Finite Group Databasebeta

← #29716698

Subgroups — #29716698

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic, complemented
2#21451624C212yesmaximal, complemented
3#21447172C212yesmaximal, complemented
4#21446983C212yesmaximal, complemented
5#21446804C212yesmaximal, complemented
6#21446228C212yesmaximal
7#21446099C212yesmaximal, complemented
8#21445807C211yesmaximal, characteristic, complemented
9C61×#2321C211yesmaximal, 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×#14519756C2212yes—
19#16840237C2212yes—
20C2×#14519692C2212yes—
21#16839698—22no—
22#16839685—24no—
23#16839674—22no—
24#16839635C2214yes—
25#16839622C2212yes—
26#16839618C2214yes—
27#16839529C2212yes—
28#16839528C2214yes—
29#16839528C2214yes—
30C2×#120×C61C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C61×#390—21no—
34C61×#388—21no—
35C61×#388—21no—
36C61×#386—21no—
37C61×#386C2211yescharacteristic
38C61×#385C2211yescharacteristic
39C61×#379C2211yescharacteristic
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#14519754—24no—
57#14519754—24no—
58#14519754—24no—
59#14519754—24no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64#14519736—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61D412yescomplemented
68C2×#11758196—22no—
69C2×#11758196—24no—
70C2×#11758196D414yescomplemented
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—24no—
73C2×C22⋊Dic61—24no—
74C2×C22⋊Dic61C2314yes—
75C2×C22⋊Dic61C2314yes—
76#14519692—24no—
77#14519684—22no—
78#14519684—24no—
79C25×C61C2311yescharacteristic
80C22×D4×C61—21no—
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61D411yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85#120×C61—22no—
86#120×C61Q814yescomplemented
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61D411yescharacteristic
105#95×C61—21no—
106#95×C61—21no—
107#95×C61—21no—
108C24×C61—21no—
109C24×C61—22no—
110C24×C61—21no—
111C24×C61—22no—
112C24×C61—81no—
113C24×C61—81no—
114C24×C61C2×D411yescharacteristic
115C24×C61C2×Q812yescomplemented
116C24×C61C2411yescharacteristic
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—24no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C2×D4×C61C2×Q812yes—
123C2×D4×C61#4112yes—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—81no—
131C22×C4×C61—81no—
132C22×C4×C61—81no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61C2×D414yes—
144C22×Dic61C2×D414yes—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—82no—
150C2×Dic122—82no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—81no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—21no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—81no—
162C22⋊C4×C61—81no—
163C22⋊C4×C61—81no—
164C22⋊C4×C61C2×Q812yescomplemented
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—81no—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171C22⋊Dic61—24no—
172C22⋊Dic61—24no—
173C22⋊Dic61—84no—
174C22⋊Dic61—84no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—82no—
182#11758196—24no—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186#11758196—84no—
187C4×Dic61—24no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—44no—
197C23×C61—44no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—42no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—84no—
208C23×C61C22⋊Q812yes—
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C22×Q811yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C23×C612−1+412yes—
214D4×C61—24no—
215D4×C61—24no—
216D4×C61—84no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—44no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—81no—
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12012yes—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61C22⋊Q812yes—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—44no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250C2×Dic61—82no—
251C2×Dic61—82no—
252Dic122—44no—
253Dic122—82no—
254Dic122—82no—
255Dic122—82no—
256#24352—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—21no—
259C22×C61—22no—
260C22×C61—21no—
261C22×C61—22no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—48no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—84no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×C22⋊Q811yescharacteristic
280C22×C61#53611yescharacteristic
281C22×C61#54812yes—
282C22×C61D4×Q812yes—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—22no—
286C4×C61—21no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61—81no—
292Dic61—44no—
293Dic61—44no—
294Dic61—84no—
295#3501—611no—
296#2626—612no—
297#2321D6111yescharacteristic
298#2062—612no—
299#2029—612no—
300#1996—612no—
301#1889—612no—
302#1867—612no—
303#1799—611no—
304C2×C61—22no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—42no—
309C2×C61—82no—
310C2×C61#240911yescharacteristic
311C2×C61#302611yescharacteristic
312C2×C61#302711yescharacteristic
313#554—1222no—
314#554—1224no—
315D4×Q8—1222no—
316#544—1221no—
317#543—612no—
318#543—1221no—
319#533—1221no—
320C2×C22⋊Q8—612no—
321C2×C22⋊Q8—1221no—
322C2×#120C2×D6111yescharacteristic
323#518—612no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C22⋊C4C2×D6111yescharacteristic
326C2×#99—612no—
327#399—1222no—
328#396—1224no—
329#393—1222no—
330#390—21no—
331#388—21no—
332#388—21no—
333#386—21no—
334#386C2×D6111yescharacteristic
335#385—614no—
336#385C2×D6111yescharacteristic
337#380—612no—
338#379—614no—
339#379C2×D6111yescharacteristic
340C4×C22⋊C4C2×D6111yescharacteristic
341#342—612no—
342#342—614no—
343#342—614no—
344C61#2435211yescharacteristic, complemented, Sylow
345C25C22×D6111yescharacteristic
346C22×Q8—612no—
347C22×Q8—1221no—
348C22×D4—21no—
349C22×D4C22×D6111yescharacteristic
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1224no—
356#125—1222no—
357#125—1224no—
358C22⋊Q8—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1224no—
361C22⋊Q8—1224no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370#120—22no—
371#120Dic12214yescomplemented
372C4×Q8—1222no—
373C4×Q8—1222no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—1224no—
377C4×D4—1224no—
378#117—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1224no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4C22⋊D6111yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—614no—
396C2×C22⋊C4—614no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4C22⋊D6111yescharacteristic
401C2×C22⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4C22×D6111yescharacteristic
404C2×C42—21no—
405C2×C42C22⋊D6111yescharacteristic
406#99—1224no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—1224no—
411#95—1222no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—81no—
417C24—81no—
418C24C2×Dic12212yescomplemented
419C24C2×C22⋊D6111yescharacteristic
420C24C23×D6111yescharacteristic
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8—4882no—
426C2×Q8—4882no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×Dic12212yes—
433C2×D4#1175822212yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1224no—
448C22×C4D4×D6112yes—
449C22×C4D4×D6112yes—
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C4⋊C4—22no—
456C4⋊C4—81no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1224no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1224no—
464C4⋊C4—1222no—
465C4⋊C4—1224no—
466C4⋊C4—1222no—
467C4⋊C4—4884no—
468C4⋊C4—4882no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—4884no—
486C22⋊C4—4884no—
487C22⋊C4C2×Dic12212yescomplemented
488C42—21no—
489C42—21no—
490C42—81no—
491C42—1224no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—44no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23#1451975612yes—
513C23C2×D4×D6112yes—
514C23C22×C22⋊D6111yescharacteristic
515C23C22×Dic12211yescharacteristic
516C23C2×#1175822211yescharacteristic
517C23#1451982412yes—
518Q8—4882no—
519Q8—4882no—
520Q8—4882no—
521Q8—2444no—
522D4—24no—
523D4—24no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—81no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4#1451974511yescharacteristic
557C2×C4#1451974511yescharacteristic
558C2×C4#1451974512yes—
559C2×C4#1451975612yes—
560C22—21no—
561C22—22no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—84no—
579C22—81no—
580C22—81no—
581C22C2×#1451974511yescharacteristic
582C22C2×#1451975611yescharacteristic
583C22#1684033311yescharacteristic
584C22#1684042112yes—
585C22D4×Dic12212yes—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—4884no—
596C4—2444no—
597C4—2444no—
598C2—21no—
599C2—21no—
600C2—22no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2#2144918411yescharacteristic
605C2#2145521511yescharacteristic
606C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2321C2
socleC22×C61C2×#120
prime radicalC2×#120×C61C22
hypercenter#2321D61
hypocenterC61#24352
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2321

Complements

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

NH ≅ G/Ntype
1. #21459622C2semidirect
2. #21451624C2semidirect
3. #21447172C2semidirect
4. #21446983C2semidirect
5. #21446804C2semidirect
7. #21446099C2semidirect
8. #21445807C2semidirect
42. C22×Dic122D4semidirect
67. C2×C4⋊Dic61D4semidirect
70. C2×#11758196D4semidirect
86. #120×C61Q8semidirect
115. C24×C61C2×Q8semidirect
164. C22⋊C4×C61C2×Q8semidirect
344. C61#24352semidirect
371. #120Dic122semidirect
418. C24C2×Dic122semidirect
487. C22⋊C4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214516242
#214471722
#214469832
#214468042
#214462282
#214460992
#214458072
C61×#23212
#2435261