Finite Group Databasebeta

← #29716680

Subgroups — #29716680

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic, complemented
2#21454106C212yesmaximal, complemented
3#21447172C212yesmaximal, complemented
4#21446983C212yesmaximal, complemented
5#21446804C212yesmaximal, complemented
6#21446228C212yesmaximal
7#21446099C212yesmaximal, complemented
8#21445807C211yesmaximal, characteristic, complemented
9C4×#120×C61C211yesmaximal, characteristic
10#16840471—24no—
11#16840469—22no—
12D4×Dic122—22no—
13#16840393—21no—
14#16840388—21no—
15#16840388C2212yes—
16#16840319—21no—
17C2×#14519756—21no—
18C2×#14519756—22no—
19C2×#14519756C2212yes—
20#16840237C2212yes—
21C2×#14519692C2212yes—
22#16839635—24no—
23#16839635C2214yes—
24#16839622—22no—
25#16839622C2212yes—
26#16839618C2214yes—
27#16839529C2212yes—
28#16839528C2214yes—
29#16839528C2214yes—
30C24×C4×C61C2211yescharacteristic
31C2×#120×C61C2211yescharacteristic
32C2×C4×D4×C61—21no—
33C2×C4×D4×C61C2211yescharacteristic
34C61×#386—21no—
35C61×#386C2211yescharacteristic
36C61×#385—21no—
37C61×#385C2211yescharacteristic
38C61×#379C2211yescharacteristic
39C4×C22⋊C4×C61—21no—
40C4×C22⋊C4×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D412yescomplemented
43#14519777—21no—
44#14519777—21no—
45#14519776—24no—
46#14519769—22no—
47#14519768—24no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519756—22no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756—22no—
56#14519756—48no—
57#14519754—24no—
58#14519754—24no—
59#14519754—24no—
60#14519754—24no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63D4×Dic61—24no—
64D4×Dic61—24no—
65#14519736—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61D412yescomplemented
69C2×#11758196—22no—
70C2×#11758196—24no—
71C2×#11758196D414yescomplemented
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—24no—
74C2×C22⋊Dic61—24no—
75C2×C22⋊Dic61C2314yes—
76C2×C22⋊Dic61C2314yes—
77#14519692—24no—
78#14519684—22no—
79#14519684—24no—
80C25×C61C2311yescharacteristic
81C22×D4×C61—21no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—22no—
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61—22no—
87C23×C4×C61—44no—
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91#120×C61—22no—
92#120×C61Q814yescomplemented
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—44no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61D411yescharacteristic
108#95×C61—21no—
109#95×C61—21no—
110#95×C61—21no—
111C24×C61—21no—
112C24×C61—22no—
113C24×C61—21no—
114C24×C61—22no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×Q812yescomplemented
119C24×C61C2411yescharacteristic
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—24no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61C2×Q812yes—
126C2×D4×C61#4112yes—
127C22×C4×C61—24no—
128C22×C4×C61—28no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—44no—
138C22×C4×C61—42no—
139C22×C4×C61—44no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—44no—
143C22×C4×C61—42no—
144C22×C4×C61—41no—
145C22×C4×C61—42no—
146C22×C4×C61—41no—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—24no—
157C22×Dic61C2×D414yes—
158C22×Dic61C2×D414yes—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—44no—
164C2×Dic122—82no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—42no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—42no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61C2×Q812yescomplemented
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—42no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184C22⋊Dic61—48no—
185C22⋊Dic61—48no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—44no—
193#11758196—24no—
194#11758196—24no—
195#11758196—24no—
196#11758196—24no—
197#11758196—48no—
198C4×Dic61—24no—
199C23×C61—24no—
200C23×C61—24no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—21no—
207C23×C61—41no—
208C23×C61—44no—
209C23×C61—41no—
210C23×C61—42no—
211C23×C61—41no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—44no—
215C23×C61—48no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—42no—
221C23×C61C22⋊Q812yescomplemented
222C23×C61C22×D411yescharacteristic
223C23×C61C22×D412yes—
224C23×C61C22×Q811yescharacteristic
225C23×C61C2×#4111yescharacteristic
226C23×C612−1+412yes—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—48no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—48no—
242C2×C4×C61—44no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—44no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—81no—
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12012yes—
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61C22⋊Q812yescomplemented
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—44no—
263C2×Dic61—44no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—82no—
268C2×Dic61—82no—
269Dic122—44no—
270Dic122—82no—
271Dic122—82no—
272Dic122—82no—
273#24350—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—21no—
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—24no—
279C22×C61—28no—
280C22×C61—22no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—44no—
291C22×C61—42no—
292C22×C61—41no—
293C22×C61—42no—
294C22×C61—41no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61C2×#12011yescharacteristic
298C22×C61C2×C22⋊Q811yescharacteristic
299C22×C61#53611yescharacteristic
300C22×C61#54812yes—
301C22×C61D4×Q812yes—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61B(4,𝔽2)12yes—
311Dic61—44no—
312Dic61—44no—
313Dic61—84no—
314#3501—611no—
315#2848—612no—
316C4×#120D6111yescharacteristic
317#2062—612no—
318#2029—612no—
319#1996—612no—
320#1889—612no—
321#1867—612no—
322#1799—611no—
323C2×C61—22no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—22no—
327C2×C61—44no—
328C2×C61—42no—
329C2×C61#240911yescharacteristic
330C2×C61C2×B(4,𝔽2)11yescharacteristic
331C2×C61#302511yescharacteristic
332C24×C4C2×D6111yescharacteristic
333#554—1222no—
334#554—1224no—
335D4×Q8—1222no—
336#544—1221no—
337#543—612no—
338#543—1221no—
339#533—1221no—
340C2×C22⋊Q8—612no—
341C2×C22⋊Q8—1221no—
342C2×C22⋊Q8—1222no—
343C2×#120C2×D6111yescharacteristic
344#518—612no—
345C2×C4×D4—21no—
346C2×C4×D4C2×D6111yescharacteristic
347C2×#99—612no—
348#386—21no—
349#386C2×D6111yescharacteristic
350#385—21no—
351#385—614no—
352#385—1224no—
353#385C2×D6111yescharacteristic
354#380—612no—
355#380—1222no—
356#379—614no—
357#379C2×D6111yescharacteristic
358C4×C22⋊C4—21no—
359C4×C22⋊C4C2×D6111yescharacteristic
360#342—612no—
361#342—614no—
362#342—614no—
363C61#2435011yescharacteristic, complemented, Sylow
364C25C22×D6111yescharacteristic
365C22×Q8—612no—
366C22×Q8—1221no—
367C22×D4—21no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—22no—
370C23×C4—21no—
371C23×C4—22no—
372C23×C4—21no—
373C23×C4—44no—
374C23×C4C22⋊D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1224no—
380#125—1222no—
381#125—1224no—
382C22⋊Q8—1224no—
383C22⋊Q8—1224no—
384C22⋊Q8—1224no—
385C22⋊Q8—1224no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—2448no—
395#120—22no—
396#120Dic12214yescomplemented
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—44no—
402C4×D4—1224no—
403C4×D4—1224no—
404#117—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1224no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22×D6111yescharacteristic
424C2×C22⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4C22×D6111yescharacteristic
426C2×C42—21no—
427C2×C42C22⋊D6111yescharacteristic
428#99—1224no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1224no—
433#95—1222no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—42no—
439C24—42no—
440C24C2×Dic12212yescomplemented
441C24C2×C22⋊D6111yescharacteristic
442C24C23×D6111yescharacteristic
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×Q8—1222no—
447C2×Q8—4882no—
448C2×Q8—2444no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C2×Dic12212yes—
455C2×D4#1175822212yes—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1224no—
481C22×C4D4×D6112yes—
482C22×C4D4×D6112yes—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—1224no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1224no—
497C4⋊C4—1222no—
498C4⋊C4—1224no—
499C4⋊C4—1222no—
500C4⋊C4—2448no—
501C4⋊C4—2444no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—1224no—
511C22⋊C4—1224no—
512C22⋊C4—1224no—
513C22⋊C4—1224no—
514C22⋊C4—1224no—
515C22⋊C4—2448no—
516C22⋊C4—2448no—
517C22⋊C4C2×Dic12212yescomplemented
518C42—21no—
519C42—21no—
520C42—42no—
521C42—1224no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—21no—
527C23—22no—
528C23—24no—
529C23—21no—
530C23—41no—
531C23—44no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—44no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23#1451975612yescomplemented
545C23C2×D4×D6112yes—
546C23C22×C22⋊D6111yescharacteristic
547C23C22×Dic12211yescharacteristic
548C23C2×#1175822211yescharacteristic
549C23#1451982412yes—
550Q8—4882no—
551Q8—4882no—
552Q8—4882no—
553Q8—2444no—
554D4—24no—
555D4—24no—
556D4—48no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—81no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—4882no—
586C2×C4—4882no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4#1451974511yescharacteristic
593C2×C4#1451974512yes—
594C2×C4#1451974511yescharacteristic
595C2×C4#1451975612yescomplemented
596C22—21no—
597C22—22no—
598C22—21no—
599C22—22no—
600C22—24no—
601C22—28no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22C2×#1451974511yescharacteristic
620C22C2×#1451975611yescharacteristic
621C22#1684033311yescharacteristic
622C22#1684042112yes—
623C22D4×Dic12212yes—
624C4—21no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—4884no—
633C4—2444no—
634C4—2444no—
635C4#1683994812yes—
636C2—21no—
637C2—21no—
638C2—22no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2#2144918411yescharacteristic
643C2C2×#1683994811yescharacteristic
644C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC4×#120×C61C2
socleC22×C61C2×#120
prime radicalC2×#120×C61C22
hypercenterC4×#120D61
hypocenterC61#24350
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
1. #21459622C2semidirect
2. #21454106C2semidirect
3. #21447172C2semidirect
4. #21446983C2semidirect
5. #21446804C2semidirect
7. #21446099C2semidirect
8. #21445807C2semidirect
42. C22×Dic122D4semidirect
68. C2×C4⋊Dic61D4semidirect
71. C2×#11758196D4semidirect
92. #120×C61Q8semidirect
118. C24×C61C2×Q8semidirect
175. C22⋊C4×C61C2×Q8semidirect
221. C23×C61C22⋊Q8semidirect
259. C2×C4×C61C22⋊Q8semidirect
363. C61#24350semidirect
396. #120Dic122semidirect
440. C24C2×Dic122semidirect
517. C22⋊C4C2×Dic122semidirect
544. C23#14519756semidirect
595. C2×C4#14519756semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214541062
#214471722
#214469832
#214468042
#214462282
#214460992
#214458072
C4×#120×C612
#2435061