Finite Group Databasebeta

← #29647706

Subgroups — #29647706

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459571C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic, complemented
3#21444394C212yesmaximal, complemented
4#21444390C211yesmaximal, characteristic, complemented
5#21444369C212yesmaximal, complemented
6#21444360C211yesmaximal, characteristic
7#21444342C212yesmaximal, complemented
8#21444338C211yesmaximal, characteristic, complemented
9#21444317C212yesmaximal, complemented
10#21444308C211yesmaximal, characteristic
11C61×#2347C211yesmaximal, characteristic, complemented
12#16840480—21no—
13#16840467—21no—
14#16840451—21no—
15#16840447—21no—
16#16840445—22no—
17#16840438—22no—
18#16840390C411yescharacteristic, complemented
19#16840383C412yescomplemented
20#16840374C412yescomplemented
21#16840338C411yescharacteristic, complemented
22#16840319C2212yes—
23#16840296C2212yes—
24#16840273C2211yescharacteristic
25C2×#14519756C2211yescharacteristic
26#16840242C411yescharacteristic, complemented
27#16840239C411yescharacteristic, complemented
28#16839823—22no—
29#16839819C2212yes—
30#16839815C2212yes—
31#16839789—21no—
32#16839726C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839718—21no—
35#16839498C2212yes—
36#16839498C2211yescharacteristic
37#16839498C2212yes—
38#16839498C2212yes—
39#16839498C2212yes—
40#16839498C2211yescharacteristic
41#16839498C2212yes—
42#16839495C2212yes—
43#16839495C2211yescharacteristic
44#16839495C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C61×#389—21no—
47C61×#388—22no—
48C61×#383C2212yes—
49C61×#380C2211yescharacteristic
50C4×C4⋊C4×C61—21no—
51C4×C4⋊C4×C61C2212yes—
52C4×C22⋊C4×C61C2211yescharacteristic
53C2×#11758222C2×C411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C2×#11758194—21no—
56C22×#92418—21no—
57#14519777C2×C412yes—
58#14519777C2×C412yes—
59#14519776—21no—
60#14519772—22no—
61#14519771—22no—
62#14519770—22no—
63#14519770—22no—
64#14519769—22no—
65#14519769C2×C412yes—
66#14519768—21no—
67#14519767—22no—
68#14519767—22no—
69#14519765C2×C412yes—
70#14519763—22no—
71#14519761—22no—
72#14519760—21no—
73#14519760—22no—
74#14519759—22no—
75#14519757—21no—
76#14519757—21no—
77#14519756—21no—
78#14519756—21no—
79#14519756—22no—
80#14519756C2×C412yes—
81#14519756C2×C411yescharacteristic
82#14519756C2312yes—
83#14519756C2311yescharacteristic
84#14519755—22no—
85#14519754—22no—
86#14519751C2×C411yescharacteristic
87#14519751C2311yescharacteristic
88#14519746—21no—
89C4×Dic122—22no—
90C4×Dic122C2×C412yes—
91C4×Dic122C2×C412yes—
92C4×Dic122C2×C412yes—
93#14519742—21no—
94#14519742—21no—
95C4×D244C2×C412yes—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519736—22no—
101#14519736C2×C411yescharacteristic
102#14519735—21no—
103#14519734—22no—
104#14519733—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×#11758196—21no—
107C4⋊C4×D61—21no—
108C2×C22⋊Dic61—21no—
109C4⋊#92418—22no—
110C4⋊#92418—22no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113C22⋊#92418C2311yescharacteristic
114C22⋊#92418C2311yescharacteristic
115C22⋊#92418C2312yes—
116#14519688—22no—
117C4×#92418—22no—
118C23×C4×C61C2311yescharacteristic
119#117×C61C2×C411yescharacteristic
120#117×C61C2×C411yescharacteristic
121#117×C61C2×C412yes—
122#117×C61D411yescharacteristic
123#117×C61D411yescharacteristic
124#117×C61D412yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C22⋊C4×C61D412yes—
132C2×C42×C61—21no—
133C2×C42×C61—22no—
134C2×C42×C61C2312yes—
135#95×C61—22no—
136#95×C61—22no—
137#95×C61C2311yescharacteristic
138#95×C61C2311yescharacteristic
139#95×C61C2312yes—
140C24×C61C2×D411yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61—42no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158#11758222—42no—
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—22no—
162C2×Dic122—21no—
163C2×Dic122—22no—
164C2×Dic122—42no—
165C2×Dic122C22×C412yes—
166C2×Dic122C22×C411yescharacteristic
167C4⋊C4×C61—42no—
168C4⋊C4×C61—44no—
169C4⋊C4×C61C22⋊C411yescharacteristic
170C4⋊C4×C61C22⋊C411yescharacteristic
171C4⋊C4×C61C22⋊C412yes—
172C4⋊C4×C61C22⋊C411yescharacteristic
173C4⋊C4×C61C22⋊C411yescharacteristic
174C4⋊C4×C61C22⋊C412yes—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—81no—
177C22⋊C4×C61C22⋊C412yes—
178C22⋊C4×C61C22⋊C412yes—
179C22⋊C4×C61C22⋊C412yes—
180C22⋊C4×C61C22⋊C412yes—
181C42×C61—42no—
182C42×C61—82no—
183C42×C61C22×C412yes—
184C42×C61C22×C412yes—
185C42×C61C2×D412yes—
186C42×C61C2×D412yes—
187C22⋊Dic61—22no—
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190C22⋊Dic61—22no—
191C22⋊Dic61—21no—
192#11758198—22no—
193#11758198—22no—
194#11758198—21no—
195#11758198—22no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61C22×C412yes—
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C411yescharacteristic
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—21no—
211#11758196—22no—
212#11758196—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—21no—
217#11758194—42no—
218C2×#92418—21no—
219C2×#92418—22no—
220C2×#92418—21no—
221C2×#92418—42no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C2×C22⋊C411yescharacteristic
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C22⋊D61—42no—
257C22⋊D61—41no—
258C22⋊D61—41no—
259C22⋊D61—81no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—42no—
266C2×Dic61—81no—
267D244—41no—
268C4×D61—42no—
269C4×D61—82no—
270Dic122—42no—
271Dic122—42no—
272Dic122—41no—
273Dic122—82no—
274#92418—41no—
275#92418—41no—
276#92418—42no—
277#17612—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61C22×C22⋊C411yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57811yescharacteristic
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—82no—
303C4×C61—81no—
304C4×C61#41012yes—
305Dic61—42no—
306Dic61—41no—
307Dic61—41no—
308Dic61—41no—
309Dic61—42no—
310#3495—611no—
311#2924—611no—
312#2347D6111yescharacteristic, complemented
313#1521—612no—
314#1520—611no—
315#1515—612no—
316#1513—611no—
317#1509—612no—
318#1508—611no—
319#1503—612no—
320#1501—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—41no—
324C2×C61—41no—
325C2×C61C2×#41011yescharacteristic
326C2×C61#289811yescharacteristic
327C2×C61#290111yescharacteristic
328D61—81no—
329#556—1221no—
330#554—1221no—
331#552—1221no—
332#552—1222no—
333#552—1221no—
334#551—1222no—
335#544—611no—
336#542—612no—
337#541—612no—
338#536—611no—
339#533—612no—
340#529—612no—
341#525—611no—
342C2×C22⋊Q8—611no—
343#520—611no—
344#519—611no—
345C2×#117C2×D6111yescharacteristic
346#433—1222no—
347#432—612no—
348#431—612no—
349#424—1221no—
350#407—611no—
351C2×C22⋊C8—611no—
352#405—1221no—
353#389—21no—
354#388—22no—
355#383C2×D6112yes—
356#380C2×D6111yescharacteristic
357C4×C4⋊C4—21no—
358C4×C4⋊C4C2×D6112yes—
359C4×C22⋊C4C2×D6111yescharacteristic
360#328—611no—
361#328—611no—
362#328—612no—
363#328—612no—
364#328—612no—
365#328—612no—
366#328—612no—
367#327—611no—
368#327—611no—
369#327—612no—
370C61#1761211yescharacteristic, complemented, Sylow
371C2×#41—611no—
372C22×Q8—611no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—1221no—
375C22×C8—1221no—
376C4⋊Q8—612no—
377C4⋊Q8—612no—
378C4⋊Q8—1221no—
379#126—1222no—
380#126—1222no—
381#126—1222no—
382#126—1222no—
383#125—612no—
384#125—1222no—
385#125—1222no—
386#125—1221no—
387#125—1222no—
388#124—612no—
389#124—1222no—
390#123—1221no—
391#123—1222no—
392#123—1222no—
393#123—1222no—
394C22⋊Q8—612no—
395C22⋊Q8—611no—
396C22⋊Q8—612no—
397C22⋊Q8—611no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—1222no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405#121—611no—
406#121—611no—
407#120—1221no—
408C4×Q8—612no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8—1222no—
414C4×D4—612no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1222no—
418C4×D4—1222no—
419#117—611no—
420#117—1222no—
421#117—1221no—
422#117—1222no—
423#117—1222no—
424#117C4×D6111yescharacteristic
425#117C4×D6111yescharacteristic
426#117C4×D6112yes—
427#117D24411yescharacteristic
428#117D24412yes—
429#117D24411yescharacteristic
430C2×C4⋊C4—21no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C22⋊D6111yescharacteristic
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4C22⋊D6112yes—
441C2×C42—21no—
442C2×C42—22no—
443C2×C42C22×D6112yes—
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446C22⋊C8—611no—
447C22⋊C8—611no—
448C22⋊C8—612no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451#97—1222no—
452C4×C8—1222no—
453#95—22no—
454#95—22no—
455#95C22×D6111yescharacteristic
456#95C22×D6111yescharacteristic
457#95C22×D6112yes—
458C24C2×C22⋊D6111yescharacteristic
459#41—2442no—
460C2×Q8—612no—
461C2×Q8—611no—
462C2×Q8—1222no—
463C2×Q8—1221no—
464C2×Q8—2442no—
465C2×D4—611no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4C2×C4×D6111yescharacteristic
479C22×C4C2×D24411yescharacteristic
480C22×C4C2×D24411yescharacteristic
481C22×C4C2×C22⋊D6112yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6112yes—
484C22×C4C23×D6111yescharacteristic
485M16—2442no—
486C2×C8—1221no—
487C2×C8—1222no—
488C2×C8—1221no—
489C2×C8—2442no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1221no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4#1175819811yescharacteristic
509C4⋊C4#1175819811yescharacteristic
510C4⋊C4#1175819812yes—
511C4⋊C4#1175819811yescharacteristic
512C4⋊C4#1175819811yescharacteristic
513C4⋊C4#1175819812yes—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—1222no—
517C22⋊C4—1221no—
518C22⋊C4—1222no—
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1221no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1221no—
525C22⋊C4—1222no—
526C22⋊C4#1175819812yes—
527C22⋊C4#1175819812yes—
528C22⋊C4#1175819812yes—
529C22⋊C4#1175819812yes—
530C42—42no—
531C42—82no—
532C42—1221no—
533C42—1222no—
534C42—1222no—
535C42—1222no—
536C42C2×C4×D6112yes—
537C42C2×C4×D6112yes—
538C42C2×D24412yes—
539C42C2×D24412yes—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—1221no—
545C23C2×#1175819811yescharacteristic
546C23C2×#1175819811yescharacteristic
547C23C22×C22⋊D6111yescharacteristic
548Q8—4882no—
549Q8—2442no—
550Q8—2441no—
551Q8—2442no—
552D4—4881no—
553D4—2441no—
554D4—2442no—
555D4—2441no—
556D4—2441no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—4882no—
581C2×C4—4881no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4C2×#1175819812yes—
585C2×C4C2×#1175819812yes—
586C2×C4C2×#1175819812yes—
587C2×C4C2×#1175819812yes—
588C2×C4C2×#1175819811yescharacteristic
589C2×C4C2×#1175819811yescharacteristic
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C22×D24411yescharacteristic
592C8—2441no—
593C8—2441no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—81no—
604C22—81no—
605C22—2441no—
606C22—2441no—
607C22—2441no—
608C22C22×#1175819811yescharacteristic
609C22#1684057311yescharacteristic
610C22#1684057711yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—82no—
620C4—81no—
621C4—2441no—
622C4—2442no—
623C4—2441no—
624C4—2441no—
625C4—2442no—
626C4#1683973512yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—4881no—
632C2C2×#1683973511yescharacteristic
633C2#2145434911yescharacteristic
634C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2347C2
socleC22×C61C22×C22⋊C4
prime radical#16840273C22
cosocleC22×C4×C61C24
hypercenter#2347D61
hypocenterC61#17612
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2347

Complements

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

NH ≅ G/Ntype
2. #21454477C2semidirect
3. #21444394C2semidirect
4. #21444390C2semidirect
5. #21444369C2semidirect
7. #21444342C2semidirect
8. #21444338C2semidirect
9. #21444317C2semidirect
11. C61×#2347C2semidirect
18. #16840390C4semidirect
19. #16840383C4semidirect
20. #16840374C4semidirect
21. #16840338C4semidirect
26. #16840242C4semidirect
27. #16840239C4semidirect
312. #2347D61semidirect
370. C61#17612semidirect

Maximal subgroups

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

subgroupindex
#214595712
#214544772
#214443942
#214443902
#214443692
#214443602
#214443422
#214443382
#214443172
#214443082
C61×#23472
#1761261