#417—612no—
343#417—612no—
344#416—1222no—
345#389—1222no—
346#385—1222no—
347#384—612no—
348#382—612no—
349#343C2×D6114yescomplemented
350#339—614no—
351C61#2086411yescharacteristic, complemented, Sylow
352C2×#41—614no—
353C2×#41—612no—
354C2×#41—614no—
355C2×#41—612no—
356C2×#41—1221no—
357C2×#41—1222no—
358C2×#41—1221no—
359C2×#41—1222no—
360C22×Q8—611no—
361C22×D4—611no—
362C22×D4—1221no—
363C23×C4—1222no—
364C23×C4—1221no—
365C23×C4D24412yes—
366C23×C4C22×D6111yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—21no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16C4×D6114yescomplemented
373C2×M16C22×D6112yes—
374C2×M16C22×D6112yes—
375C22×C8—21no—
376C22×C8—21no—
377C4×Q8—1224no—
378C4×Q8—1224no—
379C4×D4—1224no—
380C4×D4—1224no—
381C4×D4—1228no—
382C4×D4—1228no—
383#117—614no—
384#117—612no—
385#117—612no—
386#117—1224no—
387#117—1224no—
388#117—1221no—
389#117—1222no—
390#117—1221no—
391#117—1221no—
392#117—1221no—
393#117—1221no—
394#117—1221no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4D24412yes—
409C2×C4⋊C4C22×D6112yes—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—612no—
415C2×C42—1222no—
416C2×C42—1221no—
417C2×C42—1221no—
418#107—1222no—
419#107—1222no—
420#106—1222no—
421#106—1222no—
422#104—6116no—
423#103—1222no—
424#103—1222no—
425#102—1222no—
426#102—1222no—
427#95—1224no—
428C24—1221no—
429C24C2×D24411yescharacteristic
430#41—614no—
431#41—614no—
432#41—618no—
433#41—1222no—
434#41—1222no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1224no—
439#41—1224no—
440C2×Q8—612no—
441C2×Q8—614no—
442C2×Q8—1221no—
443C2×Q8—1222no—
444C2×D4—612no—
445C2×D4—614no—
446C2×D4—1222no—
447C2×D4—1224no—
448C2×D4—1222no—
449C2×D4—1224no—
450C2×D4—1221no—
451C2×D4—1222no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—82no—
455C22×C4—1228no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—4884no—
467C22×C4C2×C4×D6112yes—
468C22×C4C2×D24414yes—
469C22×C4C2×D24411yescharacteristic
470C22×C4#1175822212yes—
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4#1175822412yes—
476C22×C4Q8×D6112yes—
477C22×C4C23×D6111yescharacteristic
478M16—22no—
479M16—24no—
480M16—22no—
481M16—22no—
482M16—42no—
483M16C2×C4×D6114yescomplemented
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1228no—
495C4⋊C4—1224no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—4884no—
503C22⋊C4—1228no—
504C22⋊C4—1222no—
505C22⋊C4—1224no—
506C22⋊C4—1222no—
507C22⋊C4—2441no—
508C22⋊C4—2441no—
509C42—614no—
510C42—1228no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C42—2441no—
515C42—2441no—
516C23—41no—
517C23—41no—
518C23—1222no—
519C23—1224no—
520C23—2442no—
521C23—2442no—
522C23C4×D24412yes—
523C23#1451974711yescharacteristic
524C23#1451974711yescharacteristic
525C23#1451975712yes—
526C23C22×D24411yescharacteristic
527Q8—614no—
528Q8—1222no—
529Q8—1224no—
530D4—614no—
531D4—1224no—
532D4—1224no—
533D4—1228no—
534D4—1222no—
535D4—1224no—
536C2×C4—24no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—82no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1228no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—4888no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C4⋊C4×D6112yes—
558C2×C4C4⋊C4×D6112yes—
559C2×C4#1451973712yes—
560C2×C4#1451973712yes—
561C2×C4C4×D24412yes—
562C2×C4#1451974711yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451975512yes—
565C2×C4#1451975512yes—
566C2×C4#1451975712yes—
567C2×C4#1451975912yes—
568C2×C4#1451975912yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×Q8×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C8—24no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—42no—
584C22—42no—
585C22—1224no—
586C22—2448no—
587C22—2442no—
588C22—2442no—
589C22#1683963612yes—
590C22#1683963612yes—
591C22C2×C4×D24411yescharacteristic
592C22C2×#1451974711yescharacteristic
593C22C2×#1451975711yescharacteristic
594C4—44no—
595C4—1224no—
596C4—4881no—
597C4—4881no—
598C4—2444no—
599C4—2442no—
600C4—2442no—
601C4#1683963612yes—
602C4#1683963612yes—
603C4C2×C4⋊C4×D6111yescharacteristic
604C4C2×#1451973711yescharacteristic
605C4C2×#1451975511yescharacteristic
606C4C2×#1451975911yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—2444no—
612C2C2×#1683963611yescharacteristic
613C2#2145517911yescharacteristic
614C2#2145518611yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#343D61
hypocenterC61#20864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. #21454409C2semidirect
5. #21448443C2semidirect
6. #21446055C2semidirect
7. #21446048C2semidirect
8. #21445530C2semidirect
9. C2×C61×#343C2semidirect
25. #16839804C22semidirect
26. #16839804C22semidirect
28. #16839778C4semidirect
37. #16839631C22semidirect
38. #16839627C22semidirect
39. #16839523C22semidirect
43. C61×#343C22semidirect
70. #14519736C2×C4semidirect
96. #14519703C2×C4semidirect
109. C2×M16×C61C2×C4semidirect
153. #11758222C4⋊C4semidirect
158. C2×D244C4⋊C4semidirect
168. C2×Dic122C4⋊C4semidirect
175. M16×C61C22×C4semidirect
207. C4×Dic61C22×C4semidirect
307. C2×#343D61semidirect
349. #343C2×D61semidirect
351. C61#20864semidirect
372. C2×M16C4×D61semidirect
483. M16C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402432
#214544092
C2×#168397782
#214543392
#214484432
#214460552
#214460482
#214455302
C2×C61×#3432
#2086461