Finite Group Databasebeta

← #29648416

Subgroups — #29648416

Order 15616.

972 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic
3C2×#16839814C211yesmaximal, characteristic
4#21444520C211yesmaximal, characteristic, complemented
5#21444519C211yesmaximal, characteristic, complemented
6#21444517C211yesmaximal, characteristic, complemented
7#21444502C212yesmaximal, complemented
8#21444489C212yesmaximal, complemented
9#21444482C212yesmaximal, complemented
10#21444482C212yesmaximal, complemented
11C61×#1552C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840318C412yes—
14#16840317C2211yescharacteristic
15#16840317C2211yescharacteristic
16#16840316—21no—
17#16840316—21no—
18#16840315C412yes—
19#16840314—21no—
20#16840314—21no—
21C4⋊Q8×D61C2211yescharacteristic
22C4⋊Q8×D61C2211yescharacteristic
23#125×D61C412yes—
24C22⋊Q8×D61—21no—
25C22⋊Q8×D61—21no—
26C4×Q8×D61—21no—
27C4×Q8×D61—21no—
28#117×D61C412yes—
29C2×C42×D61C2211yescharacteristic
30#16839819C2211yescharacteristic
31#16839817—21no—
32#16839814—21no—
33#16839814—21no—
34#16839814C2212yes—
35#16839814C2212yes—
36#16839814C2212yes—
37#16839812—21no—
38#16839784C2212yes—
39C2×C4⋊#92418C2211yescharacteristic
40#16839724—21no—
41C2×#14519689—21no—
42#16839714—21no—
43C2×C8×Dic61—21no—
44#16839509C2212yescomplemented
45#16839509C2212yescomplemented
46#16839508C2212yescomplemented
47#16839508C2212yescomplemented
48#16839507C2212yescomplemented
49#16839507C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#432C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×Q8×D61—21no—
56C2×Q8×D61—21no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758189—22no—
61C22×#92418—21no—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519776—21no—
65#14519776—21no—
66#14519776C2×C412yes—
67#14519776C2×C412yes—
68#14519775—21no—
69#14519775—21no—
70#14519769C2×C412yes—
71#14519768C2×C412yes—
72#14519768C2×C412yes—
73#14519767—22no—
74#14519767—22no—
75#14519757—21no—
76#14519757—21no—
77#14519756—21no—
78#14519756—21no—
79#14519755—22no—
80#14519755—22no—
81#14519754—22no—
82#14519754—22no—
83#14519753—21no—
84#14519753—21no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91#14519742—22no—
92#14519742—22no—
93#14519742—21no—
94#14519742—21no—
95#14519736C2×C412yes—
96#14519735C2×C412yes—
97#14519735C2×C412yes—
98#14519734—22no—
99#14519733—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61C2×C412yes—
105C4⋊C4×D61C2×C412yes—
106C22⋊C4×D61—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418—21no—
120C4⋊#92418C2312yes—
121C4⋊#92418C2312yes—
122#14519706—22no—
123C22⋊#92418—21no—
124#14519689—22no—
125#14519689—22no—
126#14519689—22no—
127#14519688—21no—
128C8×Dic61—22no—
129C8×Dic61—22no—
130C8×Dic61—22no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134#125×C61C2×C412yes—
135C22⋊Q8×C61—21no—
136C22⋊Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61—21no—
139#117×C61C2×C412yes—
140C2×C42×C61C2311yescharacteristic
141C4⋊C8×C61—21no—
142C4⋊C8×C61C2312yes—
143C4⋊C8×C61C2312yes—
144C22⋊C8×C61—21no—
145#97×C61—21no—
146C23×D61C22⋊C411yescharacteristic
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155Q8×D61—42no—
156Q8×D61—42no—
157Q8×D61—44no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22×C411yescharacteristic
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—42no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C412yes—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22×C412yes—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—42no—
230#11758189—44no—
231C23×C61C2×C22⋊C411yescharacteristic
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61C2×C22⋊C411yescharacteristic
236C22×D61C2×C22⋊C411yescharacteristic
237Q8×C61—42no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61#10114yes—
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—21no—
268C4×D61—21no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61#13514yes—
279Dic122—44no—
280Dic122—42no—
281Dic122—41no—
282Dic122—41no—
283Dic122—44no—
284C8×C61—42no—
285C8×C61—42no—
286#92418—42no—
287#92418—42no—
288#17715—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×#10112yes—
292C22×C61C22×C22⋊C411yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—44no—
299C2×D61#41612yes—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61C2×#13512yes—
307C4×C61#57511yescharacteristic
308C4×C61#57511yescharacteristic
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—42no—
313Dic61—42no—
314Dic61C2×#10112yes—
315Dic61C2×#10112yes—
316Dic61#41612yes—
317C2×#533—611no—
318#2920—611no—
319C2×#431—611no—
320#1552—611no—
321#1552—611no—
322#1552—611no—
323#1552D6111yescharacteristic, complemented
324#1549—612no—
325#1546—612no—
326#1544—612no—
327#1544—612no—
328C2×C61—22no—
329C2×C61C22×#10111yescharacteristic
330C2×C61C2×#41611yescharacteristic
331C2×C61#289611yescharacteristic
332D61—22no—
333D61—41no—
334#533—611no—
335#533—611no—
336#533—612no—
337#533—612no—
338#533—611no—
339#533—1221no—
340#533—1221no—
341#533—1221no—
342#533—1221no—
343#533C2×D6111yescharacteristic
344C2×C4⋊Q8—611no—
345C2×C4⋊Q8—611no—
346C2×#125—612no—
347C2×C22⋊Q8—1221no—
348C2×C22⋊Q8—1221no—
349C2×C4×Q8—1221no—
350C2×C4×Q8—1221no—
351C2×#117—612no—
352C22×C42—611no—
353#432—611no—
354#432—1221no—
355#432C2×D6111yescharacteristic
356#431—612no—
357#431—612no—
358#431—612no—
359#431—1221no—
360#431—1221no—
361#431—1221no—
362#423—612no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8C2×D6111yescharacteristic
365#407—1221no—
366C2×C22⋊C8—1221no—
367C4×M16—1221no—
368C2×C4×C8—1221no—
369#332—612no—
370#332—612no—
371#332—612no—
372#332—612no—
373#332—612no—
374#332—612no—
375#332C2×D6112yescomplemented
376#332C2×D6112yescomplemented
377C61#1771511yescharacteristic, complemented, Sylow
378C22×Q8—1221no—
379C22×Q8—1221no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C2×M16—1222no—
384C22×C8—21no—
385C22×C8—1221no—
386C4⋊Q8—612no—
387C4⋊Q8—612no—
388C4⋊Q8—611no—
389C4⋊Q8—611no—
390C4⋊Q8—1221no—
391C4⋊Q8—1221no—
392C4⋊Q8—1221no—
393C4⋊Q8—1221no—
394C4⋊Q8C22×D6111yescharacteristic
395C4⋊Q8C22×D6111yescharacteristic
396#125—612no—
397#125—612no—
398#125—612no—
399#125—1222no—
400#125—1222no—
401#125C4×D6112yes—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8—1221no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×Q8—1221no—
426#117—612no—
427#117—612no—
428#117—612no—
429#117—1222no—
430#117—1222no—
431#117C4×D6112yes—
432C2×C4⋊C4—612no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C22⋊C4—1222no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1222no—
451C2×C42C22×D6111yescharacteristic
452C4⋊C8—21no—
453C4⋊C8—612no—
454C4⋊C8—612no—
455C4⋊C8—1222no—
456C4⋊C8—1221no—
457C4⋊C8C22×D6112yes—
458C4⋊C8C22×D6112yes—
459C22⋊C8—21no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462C22⋊C8—1222no—
463C22⋊C8—1221no—
464#97—21no—
465#97—1221no—
466C4×C8—1222no—
467C4×C8—1222no—
468C4×C8—1222no—
469C24—611no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—1222no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—2444no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—1224no—
495C22×C4—1224no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—2441no—
499C22×C4—2441no—
500C22×C4C2×C4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503M16—2444no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—2442no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4C2×C4×D6112yes—
529C4⋊C4C2×C4×D6112yes—
530C22⋊C4—22no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C42—22no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1222no—
548C42—1222no—
549C42—1222no—
550C42C2×C4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C42C23×D6111yescharacteristic
554C23—611no—
555C23—611no—
556C23—1222no—
557C23—1222no—
558C23—2441no—
559C23C22⋊C4×D6111yescharacteristic
560Q8—42no—
561Q8—41no—
562Q8—41no—
563Q8—2444no—
564Q8—2444no—
565Q8—2442no—
566Q8—2441no—
567Q8—2441no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—614no—
578C2×C4—614no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2441no—
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C8—42no—
608C8—42no—
609C8—2442no—
610C8—2442no—
611C22—21no—
612C22—21no—
613C22—612no—
614C22—1221no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—2444no—
620C22#101×D6112yes—
621C22C2×C22⋊C4×D6111yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—42no—
628C4—612no—
629C4—612no—
630C4—612no—
631C4—1221no—
632C4—1221no—
633C4—1221no—
634C4—2442no—
635C4—2442no—
636C4#135×D6112yes—
637C4#1684055811yescharacteristic
638C4#1684056111yescharacteristic
639C2—22no—
640C2—1222no—
641C2—2441no—
642C2C2×#101×D6111yescharacteristic
643C2#416×D6111yescharacteristic
644C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1552C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1552D61
hypocenterC61#17715
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
4. #21444520C2semidirect
5. #21444519C2semidirect
6. #21444517C2semidirect
7. #21444502C2semidirect
8. #21444489C2semidirect
9. #21444482C2semidirect
10. #21444482C2semidirect
11. C61×#1552C2semidirect
44. #16839509C22semidirect
45. #16839509C22semidirect
46. #16839508C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839507C22semidirect
53. C61×#332C22semidirect
54. C61×#332C22semidirect
323. #1552D61semidirect
375. #332C2×D61semidirect
376. #332C2×D61semidirect
377. C61#17715semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544492
C2×#168398142
#214445202
#214445192
#214445172
#214445022
#214444892
#214444822
C61×#15522
#1771561