Finite Group Databasebeta

← #29690023

Subgroups — #29690023

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446763C211yesmaximal, characteristic, complemented
5#21446762C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446264C214yesmaximal, complemented
8#21445837C214yesmaximal, complemented
9C61×#1980C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic, complemented
12#16840317C411yescharacteristic, complemented
13#16840310C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840294—21no—
16#16840294—21no—
17#16840293—21no—
18#16840293—21no—
19#16840291C412yescomplemented
20#16840291C412yescomplemented
21#16840272—21no—
22#16840272—21no—
23#16840269C2212yes—
24#41×Dic61C2212yes—
25C4×#11758224—21no—
26C4×#11758224—21no—
27C2×#14519734C2212yes—
28C2×C42×D61C2211yescharacteristic
29#16839817—21no—
30#16839817—21no—
31#16839813—21no—
32#16839813—21no—
33#16839785C2212yes—
34C2×#14519704C2214yes—
35#16839725C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839631—21no—
38#16839626—21no—
39#16839621—21no—
40#16839619—21no—
41C4×#11758196C2212yes—
42C4×C4⋊Dic61C2211yescharacteristic
43#16839523C2214yes—
44#16839522C2214yes—
45C61×#532C2211yescharacteristic
46C2×#104×C61C2214yes—
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C2×#11758224—21no—
50C2×#11758224—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519770—22no—
58#14519770—22no—
59#14519769C2×C412yes—
60#14519768—21no—
61#14519768—21no—
62#14519764—21no—
63#14519764—21no—
64#14519759—22no—
65#14519759—22no—
66#14519758—22no—
67#14519757—21no—
68#14519757—21no—
69#14519756—21no—
70#14519756—21no—
71#14519753C2×C412yes—
72#14519753C2×C412yes—
73#14519748—21no—
74#14519748—21no—
75#14519748C2×C412yes—
76#14519748C2×C412yes—
77#14519745—22no—
78Q8×Dic61C2×C412yes—
79Q8×Dic61C2×C412yes—
80C4×Dic122—21no—
81C4×Dic122—21no—
82#14519742—21no—
83#14519742—21no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61C2×C412yes—
89D4×Dic61C2×C412yes—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92#14519737—22no—
93#14519737—22no—
94#14519736—22no—
95#14519736C2312yes—
96#14519735—21no—
97#14519735—21no—
98#14519734—44no—
99#14519733—22no—
100#14519733—22no—
101C2×C4⋊Dic61—21no—
102C2×#11758196—22no—
103C2×#11758196—22no—
104C2×#11758198—22no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2311yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118#14519706—22no—
119#14519706—22no—
120#14519704—44no—
121#14519689—22no—
122#14519689—22no—
123#14519688—21no—
124C4×#92418—21no—
125#14519684—21no—
126#14519684—22no—
127C2×#41×C61C2312yes—
128C2×M16×C61C2312yes—
129C4⋊Q8×C61C2×C411yescharacteristic
130#127×C61C2×C411yescharacteristic
131#124×C61C2×C412yes—
132#121×C61—21no—
133#121×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C2×C4⋊C4×C61—21no—
137C2×C42×C61C2312yes—
138C2×C42×C61C2311yescharacteristic
139#104×C61—44no—
140#97×C61—21no—
141C4×C8×C61—21no—
142#95×C61—21no—
143C23×D61C2×D411yescharacteristic
144#41×C61—24no—
145#41×C61—24no—
146C2×Q8×C61C22×C412yes—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C22×C412yes—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164#11758224—44no—
165#11758224—44no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
179C2×C4×D61C22×C411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×Dic122—21no—
183C2×Dic122—21no—
184M16×C61—22no—
185M16×C61—22no—
186C2×C8×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C42×C61—24no—
193C42×C61C22×C411yescharacteristic
194C42×C61C22×C411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—44no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61—42no—
210C4⋊Dic61C22×C412yes—
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—44no—
217#11758196—44no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—24no—
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234#11758194—22no—
235#11758194—22no—
236C2×#92418—22no—
237C23×C61—22no—
238C23×C61C22×D411yescharacteristic
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61C2×C22⋊C411yescharacteristic
244C22×D61C2×C22⋊C411yescharacteristic
245Q8×C61—24no—
246D4×C61—24no—
247D4×C61—42no—
248D4×C61—42no—
249D4×C61—42no—
250D4×C61—42no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—44no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61H(5,𝔽2)12yes—
265C2×C4×C612−1+412yes—
266C22⋊D61—44no—
267C22⋊D61—44no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—44no—
275C2×Dic61—44no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—41no—
297Dic122—42no—
298Dic122—42no—
299C8×C61—42no—
300#92418—42no—
301#21900—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61#53511yescharacteristic
309C22×C61#53611yescharacteristic
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—42no—
317C2×D61—42no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—42no—
322C4×C61—82no—
323C4×C61#42112yes—
324C4×C61#52012yes—
325C4×C61#53712yes—
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—42no—
332Dic61—42no—
333Dic61—82no—
334Dic61#42112yes—
335#3444—611no—
336#2920—611no—
337#2285—611no—
338#1986—611no—
339#1986—611no—
340#1980—611no—
341#1980D6111yescharacteristic, complemented
342#1897—614no—
343#1806—614no—
344C2×C61—21no—
345C2×C61—21no—
346C2×C61—82no—
347C2×C61#231111yescharacteristic
348C2×C61C2×#42111yescharacteristic
349C2×C61C2×#42111yescharacteristic
350D61—41no—
351D61—41no—
352#533—611no—
353#533—611no—
354#533—611no—
355#532—611no—
356#532—611no—
357#532C2×D6111yescharacteristic
358#529—612no—
359#529—612no—
360#529—1221no—
361#529—1221no—
362#529—1221no—
363#529—1221no—
364#525—612no—
365#525—1221no—
366#525—1221no—
367C4×#41—612no—
368C4×#41—1221no—
369C4×#41—1221no—
370C2×#117—612no—
371C22×C42—611no—
372#432—1221no—
373#432—1221no—
374#431—1221no—
375#431—1221no—
376#423—612no—
377C2×#104—614no—
378C2×#104C2×D6114yes—
379#407—612no—
380C4×M16—611no—
381C4×M16C2×D6111yescharacteristic
382#384—1221no—
383#382—1221no—
384#380—1221no—
385#380—1221no—
386C4×C4⋊C4—612no—
387C4×C4⋊C4—611no—
388C4×C4⋊C4C2×D6111yescharacteristic
389#339—614no—
390#339—614no—
391C61#2190011yescharacteristic, complemented, Sylow
392C2×#41—1221no—
393C2×#41—1221no—
394C2×#41C22×D6112yes—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C2×M16—612no—
399C2×M16C22×D6112yes—
400C4⋊Q8—611no—
401C4⋊Q8—611no—
402C4⋊Q8C4×D6111yescharacteristic
403#127C4×D6111yescharacteristic
404#126—1222no—
405#126—1222no—
406#125—612no—
407#125—1221no—
408#125—1221no—
409#124—1221no—
410#124—1221no—
411#124C4×D6112yes—
412#123—1222no—
413#123—1222no—
414#123—1222no—
415C22⋊Q8—612no—
416C22⋊Q8—612no—
417C22⋊Q8—1221no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8—1221no—
421#121—21no—
422#121—21no—
423#121—612no—
424#121—612no—
425#121—1221no—
426#121—1221no—
427#120—1222no—
428C4×Q8—612no—
429C4×Q8—612no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8—1221no—
433C4×Q8—1221no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—612no—
437C4×D4—612no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4—1222no—
445C4×D4—1222no—
446#117—612no—
447#117—1222no—
448#117—1222no—
449#117—1222no—
450#117—1221no—
451#117—1221no—
452#117—2444no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—1222no—
455C2×C4⋊C4—1222no—
456C2×C4⋊C4—1221no—
457C2×C22⋊C4—1222no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—612no—
470C2×C42—1222no—
471C2×C42C22×D6112yes—
472C2×C42C22×D6111yescharacteristic
473C4⋊C8—1222no—
474C4⋊C8—1222no—
475#104—44no—
476#104—2444no—
477C22⋊C8—1222no—
478C22⋊C8—1222no—
479#97—21no—
480#97—1221no—
481C4×C8—21no—
482C4×C8—1221no—
483#95—21no—
484#95—1222no—
485#95—1221no—
486C24—611no—
487#41—24no—
488#41—24no—
489#41—2444no—
490#41—2444no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8C2×C4×D6112yes—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4C2×C4×D6112yes—
500C22×C4—22no—
501C22×C4—22no—
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—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4D4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527M16—22no—
528M16—22no—
529M16—1222no—
530M16—1222no—
531C2×C8—22no—
532C2×C8—1222no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—2442no—
547C4⋊C4—2444no—
548C4⋊C4—2444no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—2444no—
559C42—24no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—1222no—
573C42—1222no—
574C42—1222no—
575C42—1224no—
576C42C2×C4×D6111yescharacteristic
577C42C2×C4×D6111yescharacteristic
578C42D4×D6111yescharacteristic
579C42D4×D6111yescharacteristic
580C23—22no—
581C23—611no—
582C23—611no—
583C23—1221no—
584C23—1221no—
585C23—2441no—
586C23—2441no—
587C23C2×D4×D6111yescharacteristic
588Q8—24no—
589Q8—2442no—
590Q8—2442no—
591D4—24no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—2444no—
597D4—2444no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2444no—
629C2×C4—2441no—
630C2×C4—2444no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4C22⋊C4×D6111yescharacteristic
635C2×C4C22⋊C4×D6111yescharacteristic
636C2×C4C22⋊C4×D6111yescharacteristic
637C2×C4C22⋊C4×D6111yescharacteristic
638C2×C4C22×C4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4#1451982212yes—
641C2×C4#1451982412yes—
642C8—42no—
643C8—2442no—
644C22—21no—
645C22—21no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—1221no—
650C22—1221no—
651C22—1221no—
652C22—1221no—
653C22—1221no—
654C22—1221no—
655C22—2442no—
656C22—2442no—
657C22C2×C22⋊C4×D6111yescharacteristic
658C22#1684032511yescharacteristic
659C22#1684033211yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—42no—
664C4—82no—
665C4—612no—
666C4—1221no—
667C4—1221no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—4882no—
672C4—2442no—
673C4—2442no—
674C4#1683977712yes—
675C4#1684024112yes—
676C4#1684034112yes—
677C2—21no—
678C2—21no—
679C2—82no—
680C2—2441no—
681C2—2441no—
682C2#2144840911yescharacteristic
683C2#421×D6111yescharacteristic
684C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1980C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1980D61
hypocenterC61#21900
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #21448223C2semidirect
4. #21446763C2semidirect
5. #21446762C2semidirect
6. #21446739C2semidirect
7. #21446264C2semidirect
8. #21445837C2semidirect
9. C61×#1980C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840310C4semidirect
14. #16840310C4semidirect
19. #16840291C4semidirect
20. #16840291C4semidirect
341. #1980D61semidirect
391. C61#21900semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544502
#214482232
#214467632
#214467622
#214467392
#214462642
#214458372
C61×#19802
#2190061