Finite Group Databasebeta

← #29840165

Subgroups — #29840165

Order 15616.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2#21448789C214yesmaximal, complemented
3#21448781C214yesmaximal, complemented
4#21448758C214yesmaximal, complemented
5#21448754C214yesmaximal, complemented
6#21448429C211yesmaximal, characteristic, complemented
7#21448428C211yesmaximal, characteristic, complemented
8#21448427C211yesmaximal, characteristic, complemented
9#21448408C212yesmaximal
10#21448224C211yesmaximal, characteristic, complemented
11#21448215C212yesmaximal, complemented
12#21448214C212yesmaximal, complemented
13#382×D61C211yesmaximal, characteristic, complemented
14#21448173C212yesmaximal, complemented
15C61×#2314C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17#16840230C4116yescomplemented
18C2×#14519742C2212yes—
19C2×#14519739C2212yes—
20C2×#14519737C2212yes—
21C2×#14519737C2212yes—
22C2×#14519735C2212yes—
23C2×#14519734C2211yescharacteristic
24C2×#14519734C2211yescharacteristic
25C2×C4⋊C4×D61C2212yes—
26C22×C4×Dic61C2211yescharacteristic
27C2×C42×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839666C2214yes—
30#16839662C2214yes—
31#16839655C2214yes—
32#16839654C2214yes—
33#16839652C2214yes—
34#16839649C2214yes—
35#16839648C2214yes—
36#16839643C2214yes—
37#16839640C2214yes—
38#16839638C2214yes—
39#16839632C2211yescharacteristic
40#16839629C2214yes—
41#16839629C2214yes—
42#16839628C2211yescharacteristic
43#16839627C2212yes—
44#16839627C2212yes—
45#16839626C2214yes—
46#16839624C2214yes—
47#16839623C2214yes—
48#16839622C2212yes—
49#16839619C2212yes—
50#16839619C2212yes—
51#16839619C2214yes—
52C4⋊C4×Dic61C2212yes—
53C4⋊C4×Dic61C2212yes—
54C4⋊C4×Dic61C2214yes—
55C4×#11758196C2214yes—
56C4×C4⋊Dic61C2212yes—
57#16839613C2214yes—
58C22⋊C4×Dic61C2214yes—
59C4×#11758198C2214yes—
60C4×#11758198C2214yes—
61#16839609C2214yes—
62#16839608C2214yes—
63#16839607C2214yes—
64#95×D61C2214yes—
65C2×#117×C61C2211yescharacteristic
66C61×#388C2214yes—
67C61×#387C2214yes—
68C61×#384C2211yescharacteristic
69C61×#382C2211yescharacteristic
70C61×#380C2212yes—
71C4×C4⋊C4×C61C2212yes—
72C2×#11758222—24no—
73C2×#11758222—24no—
74C2×#11758222C2×C418yes—
75C22×Dic122C2311yescharacteristic
76C22×D244C2311yescharacteristic
77C22×C22⋊D61C2311yescharacteristic
78C22×C22⋊D61C2311yescharacteristic
79C23×Dic61C2312yes—
80C22×C4×D61C2311yescharacteristic
81C22×C4×D61C2314yes—
82C22×C4×D61C2312yes—
83C22×C4×D61C2312yes—
84C22×C4×D61C2311yescharacteristic
85#14519742C2×C4116yescomplemented
86#14519739C2×C4116yes—
87#14519737C2×C4116yes—
88#14519737C2×C4116yescomplemented
89#14519735—28no—
90#14519734C2×C418yes—
91#14519734C2×C418yescomplemented
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2312yes—
94C2×C4⋊Dic61C2314yes—
95C2×#11758196C2314yes—
96C2×#11758196C2312yes—
97C2×#11758196C2312yes—
98C4⋊C4×D61—28no—
99C2×C22⋊Dic61C2314yes—
100C2×#11758198C2314yes—
101C2×#11758198C2314yes—
102C2×#11758198C2314yes—
103C2×#11758198C2312yes—
104C2×#11758198C2312yes—
105C2×C4×Dic61C2×C418yescomplemented
106C2×C4×Dic61C2314yes—
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2314yes—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2312yes—
112C2×C4×Dic61C2312yes—
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2312yes—
115C42×D61C2×C418yes—
116C42×D61C2×C418yescomplemented
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#14519684C2314yes—
127#14519684C2314yes—
128C23×C4×C61C2311yescharacteristic
129#117×C61C2×C418yescomplemented
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2314yes—
133C2×C4⋊C4×C61C2312yes—
134C2×C22⋊C4×C61C2314yes—
135C2×C22⋊C4×C61C2312yes—
136C2×C42×C61C2314yes—
137C2×C42×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139#95×C61C2314yes—
140#95×C61C2314yes—
141#95×C61C2314yes—
142C24×C61C2411yescharacteristic
143C23×D61C2411yescharacteristic
144C23×D61C2411yescharacteristic
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61C22×C414yes—
148C22×C4×C61C2414yes—
149C22×C4×C61C2414yes—
150C22×C4×C61C2412yes—
151C22×C4×C61C2412yes—
152C22×C4×C61C2411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61C22×C414yes—
159C2×C22⋊D61C22×C414yes—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61C22×C418yes—
163C22×Dic61C2414yes—
164C22×Dic61C2414yes—
165C22×Dic61C2412yes—
166C22×Dic61C2411yescharacteristic
167C22×Dic61C2412yes—
168C22×Dic61C2411yescharacteristic
169#11758222—232no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244C22×C414yes—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—28no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×C4×D61—44no—
181C2×C4×D61—44no—
182C2×C4×D61C22×C414yes—
183C2×C4×D61C22×C418yes—
184C2×C4×D61C22×C418yes—
185C2×C4×D61C22×C414yes—
186C2×C4×D61#4114yes—
187C2×C4×D61#4114yes—
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122C22×C414yes—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—24no—
193C4⋊C4×C61—44no—
194C4⋊C4×C61C22×C418yes—
195C22⋊C4×C61—44no—
196C22⋊C4×C61C22×C418yes—
197C42×C61—44no—
198C42×C61C22×C414yes—
199C42×C61C22×C414yes—
200C22⋊Dic61—44no—
201#11758198—28no—
202#11758198—44no—
203#11758198—44no—
204#11758198C22×C418yes—
205#11758198C22×C418yes—
206C4⋊Dic61—24no—
207C4⋊Dic61—24no—
208C4⋊Dic61—44no—
209#11758196—28no—
210#11758196C22×C418yes—
211#11758196C22×C418yes—
212C4×Dic61—28no—
213C4×Dic61—44no—
214C4×Dic61—44no—
215C4×Dic61—44no—
216C4×Dic61C22×C414yes—
217C4×Dic61C22×C418yes—
218C4×Dic61C22×C418yes—
219C4×Dic61C22×C414yes—
220C4×Dic61C22×C418yes—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61C23×C412yes—
226C23×C61C2511yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61C23×C412yes—
234C22×D61C23×C412yes—
235C22×D61C2×#4112yes—
236C22×D61C2×#4112yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—28no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—24no—
250C2×C4×C61C23×C412yes—
251C2×C4×C61C23×C414yes—
252C2×C4×C61C23×C414yes—
253C2×C4×C61C23×C412yes—
254C2×C4×C61C2×#4112yes—
255C2×C4×C61C2×#4112yes—
256C22⋊D61—28no—
257C22⋊D61—28no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—48no—
271C2×Dic61C23×C414yes—
272C2×Dic61C23×C414yes—
273C2×Dic61C23×C412yes—
274C2×Dic61C23×C412yes—
275C2×Dic61C2×#4112yes—
276C2×Dic61C2×#4112yes—
277D244—28no—
278C4×D61—28no—
279C4×D61—48no—
280C4×D61—48no—
281C4×D61—88no—
282C4×D61#117116yes—
283Dic122—28no—
284#34189—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—24no—
287C22×C61—42no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C24×C411yescharacteristic
291C22×C61C22×#4111yescharacteristic
292C22×C61C22×#4111yescharacteristic
293C22×C61#58512yes—
294C22×C61#58512yes—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—24no—
301C2×D61—42no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61C2×#11714yes—
305C4×C61—22no—
306C4×C61—24no—
307C4×C61—24no—
308C4×C61—44no—
309C4×C61—44no—
310C4×C61C2×#11714yes—
311Dic61—22no—
312Dic61—24no—
313Dic61—24no—
314Dic61—44no—
315Dic61—44no—
316Dic61C2×#11714yes—
317C2×C4×#41—611no—
318#2357—614no—
319#2356—614no—
320#2353—614no—
321#2353—614no—
322#2314—611no—
323#2314—611no—
324#2314—611no—
325#2314D6111yescharacteristic, complemented
326#2311—612no—
327#2285—611no—
328#2283—612no—
329#2283—612no—
330C2×#382—611no—
331#2274—612no—
332C2×C61—42no—
333C2×C61C22×#11711yescharacteristic
334C2×C61#342812yes—
335C2×C61#351012yes—
336C2×C61#351211yescharacteristic
337C2×C61#351211yescharacteristic
338D61—24no—
339D61—42no—
340C22×#41—611no—
341C4×#41—6116no—
342C2×C4×Q8—612no—
343C2×C4×D4—612no—
344C2×C4×D4—612no—
345C2×C4×D4—612no—
346C2×#117—612no—
347C2×#117—611no—
348C2×#117—611no—
349C2×#117C2×D6111yescharacteristic
350C22×C4⋊C4—612no—
351C22×C42—611no—
352C22×C42—611no—
353C22×C42—611no—
354#391—614no—
355#390—614no—
356#388—614no—
357#388—614no—
358#388—614no—
359#388C2×D6114yes—
360#387—614no—
361#387—614no—
362#387—614no—
363#387C2×D6114yes—
364#386—614no—
365#386—614no—
366#384—611no—
367#384C2×D6111yescharacteristic
368#383—614no—
369#383—614no—
370#382—612no—
371#382—612no—
372#382—611no—
373#382—614no—
374#382C2×D6111yescharacteristic
375#381—614no—
376#381—614no—
377#380—612no—
378#380—612no—
379#380—612no—
380#380—614no—
381#380C2×D6112yes—
382C4×C4⋊C4—612no—
383C4×C4⋊C4—612no—
384C4×C4⋊C4—614no—
385C4×C4⋊C4—612no—
386C4×C4⋊C4—614no—
387C4×C4⋊C4—614no—
388C4×C4⋊C4C2×D6112yes—
389C4×C22⋊C4—614no—
390C4×C22⋊C4—614no—
391C4×C22⋊C4—614no—
392C4×C22⋊C4—614no—
393#376—614no—
394#376—614no—
395C2×#95—614no—
396C61#3418911yescharacteristic, complemented, Sylow
397C2×#41—618no—
398C2×#41—1224no—
399C2×#41—1224no—
400C22×Q8—611no—
401C22×D4—611no—
402C22×D4—611no—
403C22×D4—611no—
404C23×C4—611no—
405C23×C4—612no—
406C23×C4—611no—
407C23×C4—612no—
408C23×C4—612no—
409C23×C4—614no—
410C23×C4C22×D6111yescharacteristic
411C4×Q8—6116no—
412C4×D4—6116no—
413C4×D4—6116no—
414C4×D4—6116no—
415#117—618no—
416#117—618no—
417#117—1228no—
418#117C4×D6118yescomplemented
419C2×C4⋊C4—612no—
420C2×C4⋊C4—612no—
421C2×C4⋊C4—612no—
422C2×C4⋊C4—614no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—614no—
425C2×C4⋊C4—1228no—
426C2×C4⋊C4C22×D6112yes—
427C2×C4⋊C4C22×D6112yes—
428C2×C4⋊C4C22×D6114yes—
429C2×C4⋊C4C22×D6112yes—
430C2×C22⋊C4—612no—
431C2×C22⋊C4—612no—
432C2×C22⋊C4—614no—
433C2×C22⋊C4—614no—
434C2×C22⋊C4—614no—
435C2×C22⋊C4—614no—
436C2×C22⋊C4C22×D6114yes—
437C2×C22⋊C4C22×D6112yes—
438C2×C42—618no—
439C2×C42—611no—
440C2×C42—612no—
441C2×C42—612no—
442C2×C42—612no—
443C2×C42—611no—
444C2×C42—618no—
445C2×C42—618no—
446C2×C42—614no—
447C2×C42—614no—
448C2×C42—614no—
449C2×C42—614no—
450C2×C42C22×D6114yes—
451C2×C42C22×D6111yescharacteristic
452C2×C42C22×D6111yescharacteristic
453#95—614no—
454#95—614no—
455#95—614no—
456#95—614no—
457#95—614no—
458#95—614no—
459#95—614no—
460#95—614no—
461#95—614no—
462#95—614no—
463#95—614no—
464#95C22×D6114yes—
465#95C22×D6114yes—
466#95C22×D6114yes—
467C24—611no—
468C24—611no—
469C24C23×D6111yescharacteristic
470#41—12232no—
471C2×Q8—614no—
472C2×Q8—1222no—
473C2×Q8—1222no—
474C2×D4—614no—
475C2×D4—614no—
476C2×D4—614no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1222no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—611no—
486C22×C4—614no—
487C22×C4—614no—
488C22×C4—614no—
489C22×C4—612no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—618no—
493C22×C4—614no—
494C22×C4—618no—
495C22×C4—618no—
496C22×C4—614no—
497C22×C4—614no—
498C22×C4—1228no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—2444no—
502C22×C4—2444no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4C2×C4×D6114yes—
510C22×C4C23×D6114yes—
511C22×C4C23×D6114yes—
512C22×C4C23×D6111yescharacteristic
513C22×C4C23×D6112yes—
514C22×C4C23×D6112yes—
515C22×C4C23×D6111yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—618no—
520C4⋊C4—618no—
521C4⋊C4—1228no—
522C4⋊C4—1224no—
523C4⋊C4—1224no—
524C4⋊C4—2444no—
525C4⋊C4C2×C4×D6118yes—
526C22⋊C4—44no—
527C22⋊C4—618no—
528C22⋊C4—618no—
529C22⋊C4—1228no—
530C22⋊C4—2444no—
531C22⋊C4—2444no—
532C22⋊C4—2444no—
533C22⋊C4C2×C4×D6118yes—
534C42—44no—
535C42—614no—
536C42—614no—
537C42—618no—
538C42—618no—
539C42—618no—
540C42—1228no—
541C42—2444no—
542C42—2444no—
543C42—2444no—
544C42C2×C4×D6114yes—
545C42C2×C4×D6114yes—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—612no—
551C23—612no—
552C23—612no—
553C23—612no—
554C23—1222no—
555C23—1222no—
556C23—1221no—
557C23—1221no—
558C23—1222no—
559C23—1222no—
560C23C22×C4×D6112yes—
561C23C24×D6111yescharacteristic
562Q8—1228no—
563D4—1228no—
564D4—1228no—
565D4—1228no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—6116no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1228no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—4888no—
600C2×C4—2448no—
601C2×C4—2448no—
602C2×C4—2448no—
603C2×C4C22×C4×D6112yes—
604C2×C4C22×C4×D6114yes—
605C2×C4C22×C4×D6114yes—
606C2×C4C22×C4×D6112yes—
607C2×C4#41×D6112yes—
608C2×C4#41×D6112yes—
609C22—22no—
610C22—24no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—614no—
615C22—1222no—
616C22—1222no—
617C22—1224no—
618C22—1224no—
619C22—1222no—
620C22—1224no—
621C22—2441no—
622C22—2441no—
623C22—2442no—
624C22C23×C4×D6111yescharacteristic
625C22C2×#41×D6111yescharacteristic
626C22C2×#41×D6111yescharacteristic
627C22#1684059712yes—
628C22#1684059912yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—614no—
635C4—1222no—
636C4—1224no—
637C4—1224no—
638C4—2444no—
639C4—2444no—
640C4#117×D6114yes—
641C2—42no—
642C2—1224no—
643C2—2442no—
644C2C2×#117×D6111yescharacteristic
645C2#2145901912yes—
646C2#2145971112yes—
647C2#2145973811yescharacteristic
648C2#2145974211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2314C2
socle, cosocleC23×C61C25
prime radicalC22×#11758222C22
hypercenter#2314D61
hypocenterC61#34189
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #21448789C2semidirect
3. #21448781C2semidirect
4. #21448758C2semidirect
5. #21448754C2semidirect
6. #21448429C2semidirect
7. #21448428C2semidirect
8. #21448427C2semidirect
10. #21448224C2semidirect
11. #21448215C2semidirect
12. #21448214C2semidirect
13. #382×D61C2semidirect
14. #21448173C2semidirect
15. C61×#2314C2semidirect
17. #16840230C4semidirect
85. #14519742C2×C4semidirect
88. #14519737C2×C4semidirect
91. #14519734C2×C4semidirect
105. C2×C4×Dic61C2×C4semidirect
116. C42×D61C2×C4semidirect
129. #117×C61C2×C4semidirect
325. #2314D61semidirect
396. C61#34189semidirect
418. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
#214487892
#214487812
#214487582
#214487542
#214484292
#214484282
#214484272
#214484082
#214482242
#214482152
#214482142
#382×D612
#214481732
C61×#23142
#3418961