Finite Group Databasebeta

← #29850012

Subgroups — #29850012

Order 15616.

1918 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450425C212yesmaximal, complemented
2#21450414C212yesmaximal, complemented
3#21450403C212yesmaximal, complemented
4#21450392C212yesmaximal, complemented
5#21450377C214yesmaximal, complemented
6#21450079C214yesmaximal, complemented
7#21449119C214yesmaximal, complemented
8C4×#14519770C214yesmaximal, complemented
9#21448310C211yesmaximal, characteristic
10#21448310C211yesmaximal, characteristic
11#21448302C211yesmaximal, characteristic, complemented
12#21448222C211yesmaximal, characteristic, complemented
13#117×Dic61C211yesmaximal, characteristic, complemented
14C2×C4×C4⋊Dic61C211yesmaximal, characteristic, complemented
15C61×#2297C211yesmaximal, characteristic
16C2×#14519770C2214yes—
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4⋊Dic61C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C61⋊#401C2214yes—
23#16839704C2214yes—
24#16839703C2214yes—
25#16839702C2214yes—
26#16839690C2214yes—
27#16839690C2214yes—
28#16839687C2214yes—
29#16839685C2214yes—
30#16839669C2211yescharacteristic
31#16839669C2212yes—
32#16839669C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839665C2212yes—
35#16839665C2211yescharacteristic
36#16839660C2212yes—
37#16839660C2212yes—
38#16839656C2214yes—
39#16839655C2214yes—
40#16839650C2214yes—
41#16839649C2214yes—
42#16839647C2214yes—
43#16839644C2214yes—
44#16839642C2214yes—
45#16839642C2214yes—
46#16839635C2212yes—
47#16839635C2212yes—
48#16839631C2212yes—
49#16839629C2214yes—
50#16839627C2212yes—
51#16839620C2214yes—
52C4⋊C4×Dic61C2212yes—
53C4⋊C4×Dic61C2212yes—
54C4×#11758196C2214yes—
55C4×#11758196C2214yes—
56C4×C4⋊Dic61C2212yes—
57C4×C4⋊Dic61C2212yes—
58C4×C4⋊Dic61C2212yes—
59C4×C4⋊Dic61C2212yes—
60C22⋊C4×Dic61C2214yes—
61C4×C22⋊Dic61C2212yes—
62C4×C22⋊Dic61C2214yes—
63C4×C22⋊Dic61C2212yes—
64#16839608C2212yes—
65C2×#14519684C2212yes—
66C42×Dic61C2212yes—
67C22×#41×C61C2211yescharacteristic
68C2×#117×C61C2211yescharacteristic
69C22×C42×C61C2211yescharacteristic
70C61×#391C2212yes—
71C61×#390C2212yes—
72C61×#386C2212yes—
73C61×#386C2212yes—
74C4×C22⋊C4×C61C2214yes—
75C23×Dic61C2312yes—
76C23×Dic61C2312yes—
77#14519770—216no—
78#14519736—24no—
79#14519736—24no—
80C2×C4⋊Dic61—28no—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2312yes—
87C2×C4⋊Dic61C2312yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×C22⋊Dic61C2314yes—
93C2×C22⋊Dic61C2312yes—
94C2×C22⋊Dic61C2312yes—
95C2×C22⋊Dic61C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C22⋊Dic61C2314yes—
98C2×C4×Dic61—24no—
99C2×C4×Dic61—24no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2312yes—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2314yes—
110#14519684C2314yes—
111#14519684C2314yes—
112#14519684C2314yes—
113#14519684C2314yes—
114#14519684C2312yes—
115#14519684C2312yes—
116#14519684C2314yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2312yes—
122#14519684C2312yes—
123#14519684C2314yes—
124C2×#41×C61—24no—
125C2×#41×C61—24no—
126C2×#41×C61—24no—
127C22×Q8×C61C2311yescharacteristic
128C22×D4×C61C2312yes—
129C22×D4×C61C2311yescharacteristic
130C23×C4×C61C2311yescharacteristic
131C23×C4×C61C2312yes—
132C23×C4×C61C2311yescharacteristic
133C23×C4×C61C2311yescharacteristic
134#117×C61—24no—
135C2×C4⋊C4×C61C2312yes—
136C2×C22⋊C4×C61C2314yes—
137C2×C22⋊C4×C61C2312yes—
138C2×C22⋊C4×C61C2314yes—
139C2×C42×C61—24no—
140C2×C42×C61C2312yes—
141C2×C42×C61C2312yes—
142C2×C42×C61C2312yes—
143#95×C61C2314yes—
144#95×C61C2314yes—
145C24×C61C2412yes—
146C24×C61C2411yescharacteristic
147#41×C61—416no—
148C2×Q8×C61—22no—
149C2×Q8×C61—22no—
150C2×Q8×C61—22no—
151C2×D4×C61—24no—
152C2×D4×C61—24no—
153C2×D4×C61—24no—
154C2×D4×C61—22no—
155C2×D4×C61—22no—
156C2×D4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—24no—
159C22×C4×C61—22no—
160C22×C4×C61—24no—
161C22×C4×C61—24no—
162C22×C4×C61—24no—
163C22×C4×C61—42no—
164C22×C4×C61#4114yes—
165C22×C4×C61#4114yes—
166C22×C4×C61#4114yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2414yes—
170C22×C4×C61C2412yes—
171C22×C4×C61C2411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C22×Dic61—24no—
174C22×Dic61—24no—
175C22×Dic61—24no—
176C22×Dic61—24no—
177C22×Dic61—42no—
178C22×Dic61—42no—
179C22×Dic61C2412yes—
180C22×Dic61C2412yes—
181C22×Dic61C2412yes—
182C22×Dic61C2412yes—
183C22×Dic61C2414yes—
184C22×Dic61C2412yes—
185C22×Dic61C2412yes—
186C4⋊C4×C61—24no—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—28no—
189C22⋊C4×C61—44no—
190C42×C61—24no—
191C42×C61—24no—
192C42×C61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195C22⋊Dic61—28no—
196C22⋊Dic61—28no—
197C22⋊Dic61—44no—
198C22⋊Dic61—44no—
199C4⋊Dic61—24no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—24no—
205C4⋊Dic61#4118yescomplemented
206#11758196—28no—
207#11758196—28no—
208#11758196—44no—
209#11758196—44no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—24no—
213C4×Dic61—24no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—24no—
217C4×Dic61—24no—
218C4×Dic61—28no—
219C4×Dic61—44no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—24no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—24no—
227C23×C61C2×#4112yes—
228C23×C61C2×#4112yes—
229C23×C61C2×#4112yes—
230C23×C61C2511yescharacteristic
231Q8×C61—44no—
232D4×C61—48no—
233D4×C61—44no—
234C2×C4×C61—28no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—44no—
247C2×C4×C61—48no—
248C2×C4×C61#126116yescomplemented
249C2×C4×C61C2×#4114yes—
250C2×C4×C61C2×#4112yes—
251C2×C4×C61C2×#4112yes—
252C2×C4×C61C2×#4112yes—
253C2×C4×C61C2×#4112yes—
254C2×C4×C61C2×#4112yes—
255C2×C4×C61C2×#4112yes—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—48no—
274C2×Dic61—48no—
275C2×Dic61C2×#4114yes—
276C2×Dic61C2×#4114yes—
277#34816—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61C2×#12614yes—
288C22×C61C22×#4112yes—
289C22×C61C22×#4111yescharacteristic
290C22×C61C22×#4111yescharacteristic
291C22×C61C22×#4111yescharacteristic
292C22×C61#58512yes—
293C4×C61—24no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61—44no—
297C4×C61C2×#12614yes—
298C4×C61C2×#12614yes—
299Dic61—24no—
300Dic61—24no—
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304Dic61—42no—
305Dic61—44no—
306#2514—612no—
307#2513—612no—
308#2512—612no—
309#2511—612no—
310#2510—614no—
311#2481—614no—
312#2401—614no—
313C4×#126—614no—
314#2297—611no—
315#2297—611no—
316#2297D6111yescharacteristic
317#2296—611no—
318#2285—611no—
319C4×#117—611no—
320C2×C4×C4⋊C4—611no—
321C2×C61—24no—
322C2×C61—82no—
323C2×C61C22×#12611yescharacteristic
324C2×C61#344411yescharacteristic
325C2×C61#345511yescharacteristic
326C2×C61#347512yes—
327C2×C61#348012yes—
328C22×#41C2×D6111yescharacteristic
329C2×#126—614no—
330C2×#117—611no—
331C2×#117—611no—
332C2×#117C2×D6111yescharacteristic
333C22×C4⋊C4—612no—
334C22×C42—611no—
335C22×C42—611no—
336C22×C42C2×D6111yescharacteristic
337#401—614no—
338#400—614no—
339#400—614no—
340#400—614no—
341#397—614no—
342#397—614no—
343#397—614no—
344#396—614no—
345#391—611no—
346#391—612no—
347#391—611no—
348#391C2×D6112yes—
349#390—611no—
350#390—612no—
351#390—611no—
352#390C2×D6112yes—
353#389—612no—
354#389—612no—
355#388—614no—
356#388—614no—
357#388—614no—
358#387—614no—
359#387—614no—
360#387—614no—
361#386—614no—
362#386—614no—
363#386C2×D6112yes—
364#386C2×D6112yes—
365#385—612no—
366#385—612no—
367#384—612no—
368#383—614no—
369#382—612no—
370#380—614no—
371C4×C4⋊C4—612no—
372C4×C4⋊C4—612no—
373C4×C4⋊C4—612no—
374C4×C4⋊C4—614no—
375C4×C4⋊C4—614no—
376C4×C4⋊C4—612no—
377C4×C4⋊C4—612no—
378C4×C4⋊C4—612no—
379C4×C22⋊C4—612no—
380C4×C22⋊C4—614no—
381C4×C22⋊C4—612no—
382C4×C22⋊C4—614no—
383C4×C22⋊C4C2×D6114yes—
384#376—612no—
385C2×#95—612no—
386C43—612no—
387C61#3481611yescharacteristic, complemented, Sylow
388C2×#41—24no—
389C2×#41—24no—
390C2×#41—24no—
391C22×Q8C22×D6111yescharacteristic
392C22×D4C22×D6112yes—
393C22×D4C22×D6111yescharacteristic
394C23×C4—612no—
395C23×C4—612no—
396C23×C4C22×D6111yescharacteristic
397C23×C4C22×D6112yes—
398C23×C4C22×D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400#126—12216no—
401#117—24no—
402#117—1224no—
403#117—1224no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—614no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—1228no—
416C2×C4⋊C4C22×D6112yes—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—612no—
419C2×C22⋊C4—614no—
420C2×C22⋊C4—614no—
421C2×C22⋊C4—614no—
422C2×C22⋊C4—614no—
423C2×C22⋊C4C22×D6112yes—
424C2×C22⋊C4C22×D6114yes—
425C2×C22⋊C4C22×D6114yes—
426C2×C42—24no—
427C2×C42—614no—
428C2×C42—612no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—612no—
432C2×C42—612no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—612no—
436C2×C42—614no—
437C2×C42—1224no—
438C2×C42—1224no—
439C2×C42C22×D6112yes—
440C2×C42C22×D6112yes—
441C2×C42C22×D6112yes—
442#95—612no—
443#95—612no—
444#95—612no—
445#95—612no—
446#95—614no—
447#95—614no—
448#95—614no—
449#95—614no—
450#95—614no—
451#95—614no—
452#95—614no—
453#95—614no—
454#95—614no—
455#95—614no—
456#95C22×D6114yes—
457#95C22×D6114yes—
458C24C23×D6112yes—
459C24C23×D6111yescharacteristic
460#41—416no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—614no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1224no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4#1175822214yes—
491C22×C4#1175822414yes—
492C22×C4#1175822414yes—
493C22×C4C23×D6112yes—
494C22×C4C23×D6112yes—
495C22×C4C23×D6114yes—
496C22×C4C23×D6112yes—
497C22×C4C23×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499C4⋊C4—24no—
500C4⋊C4—618no—
501C4⋊C4—1224no—
502C4⋊C4—1224no—
503C4⋊C4—1228no—
504C4⋊C4—1224no—
505C4⋊C4—1228no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1224no—
509C4⋊C4—2444no—
510C4⋊C4—2444no—
511C22⋊C4—24no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—1228no—
515C22⋊C4—1228no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C22⋊C4—2444no—
519C22⋊C4—2444no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—1228no—
524C42—1224no—
525C42—1224no—
526C42—1222no—
527C42—1222no—
528C42—1224no—
529C42—1224no—
530C42—1222no—
531C42—1222no—
532C42—2444no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23C2×#1175822412yes—
541C23C2×#1175822412yes—
542C23C2×#1175822212yes—
543C23C24×D6111yescharacteristic
544Q8—44no—
545D4—48no—
546D4—44no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—614no—
562C2×C4—614no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1224no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1224no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1224no—
580C2×C4—2448no—
581C2×C4—2448no—
582C2×C4#14519770116yescomplemented
583C2×C4#41×D6114yes—
584C2×C4C2×#1175822412yes—
585C2×C4C2×#1175822412yes—
586C2×C4C2×#1175822412yes—
587C2×C4C2×#1175822412yes—
588C2×C4C2×#1175822212yes—
589C2×C4C2×#1175822212yes—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22C2×#1451977014yes—
600C22C2×#41×D6112yes—
601C22C22×#1175822411yescharacteristic
602C22C22×#1175822411yescharacteristic
603C22C22×#1175822211yescharacteristic
604C22#1684059812yes—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—1224no—
610C4—1224no—
611C4—2442no—
612C4—2442no—
613C4—2442no—
614C4—2442no—
615C4—2444no—
616C4C2×#1451977014yes—
617C4C2×#1451977014yes—
618C2—24no—
619C2—82no—
620C2C22×#1451977011yescharacteristic
621C2#2145911111yescharacteristic
622C2#2145920711yescharacteristic
623C2#2145938312yes—
624C2#2145941612yes—
subgroupG/H
centerC22×C4C23×D61
commutator, socle, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC61×#2297C2
prime radicalC22×#41×C61C22
hypercenter#2297D61
hypocenterC61#34816
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. #21450425C2semidirect
2. #21450414C2semidirect
3. #21450403C2semidirect
4. #21450392C2semidirect
5. #21450377C2semidirect
6. #21450079C2semidirect
7. #21449119C2semidirect
8. C4×#14519770C2semidirect
11. #21448302C2semidirect
12. #21448222C2semidirect
13. #117×Dic61C2semidirect
14. C2×C4×C4⋊Dic61C2semidirect
205. C4⋊Dic61#41semidirect
248. C2×C4×C61#126semidirect
387. C61#34816semidirect
582. C2×C4#14519770semidirect

Maximal subgroups

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

subgroupindex
#214504252
#214504142
#214504032
#214503922
#214503772
#214500792
#214491192
C4×#145197702
#214483102
#214483022
#214482222
#117×Dic612
C2×C4×C4⋊Dic612
C61×#22972
#3481661