Finite Group Databasebeta

← #29646397

Subgroups — #29646397

Order 15616.

1206 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#21454928C211yesmaximal, characteristic, complemented
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21444214C211yesmaximal, characteristic, complemented
5#21444212C212yesmaximal, complemented
6#21444205C212yesmaximal, complemented
7#21444204C211yesmaximal, characteristic, complemented
8#21444202C211yesmaximal, characteristic, complemented
9#21444200C212yesmaximal, complemented
10Q8⋊C8×D61C212yesmaximal, complemented
11C2×C61×Q8⋊C8C211yesmaximal, 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—
28C8×Dic122—21no—
29#16839886—21no—
30C8×D244—21no—
31#16839827—21no—
32C8×C22⋊D61—22no—
33C8×C22⋊D61—22no—
34#16839815C2211yescharacteristic
35#16839815C2211yescharacteristic
36#16839814C2212yes—
37#16839813C2212yes—
38C4⋊C8×D61C2212yes—
39C4×C8×D61C2212yes—
40#16839494C2211yescharacteristic
41#16839494C2211yescharacteristic
42Q8⋊#92418C2212yes—
43Q8⋊#92418C2212yes—
44#16839492C2212yes—
45#16839491C2211yescharacteristic
46#16839491C2211yescharacteristic
47#16839489C2212yes—
48C2×C4×Q8×C61C2211yescharacteristic
49C2×C4⋊C8×C61C2211yescharacteristic
50C2×C4×C8×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53#14519825—24no—
54#14519825C818yescomplemented
55C2×#11758226—21no—
56C2×#11758226C2×C412yes—
57C2×#11758222—22no—
58C2×#11758222C2×C411yescharacteristic
59C2×#11758222C2×C411yescharacteristic
60C2×Q8×D61—21no—
61C2×Q8×D61C2×C412yes—
62C2×C8×D61—22no—
63C2×C8×D61—22no—
64Q8×Dic61—21no—
65Q8×Dic61—21no—
66Q8×Dic61C2×C412yes—
67Q8×Dic61C2×C412yes—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×Dic122C2×C411yescharacteristic
71C4×Dic122D411yescharacteristic
72C4×Dic122D411yescharacteristic
73C4×Dic122C2311yescharacteristic
74#14519742—21no—
75#14519742—22no—
76#14519742—22no—
77#14519742C2×C412yes—
78C4×D244—22no—
79C4×D244—22no—
80C4×D244C2×C411yescharacteristic
81C4×D244D411yescharacteristic
82C4×D244D411yescharacteristic
83C4×D244C2311yescharacteristic
84#14519739—21no—
85#14519739—22no—
86#14519739—22no—
87#14519739C2×C412yes—
88C4×C22⋊D61—21no—
89C4×C22⋊D61—21no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92C4×C22⋊D61D412yes—
93C4×C22⋊D61D412yes—
94#14519736—22no—
95#14519736C2×C411yescharacteristic
96#14519736C2×C411yescharacteristic
97#14519735—22no—
98#14519735—21no—
99#14519735—22no—
100#14519735C2×C412yes—
101#14519734—21no—
102#14519734—22no—
103#14519734—22no—
104#14519734C2×C412yes—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61C2×C412yes—
109C42×D61—22no—
110C42×D61—21no—
111C42×D61—22no—
112C42×D61C2312yes—
113C4⋊#92418—21no—
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418—21no—
119C22⋊#92418—21no—
120#14519689—22no—
121#14519689—22no—
122C4×#92418—21no—
123C4×#92418C2312yes—
124C8×Dic61—22no—
125C8×Dic61—22no—
126C22×Q8×C61C2×C411yescharacteristic
127C22×C8×C61—21no—
128C22×C8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yes—
132C4×Q8×C61C2312yes—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C4×C8×C61—21no—
140C4×C8×C61C2311yescharacteristic
141C4×C8×C61C2311yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—22no—
144C2×Q8×C61C2×C814yescomplemented
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#11758226C2×C818yescomplemented
156Q8×D61—24no—
157Q8×D61—28no—
158Q8×D61C2×C818yescomplemented
159#11758222—22no—
160#11758222—216no—
161#11758222C2×C814yes—
162#11758222C2×C814yescomplemented
163#11758222M1614yes—
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—
183C2×C8×C61—42no—
184C2×C8×C61—81no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61C22×C412yes—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195C22⋊Dic61—21no—
196C22⋊Dic61C22⋊C412yes—
197#11758198—22no—
198#11758198—21no—
199#11758198—21no—
200#11758198—22no—
201#11758198C22⋊C412yes—
202#11758198C22⋊C412yes—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22⋊C411yescharacteristic
206C4⋊Dic61C22⋊C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C411yescharacteristic
209#11758196—21no—
210#11758196—21no—
211#11758196—22no—
212#11758196—22no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61C22×C412yes—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D412yes—
225C2×#92418—21no—
226C2×#92418—22no—
227C2×#92418—21no—
228C8×D61—44no—
229C8×D61—82no—
230C23×C61C2×C22⋊C411yescharacteristic
231C22×D61—21no—
232C22×D61C2×C22⋊C412yes—
233Q8×C61—22no—
234Q8×C61—24no—
235Q8×C61C22×C814yes—
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×C61C22×C812yes—
247C2×C4×C61C2×M1612yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—24no—
252C22⋊D61C22⋊C818yes—
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—
261D244C22⋊C814yes—
262D244C22×C812yes—
263C4×D61—28no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—24no—
267C4×D61—28no—
268C4×D61—42no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—82no—
272C4×D61—81no—
273C4×D61—82no—
274C4×D61C22×C814yes—
275C4×D61C2×M1614yes—
276Dic122—21no—
277Dic122—28no—
278Dic122C22⋊C814yes—
279Dic122C22×C812yes—
280C8×C61—22no—
281C8×C61—41no—
282C8×C61—42no—
283#92418—42no—
284#92418—41no—
285#92418—41no—
286#17439—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C2×C22⋊C812yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61C2×C22⋊C814yes—
297C4×C61—24no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61C2×C22⋊C812yes—
303C4×C61C23×C811yescharacteristic
304C4×C61C22×M1611yescharacteristic
305C4×C61#57711yescharacteristic
306C4×C61#57811yescharacteristic
307Dic61—22no—
308Dic61—22no—
309Dic61—22no—
310Dic61—41no—
311Dic61—41no—
312Dic61C2×C22⋊C814yes—
313C4×2−1+4—611no—
314#2988—611no—
315C8×#41—611no—
316#1479—612no—
317#1479—611no—
318#1477—611no—
319#1477—612no—
320#1476—611no—
321#1476—612no—
322C2×Q8⋊C8—612no—
323C2×Q8⋊C8D6111yescharacteristic, complemented
324C2×C61—22no—
325C2×C61C22×C22⋊C811yescharacteristic
326C2×C61#289811yescharacteristic
327C2×C61#290111yescharacteristic
328D61—24no—
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
347C8×Q8—1221no—
348C8×Q8—1221no—
349C8×D4—1221no—
350C8×D4—1221no—
351C8×D4—1222no—
352C8×D4—1222no—
353#431—611no—
354#431—612no—
355#431—611no—
356#431—612no—
357C2×C4⋊C8—612no—
358C2×C4⋊C8C2×D6111yescharacteristic
359C2×C4×C8—612no—
360C2×C4×C8C2×D6111yescharacteristic
361Q8⋊C8—612no—
362Q8⋊C8—612no—
363Q8⋊C8—612no—
364Q8⋊C8—611no—
365Q8⋊C8—611no—
366Q8⋊C8C2×D6112yes—
367Q8⋊C8C2×D6112yes—
368D4⋊C8—612no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371C61#1743911yescharacteristic, complemented, Sylow
3722−1+4—618no—
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
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—1222no—
385C22×C8—1222no—
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—611no—
445C4⋊C8—611no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1221no—
449C4⋊C8C22×D6112yes—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C4×C8—21no—
455C4×C8—612no—
456C4×C8—1222no—
457C4×C8—1222no—
458C4×C8—1221no—
459C4×C8C22×D6111yescharacteristic
460C4×C8C22×D6111yescharacteristic
461#41—614no—
462#41—614no—
463#41—618no—
464#41—614no—
465#41—12216no—
466#41—1224no—
467#41—1228no—
468#41—1222no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—611no—
472C2×Q8—618no—
473C2×Q8—611no—
474C2×Q8—1222no—
475C2×Q8—1224no—
476C2×Q8—1228no—
477C2×Q8C8×D6114yescomplemented
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
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—42no—
503C2×C8—81no—
504C2×C8—1222no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—4882no—
508C2×C8—2444no—
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—614no—
555Q8—612no—
556Q8—1228no—
557Q8—1221no—
558Q8C2×C8×D6114yes—
559D4—618no—
560D4—614no—
561D4—612no—
562D4—1228no—
563D4—1224no—
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—614no—
574C2×C4—614no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1228no—
578C2×C4—1228no—
579C2×C4—1221no—
580C2×C4—1224no—
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×C4C2×C8×D6112yes—
595C2×C4M16×D6112yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C8—22no—
600C8—41no—
601C8—42no—
602C8—2442no—
603C8—2441no—
604C8—2441no—
605C22—21no—
606C22—21no—
607C22—614no—
608C22—1222no—
609C22—1222no—
610C22—1222no—
611C22—2441no—
612C22—2441no—
613C22C22⋊C8×D6112yes—
614C22C2×C22⋊C4×D6111yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—24no—
619C4—42no—
620C4—614no—
621C4—1222no—
622C4—1222no—
623C4—1222no—
624C4—2441no—
625C4—2441no—
626C4C22⋊C8×D6112yes—
627C4C22×C8×D6111yescharacteristic
628C4C2×M16×D6111yescharacteristic
629C4#1684056811yescharacteristic
630C4#1684057511yescharacteristic
631C2—22no—
632C2—1224no—
633C2—2442no—
634C2C2×C22⋊C8×D6111yescharacteristic
635C2#2145434711yescharacteristic
636C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC4×C61C23×C8
FrattiniC42C23×D61
FittingC2×C61×Q8⋊C8C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519825C4
cosocleC42×C61C24
hypercenterC2×Q8⋊C8D61
hypocenterC61#17439
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #21454928C2semidirect
3. C8×#11758222C2semidirect
4. #21444214C2semidirect
5. #21444212C2semidirect
6. #21444205C2semidirect
7. #21444204C2semidirect
8. #21444202C2semidirect
9. #21444200C2semidirect
10. Q8⋊C8×D61C2semidirect
11. C2×C61×Q8⋊C8C2semidirect
54. #14519825C8semidirect
144. C2×Q8×C61C2×C8semidirect
155. #11758226C2×C8semidirect
158. Q8×D61C2×C8semidirect
162. #11758222C2×C8semidirect
323. C2×Q8⋊C8D61semidirect
371. C61#17439semidirect
477. C2×Q8C8×D61semidirect

Maximal subgroups

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

subgroupindex
C4×#145198252
#214549282
C8×#117582222
#214442142
#214442122
#214442052
#214442042
#214442022
#214442002
Q8⋊C8×D612
C2×C61×Q8⋊C82
#1743961