Finite Group Databasebeta

← #29647121

Subgroups — #29647121

Order 15616.

1128 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic, complemented
3C4×#14519735C211yesmaximal, characteristic, complemented
4#21444319C211yesmaximal, characteristic, complemented
5#21444317C211yesmaximal, characteristic, complemented
6#21444314C211yesmaximal, characteristic, complemented
7#21444307C214yesmaximal, complemented
8#21444302C214yesmaximal, complemented
9C61×#1502C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic, complemented
12#16840317C411yescharacteristic, complemented
13#16840310C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840294—21no—
16#16840294C412yescomplemented
17#16840293—21no—
18#16840293C2212yes—
19#16840291—21no—
20#16840291—21no—
21#16840272—21no—
22#16840272C2212yes—
23#16840269—21no—
24#41×Dic61—21no—
25C4×#11758224—21no—
26C4×#11758224C412yescomplemented
27C2×#14519735—21no—
28C2×C42×D61C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839814—21no—
31#16839813—21no—
32#16839812C2212yes—
33C4⋊C8×D61—21no—
34C22⋊C8×D61C2212yes—
35C4×C8×D61—21no—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×#11758198—21no—
39#16839609C2212yes—
40#16839607C2212yes—
41C42×Dic61—21no—
42#16839500C2214yes—
43#16839498C2214yes—
44#16839496C2214yes—
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#327C2214yes—
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×C8×D61—22no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519770—22no—
58#14519770—22no—
59#14519769—21no—
60#14519768—21no—
61#14519768C2×C412yes—
62#14519764—21no—
63#14519764C2×C412yes—
64#14519759—22no—
65#14519759—22no—
66#14519758—22no—
67#14519757—21no—
68#14519757C2×C412yes—
69#14519756—21no—
70#14519756C2312yes—
71#14519753—21no—
72#14519753—21no—
73#14519748—21no—
74#14519748—21no—
75#14519748—21no—
76#14519748C2×C412yes—
77#14519745—22no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2×C412yes—
82#14519742—21no—
83#14519742C2×C412yes—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61C2×C412yes—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92#14519737—22no—
93#14519737—22no—
94#14519736—21no—
95#14519736—22no—
96#14519735—28no—
97#14519735—21no—
98#14519735C2×C412yes—
99#14519733—22no—
100#14519733—22no—
101C2×C4⋊Dic61—21no—
102C2×#11758198—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—22no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C4⋊#92418—21no—
119#14519706—22no—
120C22⋊#92418C2312yes—
121#14519689—22no—
122C4×#92418—21no—
123C8×Dic61—22no—
124#14519684—22no—
125#14519684C2312yes—
126C2×#41×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128#127×C61C2×C411yescharacteristic
129#124×C61—21no—
130#121×C61—21no—
131#121×C61C2312yes—
132C4×D4×C61—21no—
133C4×D4×C61C2×C412yes—
134C2×C4⋊C4×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139C4×C8×C61—21no—
140#95×C61C2312yes—
141C23×D61C2×D411yescharacteristic
142#41×C61—42no—
143#41×C61—42no—
144C2×Q8×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162#11758224—44no—
163#11758224—44no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—24no—
167C2×C4×D61—42no—
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22×C411yescharacteristic
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×Dic122—21no—
181C2×Dic122C22×C412yes—
182C2×C8×C61—22no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—42no—
189C42×C61C22×C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198#11758198—28no—
199#11758198—22no—
200#11758198—22no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61C22×C412yes—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—28no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—44no—
217C4×Dic61—42no—
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C2×#92418—22no—
231C8×D61—44no—
232C23×C61—22no—
233C23×C61C22×D411yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—22no—
237C22×D61—41no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—42no—
241D4×C61—42no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—42no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—44no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61#13512yes—
258C2×C4×C61#13512yes—
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—44no—
262C22⋊D61—44no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—24no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—41no—
290C4×D61—41no—
291C4×D61#10414yes—
292Dic122—42no—
293Dic122—42no—
294C8×C61—42no—
295#92418—42no—
296#17533—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61C22×C22⋊C411yescharacteristic
303C22×C61C2×#13511yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—21no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61C2×#10412yes—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—44no—
317C4×C61—42no—
318C4×C61#41612yes—
319C4×C61#41612yes—
320C4×C61C2×#10412yes—
321Dic61—21no—
322Dic61—21no—
323Dic61—21no—
324Dic61—21no—
325Dic61—44no—
326Dic61—42no—
327Dic61—42no—
328Dic61#41012yes—
329Dic61C2×#10412yes—
330#3444—611no—
331C2×#431—611no—
332C4×#117—611no—
333#1503—611no—
334#1503—611no—
335#1502—611no—
336#1502D6111yescharacteristic, complemented
337#1501—614no—
338#1500—614no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61—82no—
342C2×C61C2×#41011yescharacteristic
343C2×C61C2×#41611yescharacteristic
344C2×C61C22×#10411yescharacteristic
345D61—22no—
346D61—41no—
347#533—611no—
348#533—611no—
349#533—611no—
350#532—611no—
351#532—611no—
352#532C2×D6111yescharacteristic
353#529—612no—
354#529—612no—
355#529—1221no—
356#529—1221no—
357#529—1221no—
358#529—1221no—
359#525—612no—
360#525—1221no—
361#525—1221no—
362C4×#41—612no—
363C4×#41—1221no—
364C4×#41—1221no—
365C2×#117—1221no—
366C22×C42—611no—
367#431—611no—
368#431—612no—
369#431—1221no—
370#431—1221no—
371#431C2×D6111yescharacteristic
372C2×C4⋊C8—1221no—
373C2×C22⋊C8—612no—
374C2×C4×C8—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—612no—
379C4×C22⋊C4—1221no—
380#376—612no—
381C43—1221no—
382#328—614no—
383#328—614no—
384#327—614no—
385#327C2×D6114yes—
386C61#1753311yescharacteristic, complemented, Sylow
387C2×#41—21no—
388C2×#41—612no—
389C2×#41—1221no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C22×C8—1222no—
394C4⋊Q8—611no—
395C4⋊Q8—611no—
396C4⋊Q8C4×D6111yescharacteristic
397#127C4×D6111yescharacteristic
398#126—1222no—
399#126—1222no—
400#125—612no—
401#125—1221no—
402#125—1221no—
403#124—21no—
404#124—612no—
405#124—1221no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409C22⋊Q8—612no—
410C22⋊Q8—612no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414C22⋊Q8—1221no—
415#121—21no—
416#121—612no—
417#121—1221no—
418#121—1221no—
419#121—1221no—
420#121C22×D6112yes—
421#120—1222no—
422C4×Q8—612no—
423C4×Q8—612no—
424C4×Q8—1221no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×D4—21no—
429C4×D4—612no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4C4×D6112yes—
440#117—612no—
441#117—1222no—
442#117—1222no—
443#117—1222no—
444#117—1221no—
445#117—1228no—
446#117—1221no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—1221no—
449C2×C22⋊C4—1222no—
450C2×C42—21no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—1222no—
463C2×C42—1222no—
464C2×C42—1222no—
465C2×C42—1221no—
466C2×C42C22×D6111yescharacteristic
467C4⋊C8—21no—
468C4⋊C8—1222no—
469C4⋊C8—1221no—
470C22⋊C8—612no—
471C22⋊C8—1222no—
472C22⋊C8C22×D6112yes—
473C4×C8—21no—
474C4×C8—1222no—
475C4×C8—1221no—
476#95—612no—
477#95—1222no—
478#95C22×D6112yes—
479C24—611no—
480#41—42no—
481#41—42no—
482#41—2444no—
483#41—2444no—
484C2×Q8—21no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4C2×C4×D6112yes—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—2442no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520C2×C8—22no—
521C2×C8—1222no—
522C2×C8—2444no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—612no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1224no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4C2×C4×D6112yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1228no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C42—42no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—1222no—
561C42—1222no—
562C42—1228no—
563C42—1222no—
564C42—2444no—
565C42—2442no—
566C42C2×C4×D6111yescharacteristic
567C42C2×C4×D6111yescharacteristic
568C42D4×D6111yescharacteristic
569C42D4×D6111yescharacteristic
570C23—22no—
571C23—611no—
572C23—611no—
573C23—1222no—
574C23—1221no—
575C23—1221no—
576C23—2441no—
577C23C2×D4×D6111yescharacteristic
578Q8—42no—
579Q8—2442no—
580Q8—2442no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—2444no—
587D4—2444no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—614no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1224no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2444no—
621C2×C4—2444no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4C22⋊C4×D6111yescharacteristic
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22⋊C4×D6111yescharacteristic
628C2×C4#1451979612yes—
629C2×C4#1451979812yes—
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C8—42no—
633C8—2442no—
634C22—21no—
635C22—21no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—612no—
640C22—1221no—
641C22—1221no—
642C22—1221no—
643C22—1221no—
644C22—1221no—
645C22—2442no—
646C22—2442no—
647C22C2×C22⋊C4×D6111yescharacteristic
648C22C2×#1451979611yescharacteristic
649C22C2×#1451979811yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—44no—
654C4—42no—
655C4—612no—
656C4—612no—
657C4—1221no—
658C4—1221no—
659C4—1221no—
660C4—1221no—
661C4—2444no—
662C4—2442no—
663C4—2442no—
664C4#1683975512yes—
665C4#1683975512yes—
666C4#104×D6112yes—
667C2—21no—
668C2—21no—
669C2—82no—
670C2—1222no—
671C2—2441no—
672C2#410×D6111yescharacteristic
673C2C2×#1683975511yescharacteristic
674C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17533
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #431×D61C2semidirect
3. C4×#14519735C2semidirect
4. #21444319C2semidirect
5. #21444317C2semidirect
6. #21444314C2semidirect
7. #21444307C2semidirect
8. #21444302C2semidirect
9. C61×#1502C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840310C4semidirect
14. #16840310C4semidirect
16. #16840294C4semidirect
26. C4×#11758224C4semidirect
336. #1502D61semidirect
386. C61#17533semidirect

Maximal subgroups

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

subgroupindex
#214591142
#431×D612
C4×#145197352
#214443192
#214443172
#214443142
#214443072
#214443022
C61×#15022
#1753361