Finite Group Databasebeta

← #29648662

Subgroups — #29648662

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3#423×D61C211yesmaximal, characteristic
4#21444557C211yesmaximal, characteristic, complemented
5#21444556C211yesmaximal, characteristic, complemented
6#21444556C211yesmaximal, characteristic, complemented
7#21444518C212yesmaximal, complemented
8#21444516C212yesmaximal, complemented
9#21444502C212yesmaximal, complemented
10#21444489C212yesmaximal, complemented
11C61×#1561C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840319C2211yescharacteristic
14#16840318C412yes—
15#16840317C2211yescharacteristic
16#16840317C2211yescharacteristic
17#16840316—21no—
18#16840316—21no—
19#16840315C412yes—
20#16840314—21no—
21#16840314—21no—
22C4⋊Q8×D61C2211yescharacteristic
23C4⋊Q8×D61C2211yescharacteristic
24#125×D61C412yes—
25C22⋊Q8×D61—21no—
26C22⋊Q8×D61—21no—
27C4×Q8×D61—21no—
28C4×Q8×D61—21no—
29#117×D61C412yes—
30C2×C42×D61C2211yescharacteristic
31#16839818—21no—
32#16839818C2212yes—
33#16839814—21no—
34#16839814C2212yes—
35#16839786C2211yescharacteristic
36#16839786C2211yescharacteristic
37#16839785—21no—
38#16839784—21no—
39#16839784C2212yes—
40C4⋊C8×D61—21no—
41C4⋊C8×D61C2212yes—
42#16839725—21no—
43M16×Dic61—21no—
44#16839509C2212yescomplemented
45#16839509C2212yescomplemented
46#16839508C2212yescomplemented
47#16839508C2212yescomplemented
48#16839507C2212yescomplemented
49#16839507C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C61×#423C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×Q8×D61—21no—
56C2×Q8×D61—21no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758189—22no—
63M16×D61—44no—
64C2×C8×D61—22no—
65#14519777C2311yescharacteristic
66#14519777C2311yescharacteristic
67#14519776—21no—
68#14519776—21no—
69#14519776C2×C412yes—
70#14519776C2×C412yes—
71#14519775—21no—
72#14519775—21no—
73#14519769C2×C412yes—
74#14519768C2×C412yes—
75#14519768C2×C412yes—
76#14519767—22no—
77#14519767—22no—
78#14519757—21no—
79#14519757—21no—
80#14519756—21no—
81#14519756—21no—
82#14519755—22no—
83#14519755—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122—22no—
93C4×Dic122—22no—
94#14519742—22no—
95#14519742—22no—
96#14519742—21no—
97#14519742—21no—
98#14519736C2×C412yes—
99#14519735C2×C412yes—
100#14519735C2×C412yes—
101#14519734—22no—
102#14519733—22no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61C2×C412yes—
108C4⋊C4×D61C2×C412yes—
109C22⋊C4×D61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊#92418—21no—
123C4⋊#92418—21no—
124C4⋊#92418C2312yes—
125C4⋊#92418C2312yes—
126#14519706—22no—
127#14519706—22no—
128#14519689—22no—
129#14519689—22no—
130#14519687—22no—
131C8×Dic61—22no—
132C2×M16×C61—21no—
133C2×M16×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135C4⋊Q8×C61C2311yescharacteristic
136#125×C61C2×C412yes—
137C22⋊Q8×C61—21no—
138C22⋊Q8×C61—21no—
139C4×Q8×C61—21no—
140C4×Q8×C61—21no—
141#117×C61C2×C412yes—
142C2×C42×C61C2311yescharacteristic
143C4⋊C8×C61—21no—
144C4⋊C8×C61—21no—
145C4⋊C8×C61C2312yes—
146C4⋊C8×C61C2312yes—
147C23×D61C22⋊C411yescharacteristic
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156Q8×D61—42no—
157Q8×D61—42no—
158Q8×D61—42no—
159Q8×D61—42no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183M16×C61—42no—
184M16×C61—42no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22×C412yes—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—22no—
194C42×C61—22no—
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199C22⋊Dic61—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
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×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231#11758194—42no—
232#11758194—42no—
233C2×#92418—22no—
234C2×#92418—22no—
235#11758189—44no—
236C8×D61—44no—
237C23×C61C2×C22⋊C411yescharacteristic
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—41no—
241C22×D61C2×C22⋊C411yescharacteristic
242C22×D61C2×C22⋊C411yescharacteristic
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61—41no—
246Q8×C61—41no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—21no—
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—44no—
285C4×D61—44no—
286C4×D61#10114yes—
287C4×D61AGL(1,ℤ/8)12yes—
288C4×D61#13712yes—
289Dic122—44no—
290Dic122—41no—
291Dic122—41no—
292Dic122—41no—
293Dic122—41no—
294Dic122—44no—
295C8×C61—42no—
296C8×C61—42no—
297#92418—42no—
298#92418—42no—
299#17751—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—21no—
303C22×C61C22×C22⋊C411yescharacteristic
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—44no—
309C2×D61C2×#10112yes—
310C2×D61#41712yes—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61C2×#10112yes—
317C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
318C4×C61C2×#13711yescharacteristic
319C4×C61#57511yescharacteristic
320C4×C61#57511yescharacteristic
321Dic61—21no—
322Dic61—21no—
323Dic61—21no—
324Dic61—21no—
325Dic61—42no—
326Dic61—42no—
327Dic61C2×#10112yes—
328Dic61#41712yes—
329C2×#533—611no—
330#2920—611no—
331C2×#423—611no—
332#1561—611no—
333#1561—611no—
334#1561—611no—
335#1561D6111yescharacteristic, complemented
336#1552—612no—
337#1552—612no—
338#1549—612no—
339#1546—612no—
340C2×C61—41no—
341C2×C61C22×#10111yescharacteristic
342C2×C61C2×#41711yescharacteristic
343C2×C61#289611yescharacteristic
344D61—22no—
345D61—22no—
346C22×M16—1221no—
347#533—611no—
348#533—611no—
349#533—612no—
350#533—612no—
351#533—611no—
352#533—1221no—
353#533—1221no—
354#533—1221no—
355#533—1221no—
356#533C2×D6111yescharacteristic
357C2×C4⋊Q8—611no—
358C2×C4⋊Q8—611no—
359C2×#125—612no—
360C2×C22⋊Q8—1221no—
361C2×C22⋊Q8—1221no—
362C2×C4×Q8—1221no—
363C2×C4×Q8—1221no—
364C2×#117—612no—
365C22×C42—611no—
366#432—612no—
367#432—1221no—
368#431—612no—
369#431—1221no—
370#423—611no—
371#423—611no—
372#423—612no—
373#423—1221no—
374#423—1221no—
375#423C2×D6111yescharacteristic
376#423C2×D6111yescharacteristic
377C2×C4⋊C8—612no—
378C2×C4⋊C8—1221no—
379#407—1221no—
380C4×M16—1221no—
381#332—612no—
382#332—612no—
383#332—612no—
384#332—612no—
385#332—612no—
386#332—612no—
387#332C2×D6112yescomplemented
388#332C2×D6112yescomplemented
389C61#1775111yescharacteristic, complemented, Sylow
390C22×Q8—1221no—
391C22×Q8—1221no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C2×M16—21no—
396C2×M16—21no—
397C2×M16—1222no—
398C2×M16—1221no—
399C2×M16—1221no—
400C2×M16—2444no—
401C22×C8—1222no—
402C4⋊Q8—612no—
403C4⋊Q8—612no—
404C4⋊Q8—611no—
405C4⋊Q8—611no—
406C4⋊Q8—1221no—
407C4⋊Q8—1221no—
408C4⋊Q8—1221no—
409C4⋊Q8—1221no—
410C4⋊Q8C22×D6111yescharacteristic
411C4⋊Q8C22×D6111yescharacteristic
412#125—612no—
413#125—612no—
414#125—612no—
415#125—1222no—
416#125—1222no—
417#125C4×D6112yes—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8—1221no—
425C22⋊Q8—1221no—
426C22⋊Q8—1221no—
427C22⋊Q8—1221no—
428C22⋊Q8—1221no—
429C22⋊Q8—1221no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—1222no—
433C4×Q8—1222no—
434C4×Q8—1221no—
435C4×Q8—1221no—
436C4×Q8—1222no—
437C4×Q8—1222no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1221no—
442#117—612no—
443#117—612no—
444#117—612no—
445#117—1222no—
446#117—1222no—
447#117C4×D6112yes—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C22⋊C4—1222no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
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—1222no—
467C2×C42C22×D6111yescharacteristic
468C4⋊C8—21no—
469C4⋊C8—21no—
470C4⋊C8—612no—
471C4⋊C8—612no—
472C4⋊C8—1222no—
473C4⋊C8—1222no—
474C4⋊C8—1221no—
475C4⋊C8—1221no—
476C4⋊C8C22×D6112yes—
477C4⋊C8C22×D6112yes—
478C22⋊C8—1222no—
479C22⋊C8—1222no—
480#97—1222no—
481C4×C8—1222no—
482C24—611no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—2442no—
490C2×Q8—2442no—
491C2×Q8—2442no—
492C2×Q8—2442no—
493C22×C4—611no—
494C22×C4—611no—
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—1222no—
509C22×C4—1224no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—2441no—
514C22×C4—2441no—
515C22×C4C2×C4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518M16—42no—
519M16—42no—
520M16—2444no—
521M16—2442no—
522M16—2442no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—1222no—
526C2×C8—1222no—
527C2×C8—2444no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—612no—
533C4⋊C4—612no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4C2×C4×D6112yes—
547C4⋊C4C2×C4×D6112yes—
548C22⋊C4—22no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C42—22no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—1222no—
566C42—1222no—
567C42—1222no—
568C42C2×C4×D6111yescharacteristic
569C42D4×D6111yescharacteristic
570C42D4×D6111yescharacteristic
571C42C23×D6111yescharacteristic
572C23—611no—
573C23—611no—
574C23—1222no—
575C23—1222no—
576C23—2441no—
577C23C22⋊C4×D6111yescharacteristic
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—2444no—
583Q8—2444no—
584Q8—2441no—
585Q8—2441no—
586Q8—2441no—
587Q8—2441no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—614no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2444no—
622C2×C4—2444no—
623C2×C4—2441no—
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4C22⋊C4×D6111yescharacteristic
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22×C4×D6111yescharacteristic
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C8—42no—
631C8—42no—
632C8—2442no—
633C8—2442no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—612no—
638C22—612no—
639C22—1221no—
640C22—1221no—
641C22—1221no—
642C22—1221no—
643C22—2444no—
644C22C2×C22⋊C4×D6111yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—612no—
651C4—612no—
652C4—1221no—
653C4—1221no—
654C4—1221no—
655C4—1221no—
656C4—2442no—
657C4—2442no—
658C4#101×D6112yes—
659C4AGL(1,ℤ/8)×D6111yescharacteristic
660C4#137×D6111yescharacteristic
661C4#1684055811yescharacteristic
662C4#1684056111yescharacteristic
663C2—41no—
664C2—1222no—
665C2—1222no—
666C2C2×#101×D6111yescharacteristic
667C2#417×D6111yescharacteristic
668C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1561C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1561D61
hypocenterC61#17751
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
4. #21444557C2semidirect
5. #21444556C2semidirect
6. #21444556C2semidirect
7. #21444518C2semidirect
8. #21444516C2semidirect
9. #21444502C2semidirect
10. #21444489C2semidirect
11. C61×#1561C2semidirect
44. #16839509C22semidirect
45. #16839509C22semidirect
46. #16839508C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839507C22semidirect
53. C61×#332C22semidirect
54. C61×#332C22semidirect
335. #1561D61semidirect
387. #332C2×D61semidirect
388. #332C2×D61semidirect
389. C61#17751semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544512
#423×D612
#214445572
#214445562
#214445182
#214445162
#214445022
#214444892
C61×#15612
#1775161