Finite Group Databasebeta

← #29646271

Subgroups — #29646271

Order 15616.

938 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
1C2×#16840484C211yesmaximal, characteristic
2#21454478C212yesmaximal, complemented
3#21444189C212yesmaximal, complemented
4#21444187C212yesmaximal, complemented
5#21444181C212yesmaximal, complemented
6#21444161C211yesmaximal, characteristic, complemented
7#21444156C211yesmaximal, characteristic
8#21444143C211yesmaximal, characteristic, complemented
9#21444141C211yesmaximal, characteristic, complemented
10#21444136C211yesmaximal, characteristic
11C61×#1467C211yesmaximal, characteristic, complemented
12#16840484—22no—
13#16840484—22no—
14#16840484—22no—
15#16840484—22no—
16C2×#14519775C411yescharacteristic
17C2×#14519772C411yescharacteristic
18C2×#14519772C2211yescharacteristic
19C2×#14519771C411yescharacteristic
20C2×#14519771C2211yescharacteristic
21C2×#14519766C411yescharacteristic
22C2×#14519765C411yescharacteristic
23C2×#14519755C411yescharacteristic
24C2×#14519755C2211yescharacteristic
25C2×C4×Dic122C411yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×#14519734C411yescharacteristic
29C2×#14519734C2211yescharacteristic
30#16839824—22no—
31#16839822—22no—
32#16839816C2212yes—
33#16839815C2212yes—
34#16839812C2212yes—
35#16839787—22no—
36#16839724C2212yes—
37C22⋊C8×D61C2212yes—
38#16839719—22no—
39#16839488C2211yescharacteristic
40#16839488C2211yescharacteristic
41#16839488C2212yes—
42#16839487C2212yes—
43#16839487C2212yes—
44#16839485C2212yes—
45#16839485C2212yes—
46C2×C4×Q8×C61C2211yescharacteristic
47C61×#432C2212yes—
48C61×#324C2211yescharacteristic
49C61×#324C2211yescharacteristic
50C61×#324C2212yes—
51C22×Dic122C2×C411yescharacteristic
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C2×#11758189—22no—
57C2×C8×D61—22no—
58#14519775—22no—
59#14519775—22no—
60#14519772—22no—
61#14519772—22no—
62#14519772—22no—
63#14519772—22no—
64#14519771—22no—
65#14519771—22no—
66#14519771—22no—
67#14519771—22no—
68#14519766—22no—
69#14519766—22no—
70#14519765—22no—
71#14519765—22no—
72#14519755—22no—
73#14519755—22no—
74#14519755—22no—
75#14519755—22no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519742—22no—
81#14519742—22no—
82#14519739—22no—
83#14519739—22no—
84#14519734—21no—
85#14519734—21no—
86#14519734—21no—
87#14519734—21no—
88#14519734D412yes—
89#14519734D412yes—
90#14519734D412yes—
91#14519734D412yes—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196C2×C411yescharacteristic
94C2×#11758196C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2311yescharacteristic
101C2×#11758198C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2311yescharacteristic
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61C2×C411yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C4⋊#92418—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119C22⋊#92418C2312yes—
120#14519689—22no—
121#14519689—22no—
122#14519687—22no—
123C4×#92418—22no—
124C8×Dic61—22no—
125C22×Q8×C61C2×C411yescharacteristic
126C4×Q8×C61—22no—
127C4×Q8×C61—22no—
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61C2×C411yescharacteristic
132C2×C42×C61C2×C411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—22no—
135C22⋊C8×C61C2312yes—
136C22⋊C8×C61C2312yes—
137#97×C61—22no—
138C23×D61C22×C411yescharacteristic
139C23×D61C22×C411yescharacteristic
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—41no—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—41no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—41no—
170C2×Dic122—42no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—21no—
186#11758198—21no—
187#11758198—22no—
188#11758198—22no—
189#11758198—21no—
190#11758198—21no—
191#11758198—22no—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194#11758198C22⋊C412yes—
195#11758198C22⋊C412yes—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—22no—
201#11758196—22no—
202#11758196—21no—
203#11758196—21no—
204#11758196—22no—
205#11758196—22no—
206#11758196C22⋊C411yescharacteristic
207#11758196C22⋊C411yescharacteristic
208#11758196C22⋊C411yescharacteristic
209#11758196C22⋊C411yescharacteristic
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—22no—
221C2×#92418—22no—
222#11758189—44no—
223C8×D61—44no—
224C23×C61C23×C411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61C2×C22⋊C412yes—
235Q8×C61—44no—
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—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61#13112yes—
253C2×C4×C61C22×D412yes—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C411yescharacteristic
270D244—44no—
271C4×D61—24no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—42no—
275Dic122—44no—
276Dic122—42no—
277Dic122—42no—
278C8×C61—42no—
279C8×C61—42no—
280#92418—42no—
281#92418—42no—
282#17420—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61C22×C22⋊C411yescharacteristic
286C22×C61C2×#13111yescharacteristic
287C22×C61#56811yescharacteristic
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306Dic61—22no—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61#41012yes—
314C2×#556—611no—
315#2924—612no—
316#1473—612no—
317#1473—612no—
318#1472—612no—
319#1467—611no—
320#1467D6111yescharacteristic, complemented
321#1465—611no—
322#1461—611no—
323#1461—611no—
324#1459—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61#288011yescharacteristic
328C2×C61C2×#41011yescharacteristic
329C2×C61#288911yescharacteristic
330D61—42no—
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—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×Q8C2×D6111yescharacteristic
348C2×C4×D4—611no—
349C2×#117—611no—
350C2×#117—611no—
351#433—1222no—
352#433—1222no—
353#432—612no—
354#432C2×D6112yes—
355#431—612no—
356#431—612no—
357#424—1222no—
358#407—612no—
359C2×C22⋊C8—612no—
360#405—1222no—
361#324—611no—
362#324—611no—
363#324—612no—
364#324—612no—
365#324—612no—
366#324C2×D6111yescharacteristic
367#324C2×D6111yescharacteristic
368#324C2×D6112yes—
369#323—612no—
370#323—612no—
371C61#1742011yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C22×Q8—611no—
374C22×Q8C4×D6111yescharacteristic
375C22×D4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C2×M16—1222no—
379C22×C8—1222no—
380C4⋊Q8—1222no—
381C4⋊Q8—1222no—
382#126—1222no—
383#126—1222no—
384#126—1222no—
385#126—1222no—
386#126—1222no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#124—1222no—
391#124—1222no—
392#124—1222no—
393#124—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408#117—612no—
409#117—612no—
410#117—612no—
411#117—612no—
412#117—1221no—
413#117—1221no—
414#117—1221no—
415#117—1221no—
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⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4C4×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
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⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42C4×D6111yescharacteristic
443C2×C42C4×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445C4⋊C8—22no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C22⋊C8—612no—
450C22⋊C8—612no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8C22×D6112yes—
454C22⋊C8C22×D6112yes—
455#97—22no—
456#97—1222no—
457C4×C8—1222no—
458C4×C8—1222no—
459C24—611no—
460C24—611no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—2441no—
469C2×Q8—2442no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—2441no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×C4×D6111yescharacteristic
487C22×C4C2×C4×D6111yescharacteristic
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C2×C4×D6111yescharacteristic
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492M16—2444no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—2444no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
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—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1222no—
517C22⋊C4—612no—
518C22⋊C4—612no—
519C22⋊C4—612no—
520C22⋊C4—612no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1221no—
529C22⋊C4—1221no—
530C22⋊C4—1222no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—612no—
536C42—612no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42D4×D6112yes—
544C42D4×D6112yes—
545C23—612no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23C22×C4×D6111yescharacteristic
556Q8—44no—
557Q8—2444no—
558Q8—2442no—
559Q8—2442no—
560D4—2444no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—611no—
579C2×C4—1224no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C22⋊C4×D6111yescharacteristic
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4C22⋊C4×D6112yes—
597C2×C4#1451978512yes—
598C2×C4C2×D4×D6112yes—
599C8—42no—
600C8—42no—
601C8—2442no—
602C8—2442no—
603C22—21no—
604C22—21no—
605C22—1222no—
606C22—1222no—
607C22—1222no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22C2×C22⋊C4×D6111yescharacteristic
616C22C2×#1451978511yescharacteristic
617C22#1684052811yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—612no—
627C4—1222no—
628C4—1221no—
629C4—1221no—
630C4—1221no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C2—21no—
635C2—21no—
636C2—4881no—
637C2—2442no—
638C2#2145426911yescharacteristic
639C2#410×D6111yescharacteristic
640C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1467C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1467D61
hypocenterC61#17420
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #21454478C2semidirect
3. #21444189C2semidirect
4. #21444187C2semidirect
5. #21444181C2semidirect
6. #21444161C2semidirect
8. #21444143C2semidirect
9. #21444141C2semidirect
11. C61×#1467C2semidirect
320. #1467D61semidirect
371. C61#17420semidirect

Maximal subgroups

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

subgroupindex
C2×#168404842
#214544782
#214441892
#214441872
#214441812
#214441612
#214441562
#214441432
#214441412
#214441362
C61×#14672
#1742061