Finite Group Databasebeta

← #29647453

Subgroups — #29647453

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3#21448225C211yesmaximal, characteristic, complemented
4#21444384C211yesmaximal, characteristic, complemented
5#21444382C211yesmaximal, characteristic, complemented
6#21444379C211yesmaximal, characteristic, complemented
7#21444359C214yesmaximal, complemented
8#21444341C214yesmaximal, complemented
9C61×#1517C211yesmaximal, 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#16839818—21no—
30#16839816C2212yes—
31#16839815C2211yescharacteristic
32#16839813—21no—
33#16839784—21no—
34#16839724C2212yes—
35C4×#11758189—21no—
36#16839631—21no—
37#16839628C2211yescharacteristic
38#16839621—21no—
39#16839621C2212yes—
40#16839613—21no—
41#16839613C2212yes—
42#16839500C2214yes—
43#16839498C2214yes—
44#16839496C2214yes—
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#382C2211yescharacteristic
48C61×#327C2214yes—
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—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—21no—
97#14519735—44no—
98#14519735C2×C412yes—
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×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—
122#14519687—22no—
123C4×#92418—21no—
124#14519684—21no—
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×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139#95×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—42no—
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—21no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—44no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—42no—
206C4⋊Dic61C22×C412yes—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—44no—
212#11758196—44no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—44no—
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×Dic61C22×C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C2×#92418—22no—
230#11758189—44no—
231C23×C61—22no—
232C23×C61C22×D411yescharacteristic
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61C2×C22⋊C411yescharacteristic
238C22×D61C2×C22⋊C411yescharacteristic
239Q8×C61—42no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—44no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61AGL(1,ℤ/8)12yes—
257C2×C4×C61#13712yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—44no—
261C22⋊D61—44no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—44no—
269C2×Dic61—44no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—41no—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—41no—
291Dic122—42no—
292Dic122—42no—
293C8×C61—42no—
294#92418—42no—
295#17585—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—21no—
310C2×D61—42no—
311C2×D61—42no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—42no—
317C4×C61—82no—
318C4×C61#41712yes—
319C4×C61#41712yes—
320Dic61—21no—
321Dic61—21no—
322Dic61—21no—
323Dic61—21no—
324Dic61—42no—
325Dic61—42no—
326Dic61—82no—
327Dic61#41012yes—
328Dic61#42112yes—
329#3444—611no—
330#2920—611no—
331#2285—611no—
332#1518—611no—
333#1518—611no—
334#1517—611no—
335#1517D6111yescharacteristic, complemented
336#1513—614no—
337#1509—614no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—82no—
341C2×C61C2×#41011yescharacteristic
342C2×C61C2×#41711yescharacteristic
343C2×C61C2×#42111yescharacteristic
344D61—41no—
345D61—41no—
346#533—611no—
347#533—611no—
348#533—611no—
349#532—611no—
350#532—611no—
351#532C2×D6111yescharacteristic
352#529—612no—
353#529—612no—
354#529—1221no—
355#529—1221no—
356#529—1221no—
357#529—1221no—
358#525—612no—
359#525—1221no—
360#525—1221no—
361C4×#41—612no—
362C4×#41—1221no—
363C4×#41—1221no—
364C2×#117—1221no—
365C22×C42—611no—
366#432—612no—
367#432—1221no—
368#431—611no—
369#431—1221no—
370#431C2×D6111yescharacteristic
371#423—1221no—
372#407—612no—
373C4×M16—1221no—
374#384—1221no—
375#382—611no—
376#382C2×D6111yescharacteristic
377#380—612no—
378#380—1221no—
379C4×C4⋊C4—612no—
380C4×C4⋊C4—1221no—
381#328—614no—
382#328—614no—
383#327—614no—
384#327C2×D6114yes—
385C61#1758511yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C2×#41—612no—
388C2×#41—1221no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C2×M16—1222no—
393C4⋊Q8—611no—
394C4⋊Q8—611no—
395C4⋊Q8C4×D6111yescharacteristic
396#127C4×D6111yescharacteristic
397#126—1222no—
398#126—1222no—
399#125—612no—
400#125—1221no—
401#125—1221no—
402#124—21no—
403#124—612no—
404#124—1221no—
405#123—1222no—
406#123—1222no—
407#123—1222no—
408C22⋊Q8—612no—
409C22⋊Q8—612no—
410C22⋊Q8—1221no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414#121—21no—
415#121—612no—
416#121—1221no—
417#121—1221no—
418#121—1221no—
419#121C22×D6112yes—
420#120—1222no—
421C4×Q8—612no—
422C4×Q8—612no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×D4—21no—
428C4×D4—612no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4C4×D6112yes—
439#117—612no—
440#117—1222no—
441#117—1222no—
442#117—1222no—
443#117—1221no—
444#117—1221no—
445#117—2444no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C22⋊C4—1222no—
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×C42C22×D6111yescharacteristic
465C4⋊C8—21no—
466C4⋊C8—1222no—
467C4⋊C8—1221no—
468C22⋊C8—612no—
469C22⋊C8—1222no—
470C22⋊C8C22×D6112yes—
471#97—1222no—
472C4×C8—21no—
473C4×C8—1221no—
474#95—21no—
475#95—612no—
476#95—1221no—
477#95C22×D6112yes—
478C24—611no—
479#41—42no—
480#41—42no—
481#41—2444no—
482#41—2444no—
483C2×Q8—21no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4C2×C4×D6112yes—
492C22×C4—22no—
493C22×C4—22no—
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—611no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519M16—2444no—
520C2×C8—22no—
521C2×C8—1222no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—612no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2444no—
535C4⋊C4—2444no—
536C4⋊C4—2442no—
537C4⋊C4C2×C4×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—2444no—
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—1222no—
563C42—2444no—
564C42C2×C4×D6111yescharacteristic
565C42C2×C4×D6111yescharacteristic
566C42D4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C23—22no—
569C23—611no—
570C23—611no—
571C23—1221no—
572C23—1221no—
573C23—2441no—
574C23—2441no—
575C23C2×D4×D6111yescharacteristic
576Q8—42no—
577Q8—2442no—
578Q8—2442no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—2444no—
585D4—2444no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4C22⋊C4×D6111yescharacteristic
623C2×C4C22⋊C4×D6111yescharacteristic
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4C22⋊C4×D6111yescharacteristic
626C2×C4#1451980112yes—
627C2×C4#1451980612yes—
628C2×C4C22×C4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C8—42no—
631C8—2442no—
632C22—21no—
633C22—21no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—1221no—
638C22—1221no—
639C22—1221no—
640C22—1221no—
641C22—1221no—
642C22—1221no—
643C22—2442no—
644C22—2442no—
645C22C2×C22⋊C4×D6111yescharacteristic
646C22C2×#1451980111yescharacteristic
647C22C2×#1451980611yescharacteristic
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—82no—
654C4—612no—
655C4—612no—
656C4—1221no—
657C4—1221no—
658C4—1221no—
659C4—1221no—
660C4—4882no—
661C4—2442no—
662C4—2442no—
663C4#1683976012yes—
664C4#1683976012yes—
665C2—21no—
666C2—21no—
667C2—82no—
668C2—2441no—
669C2—2441no—
670C2#410×D6111yescharacteristic
671C2C2×#1683976011yescharacteristic
672C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1517C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1517D61
hypocenterC61#17585
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. #21448225C2semidirect
4. #21444384C2semidirect
5. #21444382C2semidirect
6. #21444379C2semidirect
7. #21444359C2semidirect
8. #21444341C2semidirect
9. C61×#1517C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840310C4semidirect
14. #16840310C4semidirect
16. #16840294C4semidirect
26. C4×#11758224C4semidirect
335. #1517D61semidirect
385. C61#17585semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544472
#214482252
#214443842
#214443822
#214443792
#214443592
#214443412
C61×#15172
#1758561