Finite Group Databasebeta

← #29886968

Subgroups — #29886968

Order 15616.

2406 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840350C211yesmaximal, characteristic
2#21454111C212yesmaximal, complemented
3#21452413C214yesmaximal, complemented
4#21452389C214yesmaximal, complemented
5#21452167C214yesmaximal, complemented
6#21452158C214yesmaximal, complemented
7#21451592C212yesmaximal, complemented
8#21451299C211yesmaximal, characteristic, complemented
9#21451199C211yesmaximal, characteristic, complemented
10#21451166C212yesmaximal, complemented
11#21450307C211yesmaximal, characteristic, complemented
12#21450293C211yesmaximal, characteristic, complemented
13#21449178C212yesmaximal, complemented
14#21448417C211yesmaximal, characteristic, complemented
15C22×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
16#16840350—28no—
17C2×#14519770C2212yes—
18C2×#14519763C2212yes—
19C2×#14519760C2214yes—
20C2×#14519759C2212yes—
21C2×#14519756C2211yescharacteristic
22C2×#14519753C2214yescomplemented
23C2×#14519751C2211yescharacteristic
24C2×#14519750C2212yes—
25C2×#14519746C2212yes—
26C2×C4×C22⋊D61C2212yes—
27#16839701C2212yescomplemented
28#16839695C2214yescomplemented
29#16839693C2214yescomplemented
30#16839692C2212yescomplemented
31#16839690C2214yes—
32#16839690C2214yes—
33#16839688C2214yescomplemented
34#16839684C2214yescomplemented
35#16839682C2212yescomplemented
36#16839681C2214yescomplemented
37#16839679C2212yescomplemented
38#16839677C2214yes—
39#16839677C2214yes—
40#16839673C2214yescomplemented
41#16839669C2214yescomplemented
42#16839667C2212yescomplemented
43#16839663C2212yescomplemented
44#16839657C2214yescomplemented
45#16839655C2214yes—
46#16839655C2214yes—
47#16839652C2214yescomplemented
48#16839643C2214yescomplemented
49#16839641C2212yes—
50#16839641C2212yes—
51#16839641C2214yes—
52#16839641C2212yes—
53#16839641C2212yes—
54#16839635C2212yes—
55#16839635C2214yes—
56#16839635C2212yes—
57#16839635C2212yes—
58#16839635C2212yes—
59#16839633C2214yescomplemented
60#16839631C2212yescomplemented
61#16839627C2212yescomplemented
62#16839625C2214yes—
63#16839622C2212yes—
64#16839618C2212yes—
65C4×C22⋊Dic61C2211yescharacteristic
66C4×C22⋊Dic61C2211yescharacteristic
67C23×Q8×C61C2211yescharacteristic
68C24×C4×C61C2211yescharacteristic
69C2×C22⋊Q8×C61—28no—
70C2×C22⋊Q8×C61C2214yes—
71C2×C22⋊Q8×C61C2214yes—
72C22×C4⋊C4×C61C2211yescharacteristic
73C22×C4⋊C4×C61C2211yescharacteristic
74C22×C4⋊C4×C61C2211yescharacteristic
75C22×C22⋊C4×C61C2211yescharacteristic
76C22×C22⋊C4×C61C2211yescharacteristic
77C22×Dic122C2311yescharacteristic
78C22×D244C2311yescharacteristic
79C22×C22⋊D61C2312yes—
80C22×C22⋊D61C2312yes—
81C22×C22⋊D61C2312yes—
82C22×C4×D61C2312yes—
83#14519770—28no—
84#14519763—28no—
85#14519760—48no—
86#14519759—28no—
87#14519756—24no—
88#14519753—48no—
89#14519751—24no—
90#14519750—28no—
91#14519746—28no—
92C4×C22⋊D61—28no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2314yes—
96C2×#11758196C2312yes—
97C2×#11758196C2314yes—
98C2×#11758196C2312yes—
99C2×#11758196C2314yes—
100C2×#11758196C2314yes—
101C2×#11758196C2314yes—
102C2×C22⋊Dic61C2312yes—
103C2×C22⋊Dic61C2312yes—
104C2×C22⋊Dic61C2312yes—
105C2×C22⋊Dic61C2312yes—
106C2×C22⋊Dic61C2312yes—
107C2×C22⋊Dic61C2314yes—
108C2×C22⋊Dic61C2314yes—
109C2×#11758198C2312yes—
110C2×#11758198C2312yes—
111C2×#11758198C2314yes—
112C2×#11758198C2314yes—
113C2×#11758198C2314yes—
114C2×#11758198C2314yes—
115C2×C4×Dic61C2312yes—
116C2×C4×Dic61C2312yes—
117C2×C4×Dic61C2312yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2312yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#14519684C2314yes—
127#14519684C2314yes—
128#14519684C2312yes—
129C25×C61C2311yescharacteristic
130C22×Q8×C61—24no—
131C22×Q8×C61—44no—
132C22×Q8×C61C2314yes—
133C23×C4×C61—24no—
134C23×C4×C61—44no—
135C23×C4×C61C2311yescharacteristic
136C23×C4×C61C2314yes—
137C23×C4×C61C2311yescharacteristic
138C23×C4×C61C2311yescharacteristic
139C23×C4×C61C2311yescharacteristic
140C23×C4×C61C2311yescharacteristic
141C23×C4×C61C2311yescharacteristic
142C22⋊Q8×C61—216no—
143C22⋊Q8×C61—216no—
144C2×C4⋊C4×C61—24no—
145C2×C4⋊C4×C61—24no—
146C2×C4⋊C4×C61—24no—
147C2×C4⋊C4×C61C2312yes—
148C2×C4⋊C4×C61C2314yes—
149C2×C4⋊C4×C61C2312yes—
150C2×C4⋊C4×C61C2312yes—
151C2×C4⋊C4×C61C2312yes—
152C2×C22⋊C4×C61—24no—
153C2×C22⋊C4×C61D418yes—
154C2×C22⋊C4×C61C2314yes—
155C2×C22⋊C4×C61C2312yes—
156C2×C22⋊C4×C61C2312yes—
157C24×C61—24no—
158C24×C61—42no—
159C24×C61—42no—
160C24×C61C2×D414yes—
161C24×C61C2412yes—
162C24×C61C2411yescharacteristic
163C23×D61C2412yes—
164C2×Q8×C61—28no—
165C2×Q8×C61—28no—
166C2×Q8×C61—44no—
167C2×Q8×C61—416no—
168C22×C4×C61—22no—
169C22×C4×C61—24no—
170C22×C4×C61—22no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—28no—
174C22×C4×C61—28no—
175C22×C4×C61—44no—
176C22×C4×C61—42no—
177C22×C4×C61—42no—
178C22×C4×C61—42no—
179C22×C4×C61—42no—
180C22×C4×C61—42no—
181C22×C4×C61—416no—
182C22×C4×C61C2×D414yes—
183C22×C4×C61C2×D414yes—
184C22×C4×C61C2412yes—
185C22×C4×C61C2412yes—
186C22×C4×C61C2412yes—
187C22×C4×C61C2412yes—
188C22×C4×C61C2411yescharacteristic
189C22×C4×C61C2411yescharacteristic
190C22×C4×C61C2412yes—
191C2×C22⋊D61—24no—
192C2×C22⋊D61—28no—
193C2×C22⋊D61—24no—
194C2×C22⋊D61—24no—
195C2×C22⋊D61—44no—
196C2×C22⋊D61—44no—
197C22×Dic61C2412yes—
198C22×Dic61C2412yes—
199C22×Dic61C2412yes—
200C22×Dic61C2414yes—
201C22×Dic61C2414yes—
202C2×D244—22no—
203C2×D244—42no—
204C2×C4×D61—24no—
205C2×C4×D61—44no—
206C2×Dic122—22no—
207C2×Dic122—42no—
208C4⋊C4×C61—24no—
209C4⋊C4×C61—24no—
210C4⋊C4×C61—28no—
211C4⋊C4×C61—24no—
212C4⋊C4×C61—24no—
213C22⋊C4×C61—28no—
214C22⋊C4×C61C2×D418yes—
215C22⋊C4×C61C2×D418yes—
216C22⋊Dic61—24no—
217C22⋊Dic61—24no—
218C22⋊Dic61—24no—
219C22⋊Dic61—24no—
220C22⋊Dic61—24no—
221C22⋊Dic61—44no—
222C22⋊Dic61—44no—
223#11758198—24no—
224#11758198—24no—
225#11758198—28no—
226#11758198—44no—
227#11758198—44no—
228#11758198—44no—
229C4⋊Dic61—22no—
230C4⋊Dic61—22no—
231C4⋊Dic61—44no—
232#11758196—24no—
233#11758196—24no—
234#11758196—28no—
235#11758196—44no—
236#11758196—44no—
237#11758196—44no—
238C4×Dic61—24no—
239C4×Dic61—42no—
240C4×Dic61—42no—
241C23×C61—22no—
242C23×C61—22no—
243C23×C61—28no—
244C23×C61—24no—
245C23×C61—24no—
246C23×C61—24no—
247C23×C61—44no—
248C23×C61—44no—
249C23×C61—416no—
250C23×C61C22×D414yes—
251C23×C61C22×D412yes—
252C23×C612−1+414yes—
253C23×C61C2511yescharacteristic
254C22×D61—22no—
255C22×D61—24no—
256C22×D61—24no—
257C22×D61—24no—
258Q8×C61—416no—
259C2×C4×C61—28no—
260C2×C4×C61—22no—
261C2×C4×C61—24no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—24no—
269C2×C4×C61—24no—
270C2×C4×C61—24no—
271C2×C4×C61—44no—
272C2×C4×C61—44no—
273C2×C4×C61—416no—
274C2×C4×C61—44no—
275C2×C4×C61—44no—
276C2×C4×C61—44no—
277C2×C4×C61C22×D414yes—
278C2×C4×C61C22×D412yes—
279C2×C4×C61C22×D412yes—
280C2×C4×C61H(5,𝔽2)14yes—
281C22⋊D61—48no—
282C22⋊D61—84no—
283C22⋊D61—84no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—22no—
287C2×Dic61—24no—
288C2×Dic61—24no—
289C2×Dic61—24no—
290C2×Dic61—24no—
291C2×Dic61—24no—
292C2×Dic61—24no—
293C2×Dic61—24no—
294C2×Dic61—24no—
295C2×Dic61—24no—
296D244—82no—
297C4×D61—84no—
298Dic122—82no—
299#36716—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—28no—
303C22×C61—24no—
304C22×C61—24no—
305C22×C61—24no—
306C22×C61—416no—
307C22×C61—48no—
308C22×C61#53614yes—
309C22×C61C23×D411yescharacteristic
310C22×C61C2×H(5,𝔽2)12yes—
311C22×C61C2×2−1+412yes—
312C22×C61#58512yes—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—48no—
317C4×C61—24no—
318C4×C61—24no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—42no—
324Dic61—42no—
325Dic61—42no—
326Dic61—42no—
327Dic61—44no—
328Dic61—44no—
329C2×#538—611no—
330C22×C22⋊Q8D6111yescharacteristic, complemented
331#2849—612no—
332#2700—614no—
333#2698—614no—
334#2681—614no—
335#2680—614no—
336#2622—612no—
337#2591—611no—
338#2583—611no—
339#2579—612no—
340#2504—611no—
341#2502—611no—
342#2409—612no—
343#2312—611no—
344C2×C61—24no—
345C2×C61—44no—
346C2×C61C2×#53611yescharacteristic
347C2×C61#348411yescharacteristic
348C2×C61#348511yescharacteristic
349C2×C61#354114yes—
350D61—82no—
351C23×Q8C2×D6111yescharacteristic
352C24×C4C2×D6111yescharacteristic
353#538—1228no—
354C2×#126—612no—
355C2×#124—612no—
356C2×#123—614no—
357C2×#123—612no—
358C2×C22⋊Q8—28no—
359C2×C22⋊Q8—611no—
360C2×C22⋊Q8—614no—
361C2×C22⋊Q8C2×D6114yes—
362C2×C22⋊Q8C2×D6114yes—
363C2×#121—611no—
364C2×#121—612no—
365C2×#120—612no—
366C2×C4×D4—612no—
367C22×C4⋊C4C2×D6111yescharacteristic
368C22×C4⋊C4C2×D6111yescharacteristic
369C22×C4⋊C4C2×D6111yescharacteristic
370C22×C22⋊C4C2×D6111yescharacteristic
371C22×C22⋊C4C2×D6111yescharacteristic
372#400—612no—
373#398—612no—
374#398—614no—
375#398—614no—
376#397—614no—
377#397—614no—
378#397—614no—
379#396—614no—
380#395—614no—
381#395—612no—
382#394—612no—
383#394—614no—
384#394—614no—
385#393—614no—
386#391—614no—
387#391—612no—
388#390—612no—
389#389—614no—
390#388—614no—
391#388—614no—
392#388—614no—
393#387—614no—
394#386—612no—
395#386—612no—
396#386—612no—
397#386—614no—
398#386—612no—
399#385—612no—
400#385—612no—
401#385—612no—
402#385—614no—
403#385—612no—
404#385—614no—
405#384—612no—
406#382—612no—
407#381—614no—
408#380—612no—
409#379—612no—
410C4×C22⋊C4—611no—
411C4×C22⋊C4—611no—
412C61#3671611yescharacteristic, complemented, Sylow
413C25C22×D6111yescharacteristic
414C22×Q8—24no—
415C22×Q8—44no—
416C22×Q8—611no—
417C22×Q8C22×D6114yes—
418C22×D4—612no—
419C22×D4—612no—
420C22×D4—612no—
421C22×D4—611no—
422C23×C4—24no—
423C23×C4—44no—
424C23×C4—612no—
425C23×C4C22×D6111yescharacteristic
426C23×C4C22×D6114yes—
427C23×C4C22×D6111yescharacteristic
428C23×C4C22×D6111yescharacteristic
429C23×C4C22×D6111yescharacteristic
430C23×C4C22×D6111yescharacteristic
431C23×C4C22×D6111yescharacteristic
432#126—1228no—
433#124—1228no—
434#123—1228no—
435#123—2448no—
436C22⋊Q8—216no—
437C22⋊Q8—216no—
438C22⋊Q8—1224no—
439C22⋊Q8—2448no—
440#121—1224no—
441#121—1228no—
442#120—1228no—
443C4×D4—1228no—
444C2×C4⋊C4—24no—
445C2×C4⋊C4—24no—
446C2×C4⋊C4—24no—
447C2×C4⋊C4—612no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—614no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4—614no—
454C2×C4⋊C4—614no—
455C2×C4⋊C4—614no—
456C2×C4⋊C4C22×D6112yes—
457C2×C4⋊C4C22×D6112yes—
458C2×C4⋊C4C22×D6114yes—
459C2×C4⋊C4C22×D6112yes—
460C2×C4⋊C4C22×D6112yes—
461C2×C22⋊C4—24no—
462C2×C22⋊C4—612no—
463C2×C22⋊C4—612no—
464C2×C22⋊C4—612no—
465C2×C22⋊C4—612no—
466C2×C22⋊C4—612no—
467C2×C22⋊C4—612no—
468C2×C22⋊C4—612no—
469C2×C22⋊C4—614no—
470C2×C22⋊C4—614no—
471C2×C22⋊C4—614no—
472C2×C22⋊C4—614no—
473C2×C22⋊C4—614no—
474C2×C22⋊C4—614no—
475C2×C22⋊C4C22⋊D6118yes—
476C2×C22⋊C4C22×D6114yes—
477C2×C22⋊C4C22×D6112yes—
478C2×C22⋊C4C22×D6112yes—
479C2×C42—612no—
480C2×C42—612no—
481C2×C42—612no—
482#95—612no—
483#95—612no—
484#95—614no—
485#95—614no—
486#95—614no—
487#95—614no—
488#95—614no—
489#95—614no—
490#95—614no—
491#95—614no—
492#95—614no—
493C24—24no—
494C24—42no—
495C24—42no—
496C24—612no—
497C24C2×C22⋊D6114yes—
498C24C23×D6112yes—
499C24C23×D6111yescharacteristic
500C2×Q8—28no—
501C2×Q8—28no—
502C2×Q8—44no—
503C2×Q8—416no—
504C2×Q8—1222no—
505C2×Q8—2442no—
506C2×D4—1224no—
507C2×D4—1224no—
508C2×D4—1224no—
509C2×D4—1228no—
510C2×D4—1222no—
511C2×D4—2442no—
512C2×D4—2444no—
513C2×D4—2444no—
514C22×C4—22no—
515C22×C4—24no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—28no—
520C22×C4—28no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4—42no—
527C22×C4—416no—
528C22×C4—612no—
529C22×C4—614no—
530C22×C4—614no—
531C22×C4—612no—
532C22×C4—612no—
533C22×C4—1224no—
534C22×C4—2444no—
535C22×C4C2×C22⋊D6114yes—
536C22×C4C2×C22⋊D6114yes—
537C22×C4C23×D6112yes—
538C22×C4C23×D6112yes—
539C22×C4C23×D6112yes—
540C22×C4C23×D6111yescharacteristic
541C22×C4C23×D6111yescharacteristic
542C22×C4C23×D6112yes—
543C22×C4C23×D6112yes—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—28no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—1228no—
554C4⋊C4—2444no—
555C4⋊C4—2444no—
556C4⋊C4—2444no—
557C4⋊C4—2444no—
558C22⋊C4—28no—
559C22⋊C4—1224no—
560C22⋊C4—1224no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C22⋊C4—1224no—
565C22⋊C4—1224no—
566C22⋊C4—1228no—
567C22⋊C4—2444no—
568C22⋊C4—2444no—
569C22⋊C4—2444no—
570C22⋊C4—2444no—
571C22⋊C4—2444no—
572C22⋊C4C2×C22⋊D6118yes—
573C22⋊C4C2×C22⋊D6118yes—
574C42—1224no—
575C42—2442no—
576C42—2442no—
577C23—22no—
578C23—22no—
579C23—28no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—44no—
584C23—44no—
585C23—416no—
586C23—1222no—
587C23—1224no—
588C23—1224no—
589C23—1224no—
590C23C22×C22⋊D6114yes—
591C23C22×C22⋊D6112yes—
592C23#1451982514yes—
593C23C24×D6111yescharacteristic
594Q8—416no—
595Q8—4882no—
596D4—4882no—
597D4—4884no—
598D4—4884no—
599D4—2448no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1224no—
622C2×C4—1224no—
623C2×C4—1224no—
624C2×C4—1224no—
625C2×C4—1224no—
626C2×C4—1224no—
627C2×C4—1224no—
628C2×C4—1224no—
629C2×C4—1224no—
630C2×C4—4884no—
631C2×C4C22×C22⋊D6114yes—
632C2×C4C22×C22⋊D6112yes—
633C2×C4C22×C22⋊D6112yes—
634C2×C4#1451982214yes—
635C22—22no—
636C22—28no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—416no—
642C22—48no—
643C22—2442no—
644C22—2448no—
645C22—2442no—
646C22—2442no—
647C22#1684033614yes—
648C22C23×C22⋊D6111yescharacteristic
649C22C2×#1451982212yes—
650C22C2×#1451982512yes—
651C22#1684059812yes—
652C4—22no—
653C4—22no—
654C4—24no—
655C4—24no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—2442no—
660C4—2444no—
661C4—2444no—
662C4—2442no—
663C4—2442no—
664C2—24no—
665C2—44no—
666C2—4882no—
667C2C2×#1684033611yescharacteristic
668C2#2145945411yescharacteristic
669C2#2145946311yescharacteristic
670C2#2146011614yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC22×C22⋊Q8×C61C2
prime radicalC2×#16840350C2
hypercenterC22×C22⋊Q8D61
hypocenterC61#36716
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #21454111C2semidirect
3. #21452413C2semidirect
4. #21452389C2semidirect
5. #21452167C2semidirect
6. #21452158C2semidirect
7. #21451592C2semidirect
8. #21451299C2semidirect
9. #21451199C2semidirect
10. #21451166C2semidirect
11. #21450307C2semidirect
12. #21450293C2semidirect
13. #21449178C2semidirect
14. #21448417C2semidirect
15. C22×C22⋊Q8×C61C2semidirect
22. C2×#14519753C22semidirect
27. #16839701C22semidirect
28. #16839695C22semidirect
29. #16839693C22semidirect
30. #16839692C22semidirect
33. #16839688C22semidirect
34. #16839684C22semidirect
35. #16839682C22semidirect
36. #16839681C22semidirect
37. #16839679C22semidirect
40. #16839673C22semidirect
41. #16839669C22semidirect
42. #16839667C22semidirect
43. #16839663C22semidirect
44. #16839657C22semidirect
47. #16839652C22semidirect
48. #16839643C22semidirect
59. #16839633C22semidirect
60. #16839631C22semidirect
61. #16839627C22semidirect
330. C22×C22⋊Q8D61semidirect
412. C61#36716semidirect

Maximal subgroups

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

subgroupindex
C2×#168403502
#214541112
#214524132
#214523892
#214521672
#214521582
#214515922
#214512992
#214511992
#214511662
#214503072
#214502932
#214491782
#214484172
C22×C22⋊Q8×C612
#3671661