Finite Group Databasebeta

← #30610483

Subgroups — #30610483

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
1#21462738C211yesmaximal, characteristic, complemented
2#21462628C211yesmaximal, characteristic, complemented
3#21461161C212yesmaximal, complemented
4#21461154C212yesmaximal, complemented
5#21461150C212yesmaximal, complemented
6#21461147C212yesmaximal, complemented
7#21461121C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92420C211yesmaximal, characteristic, complemented
112−1+4×Dic61C211yesmaximal, characteristic
12#16840813—21no—
13#16840813—21no—
14#16840797—21no—
15#16840797—21no—
16#16840793—22no—
17#16840793—22no—
18#16840783C2212yes—
19#16840779C2212yes—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22#16840765C2212yes—
23C2×C4⋊#92420C2211yescharacteristic
24C4×#11758217C2212yes—
25C2×C4×#92420C2211yescharacteristic
26Q8⋊#92420C2212yes—
27Q8⋊#92420C2212yes—
28Q8⋊#92420C2211yescharacteristic
29Q8⋊#92420C2211yescharacteristic
30Q8⋊#92420C2212yes—
31Q8⋊#92420C2212yes—
32Q8⋊#92420C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840245—22no—
37#16840245—21no—
38#16840245C412yes—
39#16840245C411yescharacteristic
40#16840245C411yescharacteristic
41#16840240—21no—
42#16840240C412yes—
43#16840240C411yescharacteristic
44#41×Dic61—22no—
45#41×Dic61—21no—
46#41×Dic61C2211yescharacteristic
47#41×Dic61C2212yes—
48#41×Dic61C2211yescharacteristic
49C2×Q8×Dic61—21no—
50C2×Q8×Dic61C2211yescharacteristic
51C2×Q8×Dic61C2212yes—
52C2×2−1+4×C61C411yescharacteristic
53C2×#11758217—22no—
54C2×#11758217—22no—
55C22×#92420—21no—
56C22×#92420—21no—
57#14519844—22no—
58#14519844—22no—
59C4⋊#92420—21no—
60C4⋊#92420—21no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64C22⋊#92420—21no—
65C22⋊#92420—21no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75Q8×Dic61—21no—
76Q8×Dic61—22no—
77Q8×Dic61—21no—
78Q8×Dic61—22no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81Q8×Dic61—22no—
82Q8×Dic61—21no—
83Q8×Dic61—22no—
84Q8×Dic61C2×C412yes—
85Q8×Dic61C2×C412yes—
86Q8×Dic61C2×C412yes—
87Q8×Dic61C2×C411yescharacteristic
88Q8×Dic61C2×C412yes—
89Q8×Dic61D411yescharacteristic
90Q8×Dic61D411yescharacteristic
91Q8×Dic61C2312yes—
92Q8×Dic61C2311yescharacteristic
93D4×Dic61—22no—
94D4×Dic61—21no—
95D4×Dic61—21no—
96D4×Dic61—22no—
97D4×Dic61—21no—
98D4×Dic61—22no—
99D4×Dic61—22no—
100D4×Dic61—22no—
101D4×Dic61—22no—
102D4×Dic61C2×C411yescharacteristic
103D4×Dic61C2×C412yes—
104D4×Dic61D412yes—
105D4×Dic61D411yescharacteristic
106D4×Dic61D412yes—
107D4×Dic61D411yescharacteristic
108D4×Dic61C2311yescharacteristic
109#14519736—22no—
110#14519736—22no—
111#14519736—22no—
112#14519736—22no—
113#14519736—21no—
114#14519736—21no—
115#14519736—22no—
116#14519736C2×C412yes—
117#14519736C2×C411yescharacteristic
118#14519736C2×C412yes—
119#14519736C2×C411yescharacteristic
120C2×C4⋊Dic61—21no—
121C2×C4⋊Dic61—22no—
122C2×C4⋊Dic61—22no—
123C2×C4⋊Dic61—21no—
124C2×C4⋊Dic61C2×C412yes—
125C2×C4⋊Dic61C2×C411yescharacteristic
126C2×C4×Dic61—21no—
127C2×C4×Dic61—22no—
128C2×C4×Dic61—21no—
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2312yes—
1322−1+4×C61—24no—
1332−1+4×C61—24no—
134C2×#41×C61—21no—
135C2×#41×C61—22no—
136C2×#41×C61C2×C412yes—
137C2×#41×C61C2×C411yescharacteristic
138C2×#41×C61C2×C411yescharacteristic
139C22×Q8×C61—21no—
140C22×Q8×C61C2×C412yes—
141C22×Q8×C61C2×C411yescharacteristic
142#41×C61—216no—
143#41×C61—28no—
144#41×C61—24no—
145#41×C61—24no—
146#41×C61—22no—
147#41×C61—22no—
148#41×C61—22no—
149#41×C61M1614yescomplemented
150C2×Q8×C61—28no—
151C2×Q8×C61—24no—
152C2×Q8×C61—24no—
153C2×Q8×C61—22no—
154C2×Q8×C61—21no—
155C2×Q8×C61—22no—
156C2×Q8×C61C22⋊C411yescharacteristic
157C2×Q8×C61M1614yescomplemented
158C2×Q8×C61C22×C411yescharacteristic
159C2×Q8×C61C22×C412yes—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61C22⋊C412yes—
163C2×D4×C61C22⋊C411yescharacteristic
164C2×D4×C61C22×C411yescharacteristic
165C22×C4×C61—22no—
166C22×C4×C61—21no—
167C22×C4×C61—22no—
168C22×C4×C61—21no—
169C22×C4×C61C22×C412yes—
170C22×C4×C61C22×C411yescharacteristic
171C22×Dic61—22no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—22no—
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D412yes—
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D412yes—
180#11758217—44no—
181#11758217—82no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—21no—
186C2×#92420—22no—
187C2×#92420—22no—
188C2×#92420—42no—
189C2×#92420—81no—
190C22⋊Dic61—22no—
191C22⋊Dic61—21no—
192C22⋊Dic61—21no—
193C22⋊Dic61—22no—
194C22⋊Dic61—21no—
195C22⋊Dic61C22⋊C412yes—
196C22⋊Dic61C22⋊C412yes—
197C22⋊Dic61C22⋊C412yes—
198C4⋊Dic61—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22⋊C412yes—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C412yes—
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22×C411yescharacteristic
212C4⋊Dic61C22×C412yes—
213C4⋊Dic61C22×C411yescharacteristic
214C4×Dic61—22no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—21no—
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C412yes—
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D412yes—
228C4×Dic61C2×D412yes—
229C4×Dic61C2411yescharacteristic
230C23×C61—21no—
231C23×C61C2×C22⋊C412yes—
232C23×C61C2×C22⋊C411yescharacteristic
233Q8×C61—28no—
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—24no—
238Q8×C61C2×M1612yes—
239Q8×C61C2×M1614yes—
240D4×C61—28no—
241D4×C61—24no—
242D4×C61—24no—
243D4×C61—21no—
244D4×C61—22no—
245D4×C61C2×M1612yes—
246C2×C4×C61—28no—
247C2×C4×C61—28no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×M1612yes—
257C2×C4×C61C2×M1612yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—42no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—81no—
271C2×Dic61—82no—
272C2×Dic61—81no—
273C2×Dic61—82no—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C22×D411yescharacteristic
279C2×Dic61C22×D411yescharacteristic
280#92420—22no—
281#92420—41no—
282#92420—42no—
283#92420—42no—
284#92420—41no—
285#92420—41no—
286#17454—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61#40712yes—
296C22×C61C22×C22⋊C411yescharacteristic
297C4×C61—24no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61#40712yes—
301C4×C61C22×M1611yescharacteristic
302C4×C61C22×M1611yescharacteristic
303Dic61—21no—
304Dic61—22no—
305Dic61—21no—
306Dic61—21no—
307Dic61—22no—
308Dic61—41no—
309Dic61—42no—
310Dic61—41no—
311Dic61#57711yescharacteristic
312Dic61#57811yescharacteristic
313C4×2−1+4—611no—
314#2989—611no—
315#2965—611no—
316#1491—612no—
317#1489—612no—
318#1488—612no—
319#1487—612no—
320#1479—611no—
321#1477—611no—
322#1476—611no—
323C2×Q8⋊C8—611no—
324C2×C61—22no—
325C2×C61—42no—
326C2×C61—42no—
327C2×C61C2×#40711yescharacteristic
328C2×C61#289811yescharacteristic
329C2×C61#290111yescharacteristic
330C2×2−1+4#692311yescharacteristic
331#520—611no—
332#520—612no—
333#520—611no—
334#520—1222no—
335#520—1221no—
336#519—612no—
337#519—611no—
338#519—1221no—
339C4×#41—611no—
340C4×#41—611no—
341C4×#41—612no—
342C4×#41—1222no—
343C4×#41—1221no—
344C2×C4×Q8—611no—
345C2×C4×Q8—612no—
346C2×C4×Q8—1221no—
347#446—1221no—
348#446—1221no—
349#436—1221no—
350#436—1221no—
351#435—1222no—
352#435—1222no—
353#432—612no—
354#431—611no—
355#431—612no—
356#431—611no—
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⋊C8—612no—
367Q8⋊C8—612no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—612no—
371C61#1745411yescharacteristic, complemented, Sylow
3722−1+4—24no—
3732−1+4—24no—
374C2×#41—21no—
375C2×#41—22no—
376C2×#41C2×#692312yes—
377C2×#41C2×#692311yescharacteristic
378C2×#41C2×#692311yescharacteristic
379C22×Q8—21no—
380C22×Q8C2×#692312yes—
381C22×Q8C2×#692311yescharacteristic
382C2×M16—1222no—
383C2×M16—1222no—
384C22×C8—1221no—
385C22×C8—1221no—
386C4×Q8—611no—
387C4×Q8—611no—
388C4×Q8—612no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—1221no—
396C4×Q8—1222no—
397C4×Q8—1221no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—612no—
405C4×D4—611no—
406C4×D4—612no—
407C4×D4—611no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—612no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1222no—
416C4×D4—1221no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1221no—
420#117—612no—
421#117—611no—
422#117—612no—
423#117—611no—
424#117—1221no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428#117—1222no—
429#117—1222no—
430#117—1221no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1222no—
437C2×C42—611no—
438C2×C42—612no—
439C2×C42—1221no—
440C2×C42—1222no—
441C2×C42—1221no—
442C2×C42—1222no—
443C4⋊C8—612no—
444C4⋊C8—611no—
445C4⋊C8—611no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454#97—1222no—
455#97—1222no—
456C4×C8—611no—
457C4×C8—611no—
458C4×C8—612no—
459C4×C8—1221no—
460C4×C8—1221no—
461#41—216no—
462#41—28no—
463#41—24no—
464#41—24no—
465#41—22no—
466#41—22no—
467#41—22no—
468#41#1175821714yescomplemented
469C2×Q8—28no—
470C2×Q8—24no—
471C2×Q8—24no—
472C2×Q8—22no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8#1175821714yescomplemented
476C2×Q8C22⋊#692311yescharacteristic
477C2×Q8C22×#692311yescharacteristic
478C2×Q8C22×#692312yes—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4C22⋊#692312yes—
482C2×D4C22⋊#692311yescharacteristic
483C2×D4C22×#692311yescharacteristic
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—612no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4C22×#692312yes—
498C22×C4C22×#692311yescharacteristic
499M16—4882no—
500M16—2444no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1222no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1222no—
507C2×C8—4881no—
508C2×C8—2442no—
509C4⋊C4—612no—
510C4⋊C4—611no—
511C4⋊C4—612no—
512C4⋊C4—611no—
513C4⋊C4—611no—
514C4⋊C4—612no—
515C4⋊C4—611no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C22⋊C4—612no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—1221no—
529C22⋊C4—1222no—
530C22⋊C4—1221no—
531C22⋊C4—1222no—
532C22⋊C4—1221no—
533C42—611no—
534C42—611no—
535C42—612no—
536C42—612no—
537C42—611no—
538C42—611no—
539C42—612no—
540C42—1221no—
541C42—1222no—
542C42—1222no—
543C42—1222no—
544C42—1222no—
545C42—1221no—
546C42—1222no—
547C42—1222no—
548C42—1221no—
549C23—21no—
550C23C2×C22⋊#692312yes—
551C23C2×C22⋊#692311yescharacteristic
552Q8—28no—
553Q8—24no—
554Q8—22no—
555Q8—21no—
556Q8—22no—
557Q8C2×#1175821712yes—
558Q8C2×#1175821714yes—
559D4—28no—
560D4—24no—
561D4—24no—
562D4—22no—
563D4—21no—
564D4C2×#1175821712yes—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—612no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—611no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—4881no—
587C2×C4—4882no—
588C2×C4—4881no—
589C2×C4—4882no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2442no—
594C2×C4C2×C22⋊#692312yes—
595C2×C4C2×C22⋊#692311yescharacteristic
596C2×C4C2×#1175821712yes—
597C2×C4C2×#1175821712yes—
598C2×C4C23×#692311yescharacteristic
599C8—1222no—
600C8—2441no—
601C8—2442no—
602C8—2441no—
603C8—2441no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22#1684073212yes—
614C22C22×C22⋊#692311yescharacteristic
615C4—24no—
616C4—22no—
617C4—22no—
618C4—611no—
619C4—611no—
620C4—1221no—
621C4—1222no—
622C4—1221no—
623C4—1221no—
624C4—1222no—
625C4—2441no—
626C4—2442no—
627C4—2441no—
628C4#1684073212yes—
629C4C22×#1175821711yescharacteristic
630C4C22×#1175821711yescharacteristic
631C2—22no—
632C2—42no—
633C2—42no—
634C2C2×#1684073211yescharacteristic
635C2#2146244111yescharacteristic
636C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC2×2−1+4×C61C4
socleC22×C61C22×C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×2−1+4#6923
hypocenterC61#17454
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #21462738C2semidirect
2. #21462628C2semidirect
3. #21461161C2semidirect
4. #21461154C2semidirect
5. #21461150C2semidirect
6. #21461147C2semidirect
7. #21461121C2semidirect
8. #21461114C2semidirect
9. #21461110C2semidirect
10. C2×Q8⋊#92420C2semidirect
149. #41×C61M16semidirect
157. C2×Q8×C61M16semidirect
371. C61#17454semidirect
468. #41#11758217semidirect
475. C2×Q8#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627382
#214626282
#214611612
#214611542
#214611502
#214611472
#214611212
#214611142
#214611102
C2×Q8⋊#924202
2−1+4×Dic612
#1745461