Finite Group Databasebeta

← #29646214

Subgroups — #29646214

Order 15616.

944 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×#16840451C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic, complemented
3#21444184C211yesmaximal, characteristic, complemented
4#21444182C211yesmaximal, characteristic, complemented
5#21444176C212yesmaximal, complemented
6#21444174C212yesmaximal, complemented
7#21444155C211yesmaximal, characteristic
8#21444151C212yesmaximal, complemented
9#21444135C211yesmaximal, characteristic
10#21444131C212yesmaximal, complemented
11C61×#2924C211yesmaximal, characteristic, complemented
12#16840451—21no—
13#16840451—21no—
14#16840451—22no—
15#16840451—21no—
16#16840451—21no—
17#16840451—22no—
18C2×#14519770C412yes—
19C2×#14519769C2212yes—
20C2×#14519768C411yescharacteristic
21C2×#14519759C412yes—
22C2×#14519757C411yescharacteristic
23C2×#14519756C2211yescharacteristic
24C2×#14519742C411yescharacteristic
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×#14519734C2212yes—
27C2×C4⋊C4×D61C411yescharacteristic
28#16839823—22no—
29#16839819C2212yes—
30#16839815C2212yes—
31#16839789—21no—
32#16839726C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839718—21no—
35#16839487C2212yes—
36#16839487C2212yes—
37#16839487C2211yescharacteristic
38#16839487C2211yescharacteristic
39#16839487C2212yes—
40#16839487C2212yes—
41#16839487C2212yes—
42#16839485C2211yescharacteristic
43#16839485C2212yes—
44#16839485C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yes—
48C61×#431C2212yes—
49C61×#424—21no—
50C61×#407C2211yescharacteristic
51C2×C22⋊C8×C61C2211yescharacteristic
52C61×#405—21no—
53C22×Dic122C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2×C411yescharacteristic
57C2×#11758194—21no—
58C22×#92418—21no—
59#14519770—22no—
60#14519770—22no—
61#14519770—22no—
62#14519770—22no—
63#14519769—22no—
64#14519769—22no—
65#14519769—22no—
66#14519769—22no—
67#14519768—22no—
68#14519768—21no—
69#14519768—21no—
70#14519759—22no—
71#14519759—22no—
72#14519759—22no—
73#14519759—22no—
74#14519757—22no—
75#14519757—21no—
76#14519757—21no—
77#14519756—21no—
78#14519756—21no—
79#14519756—21no—
80#14519756—21no—
81#14519742—22no—
82#14519742—21no—
83#14519742—21no—
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86C4×C22⋊D61—21no—
87C4×C22⋊D61—21no—
88#14519734—22no—
89#14519734—22no—
90#14519734—22no—
91#14519734—22no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×C4⋊Dic61C2×C412yes—
94C2×C4⋊Dic61C2×C412yes—
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×#11758196C2×C412yes—
98C2×#11758196C2312yes—
99C2×#11758196C2311yescharacteristic
100C4⋊C4×D61—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C2×C22⋊Dic61C2×C411yescharacteristic
104C2×C22⋊Dic61C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2311yescharacteristic
107C2×C4×Dic61C2×C412yes—
108C2×C4×Dic61C2×C411yescharacteristic
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
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121#117×C61D411yescharacteristic
122#117×C61D411yescharacteristic
123#117×C61D412yes—
124#117×C61D411yescharacteristic
125#117×C61D411yescharacteristic
126#117×C61D412yes—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C22⋊C4×C61C2×C412yes—
130C2×C42×C61C2312yes—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2312yes—
138#97×C61—22no—
139C4×C8×C61—22no—
140C24×C61C22×C411yescharacteristic
141C23×D61C22×C411yescharacteristic
142C22×C4×C61—42no—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C412yes—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—41no—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C411yescharacteristic
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—42no—
170M16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—22no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22⋊C411yescharacteristic
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C4⋊C4×C61C22⋊C411yescharacteristic
180C4⋊C4×C61C22⋊C411yescharacteristic
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61C2×D412yes—
186C42×C61C2×D412yes—
187C42×C61C2×D412yes—
188C42×C61C2×D412yes—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—21no—
212#11758196—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—21no—
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×C61C23×C411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61#13112yes—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C4×D61—44no—
266C4×D61—44no—
267C4×D61—42no—
268C4×D61—42no—
269Dic122—42no—
270Dic122—42no—
271C8×C61—41no—
272C8×C61—41no—
273C8×C61—42no—
274#92418—41no—
275#92418—41no—
276#92418—42no—
277#17413—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—22no—
292C2×D61—42no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—42no—
296C4×C61—22no—
297C4×C61—22no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61#41012yes—
304Dic61—42no—
305Dic61—41no—
306Dic61—42no—
307Dic61—41no—
308Dic61—41no—
309C2×#552—611no—
310#2924—611no—
311#2924D6111yescharacteristic, complemented
312#1472—611no—
313#1472—611no—
314#1470—612no—
315#1470—612no—
316#1465—611no—
317#1464—612no—
318#1459—611no—
319#1458—612no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—41no—
323C2×C61—41no—
324C2×C61#288011yescharacteristic
325C2×C61C2×#41011yescharacteristic
326C2×C61#288911yescharacteristic
327D61—42no—
328#552—1221no—
329#552—1221no—
330#552—1222no—
331#552—1221no—
332#552—1221no—
333#552—1222no—
334C2×#126—612no—
335C2×#125—611no—
336C2×#125—612no—
337C2×#123—612no—
338C2×C22⋊Q8—611no—
339C2×C22⋊Q8—611no—
340C2×C4×Q8—611no—
341C2×C4×D4—611no—
342C2×#117—612no—
343C2×#117C2×D6111yescharacteristic
344C22×C4⋊C4—611no—
345#433—22no—
346#433—1222no—
347#432—612no—
348#432C2×D6112yes—
349#431—612no—
350#431C2×D6112yes—
351#424—21no—
352#424—1221no—
353#407—611no—
354#407C2×D6111yescharacteristic
355C2×C22⋊C8—611no—
356C2×C22⋊C8C2×D6111yescharacteristic
357#405—21no—
358#405—1221no—
359#324—612no—
360#324—612no—
361#324—611no—
362#324—611no—
363#324—612no—
364#324—612no—
365#324—612no—
366#323—611no—
367#323—612no—
368#323—611no—
369C61#1741311yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×D4—611no—
372C23×C4—612no—
373C23×C4—611no—
374C23×C4C22×D6111yescharacteristic
375C2×M16—21no—
376C2×M16—1221no—
377C22×C8—21no—
378C22×C8—1221no—
379#126—1222no—
380#126—1222no—
381#126—1222no—
382#126—1222no—
383#125—1222no—
384#125—1221no—
385#125—1221no—
386#125—1222no—
387#125—1222no—
388#125—1222no—
389#125—1222no—
390#123—1222no—
391#123—1222no—
392#123—1222no—
393#123—1222no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1221no—
397C22⋊Q8—1221no—
398C22⋊Q8—1222no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412#117D24411yescharacteristic
413#117D24411yescharacteristic
414#117D24412yes—
415#117C22⋊D6111yescharacteristic
416#117C22⋊D6111yescharacteristic
417#117C22⋊D6112yes—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—612no—
421C2×C4⋊C4—612no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C4×D6111yescharacteristic
430C2×C4⋊C4C4×D6111yescharacteristic
431C2×C22⋊C4—611no—
432C2×C22⋊C4—612no—
433C2×C22⋊C4—612no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4C4×D6112yes—
436C2×C42—612no—
437C2×C42—611no—
438C2×C42C22×D6112yes—
439C4⋊C8—22no—
440C4⋊C8—22no—
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—22no—
454#97—1222no—
455C4×C8—22no—
456C4×C8—1222no—
457C24—611no—
458C24C2×C4×D6111yescharacteristic
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—2442no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—2441no—
466C22×C4—42no—
467C22×C4—611no—
468C22×C4—612no—
469C22×C4—611no—
470C22×C4—612no—
471C22×C4—611no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4C2×C4×D6112yes—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×C4×D6112yes—
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊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—1222no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4#1175819812yes—
514C4⋊C4#1175819811yescharacteristic
515C4⋊C4#1175819811yescharacteristic
516C4⋊C4#1175819812yes—
517C4⋊C4#1175819811yescharacteristic
518C4⋊C4#1175819811yescharacteristic
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4#1175819812yes—
528C22⋊C4#1175819812yes—
529C22⋊C4#1175819812yes—
530C22⋊C4#1175819812yes—
531C42—1222no—
532C42—1222no—
533C42—1221no—
534C42—1221no—
535C42C2×D24412yes—
536C42C2×D24412yes—
537C42C2×C22⋊D6112yes—
538C42C2×C22⋊D6112yes—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—612no—
544C23—1221no—
545C23—1221no—
546C23—2441no—
547C23—2441no—
548C23C2×#1175819811yescharacteristic
549C23C2×#1175819811yescharacteristic
550C23C22×C4×D6111yescharacteristic
551Q8—2442no—
552Q8—2442no—
553D4—2442no—
554D4—2442no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2442no—
580C2×C4C2×#1175819812yes—
581C2×C4C2×#1175819812yes—
582C2×C4C2×#1175819811yescharacteristic
583C2×C4C2×#1175819812yes—
584C2×C4C2×#1175819812yes—
585C2×C4C2×#1175819811yescharacteristic
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C22×D24411yescharacteristic
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—1222no—
605C22—2442no—
606C22—2442no—
607C22—2441no—
608C22—2441no—
609C22C22×#1175819811yescharacteristic
610C22#131×D6111yescharacteristic
611C22#1684052711yescharacteristic
612C4—22no—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—22no—
618C4—22no—
619C4—2442no—
620C4—2441no—
621C4—2441no—
622C4—2442no—
623C4—2441no—
624C4#1683973512yes—
625C2—21no—
626C2—21no—
627C2—41no—
628C2—41no—
629C2—2442no—
630C2#2145426611yescharacteristic
631C2C2×#1683973511yescharacteristic
632C2#2145430111yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2924C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#2924D61
hypocenterC61#17413
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #21454477C2semidirect
3. #21444184C2semidirect
4. #21444182C2semidirect
5. #21444176C2semidirect
6. #21444174C2semidirect
8. #21444151C2semidirect
10. #21444131C2semidirect
11. C61×#2924C2semidirect
311. #2924D61semidirect
369. C61#17413semidirect

Maximal subgroups

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

subgroupindex
C2×#168404512
#214544772
#214441842
#214441822
#214441762
#214441742
#214441552
#214441512
#214441352
#214441312
C61×#29242
#1741361