Finite Group Databasebeta

← #29646269

Subgroups — #29646269

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840483C211yesmaximal, characteristic
2#21454479C211yesmaximal, characteristic, complemented
3#21454473C211yesmaximal, characteristic, complemented
4#21444187C212yesmaximal, complemented
5#21444187C212yesmaximal, complemented
6#21444183C211yesmaximal, characteristic, complemented
7#21444161C212yesmaximal, complemented
8#21444155C211yesmaximal, characteristic
9#21444141C212yesmaximal, complemented
10#21444135C211yesmaximal, characteristic
11C61×#1472C211yesmaximal, characteristic, complemented
12#16840483—21no—
13#16840483—21no—
14#16840483—21no—
15#16840483—21no—
16#16840483—22no—
17#16840483—22no—
18C2×#14519776C411yescharacteristic
19C2×#14519772C2212yes—
20C2×#14519770C412yes—
21C2×#14519763C412yes—
22C2×#14519753C411yescharacteristic
23C2×Q8×Dic61C411yescharacteristic
24C2×#14519742C2212yes—
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×#14519736C2211yescharacteristic
27C2×#14519735C411yescharacteristic
28#16839820—22no—
29#16839820—22no—
30#16839816C2212yes—
31#16839816C2212yes—
32#16839812C2212yes—
33#16839812C2212yes—
34#16839790—21no—
35#16839788—21no—
36#16839726C2211yescharacteristic
37C2×C22⋊#92418C2211yescharacteristic
38#16839720—21no—
39#16839717—21no—
40#16839488C2212yes—
41#16839488C2212yes—
42#16839487C2211yescharacteristic
43#16839487C2211yescharacteristic
44#16839487C2212yes—
45#16839485C2211yescharacteristic
46#16839485C2211yescharacteristic
47#16839485C2212yes—
48C2×C22⋊Q8×C61C2211yescharacteristic
49C61×#407C2211yescharacteristic
50C2×C22⋊C8×C61C2211yescharacteristic
51C61×#324C2212yes—
52C61×#324C2212yes—
53C22×Dic122C2×C412yes—
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C2×#11758194—21no—
57C22×#92418—21no—
58#14519776—22no—
59#14519776—21no—
60#14519776—21no—
61#14519772—22no—
62#14519772—22no—
63#14519772—22no—
64#14519772—22no—
65#14519770—22no—
66#14519770—22no—
67#14519770—22no—
68#14519770—22no—
69#14519763—22no—
70#14519763—22no—
71#14519763—22no—
72#14519763—22no—
73#14519753—22no—
74#14519753—21no—
75#14519753—21no—
76Q8×Dic61—22no—
77Q8×Dic61—21no—
78Q8×Dic61—21no—
79#14519742—22no—
80#14519742—22no—
81#14519742—22no—
82#14519742—22no—
83C4×C22⋊D61—21no—
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86C4×C22⋊D61—21no—
87#14519736D411yescharacteristic
88#14519736D411yescharacteristic
89#14519736D411yescharacteristic
90#14519736D411yescharacteristic
91#14519736D412yes—
92#14519736D412yes—
93#14519735—22no—
94#14519735—21no—
95#14519735—21no—
96C2×C4⋊Dic61C2×C412yes—
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×#11758196C2×C412yes—
100C2×#11758196C2×C412yes—
101C2×#11758196C2311yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×C22⋊Dic61C2×C412yes—
104C2×#11758198C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2311yescharacteristic
107C2×C4×Dic61C2×C412yes—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C412yes—
110C2×C4×Dic61C2312yes—
111#14519706—22no—
112#14519706—22no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418—21no—
116C22⋊#92418—21no—
117C22⋊#92418C2311yescharacteristic
118C22⋊#92418C2311yescharacteristic
119C22⋊#92418C2312yes—
120#14519687—22no—
121#14519687—22no—
122C8×Dic61—22no—
123C8×Dic61—22no—
124C22×Q8×C61C2×C411yescharacteristic
125C23×C4×C61C2311yescharacteristic
126C2×M16×C61—21no—
127C22×C8×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61—21no—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C4⋊C4×C61C2312yes—
134C2×C22⋊C4×C61C2×C412yes—
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61C2311yescharacteristic
139C22⋊C8×C61C2312yes—
140C24×C61C22×C411yescharacteristic
141C23×D61C22×C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—41no—
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C412yes—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—42no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—24no—
169C2×Dic122—42no—
170M16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—22no—
174C2×C8×C61—42no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊Dic61—21no—
182C22⋊Dic61—21no—
183C22⋊Dic61C22⋊C412yes—
184C22⋊Dic61C22⋊C412yes—
185C22⋊Dic61C22⋊C412yes—
186C22⋊Dic61C22⋊C412yes—
187#11758198—21no—
188#11758198—21no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22⋊C412yes—
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C412yes—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
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×C61C23×C411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—41no—
233C22×D61—41no—
234Q8×C61—42no—
235Q8×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—42no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61#13112yes—
267C4×D61—42no—
268C4×D61—42no—
269Dic122—44no—
270Dic122—44no—
271C8×C61—41no—
272C8×C61—41no—
273C8×C61—42no—
274#92418—41no—
275#92418—41no—
276#92418—42no—
277#17420—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×C61C2×#13111yescharacteristic
290C22×C61#56811yescharacteristic
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—81no—
295C2×D61—81no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61#41012yes—
302Dic61—22no—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—41no—
308Dic61—41no—
309C2×#556—611no—
310#2924—611no—
311#2924—611no—
312#1473—612no—
313#1473—612no—
314#1472—611no—
315#1472D6111yescharacteristic, complemented
316#1467—612no—
317#1465—611no—
318#1461—612no—
319#1459—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—41no—
323C2×C61—41no—
324C2×C61#288011yescharacteristic
325C2×C61C2×#41011yescharacteristic
326C2×C61#288911yescharacteristic
327D61—81no—
328#556—1221no—
329#556—1221no—
330#556—1221no—
331#556—1221no—
332#556—1222no—
333#556—1222no—
334C2×C4⋊Q8—611no—
335C2×#126—612no—
336C2×#126—612no—
337C2×#124—612no—
338C2×C22⋊Q8—611no—
339C2×C22⋊Q8C2×D6111yescharacteristic
340C2×C4×Q8—611no—
341C2×C4×Q8—612no—
342C2×C4×D4—611no—
343C2×#117—611no—
344C2×#117—611no—
345#433—1222no—
346#433—1222no—
347#432—612no—
348#432—612no—
349#431—612no—
350#431—612no—
351#424—1221no—
352#424—1221no—
353#407—611no—
354#407C2×D6111yescharacteristic
355C2×C22⋊C8—611no—
356C2×C22⋊C8C2×D6111yescharacteristic
357#405—1221no—
358#405—1221no—
359#324—612no—
360#324—612no—
361#324—611no—
362#324—611no—
363#324—612no—
364#324C2×D6112yes—
365#324C2×D6112yes—
366#323—611no—
367#323—611no—
368#323—612no—
369C61#1742011yescharacteristic, complemented, Sylow
370C22×Q8—612no—
371C22×Q8C4×D6111yescharacteristic
372C22×D4—611no—
373C23×C4—611no—
374C23×C4C22×D6111yescharacteristic
375C2×M16—21no—
376C2×M16—1221no—
377C22×C8—21no—
378C22×C8—1221no—
379C4⋊Q8—1222no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
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—21no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—1222no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1221no—
411C4×D4—1221no—
412#117—611no—
413#117—611no—
414#117—611no—
415#117—611no—
416#117—612no—
417#117—612no—
418#117—1222no—
419#117—1221no—
420#117—1221no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4C4×D6111yescharacteristic
428C2×C4⋊C4C22×D6112yes—
429C2×C22⋊C4—612no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—612no—
432C2×C22⋊C4—612no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4C4×D6112yes—
435C2×C42—612no—
436C2×C42—612no—
437C2×C42—611no—
438C2×C42—612no—
439C4⋊C8—1222no—
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442C4⋊C8—1222no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—611no—
446C22⋊C8—611no—
447C22⋊C8—612no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8C22×D6111yescharacteristic
451C22⋊C8C22×D6111yescharacteristic
452C22⋊C8C22×D6112yes—
453#97—1222no—
454#97—1222no—
455C4×C8—1222no—
456C4×C8—1222no—
457C24—611no—
458C24C2×C4×D6111yescharacteristic
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8—1222no—
463C2×Q8—1222no—
464C2×Q8—1224no—
465C2×Q8—2442no—
466C2×D4—1222no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—2441no—
470C22×C4—42no—
471C22×C4—611no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—2442no—
479C22×C4C2×C4×D6112yes—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×C4×D6112yes—
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487M16—42no—
488M16—2442no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—42no—
493C2×C8—1221no—
494C2×C8—1222no—
495C2×C8—1221no—
496C2×C8—2442no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—612no—
502C4⋊C4—611no—
503C4⋊C4—611no—
504C4⋊C4—612no—
505C4⋊C4—611no—
506C4⋊C4—611no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—612no—
518C22⋊C4—612no—
519C22⋊C4—612no—
520C22⋊C4—612no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1221no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1221no—
529C42—612no—
530C42—612no—
531C42—612no—
532C42—612no—
533C42—1222no—
534C42—1222no—
535C42—1222no—
536C42—1222no—
537C42—1221no—
538C42—1221no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—2441no—
547C23—2441no—
548C23C22⋊C4×D6111yescharacteristic
549C23C22⋊C4×D6111yescharacteristic
550C23C22×C4×D6111yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—2444no—
554Q8—2444no—
555D4—2442no—
556D4—2442no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C22⋊C4×D6111yescharacteristic
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C8—41no—
589C8—41no—
590C8—42no—
591C8—2441no—
592C8—2441no—
593C8—2442no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—81no—
603C22—81no—
604C22—4881no—
605C22—4881no—
606C22—2441no—
607C22—2441no—
608C22—2441no—
609C22C2×C22⋊C4×D6111yescharacteristic
610C22#131×D6111yescharacteristic
611C22#1684052711yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—1222no—
618C4—1222no—
619C4—1222no—
620C4—1222no—
621C4—1222no—
622C4—2441no—
623C4—2441no—
624C4#1683973412yes—
625C2—21no—
626C2—21no—
627C2—41no—
628C2—41no—
629C2—4881no—
630C2#2145426711yescharacteristic
631C2C2×#1683973411yescharacteristic
632C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1472C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1472D61
hypocenterC61#17420
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
2. #21454479C2semidirect
3. #21454473C2semidirect
4. #21444187C2semidirect
5. #21444187C2semidirect
6. #21444183C2semidirect
7. #21444161C2semidirect
9. #21444141C2semidirect
11. C61×#1472C2semidirect
315. #1472D61semidirect
369. C61#17420semidirect

Maximal subgroups

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

subgroupindex
C2×#168404832
#214544792
#214544732
#214441872
#214441832
#214441612
#214441552
#214441412
#214441352
C61×#14722
#1742061