Finite Group Databasebeta

← #29647218

Subgroups — #29647218

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459868C211yesmaximal, characteristic
2#21454854C211yesmaximal, characteristic, complemented
3#21444342C211yesmaximal, characteristic, complemented
4#21444338C212yesmaximal, complemented
5#21444330C211yesmaximal, characteristic, complemented
6#21444321C212yesmaximal, complemented
7#21444317C211yesmaximal, characteristic, complemented
8#21444308C212yesmaximal, complemented
9#21444303C211yesmaximal, characteristic, complemented
10C2×#16839498C212yesmaximal, complemented
11C61×#2275C211yesmaximal, characteristic, complemented
12#16840482C411yescharacteristic, complemented
13#16840467—21no—
14#16840452C412yescomplemented
15#16840438—22no—
16D4×Dic122C411yescharacteristic, complemented
17#16840421C412yescomplemented
18#16840417C411yescharacteristic, complemented
19#16840416C411yescharacteristic, complemented
20#16840406—21no—
21#16840377—21no—
22#16840354—22no—
23#16840319C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840273C2212yes—
26C2×#14519756C2212yes—
27#16840241—21no—
28#16839840—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic
32#16839815C2211yescharacteristic
33#16839726C2212yes—
34C2×C22⋊#92418C2212yes—
35#16839498C2211yescharacteristic
36#16839498C2212yes—
37#16839498C2212yes—
38#16839498C2212yes—
39#16839498C2211yescharacteristic
40#16839498C2212yes—
41#16839498C2212yes—
42#16839495C2211yescharacteristic
43#16839495C2211yescharacteristic
44#16839495C2212yes—
45C2×C4×D4×C61—21no—
46C2×C4×D4×C61C2211yescharacteristic
47C4×C4⋊C4×C61—21no—
48C4×C4⋊C4×C61C2211yescharacteristic
49C4×C22⋊C4×C61—22no—
50C4×C22⋊C4×C61C2212yes—
51C61×#376C2211yescharacteristic
52C2×#95×C61C2212yes—
53C2×#11758222C2×C412yescomplemented
54C2×#11758224—21no—
55C22×Dic122C2×C412yescomplemented
56C2×#11758194—22no—
57C22×#92418—22no—
58#14519777C2×C411yescharacteristic, complemented
59#14519777C2×C411yescharacteristic, complemented
60#14519776—21no—
61#14519771—21no—
62#14519770—22no—
63#14519770—22no—
64#14519769—21no—
65#14519769C2×C411yescharacteristic, complemented
66#14519767—22no—
67#14519765C2×C411yescharacteristic, complemented
68#14519764—21no—
69#14519763—22no—
70#14519761—22no—
71#14519760—22no—
72#14519760—21no—
73#14519759—22no—
74#14519758—22no—
75#14519757—21no—
76#14519756—22no—
77#14519756—21no—
78#14519756—22no—
79#14519756C2×C412yescomplemented
80#14519756C2×C412yescomplemented
81#14519756C2×C412yescomplemented
82#14519756C2311yescharacteristic
83#14519756C2312yes—
84#14519755—22no—
85#14519754—22no—
86#14519751C2×C412yescomplemented
87#14519751C2311yescharacteristic
88#14519749—22no—
89#14519748—21no—
90#14519746—22no—
91C4×Dic122—21no—
92C4×Dic122C2×C412yescomplemented
93C4×Dic122C2×C411yescharacteristic, complemented
94#14519742—21no—
95C4×D244C2×C411yescharacteristic, complemented
96D4×Dic61—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519736—21no—
102#14519736C2×C412yescomplemented
103#14519734—21no—
104#14519733—22no—
105C2×C4⋊Dic61C2×C412yescomplemented
106C2×#11758196—22no—
107C4⋊C4×D61—21no—
108C2×C22⋊Dic61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111C22⋊#92418—22no—
112C22⋊#92418—22no—
113C22⋊#92418C2311yescharacteristic
114C22⋊#92418C2312yes—
115C22⋊#92418C2311yescharacteristic
116#14519688—21no—
117C4×#92418—21no—
118C22×D4×C61D411yescharacteristic
119C23×C4×C61—22no—
120C23×C4×C61C2312yes—
121C4×D4×C61—28no—
122C4×D4×C61C2×C412yes—
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127C4×D4×C61D411yescharacteristic
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C22⋊C4×C61—22no—
131C2×C22⋊C4×C61D412yes—
132C2×C42×C61—21no—
133C2×C42×C61—21no—
134C2×C42×C61—22no—
135C2×C42×C61C2311yescharacteristic
136#95×C61—22no—
137#95×C61—21no—
138#95×C61C2311yescharacteristic
139#95×C61C2312yes—
140#95×C61C2311yescharacteristic
141C24×C61C2×D412yes—
142C2×D4×C61—24no—
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C2×D4×C61C22⋊C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—28no—
150C22×C4×C61—82no—
151C22×C4×C61—82no—
152C22×C4×C61—82no—
153C22×C4×C61C22×C412yes—
154C22×C4×C61C22×C412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D412yes—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—22no—
162C2×C22⋊D61—21no—
163C2×C22⋊D61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166#11758224—44no—
167#11758222—82no—
168C2×D244C22×C411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—22no—
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—82no—
174C2×Dic122C22×C411yescharacteristic
175C2×Dic122C22×C412yes—
176C4⋊C4×C61—24no—
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C4⋊C4×C61C22⋊C411yescharacteristic
180C22⋊C4×C61—28no—
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61—24no—
186C42×C61—81no—
187C42×C61—82no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—21no—
197C22⋊Dic61—22no—
198#11758198—21no—
199#11758198—21no—
200#11758198—22no—
201#11758198—21no—
202#11758198—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—22no—
211#11758196—22no—
212#11758196—21no—
213#11758196—22no—
214#11758196—21no—
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218#11758196—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—21no—
222#11758194—82no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—22no—
226C2×#92418—82no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—24no—
230C23×C61—42no—
231C23×C61C2×C22⋊C412yes—
232C23×C61C2×C22⋊C412yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—21no—
235D4×C61—24no—
236D4×C61—22no—
237D4×C61#10414yes—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—88no—
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C22⋊D61—41no—
266C22⋊D61—41no—
267C22⋊D61—44no—
268C22⋊D61—82no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—44no—
275C2×Dic61—82no—
276D244—41no—
277C4×D61—42no—
278C4×D61—82no—
279Dic122—42no—
280Dic122—41no—
281Dic122—42no—
282Dic122—82no—
283#92418—41no—
284#92418—42no—
285#92418—41no—
286#17548—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—21no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—44no—
295C22×C61—44no—
296C22×C61C2×#10414yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57811yescharacteristic
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61—44no—
312C4×C61—42no—
313C4×C61C2×#10412yes—
314Dic61—41no—
315Dic61—42no—
316Dic61—41no—
317Dic61—42no—
318Dic61—41no—
319#3520—611no—
320#2974—611no—
321#2275D6111yescharacteristic, complemented
322#1509—611no—
323#1508—612no—
324#1506—611no—
325#1504—612no—
326#1503—611no—
327#1501—612no—
328#1500—611no—
329C2×#328—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—24no—
333C2×C61—42no—
334C2×C61#288311yescharacteristic
335C2×C61#289811yescharacteristic
336C2×C61C22×#10411yescharacteristic
337D61—81no—
338#556—611no—
339#554—1221no—
340#552—612no—
341#551—1222no—
342D4×Q8—611no—
343#548—612no—
344#547—611no—
345#547—611no—
346#546—1221no—
347#542—1221no—
348#539—1222no—
349#533—611no—
350#529—611no—
351#525—612no—
352C2×C22⋊Q8—612no—
353#520—1221no—
354C2×C4×D4—21no—
355C2×C4×D4C2×D6111yescharacteristic
356#436—1221no—
357#435—1222no—
358C8×D4—1221no—
359#432—611no—
360#431—611no—
361#407—612no—
362C2×C22⋊C8—612no—
363C4×C4⋊C4—21no—
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—22no—
366C4×C22⋊C4C2×D6112yes—
367#376C2×D6111yescharacteristic
368C2×#95C2×D6112yes—
369#328—611no—
370#328—612no—
371#328—612no—
372#328—612no—
373#328—611no—
374#328—612no—
375#328—612no—
376#327—611no—
377#327—611no—
378#327—612no—
379C61#1754811yescharacteristic, complemented, Sylow
380C2×#41—612no—
381C2×#41—1221no—
382C22×Q8—612no—
383C22×D4C22⋊D6111yescharacteristic
384C23×C4—22no—
385C23×C4C22×D6112yes—
386C2×M16—1222no—
387C22×C8—1222no—
388C4⋊Q8—611no—
389C4⋊Q8—611no—
390C4⋊Q8—1221no—
391#126—1222no—
392#126—1222no—
393#126—1221no—
394#125—611no—
395#125—1222no—
396#125—1221no—
397#124—611no—
398#124—1221no—
399#124—1222no—
400#123—1222no—
401#123—1222no—
402#123—1221no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—612no—
407C22⋊Q8—612no—
408C22⋊Q8—611no—
409C22⋊Q8—612no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1221no—
413C22⋊Q8—1222no—
414C22⋊Q8—1221no—
415C22⋊Q8—1222no—
416#121—612no—
417#121—611no—
418#121—1222no—
419#121—1221no—
420#120—1222no—
421C4×Q8—611no—
422C4×Q8—612no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×D4—28no—
426C4×D4—611no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4C4×D6111yescharacteristic
433C4×D4C4×D6111yescharacteristic
434C4×D4C4×D6112yes—
435C4×D4D24411yescharacteristic
436C4×D4D24412yes—
437C4×D4D24411yescharacteristic
438#117—612no—
439#117—1221no—
440#117—1222no—
441#117—1221no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22⋊D6111yescharacteristic
447C2×C22⋊C4—22no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4C22⋊D6112yes—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—22no—
453C2×C42C22×D6111yescharacteristic
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C22⋊C8—611no—
457C22⋊C8—612no—
458C22⋊C8—611no—
459C22⋊C8—1222no—
460C22⋊C8—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—22no—
464#95—21no—
465#95C22×D6111yescharacteristic
466#95C22×D6112yes—
467#95C22×D6111yescharacteristic
468C24C2×C22⋊D6112yes—
469#41—4882no—
470#41—2444no—
471C2×Q8—611no—
472C2×Q8—612no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—4882no—
476C2×D4—24no—
477C2×D4—611no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1222no—
481C2×D4#1175819812yes—
482C2×D4#1175819811yescharacteristic
483C2×D4#1175819811yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4C2×C4×D6112yes—
496C22×C4C2×C4×D6112yes—
497C22×C4C2×D24412yes—
498C22×C4C2×D24412yes—
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6112yes—
502C22×C4C23×D6111yescharacteristic
503M16—4882no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1222no—
507C2×C8—4882no—
508C4⋊C4—24no—
509C4⋊C4—611no—
510C4⋊C4—612no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4#1175819811yescharacteristic
526C4⋊C4#1175819812yes—
527C4⋊C4#1175819811yescharacteristic
528C22⋊C4—28no—
529C22⋊C4—1221no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1222no—
539C22⋊C4—1221no—
540C22⋊C4#1175819812yes—
541C22⋊C4#1175819812yes—
542C22⋊C4#1175819812yes—
543C22⋊C4#1175819812yes—
544C42—24no—
545C42—81no—
546C42—82no—
547C42—1222no—
548C42—1222no—
549C42—1221no—
550C42C2×C4×D6111yescharacteristic
551C42C2×C4×D6111yescharacteristic
552C42C2×D24411yescharacteristic
553C42C2×D24411yescharacteristic
554C23—24no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—1221no—
559C23C2×#1175819812yes—
560C23C2×#1175819812yes—
561C23C22×C22⋊D6111yescharacteristic
562Q8—4882no—
563Q8—2441no—
564Q8—2442no—
565Q8—2442no—
566D4—24no—
567D4—22no—
568D4—4882no—
569D4—2441no—
570D4—2444no—
571D4—2441no—
572D4—2441no—
573D4#1451970514yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—1221no—
598C2×C4—4882no—
599C2×C4—4882no—
600C2×C4—2444no—
601C2×C4—2442no—
602C2×C4C2×#1175819811yescharacteristic
603C2×C4C2×#1175819812yes—
604C2×C4C2×#1175819811yescharacteristic
605C2×C4C2×#1175819811yescharacteristic
606C2×C4C2×#1175819812yes—
607C2×C4C2×#1175819811yescharacteristic
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C22×D24411yescharacteristic
610C8—2441no—
611C8—2442no—
612C8—2441no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—2441no—
623C22—2441no—
624C22—2441no—
625C22C2×#1451970514yes—
626C22C22×#1175819811yescharacteristic
627C22#1684057311yescharacteristic
628C22#1684057711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—2441no—
640C4—2441no—
641C4—2442no—
642C4—2442no—
643C4—2441no—
644C4C2×#1451970512yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—4881no—
650C2#2145427911yescharacteristic
651C2#2145434911yescharacteristic
652C2C22×#1451970511yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2275C2
socleC22×C61C22×C22⋊C4
prime radical#21459868C2
cosocleC22×C4×C61C24
hypercenter#2275D61
hypocenterC61#17548
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #21454854C2semidirect
3. #21444342C2semidirect
4. #21444338C2semidirect
5. #21444330C2semidirect
6. #21444321C2semidirect
7. #21444317C2semidirect
8. #21444308C2semidirect
9. #21444303C2semidirect
10. C2×#16839498C2semidirect
11. C61×#2275C2semidirect
12. #16840482C4semidirect
14. #16840452C4semidirect
16. D4×Dic122C4semidirect
17. #16840421C4semidirect
18. #16840417C4semidirect
19. #16840416C4semidirect
53. C2×#11758222C2×C4semidirect
55. C22×Dic122C2×C4semidirect
58. #14519777C2×C4semidirect
59. #14519777C2×C4semidirect
65. #14519769C2×C4semidirect
67. #14519765C2×C4semidirect
79. #14519756C2×C4semidirect
80. #14519756C2×C4semidirect
81. #14519756C2×C4semidirect
86. #14519751C2×C4semidirect
92. C4×Dic122C2×C4semidirect
93. C4×Dic122C2×C4semidirect
95. C4×D244C2×C4semidirect
102. #14519736C2×C4semidirect
105. C2×C4⋊Dic61C2×C4semidirect
321. #2275D61semidirect
379. C61#17548semidirect

Maximal subgroups

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

subgroupindex
#214598682
#214548542
#214443422
#214443382
#214443302
#214443212
#214443172
#214443082
#214443032
C2×#168394982
C61×#22752
#1754861