Finite Group Databasebeta

← #29680904

Subgroups — #29680904

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459022C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic
4#21448442C211yesmaximal, characteristic, complemented
5#21446054C214yesmaximal, complemented
6#21446047C214yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C2×C61×#421C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11#16840245—21no—
12#16840245—21no—
13#16840245C2212yes—
14#16840245C2212yes—
15#16840240—21no—
16#16840237—22no—
17#16840237—21no—
18#41×Dic61—22no—
19#41×Dic61—22no—
20C2×#14519736—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C22×C4⋊Dic61C2211yescharacteristic
24#16839804—22no—
25#16839804C2214yescomplemented
26C2×#14519711—21no—
27C2×#14519709—21no—
28#16839763—22no—
29#16839763C2214yes—
30C2×#14519701—21no—
31C2×#14519698—21no—
32#16839661—22no—
33#16839637—22no—
34#16839632C2212yescomplemented
35#16839628C2212yescomplemented
36#16839530C2212yescomplemented
37#16839530C2212yescomplemented
38#16839522C2214yescomplemented
39C22×#41×C61C2211yescharacteristic
40C22×M16×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#421C418yescomplemented
43C2×#104×C61C2214yes—
44C23×Dic61—21no—
45C2×#11758194—21no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C2×#11758194C2312yes—
49C22×#92418—21no—
50Q8×Dic61—22no—
51Q8×Dic61—22no—
52Q8×Dic61—22no—
53Q8×Dic61—22no—
54D4×Dic61—24no—
55D4×Dic61—24no—
56D4×Dic61—22no—
57D4×Dic61—22no—
58D4×Dic61—22no—
59D4×Dic61—22no—
60D4×Dic61—24no—
61D4×Dic61—24no—
62#14519736—24no—
63#14519736—24no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736—22no—
71#14519736—22no—
72#14519736D412yes—
73#14519736C2312yes—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—24no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2312yes—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C22⋊Dic61—24no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61D412yes—
90C2×C4×Dic61—24no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D412yes—
93#14519711—24no—
94#14519709—24no—
95#14519701—24no—
96#14519698—24no—
97#14519684—24no—
98C2×#41×C61—21no—
99C2×#41×C61—21no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61C2×C414yes—
103C2×#41×C61D412yes—
104C2×#41×C61Q814yes—
105C2×#41×C61C2312yes—
106C22×Q8×C61D411yescharacteristic
107C22×D4×C61—21no—
108C22×D4×C61D411yescharacteristic
109C23×C4×C61—21no—
110C23×C4×C61C2311yescharacteristic
111C2×M16×C61—21no—
112C2×M16×C61—22no—
113C2×M16×C61C2×C414yescomplemented
114C2×M16×C61C2312yes—
115C22×C8×C61—21no—
116#117×C61—22no—
117#117×C61C2×C414yescomplemented
118C2×C4⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2312yes—
121#104×C61C2×C4116yescomplemented
122C24×C61—21no—
123C24×C61C2×D411yescharacteristic
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—24no—
128#41×C61—24no—
129#41×C61—22no—
130#41×C61—22no—
131#41×C61C4⋊C418yescomplemented
132#41×C61C22×C414yes—
133#41×C61C2×Q814yes—
134C2×Q8×C61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61C4⋊C414yescomplemented
137C2×Q8×C61C2×D412yes—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—24no—
141C2×D4×C61—22no—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61C4⋊C414yescomplemented
145C2×D4×C61C2×D412yes—
146C22×C4×C61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—22no—
149C22×C4×C61—24no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×Q812yes—
159C22×C4×C61#4112yes—
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—22no—
167C22×Dic61—41no—
168C22×Dic61—41no—
169C22×Dic61—82no—
170C22×Dic61C2×D412yes—
171C22×Dic61C2×D412yes—
172M16×C61—24no—
173M16×C61—22no—
174M16×C61C22×C414yes—
175C2×C8×C61—24no—
176C2×C8×C61—22no—
177C4⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C42×C61—22no—
180C42×C61C22×C414yes—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C4⋊Dic61—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61—84no—
200C4⋊Dic61C2×D412yes—
201C4⋊Dic61C2×D412yes—
202C4×Dic61—24no—
203C4×Dic61—24no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208#11758194—22no—
209#11758194—22no—
210#11758194—42no—
211C2×#92418—22no—
212C2×#92418—42no—
213C23×C61—22no—
214C23×C61—24no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—81no—
218C23×C61—81no—
219C23×C61C4×D412yes—
220C23×C61#12011yescharacteristic
221C23×C61#12011yescharacteristic
222C23×C61C22⋊Q812yes—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—24no—
226Q8×C61C2×C4⋊C414yes—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—28no—
230D4×C61—22no—
231D4×C61—24no—
232D4×C61C2×C4⋊C414yes—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—28no—
238C2×C4×C61—24no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C2×C4⋊C412yes—
242C2×C4×C61C2×C4⋊C412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61C22⋊Q812yes—
249C2×C4×C61C22⋊Q812yes—
250C2×C4×C61C22⋊Q812yes—
251C2×C4×C61#12312yes—
252C2×C4×C61#12312yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—44no—
266C2×Dic61—42no—
267C2×Dic61—81no—
268C2×Dic61—81no—
269C2×Dic61—84no—
270C2×Dic61—84no—
271C2×Dic61—82no—
272C2×Dic61—82no—
273C8×C61—24no—
274#92418—42no—
275#20865—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—24no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—84no—
288C22×C61#38512yes—
289C22×C61#38512yes—
290C22×C61C2×C4×D411yescharacteristic
291C22×C61C2×#12011yescharacteristic
292C22×C61C2×C22⋊Q811yescharacteristic
293C4×C61—24no—
294C4×C61—44no—
295C4×C61#38512yes—
296C4×C61#38512yes—
297C4×C61C22×C4⋊C411yescharacteristic
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C2×C22⋊Q811yescharacteristic
300C4×C61C2×#12311yescharacteristic
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304Dic61—42no—
305Dic61—82no—
306#3428—611no—
307C2×#428—611no—
308C2×#421D6111yescharacteristic
309C2×#417—611no—
310#2317—611no—
311#1860—614no—
312#1859—614no—
313#1742—611no—
314#1733—611no—
315C2×C61—21no—
316C2×C61—22no—
317C2×C61—21no—
318C2×C61—22no—
319C2×C61—44no—
320C2×C61C2×#38511yescharacteristic
321C2×C61#302111yescharacteristic
322C2×C61#302211yescharacteristic
323C22×#41C2×D6111yescharacteristic
324C22×M16—611no—
325C22×M16C2×D6111yescharacteristic
326#520—612no—
327#520—612no—
328#520—1221no—
329#520—1221no—
330#519—1221no—
331#518—1222no—
332#518—1221no—
333C4×#41—1222no—
334C4×#41—1222no—
335C2×#117—611no—
336C2×#117—611no—
337C2×#117—1222no—
338C2×#117C2×D6111yescharacteristic
339C22×C4⋊C4—611no—
340#428—614no—
341#428—1222no—
342C2×#107—1221no—
343C2×#106—1221no—
344#421Dic6118yescomplemented
345C2×#104C2×D6114yes—
346#417—614no—
347#417—1222no—
348C2×#103—1221no—
349C2×#102—1221no—
350#389—1222no—
351#385—1222no—
352#384—612no—
353#382—612no—
354#343—612no—
355#343—612no—
356#339—614no—
357C61#2086511yescharacteristic, complemented, Sylow
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41Dic12214yes—
363C2×#41D24412yes—
364C2×#41C2×Dic6114yes—
365C2×#41C22×D6112yes—
366C22×Q8D24411yescharacteristic
367C22×D4—21no—
368C22×D4D24411yescharacteristic
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—612no—
375C2×M16—1221no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16C2×Dic6114yescomplemented
379C2×M16C22×D6112yes—
380C22×C8—21no—
381C22×C8—1221no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×D4—1224no—
387C4×D4—1224no—
388C4×D4—1224no—
389C4×D4—1224no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394#117—22no—
395#117—612no—
396#117—612no—
397#117—1222no—
398#117—1222no—
399#117—1224no—
400#117—1224no—
401#117—1221no—
402#117—1221no—
403#117—1221no—
404#117—1221no—
405#117—1221no—
406#117—1221no—
407#117C2×Dic6114yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1224no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1221no—
426C2×C42—612no—
427C2×C42—1224no—
428C2×C42—1221no—
429C2×C42C22×D6112yes—
430#107—1224no—
431#106—1224no—
432#104C2×Dic61116yescomplemented
433#103—1224no—
434#102—1224no—
435#95—1224no—
436C24—21no—
437C24D4×D6111yescharacteristic
438#41—22no—
439#41—22no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41C4⋊Dic6118yescomplemented
446#41C2×Dic12214yes—
447#41C22×Dic6114yes—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8C4⋊Dic6114yescomplemented
451C2×Q8C2×D24412yes—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4C4⋊Dic6114yescomplemented
459C2×D4C2×D24412yes—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—4882no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4C2×Dic12212yes—
477C22×C4C2×D24411yescharacteristic
478C22×C4C2×D24411yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4#1175822412yes—
483C22×C4#1175822412yes—
484C22×C4C22×Dic6112yes—
485C22×C4C23×D6111yescharacteristic
486M16—22no—
487M16—24no—
488M16—1222no—
489M16—1222no—
490M16—2442no—
491M16C22×Dic6114yes—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—1222no—
495C2×C8—2442no—
496C4⋊C4—24no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1224no—
500C4⋊C4—1224no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—4884no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C22⋊C4—24no—
513C22⋊C4—1224no—
514C22⋊C4—1224no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C42—22no—
520C42—1224no—
521C42—1224no—
522C42—1222no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42C22×Dic6114yes—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23D4×Dic6112yes—
534C23#1451974711yescharacteristic
535C23#1451974711yescharacteristic
536C23#1451975412yes—
537C23C2×D4×D6111yescharacteristic
538Q8—22no—
539Q8—24no—
540Q8C2×C4⋊Dic6114yes—
541D4—24no—
542D4—24no—
543D4—28no—
544D4—22no—
545D4—24no—
546D4C2×C4⋊Dic6114yes—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—4882no—
560C2×C4—4882no—
561C2×C4—4881no—
562C2×C4—4881no—
563C2×C4—4884no—
564C2×C4—4884no—
565C2×C4—2444no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4C2×C4⋊Dic6112yes—
570C2×C4C2×C4⋊Dic6112yes—
571C2×C4D4×Dic6112yes—
572C2×C4D4×Dic6112yes—
573C2×C4D4×Dic6112yes—
574C2×C4#1451974711yescharacteristic
575C2×C4#1451974711yescharacteristic
576C2×C4#1451975412yes—
577C2×C4#1451975412yes—
578C2×C4#1451975412yes—
579C2×C4#1451976112yes—
580C2×C4#1451976112yes—
581C2×C4C23×Dic6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×D24411yescharacteristic
584C2×C4C22×Dic12211yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822411yescharacteristic
587C8—24no—
588C8—2442no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1683963712yes—
602C22#1683963712yes—
603C22C2×D4×Dic6111yescharacteristic
604C22C2×#1451974711yescharacteristic
605C22C2×#1451975411yescharacteristic
606C4—24no—
607C4—44no—
608C4—4882no—
609C4—2442no—
610C4—2442no—
611C4—2442no—
612C4—2442no—
613C4#1683963712yes—
614C4#1683963712yes—
615C4C22×C4⋊Dic6111yescharacteristic
616C4C2×D4×Dic6111yescharacteristic
617C4C2×#1451975411yescharacteristic
618C4C2×#1451976111yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—44no—
624C2C2×#1683963711yescharacteristic
625C2#2145519211yescharacteristic
626C2#2145519911yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#421D61
hypocenterC61#20865
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
4. #21448442C2semidirect
5. #21446054C2semidirect
6. #21446047C2semidirect
7. #21445557C2semidirect
8. #21445529C2semidirect
25. #16839804C22semidirect
34. #16839632C22semidirect
35. #16839628C22semidirect
36. #16839530C22semidirect
37. #16839530C22semidirect
38. #16839522C22semidirect
42. C61×#421C4semidirect
113. C2×M16×C61C2×C4semidirect
117. #117×C61C2×C4semidirect
121. #104×C61C2×C4semidirect
131. #41×C61C4⋊C4semidirect
136. C2×Q8×C61C4⋊C4semidirect
144. C2×D4×C61C4⋊C4semidirect
344. #421Dic61semidirect
357. C61#20865semidirect
378. C2×M16C2×Dic61semidirect
407. #117C2×Dic61semidirect
432. #104C2×Dic61semidirect
445. #41C4⋊Dic61semidirect
450. C2×Q8C4⋊Dic61semidirect
458. C2×D4C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
#214590222
C2×#168398042
C2×#168397632
#214484422
#214460542
#214460472
#214455572
#214455292
C2×C61×#4212
#2086561