Finite Group Databasebeta

← #29647476

Subgroups — #29647476

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3#382×D61C211yesmaximal, characteristic, complemented
4#21444383C211yesmaximal, characteristic, complemented
5#21444380C211yesmaximal, characteristic, complemented
6#21444378C211yesmaximal, characteristic, complemented
7#21444356C214yesmaximal, complemented
8#21444322C214yesmaximal, complemented
9C61×#1518C211yesmaximal, characteristic, complemented
10#16840317C411yescharacteristic, complemented
11#16840317C411yescharacteristic, complemented
12#16840313C411yescharacteristic, complemented
13#16840313C411yescharacteristic, complemented
14#16840312C2211yescharacteristic
15#16840299—21no—
16#16840299C412yescomplemented
17#16840298—21no—
18#16840298C2212yes—
19#16840297—21no—
20#16840297—21no—
21#16840275—21no—
22#16840275C2212yes—
23#16840274—21no—
24C4×#11758226—21no—
25C4×#11758226C412yescomplemented
26#16840230—21no—
27C2×C4⋊C4×D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839818—21no—
30#16839816C2212yes—
31#16839815C2211yescharacteristic
32#16839813—21no—
33#16839784—21no—
34#16839724C2212yes—
35C4×#11758189—21no—
36#16839628C2211yescharacteristic
37#16839627—21no—
38#16839626—21no—
39#16839626C2212yes—
40#95×D61—21no—
41#95×D61C2212yes—
42#16839499C2214yes—
43#16839497C2214yes—
44#16839495C2214yes—
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#382C2211yescharacteristic
48C61×#328C2214yes—
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61—22no—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C2×#11758189—22no—
57#14519777C2×C411yescharacteristic, complemented
58#14519774C2×C411yescharacteristic, complemented
59#14519772—22no—
60#14519772—22no—
61#14519768—21no—
62#14519768C2×C412yescomplemented
63#14519766—21no—
64#14519766C2×C412yescomplemented
65#14519765—21no—
66#14519762—22no—
67#14519762—22no—
68#14519761—22no—
69#14519757—21no—
70#14519757—21no—
71#14519753—21no—
72#14519753C2×C412yes—
73#14519752—21no—
74#14519752—21no—
75#14519752—21no—
76#14519752C2×C412yes—
77#14519751—21no—
78#14519751C2312yes—
79#14519747—22no—
80Q8×Dic61—21no—
81Q8×Dic61C2×C412yescomplemented
82#14519742—21no—
83#14519742—21no—
84C4×D244—22no—
85C4×D244—22no—
86C4×D244—21no—
87C4×D244C2×C412yescomplemented
88#14519739—22no—
89#14519739—22no—
90#14519739—21no—
91#14519739—21no—
92#14519739—21no—
93#14519739C2×C412yescomplemented
94#14519737—22no—
95#14519737—22no—
96#14519735—21no—
97#14519735—22no—
98#14519735—22no—
99#14519735C2×C412yes—
100#14519734—22no—
101C2×C4⋊Dic61—21no—
102C2×#11758196—22no—
103C4⋊C4×D61—44no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61—22no—
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C4⋊#92418—21no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519687—22no—
121C4×#92418—21no—
122#14519684—21no—
123#14519684—22no—
124#14519684—22no—
125#14519684C2312yes—
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127C4⋊Q8×C61C2×C411yescharacteristic, complemented
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2312yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2×C412yescomplemented
133#117×C61—21no—
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
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—22no—
149C22×Dic61—22no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—44no—
154#11758226—44no—
155#11758222—42no—
156#11758222—42no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C412yes—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
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×D61C22×C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122—21no—
185C2×C8×C61—22no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—42no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—22no—
194C42×C61—22no—
195C42×C61C22×C411yescharacteristic
196C42×C61C22×C411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61—42no—
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—44no—
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×Dic61C22×C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C2×#92418—22no—
230#11758189—44no—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61C2×C22⋊C411yescharacteristic
238C22×D61C2×C22⋊C411yescharacteristic
239Q8×C61—42no—
240Q8×C61—42no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61AGL(1,ℤ/8)12yes—
253C2×C4×C61#13712yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C22⋊D61—42no—
257C22⋊D61—42no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272D244—42no—
273D244—42no—
274D244—44no—
275D244—44no—
276D244—42no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288C4×D61—41no—
289C4×D61—44no—
290C4×D61—44no—
291C4×D61—44no—
292C4×D61—88no—
293Dic122—42no—
294C8×C61—42no—
295#92418—42no—
296#17588—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61C2×#13711yescharacteristic
302C2×D61—24no—
303C2×D61—24no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61#41012yes—
312C2×D61#42112yes—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—82no—
320C4×C61#41712yes—
321C4×C61#41712yes—
322Dic61—21no—
323Dic61—21no—
324Dic61—21no—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—42no—
329Dic61—82no—
330#3444—611no—
331#2920—611no—
332C2×#382—611no—
333#1518—611no—
334#1518D6111yescharacteristic, complemented
335#1517—611no—
336#1517—611no—
337#1512—614no—
338#1504—614no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61C2×#41011yescharacteristic
342C2×C61C2×#41711yescharacteristic
343C2×C61C2×#42111yescharacteristic
344D61—22no—
345D61—22no—
346D61—82no—
347#533—611no—
348#533—611no—
349#533C2×D6111yescharacteristic
350#532—611no—
351#532—611no—
352#532—611no—
353#529—612no—
354#529—612no—
355#529—1221no—
356#529—1221no—
357#529—1221no—
358#529—1221no—
359#525—612no—
360#525—1221no—
361#525—1221no—
362C4×#41—612no—
363C4×#41—1221no—
364C4×#41—1221no—
365C22×C4⋊C4—1221no—
366C22×C42—611no—
367#432—612no—
368#432—1221no—
369#431—611no—
370#431—1221no—
371#431C2×D6111yescharacteristic
372#423—1221no—
373#407—612no—
374C4×M16—1221no—
375#382—611no—
376#382—612no—
377#382—1221no—
378#382—1221no—
379#382C2×D6111yescharacteristic
380C2×#95—612no—
381C2×#95—1221no—
382#328—614no—
383#328C2×D6114yes—
384#327—614no—
385#327—614no—
386C61#1758811yescharacteristic, complemented, Sylow
387C2×#41—612no—
388C2×#41—1221no—
389C2×#41—1221no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—1222no—
394C2×M16—1222no—
395C4⋊Q8—611no—
396C4⋊Q8C4×D6111yescharacteristic, complemented
397C4⋊Q8C4×D6111yescharacteristic, complemented
398#127—611no—
399#126—1222no—
400#126—1222no—
401#125—21no—
402#125—612no—
403#125—1221no—
404#124—612no—
405#124—1221no—
406#124—1221no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410C22⋊Q8—21no—
411C22⋊Q8—612no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414C22⋊Q8—1221no—
415C22⋊Q8C22×D6112yes—
416#121—612no—
417#121—612no—
418#121—1221no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#120—1222no—
423C4×Q8—21no—
424C4×Q8—612no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×Q8C4×D6112yescomplemented
429C4×D4—612no—
430C4×D4—612no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1221no—
435C4×D4—1221no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1222no—
440C4×D4—1221no—
441#117—21no—
442#117—612no—
443#117—1222no—
444#117—1222no—
445#117—1222no—
446#117—1221no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—2444no—
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×C42C22×D6111yescharacteristic
464C4⋊C8—21no—
465C4⋊C8—1222no—
466C4⋊C8—1221no—
467C22⋊C8—612no—
468C22⋊C8—1222no—
469C22⋊C8C22×D6112yes—
470#97—1222no—
471C4×C8—21no—
472C4×C8—1221no—
473#95—21no—
474#95—612no—
475#95—1222no—
476#95—1222no—
477#95—1221no—
478#95C22×D6112yes—
479C24—611no—
480#41—2444no—
481#41—2444no—
482#41—2442no—
483#41—2442no—
484C2×Q8—21no—
485C2×Q8—1221no—
486C2×Q8C2×C4×D6112yes—
487C2×D4—612no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1221no—
492C2×D4—1221no—
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—1222no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2444no—
518C22×C4—2444no—
519C22×C4—2444no—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523M16—2444no—
524C2×C8—22no—
525C2×C8—1222no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—42no—
532C4⋊C4—612no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1221no—
538C4⋊C4—2444no—
539C4⋊C4—2442no—
540C4⋊C4C2×C4×D6112yes—
541C22⋊C4—22no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C42—22no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—1222no—
564C42—1222no—
565C42C2×C4×D6111yescharacteristic
566C42C2×C4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C42D4×D6111yescharacteristic
569C23—611no—
570C23—611no—
571C23—1222no—
572C23—1222no—
573C23—1221no—
574C23—1221no—
575C23—1222no—
576C23C2×D4×D6111yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—2442no—
580D4—2442no—
581D4—2442no—
582D4—2444no—
583D4—2444no—
584D4—2442no—
585D4—2442no—
586D4—2442no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—611no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—4888no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2444no—
624C2×C4C22⋊C4×D6111yescharacteristic
625C2×C4C22⋊C4×D6111yescharacteristic
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22⋊C4×D6111yescharacteristic
628C2×C4#1451980412yes—
629C2×C4#1451980912yes—
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C8—42no—
633C8—2442no—
634C22—21no—
635C22—21no—
636C22—612no—
637C22—612no—
638C22—1224no—
639C22—1224no—
640C22—1221no—
641C22—1221no—
642C22—1221no—
643C22—1221no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22C2×C22⋊C4×D6111yescharacteristic
648C22C2×#1451980411yescharacteristic
649C22C2×#1451980911yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—1221no—
658C4—1221no—
659C4—1221no—
660C4—1221no—
661C4—1221no—
662C4—1221no—
663C4—4882no—
664C4—2442no—
665C4#1683976212yes—
666C4#1683976212yes—
667C2—21no—
668C2—21no—
669C2—1222no—
670C2—1222no—
671C2—4882no—
672C2#410×D6111yescharacteristic
673C2C2×#1683976211yescharacteristic
674C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1518C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC22×C4×C61C24
hypercenter#1518D61
hypocenterC61#17588
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. #382×D61C2semidirect
4. #21444383C2semidirect
5. #21444380C2semidirect
6. #21444378C2semidirect
7. #21444356C2semidirect
8. #21444322C2semidirect
9. C61×#1518C2semidirect
10. #16840317C4semidirect
11. #16840317C4semidirect
12. #16840313C4semidirect
13. #16840313C4semidirect
16. #16840299C4semidirect
25. C4×#11758226C4semidirect
57. #14519777C2×C4semidirect
58. #14519774C2×C4semidirect
62. #14519768C2×C4semidirect
64. #14519766C2×C4semidirect
81. Q8×Dic61C2×C4semidirect
87. C4×D244C2×C4semidirect
93. #14519739C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
127. C4⋊Q8×C61C2×C4semidirect
132. C4×Q8×C61C2×C4semidirect
334. #1518D61semidirect
386. C61#17588semidirect
396. C4⋊Q8C4×D61semidirect
397. C4⋊Q8C4×D61semidirect
428. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544472
#382×D612
#214443832
#214443802
#214443782
#214443562
#214443222
C61×#15182
#1758861