Finite Group Databasebeta

← #29680890

Subgroups — #29680890

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×#16840245C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C2×#16839779C211yesmaximal, characteristic
4#21448439C211yesmaximal, characteristic, complemented
5#21446050C214yesmaximal, complemented
6#21446043C214yesmaximal, complemented
7C2×#16839530C211yesmaximal, characteristic, complemented
8#21445530C211yesmaximal, characteristic, complemented
9C61×#2896C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11#16840245—22no—
12#16840245—22no—
13#16840245—22no—
14#16840245C2214yes—
15C2×Q8×Dic61—21no—
16C2×D4×Dic61—21no—
17C2×D4×Dic61—22no—
18C2×#14519736—21no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23#16839805—21no—
24#16839805—21no—
25#16839802C2212yescomplemented
26#16839802C2212yescomplemented
27#16839798—22no—
28#16839779C418yescomplemented
29C2×#14519704C2214yes—
30#16839660—22no—
31#16839634—22no—
32#16839631C2212yescomplemented
33#16839627C2212yescomplemented
34#16839530C2214yescomplemented
35#16839523C2214yescomplemented
36C22×#41×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#419—21no—
40C61×#418—21no—
41C61×#417C2212yes—
42C61×#417C2212yes—
43C61×#416—22no—
44C23×Dic61—22no—
45C23×Dic61D412yes—
46C2×#11758194—21no—
47C2×#11758194—22no—
48C2×#11758194C2×C414yescomplemented
49C2×#11758194C2312yes—
50C22×#92418—21no—
51Q8×Dic61—24no—
52Q8×Dic61—24no—
53D4×Dic61—28no—
54D4×Dic61—24no—
55D4×Dic61—24no—
56D4×Dic61—28no—
57#14519736—24no—
58#14519736—21no—
59#14519736—21no—
60#14519736—22no—
61#14519736—24no—
62#14519736C2×C414yescomplemented
63#14519736D412yes—
64#14519736C2312yes—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—24no—
73C2×C4⋊Dic61D412yes—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—24no—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2312yes—
84#14519710—22no—
85#14519710—22no—
86#14519708—22no—
87#14519708—22no—
88#14519704C2×C4116yescomplemented
89#14519684—24no—
90C2×#41×C61—21no—
91C2×#41×C61—21no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61C2×C414yes—
95C2×#41×C61D412yes—
96C2×#41×C61Q814yes—
97C2×#41×C61C2312yes—
98C22×Q8×C61D411yescharacteristic
99C22×D4×C61—21no—
100C22×D4×C61D411yescharacteristic
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×M16×C61—21no—
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C2×M16×C61C2312yes—
107C22×C8×C61—21no—
108#117×C61—21no—
109#117×C61—21no—
110#117×C61—21no—
111#117×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116#103×C61—22no—
117#103×C61—22no—
118#102×C61—22no—
119#102×C61—22no—
120C24×C61—21no—
121C24×C61C2×D411yescharacteristic
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61C4⋊C418yescomplemented
130#41×C61C22×C414yes—
131#41×C61C2×Q814yes—
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61C4⋊C414yescomplemented
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—24no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61C4⋊C414yescomplemented
143C2×D4×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—24no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61C22×C412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×Q812yes—
158C22×C4×C61#4112yes—
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—28no—
166C22×Dic61—22no—
167C22×Dic61—82no—
168C22×Dic61—84no—
169C22×Dic61C2×D414yes—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—42no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—41no—
182C42×C61—41no—
183C22⋊Dic61—28no—
184C22⋊Dic61—24no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C4⋊Dic61—28no—
188C4⋊Dic61—24no—
189C4⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C2×D412yes—
196C4⋊Dic61C2×D412yes—
197#11758196—84no—
198C4×Dic61—28no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C22×C414yescomplemented
203#11758194—24no—
204#11758194—22no—
205#11758194C22×C414yescomplemented
206C2×#92418—24no—
207C2×#92418—22no—
208C23×C61—22no—
209C23×C61—24no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61C4×D412yes—
215C23×C61#12011yescharacteristic
216C23×C61#12011yescharacteristic
217C23×C61C22⋊Q812yes—
218C23×C61C22×D411yescharacteristic
219Q8×C61—22no—
220Q8×C61—24no—
221Q8×C61C2×C4⋊C414yes—
222D4×C61—24no—
223D4×C61—24no—
224D4×C61—28no—
225D4×C61—22no—
226D4×C61—24no—
227D4×C61C2×C4⋊C414yes—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—28no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C2×C4⋊C412yes—
238C2×C4×C61C2×C4⋊C412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61C22⋊Q812yes—
245C2×C4×C61C22⋊Q812yes—
246C2×C4×C61C22⋊Q812yes—
247C2×C4×C61#12312yes—
248C2×C4×C61#12312yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—44no—
263C2×Dic61—42no—
264C2×Dic61—44no—
265C2×Dic61—82no—
266C2×Dic61—88no—
267C8×C61—42no—
268#92418—24no—
269#20864—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—24no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—48no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61#38512yes—
283C22×C61#38512yes—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×#12011yescharacteristic
286C22×C61C2×C22⋊Q811yescharacteristic
287C4×C61—24no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61#38512yes—
291C4×C61#38512yes—
292C4×C61C22×C4⋊C411yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C22⋊Q811yescharacteristic
295C4×C61C2×#12311yescharacteristic
296Dic61—44no—
297Dic61—44no—
298Dic61—42no—
299Dic61—42no—
300C2×#520—611no—
301#2911—611no—
302C2×#421—611no—
303#2896D6111yescharacteristic
304#2317—611no—
305#1860—614no—
306#1859—614no—
307C2×#343—611no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61—44no—
314C2×C61C2×#38511yescharacteristic
315C2×C61#301911yescharacteristic
316C2×C61#302011yescharacteristic
317C22×#41C2×D6111yescharacteristic
318C22×M16—611no—
319C22×M16C2×D6111yescharacteristic
320#520—614no—
321#520—1222no—
322#520—1222no—
323#520—1222no—
324C2×C4×Q8—1221no—
325C2×C4×D4—1222no—
326C2×C4×D4—1221no—
327C2×#117—611no—
328C2×#117—611no—
329C2×#117—1221no—
330C2×#117C2×D6111yescharacteristic
331C22×C4⋊C4—611no—
332C22×C4⋊C4—1221no—
333#428—612no—
334#428—612no—
335#428—1221no—
336#428—1221no—
337#427—1222no—
338#421—618no—
339C2×#104—614no—
340#419—21no—
341#418—21no—
342#417C2×D6112yes—
343#417C2×D6112yes—
344#416—22no—
345#389—1222no—
346#385—1222no—
347#384—612no—
348#382—612no—
349#343—614no—
350#339—614no—
351C61#2086411yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41Dic12214yes—
357C2×#41C4×D6114yes—
358C2×#41C22⋊D6112yes—
359C2×#41C22×D6112yes—
360C22×Q8C22⋊D6111yescharacteristic
361C22×D4—21no—
362C22×D4C22⋊D6111yescharacteristic
363C23×C4—21no—
364C23×C4—612no—
365C23×C4—1222no—
366C23×C4C22×D6111yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—614no—
371C2×M16—612no—
372C2×M16—1221no—
373C2×M16—1222no—
374C2×M16C22×D6112yes—
375C22×C8—21no—
376C22×C8—1221no—
377C4×Q8—1224no—
378C4×Q8—1224no—
379C4×D4—1228no—
380C4×D4—1228no—
381C4×D4—1224no—
382C4×D4—1224no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—614no—
388#117—612no—
389#117—612no—
390#117—1224no—
391#117—1224no—
392#117—1221no—
393#117—1222no—
394#117—1221no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1224no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—21no—
415C2×C42—612no—
416C2×C42—1222no—
417C2×C42—1221no—
418#107—1222no—
419#107—1222no—
420#106—1222no—
421#106—1222no—
422#104—6116no—
423#103—22no—
424#103—22no—
425#102—22no—
426#102—22no—
427#95—1224no—
428C24—21no—
429C24D4×D6111yescharacteristic
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41#1175819618yescomplemented
438#41C2×Dic12214yes—
439#41C2×C4×D6114yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8#1175819614yescomplemented
443C2×Q8C2×C22⋊D6112yes—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4#1175819614yescomplemented
451C2×D4C2×C22⋊D6112yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—614no—
460C22×C4—1228no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—4884no—
467C22×C4—4882no—
468C22×C4C2×Dic12212yes—
469C22×C4C2×C4×D6112yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4#1175822412yes—
474C22×C4#1175822412yes—
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478M16—22no—
479M16—22no—
480M16—42no—
481M16—614no—
482M16—1222no—
483M16—1224no—
484C2×C8—22no—
485C2×C8—42no—
486C2×C8—1222no—
487C2×C8—1224no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1228no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—4884no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—1228no—
506C22⋊C4—1222no—
507C22⋊C4—1224no—
508C22⋊C4—1222no—
509C42—41no—
510C42—41no—
511C42—614no—
512C42—1228no—
513C42—1222no—
514C42—1222no—
515C42—1222no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23#1451973712yes—
523C23#1451974511yescharacteristic
524C23#1451974511yescharacteristic
525C23#1451975412yes—
526C23C2×D4×D6111yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8C2×#1175819614yes—
530D4—24no—
531D4—24no—
532D4—28no—
533D4—22no—
534D4—24no—
535D4C2×#1175819614yes—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—4882no—
550C2×C4—4888no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4C2×#1175819612yes—
558C2×C4C2×#1175819612yes—
559C2×C4#1451973712yes—
560C2×C4#1451973712yes—
561C2×C4#1451973712yes—
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974511yescharacteristic
564C2×C4#1451975412yes—
565C2×C4#1451975412yes—
566C2×C4#1451975412yes—
567C2×C4#1451975812yes—
568C2×C4#1451975812yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×Dic12211yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C8—42no—
576C8—1224no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1683963412yes—
590C22#1683963412yes—
591C22C2×#1451973711yescharacteristic
592C22C2×#1451974511yescharacteristic
593C22C2×#1451975411yescharacteristic
594C4—24no—
595C4—81no—
596C4—81no—
597C4—2444no—
598C4—2442no—
599C4—2444no—
600C4—2442no—
601C4#1683963412yes—
602C4#1683963412yes—
603C4C22×#1175819611yescharacteristic
604C4C2×#1451973711yescharacteristic
605C4C2×#1451975411yescharacteristic
606C4C2×#1451975811yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—44no—
612C2C2×#1683963411yescharacteristic
613C2#2145517411yescharacteristic
614C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#2896D61
hypocenterC61#20864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
4. #21448439C2semidirect
5. #21446050C2semidirect
6. #21446043C2semidirect
7. C2×#16839530C2semidirect
8. #21445530C2semidirect
25. #16839802C22semidirect
26. #16839802C22semidirect
28. #16839779C4semidirect
32. #16839631C22semidirect
33. #16839627C22semidirect
34. #16839530C22semidirect
35. #16839523C22semidirect
48. C2×#11758194C2×C4semidirect
62. #14519736C2×C4semidirect
88. #14519704C2×C4semidirect
129. #41×C61C4⋊C4semidirect
134. C2×Q8×C61C4⋊C4semidirect
142. C2×D4×C61C4⋊C4semidirect
202. C4×Dic61C22×C4semidirect
205. #11758194C22×C4semidirect
351. C61#20864semidirect
437. #41#11758196semidirect
442. C2×Q8#11758196semidirect
450. C2×D4#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
#214544082
C2×#168397792
#214484392
#214460502
#214460432
C2×#168395302
#214455302
C61×#28962
#2086461