Finite Group Databasebeta

← #29646502

Subgroups — #29646502

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519825C211yesmaximal, characteristic
2#21454932C211yesmaximal, characteristic, complemented
3#21454780C211yesmaximal, characteristic, complemented
4#21444264C212yesmaximal, complemented
5#21444257C212yesmaximal, complemented
6#21444252C212yesmaximal, complemented
7#21444246C212yesmaximal, complemented
8#21444207C211yesmaximal, characteristic, complemented
9#21444199C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1479C211yesmaximal, characteristic, complemented
12C2×#14519825C411yescharacteristic
13#16840244—21no—
14#16840244C412yes—
15#16840243—22no—
16#16840243C411yescharacteristic
17#16840243C411yescharacteristic
18#16840240C411yescharacteristic
19#16840238—21no—
20#16840238C412yes—
21C4×#11758226—21no—
22C4×#11758226C2212yes—
23C4×#11758222—22no—
24C4×#11758222C2211yescharacteristic
25C4×#11758222C2211yescharacteristic
26C4×Q8×D61—21no—
27C4×Q8×D61C2212yes—
28#16839892—21no—
29#16839889—21no—
30#16839841—21no—
31#16839839—21no—
32#16839830—22no—
33#16839830—22no—
34#16839818C2212yes—
35#16839813C2212yes—
36#16839784C2212yes—
37C2×C4⋊#92418C2211yescharacteristic
38C4×#11758189C2212yes—
39C2×C4×#92418C2211yescharacteristic
40#16839494C2212yes—
41Q8⋊#92418C2212yes—
42Q8⋊#92418C2212yes—
43#16839492C2211yescharacteristic
44#16839492C2211yescharacteristic
45#16839491C2212yes—
46#16839489C2211yescharacteristic
47#16839489C2211yescharacteristic
48C2×C4×Q8×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C61×#431C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53#14519825—24no—
54#14519825—24no—
55C2×#11758226—21no—
56C2×#11758226C2×C412yes—
57C2×#11758222—22no—
58C2×#11758222C2×C411yescharacteristic
59C2×#11758222C2×C411yescharacteristic
60C2×Q8×D61—21no—
61C2×Q8×D61C2×C412yes—
62C2×#11758189—22no—
63C2×#11758189—22no—
64C22×#92418—21no—
65C22×#92418—21no—
66Q8×Dic61—21no—
67Q8×Dic61—21no—
68Q8×Dic61C2×C412yes—
69Q8×Dic61C2×C412yes—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×Dic122C2×C411yescharacteristic
73C4×Dic122D411yescharacteristic
74C4×Dic122D411yescharacteristic
75C4×Dic122C2311yescharacteristic
76#14519742—21no—
77#14519742—22no—
78#14519742—22no—
79#14519742C2×C412yes—
80C4×D244—22no—
81C4×D244—22no—
82C4×D244C2×C411yescharacteristic
83C4×D244D411yescharacteristic
84C4×D244D411yescharacteristic
85C4×D244C2311yescharacteristic
86#14519739—21no—
87#14519739—22no—
88#14519739—22no—
89#14519739C2×C412yes—
90C4×C22⋊D61—21no—
91C4×C22⋊D61—21no—
92C4×C22⋊D61—22no—
93C4×C22⋊D61—22no—
94C4×C22⋊D61D412yes—
95C4×C22⋊D61D412yes—
96#14519736—22no—
97#14519736C2×C411yescharacteristic
98#14519736C2×C411yescharacteristic
99#14519735—22no—
100#14519735—21no—
101#14519735—22no—
102#14519735C2×C412yes—
103#14519734—21no—
104#14519734—22no—
105#14519734—22no—
106#14519734C2×C412yes—
107C4⋊C4×D61—22no—
108C4⋊C4×D61—21no—
109C4⋊C4×D61—22no—
110C4⋊C4×D61C2×C412yes—
111C42×D61—22no—
112C42×D61—21no—
113C42×D61—22no—
114C42×D61C2312yes—
115C4⋊#92418—21no—
116C4⋊#92418C2312yes—
117#14519706—22no—
118#14519706—22no—
119#14519689—22no—
120#14519689—22no—
121#14519687—22no—
122#14519687—22no—
123C4×#92418—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C22×Q8×C61C2×C411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2×C412yes—
130C4×Q8×C61C2312yes—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61—21no—
139C22⋊C8×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2312yes—
142C2×Q8×C61—21no—
143C2×Q8×C61—22no—
144C2×Q8×C61M1614yescomplemented
145C2×Q8×C61C22×C412yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—21no—
152C2×C22⋊D61C22⋊C412yes—
153#11758226—28no—
154#11758226—24no—
155#11758226—24no—
156Q8×D61—28no—
157Q8×D61—24no—
158Q8×D61—24no—
159#11758222—216no—
160#11758222—22no—
161#11758222—22no—
162#11758222—22no—
163#11758222M1614yescomplemented
164C2×D244—22no—
165C2×D244C22⋊C411yescharacteristic
166C2×D244C22×C411yescharacteristic
167C2×C4×D61—22no—
168C2×C4×D61—21no—
169C2×C4×D61—22no—
170C2×C4×D61—21no—
171C2×C4×D61—22no—
172C2×C4×D61—21no—
173C2×C4×D61—22no—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×Dic122—22no—
178C2×Dic122C22⋊C411yescharacteristic
179C2×Dic122C22×C411yescharacteristic
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193C22⋊Dic61—21no—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—21no—
197#11758198—21no—
198#11758198—22no—
199#11758198C22⋊C412yes—
200#11758198C22⋊C412yes—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207#11758196—21no—
208#11758196—21no—
209#11758196—22no—
210#11758196—22no—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—22no—
226C2×#92418—42no—
227C2×#92418—81no—
228#11758189—44no—
229#11758189—82no—
230C23×C61C2×C22⋊C411yescharacteristic
231C22×D61—21no—
232C22×D61C2×C22⋊C412yes—
233Q8×C61—22no—
234Q8×C61—24no—
235Q8×C61C2×M1614yes—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—81no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×M1612yes—
247C2×C4×C61C2×M1612yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—24no—
252C22⋊D61—24no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259D244—28no—
260D244—21no—
261D244—22no—
262D244C2×M1612yes—
263C4×D61—28no—
264C4×D61—28no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—24no—
270C4×D61—42no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—82no—
274C4×D61—81no—
275C4×D61—82no—
276Dic122—28no—
277Dic122—21no—
278Dic122—22no—
279Dic122C2×M1612yes—
280C8×C61—42no—
281C8×C61—41no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#92418—42no—
286#17454—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61#40712yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—41no—
296C2×D61—41no—
297C4×C61—24no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61#40712yes—
303C4×C61C22×M1611yescharacteristic
304C4×C61C22×M1611yescharacteristic
305C4×C61#57711yescharacteristic
306C4×C61#57811yescharacteristic
307Dic61—22no—
308Dic61—22no—
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313C4×2−1+4—611no—
314#2989—611no—
315#2965—611no—
316#1491—612no—
317#1489—612no—
318#1488—612no—
319#1487—612no—
320#1479D6111yescharacteristic, complemented
321#1477—611no—
322#1476—611no—
323C2×Q8⋊C8—611no—
324C2×C61—22no—
325C2×C61C2×#40711yescharacteristic
326C2×C61#289811yescharacteristic
327C2×C61#290111yescharacteristic
328D61—42no—
329D61—42no—
330C2×2−1+4—611no—
331#520—611no—
332#520—612no—
333#520—611no—
334#520—1222no—
335#520—1221no—
336#519—611no—
337#519—612no—
338#519—1221no—
339C4×#41—611no—
340C4×#41—612no—
341C4×#41—611no—
342C4×#41—1222no—
343C4×#41—1221no—
344C2×C4×Q8—612no—
345C2×C4×Q8—1221no—
346C2×C4×Q8C2×D6111yescharacteristic
347#446—1221no—
348#446—1221no—
349#436—1221no—
350#436—1221no—
351#435—1222no—
352#435—1222no—
353#432—612no—
354#431—612no—
355#431C2×D6111yescharacteristic
356#431C2×D6111yescharacteristic
357#423—612no—
358C2×C4⋊C8—611no—
359C4×M16—612no—
360C2×C4×C8—611no—
361Q8⋊C8—612no—
362Q8⋊C8—612no—
363Q8⋊C8—611no—
364Q8⋊C8—611no—
365Q8⋊C8—612no—
366Q8⋊C8C2×D6112yes—
367Q8⋊C8C2×D6112yes—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—612no—
371C61#1745411yescharacteristic, complemented, Sylow
3722−1+4—1224no—
3732−1+4—1224no—
374C2×#41—611no—
375C2×#41—612no—
376C2×#41—611no—
377C2×#41—1222no—
378C2×#41—1221no—
379C22×Q8—612no—
380C22×Q8—1221no—
381C22×Q8C4×D6111yescharacteristic
382C2×M16—1222no—
383C2×M16—1222no—
384C22×C8—1221no—
385C22×C8—1221no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C4×D6112yes—
403C4×Q8C22×D6112yes—
404C4×D4—612no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—611no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—611no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1221no—
417C4×D4—1222no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—611no—
421#117—612no—
422#117—611no—
423#117—612no—
424#117—1222no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428#117—1222no—
429#117—1221no—
430#117—1221no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—612no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4C4×D6111yescharacteristic
437C2×C42—21no—
438C2×C42—612no—
439C2×C42—1222no—
440C2×C42—1222no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443C4⋊C8—21no—
444C4⋊C8—612no—
445C4⋊C8—1222no—
446C4⋊C8—1222no—
447C4⋊C8—1221no—
448C4⋊C8C22×D6111yescharacteristic
449C4⋊C8C22×D6111yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—21no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454#97—1222no—
455#97—1222no—
456C4×C8—21no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—1221no—
460C4×C8C22×D6112yes—
461#41—614no—
462#41—12216no—
463#41—1224no—
464#41—1228no—
465#41—1224no—
466#41—1222no—
467#41—1222no—
468#41—1222no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—611no—
472C2×Q8—611no—
473C2×Q8—1222no—
474C2×Q8—1224no—
475C2×Q8—1228no—
476C2×Q8—1224no—
477C2×Q8#1175818914yescomplemented
478C2×Q8C2×C4×D6112yes—
479C2×D4—612no—
480C2×D4—611no—
481C2×D4—611no—
482C2×D4—1222no—
483C2×D4—1221no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4C2×C4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499M16—4882no—
500M16—2444no—
501C2×C8—22no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1222no—
507C2×C8—4881no—
508C2×C8—2442no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—612no—
513C4⋊C4—611no—
514C4⋊C4—612no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4C2×C4×D6112yes—
525C22⋊C4—612no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1221no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C42—22no—
534C42—22no—
535C42—612no—
536C42—612no—
537C42—612no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C42—1222no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42C2×C4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42D4×D6111yescharacteristic
548C42C23×D6111yescharacteristic
549C23—612no—
550C23—1221no—
551C23C22⋊C4×D6111yescharacteristic
552Q8—24no—
553Q8—22no—
554Q8—612no—
555Q8—1228no—
556Q8—1222no—
557Q8—1221no—
558Q8C2×#1175818914yes—
559D4—612no—
560D4—1228no—
561D4—1224no—
562D4—1224no—
563D4—1222no—
564D4—1221no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—81no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—1228no—
576C2×C4—1228no—
577C2×C4—1221no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—4881no—
586C2×C4—4882no—
587C2×C4—4882no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C22⋊C4×D6111yescharacteristic
592C2×C4C22⋊C4×D6111yescharacteristic
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4M16×D6112yes—
595C2×C4C2×#1175818912yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C8—42no—
600C8—41no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—1222no—
611C22—2441no—
612C22—2441no—
613C22#1683972412yes—
614C22C2×C22⋊C4×D6111yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—24no—
619C4—42no—
620C4—1222no—
621C4—1222no—
622C4—1222no—
623C4—1222no—
624C4—2441no—
625C4—2441no—
626C4#1683972412yes—
627C4C2×M16×D6111yescharacteristic
628C4C22×#1175818911yescharacteristic
629C4#1684057111yescharacteristic
630C4#1684057811yescharacteristic
631C2—22no—
632C2—2442no—
633C2—2442no—
634C2C2×#1683972411yescharacteristic
635C2#2145435111yescharacteristic
636C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1479C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519825C4
cosocleC42×C61C24
hypercenter#1479D61
hypocenterC61#17454
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #21454932C2semidirect
3. #21454780C2semidirect
4. #21444264C2semidirect
5. #21444257C2semidirect
6. #21444252C2semidirect
7. #21444246C2semidirect
8. #21444207C2semidirect
9. #21444199C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#1479C2semidirect
144. C2×Q8×C61M16semidirect
163. #11758222M16semidirect
320. #1479D61semidirect
371. C61#17454semidirect
477. C2×Q8#11758189semidirect

Maximal subgroups

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

subgroupindex
C4×#145198252
#214549322
#214547802
#214442642
#214442572
#214442522
#214442462
#214442072
#214441992
C2×Q8⋊#924182
C61×#14792
#1745461