Finite Group Databasebeta

← #29648642

Subgroups — #29648642

Order 15616.

952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3C2×#16839818C211yesmaximal, characteristic
4#21444543C212yesmaximal, complemented
5#21444534C212yesmaximal, complemented
6#21444528C212yesmaximal, complemented
7#21444526C212yesmaximal, complemented
8#21444510C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444507C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840318—21no—
14#16840317C2211yescharacteristic
15#16840317C2211yescharacteristic
16#16840316—21no—
17#16840316C412yes—
18#16840315—21no—
19#16840314—21no—
20#16840314C412yes—
21C4⋊Q8×D61C2211yescharacteristic
22C4⋊Q8×D61C2211yescharacteristic
23#125×D61—21no—
24C22⋊Q8×D61—21no—
25C22⋊Q8×D61C412yes—
26C4×Q8×D61—21no—
27C4×Q8×D61C412yes—
28#117×D61—21no—
29C2×C42×D61C2211yescharacteristic
30#16839818—21no—
31#16839818—21no—
32#16839818—21no—
33#16839818C2212yes—
34#16839818C2212yes—
35#16839818C2212yes—
36#16839814—21no—
37#16839814C2212yes—
38#16839786C2211yescharacteristic
39C2×C4⋊#92418C2211yescharacteristic
40#16839725—21no—
41C2×#14519689—21no—
42M16×Dic61—21no—
43C2×#14519687—21no—
44#16839511C2212yescomplemented
45#16839510C2212yescomplemented
46#16839510C2212yescomplemented
47#16839509C2212yescomplemented
48#16839508C2212yescomplemented
49#16839507C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#333C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×Q8×D61—21no—
56C2×Q8×D61C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758194—21no—
61C22×#92418—21no—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519776C2×C412yes—
68#14519775—21no—
69#14519775C2×C412yes—
70#14519769—21no—
71#14519768—21no—
72#14519768—21no—
73#14519767—22no—
74#14519767—22no—
75#14519757—21no—
76#14519757C2×C412yes—
77#14519756—21no—
78#14519756C2×C412yes—
79#14519755—22no—
80#14519755—22no—
81#14519754—22no—
82#14519754—22no—
83#14519753—21no—
84#14519753C2×C412yes—
85Q8×Dic61—21no—
86Q8×Dic61C2×C412yes—
87C4×Dic122—21no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122C2×C412yes—
91#14519742—22no—
92#14519742—22no—
93#14519742—21no—
94#14519742C2×C412yes—
95#14519736—21no—
96#14519735—21no—
97#14519735—21no—
98#14519734—22no—
99#14519733—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
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⋊#92418—21no—
121C4⋊#92418C2312yes—
122C4⋊#92418C2312yes—
123#14519689—22no—
124#14519689—22no—
125#14519689—22no—
126#14519689—22no—
127#14519687—22no—
128#14519687—22no—
129#14519687—22no—
130C8×Dic61—22no—
131C2×M16×C61—21no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135#125×C61—21no—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61C2×C412yes—
138C4×Q8×C61—21no—
139C4×Q8×C61C2×C412yes—
140#117×C61—21no—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61—21no—
144C4⋊C8×C61C2312yes—
145C4⋊C8×C61C2312yes—
146C23×D61C22⋊C411yescharacteristic
147C2×Q8×C61—21no—
148C2×Q8×C61C22×C412yes—
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—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122C22×C412yes—
181M16×C61—42no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—42no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C22⋊C4×C61—22no—
192C42×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197C22⋊Dic61—22no—
198#11758198—22no—
199#11758198—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229#11758194—42no—
230C2×#92418—22no—
231C2×#92418—22no—
232C2×#92418—42no—
233C23×C61C2×C22⋊C411yescharacteristic
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—41no—
237C22×D61C2×C22⋊C411yescharacteristic
238C22×D61C2×C22⋊C411yescharacteristic
239Q8×C61—42no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61#13712yes—
251C2×C4×C61#13712yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—21no—
269C4×D61—21no—
270C4×D61—21no—
271C4×D61—21no—
272C4×D61—21no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—21no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—44no—
283C4×D61—44no—
284Dic122—44no—
285Dic122—44no—
286Dic122—41no—
287Dic122—41no—
288Dic122—42no—
289C8×C61—42no—
290C8×C61—42no—
291#92418—42no—
292#92418—42no—
293#17748—611nomaximal, Sylow
294C22×C61—21no—
295C22×C61—21no—
296C22×C61#41912yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—44no—
304C2×D61#41312yes—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61#41912yes—
311C4×C61C2×#13711yescharacteristic
312C4×C61C2×#13711yescharacteristic
313C4×C61#57511yescharacteristic
314C4×C61#57511yescharacteristic
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—42no—
322Dic61—42no—
323C2×#533—611no—
324#2920—611no—
325C2×#432—611no—
326#1558—612no—
327#1556—612no—
328#1554—612no—
329#1554—612no—
330#1550—611no—
331#1550—611no—
332#1550—611no—
333#1550D6111yescharacteristic, complemented
334C2×C61—22no—
335C2×C61C2×#41311yescharacteristic
336C2×C61C2×#41911yescharacteristic
337C2×C61#289611yescharacteristic
338D61—22no—
339D61—41no—
340#533—611no—
341#533—611no—
342#533—612no—
343#533—612no—
344#533—611no—
345#533—1221no—
346#533—1221no—
347#533—1221no—
348#533—1221no—
349#533C2×D6111yescharacteristic
350C2×C4⋊Q8—611no—
351C2×C4⋊Q8—611no—
352C2×#125—1221no—
353C2×C22⋊Q8—612no—
354C2×C22⋊Q8—1221no—
355C2×C4×Q8—612no—
356C2×C4×Q8—1221no—
357C2×#117—1221no—
358C22×C42—611no—
359#432—612no—
360#432—612no—
361#432—612no—
362#432—1221no—
363#432—1221no—
364#432—1221no—
365#431—612no—
366#431—1221no—
367#423—611no—
368#423C2×D6111yescharacteristic
369C2×C4⋊C8—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
371#407—1221no—
372C2×C22⋊C8—1221no—
373C4×M16—1221no—
374C2×#97—1221no—
375#333—612no—
376#333—612no—
377#333—612no—
378#333C2×D6112yescomplemented
379#332—612no—
380#332—612no—
381#332—612no—
382#332C2×D6112yescomplemented
383C61#1774811yescharacteristic, complemented, Sylow
384C22×Q8—612no—
385C22×Q8—1221no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C2×M16—21no—
390C2×M16—1221no—
391C22×C8—21no—
392C22×C8—1221no—
393C4⋊Q8—612no—
394C4⋊Q8—612no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8—1221no—
400C4⋊Q8—1221no—
401C4⋊Q8C22×D6111yescharacteristic
402C4⋊Q8C22×D6111yescharacteristic
403#125—21no—
404#125—1222no—
405#125—1222no—
406#125—1221no—
407#125—1221no—
408#125—1221no—
409C22⋊Q8—21no—
410C22⋊Q8—612no—
411C22⋊Q8—612no—
412C22⋊Q8—612no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1221no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8C4×D6112yes—
421C4×Q8—21no—
422C4×Q8—612no—
423C4×Q8—612no—
424C4×Q8—612no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1221no—
429C4×Q8—1222no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8C4×D6112yes—
433#117—21no—
434#117—1222no—
435#117—1222no—
436#117—1221no—
437#117—1221no—
438#117—1221no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1221no—
445C2×C22⋊C4—1222no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—1222no—
458C2×C42C22×D6111yescharacteristic
459C4⋊C8—21no—
460C4⋊C8—21no—
461C4⋊C8—612no—
462C4⋊C8—612no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465C4⋊C8C22×D6112yes—
466C4⋊C8C22×D6112yes—
467C22⋊C8—1222no—
468C22⋊C8—1222no—
469C22⋊C8—1222no—
470C22⋊C8—1222no—
471#97—1222no—
472#97—1222no—
473#97—1222no—
474C4×C8—1222no—
475C24—611no—
476C2×Q8—21no—
477C2×Q8—612no—
478C2×Q8—1222no—
479C2×Q8—1222no—
480C2×Q8—1221no—
481C2×Q8—2442no—
482C2×Q8—2442no—
483C2×Q8—2444no—
484C2×Q8C2×C4×D6112yes—
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—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—2441no—
505C22×C4—2441no—
506C22×C4—2442no—
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510M16—42no—
511M16—2442no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—42no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—2442no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4C2×C4×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C42—22no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—1222no—
556C42—1222no—
557C42—1222no—
558C42C2×C4×D6111yescharacteristic
559C42D4×D6111yescharacteristic
560C42D4×D6111yescharacteristic
561C42C23×D6111yescharacteristic
562C23—611no—
563C23—611no—
564C23—1222no—
565C23—1222no—
566C23—2441no—
567C23C22⋊C4×D6111yescharacteristic
568Q8—42no—
569Q8—41no—
570Q8—41no—
571Q8—2442no—
572Q8—2444no—
573Q8—2444no—
574Q8—2441no—
575Q8—2441no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C22⋊C4×D6111yescharacteristic
611C2×C4C22⋊C4×D6111yescharacteristic
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4#1451980612yes—
614C2×C4#1451980912yes—
615C2×C4C22×C4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C8—42no—
619C8—42no—
620C8—2442no—
621C8—2442no—
622C22—21no—
623C22—21no—
624C22—612no—
625C22—1221no—
626C22—1221no—
627C22—1221no—
628C22—1221no—
629C22—1221no—
630C22—2444no—
631C22#1683976912yes—
632C22C2×C22⋊C4×D6111yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—42no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—1221no—
643C4—1221no—
644C4—2442no—
645C4—2442no—
646C4#1683976912yes—
647C4C2×#1451980611yescharacteristic
648C4C2×#1451980911yescharacteristic
649C4#1684055811yescharacteristic
650C4#1684056111yescharacteristic
651C2—22no—
652C2—1222no—
653C2—2441no—
654C2#413×D6111yescharacteristic
655C2C2×#1683976911yescharacteristic
656C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1550D61
hypocenterC61#17748
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
4. #21444543C2semidirect
5. #21444534C2semidirect
6. #21444528C2semidirect
7. #21444526C2semidirect
8. #21444510C2semidirect
9. #21444509C2semidirect
10. #21444507C2semidirect
11. C61×#1550C2semidirect
44. #16839511C22semidirect
45. #16839510C22semidirect
46. #16839510C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
53. C61×#333C22semidirect
54. C61×#332C22semidirect
333. #1550D61semidirect
378. #333C2×D61semidirect
382. #332C2×D61semidirect
383. C61#17748semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544512
C2×#168398182
#214445432
#214445342
#214445282
#214445262
#214445102
#214445092
#214445072
C61×#15502
#1774861