Finite Group Databasebeta

← #30610358

Subgroups — #30610358

Order 15616.

916 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462719C212yesmaximal, complemented
2#21461103C212yesmaximal, complemented
3#21461100C212yesmaximal, complemented
4#21461088C212yesmaximal
5#21461082C211yesmaximal, characteristic, complemented
6#21461073C211yesmaximal, characteristic, complemented
7#21461070C211yesmaximal, characteristic, complemented
8#21461062C211yesmaximal, characteristic, complemented
9#21461053C211yesmaximal, characteristic, complemented
10#21461050C211yesmaximal, characteristic, complemented
11C2×#16840483C211yesmaximal, characteristic
12#16840814—22no—
13#16840814—22no—
14#16840812—22no—
15#16840811—22no—
16#16840783C2212yes—
17#16840782C2212yes—
18#16840782C2212yes—
19#16840779C2212yes—
20#16840778C2212yes—
21#16840778C2212yes—
22#16840607C2211yescharacteristic
23#16840607C2211yescharacteristic
24#16840607C2211yescharacteristic
25#16840607C2211yescharacteristic
26#16840607C2212yes—
27#16840607C2212yes—
28#16840606C2212yes—
29#16840606C2212yes—
30#16840605C2212yes—
31#16840605C2212yes—
32#16840483—22no—
33#16840483—22no—
34#16840483—22no—
35#16840483—22no—
36C2×#14519776C411yescharacteristic
37C2×#14519772C411yescharacteristic
38C2×#14519772C2211yescharacteristic
39C2×#14519770C411yescharacteristic
40C2×#14519770C2211yescharacteristic
41C2×#14519763C411yescharacteristic
42C2×#14519763C2211yescharacteristic
43C2×#14519753C411yescharacteristic
44C2×Q8×Dic61C2211yescharacteristic
45C2×#14519742C411yescharacteristic
46C2×#14519742C2211yescharacteristic
47C2×C4×C22⋊D61C411yescharacteristic
48C2×#14519736C2211yescharacteristic
49C2×#14519735C2211yescharacteristic
50C2×C22⋊Q8×C61C411yescharacteristic
51#14519844—22no—
52#14519844—22no—
53#14519844—22no—
54C4⋊#92420—22no—
55C4⋊#92420—22no—
56C4⋊#92420—22no—
57C22⋊#92420C2312yes—
58C22⋊#92420C2312yes—
59C22⋊#92420C2312yes—
60C22⋊#92420C2312yes—
61#14519832—22no—
62#14519832—22no—
63#14519832—22no—
64C4×#92420—22no—
65C4×#92420—22no—
66C4×#92420—22no—
67C22×Dic122C2×C411yescharacteristic
68C22×Dic122C2×C411yescharacteristic
69C22×C22⋊D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71#14519776—22no—
72#14519776—22no—
73#14519772—22no—
74#14519772—22no—
75#14519772—22no—
76#14519772—22no—
77#14519770—22no—
78#14519770—22no—
79#14519770—22no—
80#14519770—22no—
81#14519763—22no—
82#14519763—22no—
83#14519763—22no—
84#14519763—22no—
85#14519753—22no—
86#14519753—22no—
87Q8×Dic61—22no—
88Q8×Dic61—22no—
89#14519742—21no—
90#14519742—21no—
91#14519742—21no—
92#14519742—21no—
93#14519742D412yes—
94#14519742D412yes—
95#14519742D412yes—
96#14519742D412yes—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—21no—
100C4×C22⋊D61—21no—
101#14519736—22no—
102#14519736—22no—
103#14519735—22no—
104#14519735—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×C4⋊Dic61C2311yescharacteristic
108C2×C4⋊Dic61C2311yescharacteristic
109C2×#11758196C2×C411yescharacteristic
110C2×#11758196C2×C411yescharacteristic
111C2×#11758196C2×C411yescharacteristic
112C2×#11758196C2×C411yescharacteristic
113C2×#11758196C2×C411yescharacteristic
114C2×C22⋊Dic61C2×C411yescharacteristic
115C2×C22⋊Dic61C2×C411yescharacteristic
116C2×C22⋊Dic61C2311yescharacteristic
117C2×#11758198C2×C411yescharacteristic
118C2×#11758198C2×C411yescharacteristic
119C2×#11758198C2×C411yescharacteristic
120C2×#11758198C2×C411yescharacteristic
121C2×#11758198C2311yescharacteristic
122C2×C4×Dic61C2×C411yescharacteristic
123C2×C4×Dic61C2×C411yescharacteristic
124C2×C4×Dic61C2×C411yescharacteristic
125C2×C4×Dic61C2×C411yescharacteristic
126C2×C4×Dic61C2311yescharacteristic
127C2×C4×Dic61C2311yescharacteristic
128C2×C4×Dic61C2311yescharacteristic
129C22×Q8×C61C2×C411yescharacteristic
130C23×C4×C61C2×C411yescharacteristic
131C22⋊Q8×C61—22no—
132C22⋊Q8×C61—22no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C4⋊C4×C61C2×C411yescharacteristic
136C2×C22⋊C4×C61C2×C411yescharacteristic
137C2×C22⋊C4×C61C2×C411yescharacteristic
138C24×C61C22×C411yescharacteristic
139C23×D61C22×C411yescharacteristic
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—41no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61—41no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—42no—
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C411yescharacteristic
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C2411yescharacteristic
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122C22⋊C412yes—
172C2×Dic122C22⋊C412yes—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420—22no—
176C2×#92420—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22⋊C412yes—
181C4⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—22no—
188#11758198—22no—
189#11758198—21no—
190#11758198—21no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198#11758196—22no—
199#11758196—21no—
200#11758196—21no—
201#11758196—22no—
202#11758196C22⋊C411yescharacteristic
203#11758196C22⋊C411yescharacteristic
204#11758196C22⋊C411yescharacteristic
205#11758196C22⋊C411yescharacteristic
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C23×C411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—41no—
232C22×D61—41no—
233Q8×C61—44no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61#13112yes—
253C22⋊D61—42no—
254C22⋊D61—42no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C22×D412yes—
272C4×D61—42no—
273C4×D61—42no—
274Dic122—24no—
275Dic122—42no—
276Dic122—42no—
277Dic122—42no—
278#92420—42no—
279#92420—42no—
280#92420—42no—
281#92420—42no—
282#17428—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C22×C22⋊C411yescharacteristic
291C22×C61C2×#13111yescharacteristic
292C22×C61#56811yescharacteristic
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—81no—
297C2×D61—81no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305Dic61—21no—
306Dic61—21no—
307Dic61—22no—
308Dic61—22no—
309Dic61—21no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61#41312yes—
314C2×#556—611no—
315#2984—612no—
316#1473—612no—
317#1473—612no—
318#1469—612no—
319#1467—611no—
320#1464—611no—
321#1464—611no—
322#1461—611no—
323#1458—611no—
324#1458—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—81no—
328C2×C61#288011yescharacteristic
329C2×C61#288411yescharacteristic
330C2×C61C2×#41311yescharacteristic
331D61—81no—
332#556—1222no—
333#556—1222no—
334#556—1222no—
335#556—1222no—
336C2×C4⋊Q8—611no—
337C2×#126—611no—
338C2×#126—611no—
339C2×#126—611no—
340C2×#126—611no—
341C2×#124—611no—
342C2×#124—611no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8#692311yescharacteristic
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×D4—611no—
349C2×#117—611no—
350C2×#117—611no—
351#446—1222no—
352#446—1222no—
353#446—1222no—
354C8×Q8—1222no—
355#432—612no—
356#432—612no—
357#432—612no—
358#431—612no—
359#431—612no—
360#431—612no—
361#324—611no—
362#324—611no—
363#324—611no—
364#324—611no—
365#324—612no—
366#324—612no—
367#323—612no—
368#323—612no—
369#323—612no—
370#323—612no—
371C61#1742811yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C22×Q8—611no—
374C22×Q8C2×#692311yescharacteristic
375C22×D4—611no—
376C23×C4—611no—
377C23×C4C2×#692311yescharacteristic
378C4⋊Q8—1222no—
379C4⋊Q8—1222no—
380#126—1222no—
381#126—1222no—
382#126—1222no—
383#126—1222no—
384#126—1222no—
385#126—1222no—
386#126—1222no—
387#126—1222no—
388#124—1222no—
389#124—1222no—
390#124—1222no—
391#124—1222no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—612no—
399C4×Q8—612no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—1221no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413#117—1222no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4C2×#692311yescharacteristic
424C2×C4⋊C4C2×#692311yescharacteristic
425C2×C4⋊C4C2×#692311yescharacteristic
426C2×C22⋊C4—611no—
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4C2×#692311yescharacteristic
435C2×C22⋊C4C2×#692311yescharacteristic
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C4⋊C8—1222no—
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C22⋊C8—612no—
450C22⋊C8—612no—
451C22⋊C8—612no—
452C22⋊C8—612no—
453#97—1222no—
454#97—1222no—
455#97—1222no—
456C4×C8—1222no—
457C4×C8—1222no—
458C4×C8—1222no—
459C24—611no—
460C24C22×#692311yescharacteristic
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—612no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—2442no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—2441no—
488C22×C4C22×#692311yescharacteristic
489C22×C4C22×#692311yescharacteristic
490C22×C4C22×#692311yescharacteristic
491C22×C4C22×#692311yescharacteristic
492C22×C4C22×#692311yescharacteristic
493C22×C4C22×#692311yescharacteristic
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—611no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4C22⋊#692312yes—
516C4⋊C4C22⋊#692312yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1222no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1222no—
529C42—612no—
530C42—612no—
531C42—612no—
532C42—612no—
533C42—612no—
534C42—612no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—2441no—
552C23—2441no—
553C23C23×#692311yescharacteristic
554Q8—44no—
555Q8—1224no—
556Q8—2442no—
557Q8—2442no—
558Q8—2442no—
559D4—2442no—
560D4—2442no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×C22⋊#692311yescharacteristic
595C2×C4C2×C22⋊#692311yescharacteristic
596C2×C4C2×C22⋊#692312yes—
597C2×C4C2×C22⋊#692312yes—
598C2×C4#1451986412yes—
599C8—2442no—
600C8—2442no—
601C8—2442no—
602C8—2442no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—4881no—
611C22—4881no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22C22×C22⋊#692311yescharacteristic
616C22C2×#1451986411yescharacteristic
617C22#1684084011yescharacteristic
618C4—22no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—612no—
626C4—1221no—
627C4—1221no—
628C4—1222no—
629C4—1221no—
630C4—1222no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C2—21no—
635C2—21no—
636C2—81no—
637C2—4881no—
638C2#2146240711yescharacteristic
639C2#2146242011yescharacteristic
640C2D61⋊#41311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#17428
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462719C2semidirect
2. #21461103C2semidirect
3. #21461100C2semidirect
5. #21461082C2semidirect
6. #21461073C2semidirect
7. #21461070C2semidirect
8. #21461062C2semidirect
9. #21461053C2semidirect
10. #21461050C2semidirect
371. C61#17428semidirect

Maximal subgroups

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

subgroupindex
#214627192
#214611032
#214611002
#214610882
#214610822
#214610732
#214610702
#214610622
#214610532
#214610502
C2×#168404832
#1742861