Finite Group Databasebeta

← #29840155

Subgroups — #29840155

Order 15616.

2542 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2#21448773C214yesmaximal, complemented
3#21448765C214yesmaximal, complemented
4#21448747C214yesmaximal, complemented
5#21448743C214yesmaximal, complemented
6#21448429C211yesmaximal, characteristic, complemented
7#21448428C211yesmaximal, characteristic, complemented
8#21448427C211yesmaximal, characteristic, complemented
9#21448408C212yesmaximal
10#21448225C211yesmaximal, characteristic, complemented
11#21448223C212yesmaximal, complemented
12#21448221C212yesmaximal, complemented
13#384×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×#14519734C2214yes—
24C2×#14519734C2211yescharacteristic
25C2×#14519734C2211yescharacteristic
26C2×#14519733C2214yes—
27C2×C4⋊C4×D61C2212yes—
28C2×C4⋊C4×D61C2214yes—
29C22×C4×Dic61C2211yescharacteristic
30C2×C42×D61C2211yescharacteristic
31C2×C42×D61C2211yescharacteristic
32#16839666C2214yes—
33#16839662C2214yes—
34#16839655C2214yes—
35#16839654C2214yes—
36#16839652C2214yes—
37#16839649C2214yes—
38#16839648C2214yes—
39#16839643C2214yes—
40#16839640C2214yes—
41#16839638C2214yes—
42#16839632C2211yescharacteristic
43#16839631C2212yes—
44#16839631C2214yes—
45#16839631C2212yes—
46#16839628C2211yescharacteristic
47#16839626C2214yes—
48#16839626C2214yes—
49#16839622C2212yes—
50#16839621C2214yes—
51#16839621C2214yes—
52#16839620C2214yes—
53#16839619C2212yes—
54#16839619C2212yes—
55C4⋊C4×Dic61C2212yes—
56C4⋊C4×Dic61C2212yes—
57C4×#11758196C2214yes—
58C4×#11758196C2214yes—
59C4×C4⋊Dic61C2212yes—
60#16839613C2214yes—
61#16839613C2214yes—
62#16839613C2214yes—
63C4×#11758198C2214yes—
64#16839609C2214yes—
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×D61C2312yes—
81C22×C4×D61C2312yes—
82C22×C4×D61C2311yescharacteristic
83C22×C4×D61C2314yes—
84C22×C4×D61C2311yescharacteristic
85#14519742C2×C4116yescomplemented
86#14519739C2×C4116yes—
87#14519737C2×C4116yes—
88#14519737C2×C4116yescomplemented
89#14519735—28no—
90#14519734—48no—
91#14519734C2×C418yes—
92#14519734C2×C418yescomplemented
93#14519733—48no—
94C2×C4⋊Dic61C2312yes—
95C2×C4⋊Dic61C2312yes—
96C2×C4⋊Dic61C2314yes—
97C2×#11758196C2314yes—
98C2×#11758196C2314yes—
99C2×#11758196C2314yes—
100C2×#11758196C2314yes—
101C2×#11758196C2314yes—
102C2×#11758196C2312yes—
103C2×#11758196C2312yes—
104C4⋊C4×D61—28no—
105C4⋊C4×D61—48no—
106C2×C22⋊Dic61C2314yes—
107C2×#11758198C2314yes—
108C2×#11758198C2314yes—
109C2×#11758198C2314yes—
110C2×#11758198C2312yes—
111C2×#11758198C2312yes—
112C2×C4×Dic61C2×C418yescomplemented
113C2×C4×Dic61C2314yes—
114C2×C4×Dic61C2314yes—
115C2×C4×Dic61C2314yes—
116C2×C4×Dic61C2314yes—
117C2×C4×Dic61C2311yescharacteristic
118C2×C4×Dic61C2312yes—
119C2×C4×Dic61C2312yes—
120C2×C4×Dic61C2311yescharacteristic
121C2×C4×Dic61C2312yes—
122C42×D61C2×C418yes—
123C42×D61C2×C418yescomplemented
124#14519684C2314yes—
125#14519684C2314yes—
126#14519684C2314yes—
127#14519684C2314yes—
128#14519684C2314yes—
129#14519684C2314yes—
130#14519684C2314yes—
131C23×C4×C61C2311yescharacteristic
132#117×C61C2×C418yescomplemented
133C2×C4⋊C4×C61C2312yes—
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2314yes—
136C2×C4⋊C4×C61C2312yes—
137C2×C22⋊C4×C61C2314yes—
138C2×C22⋊C4×C61C2312yes—
139C2×C42×C61C2314yes—
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142#95×C61C2314yes—
143#95×C61C2314yes—
144#95×C61C2314yes—
145C24×C61C2411yescharacteristic
146C23×D61C2411yescharacteristic
147C23×D61C2411yescharacteristic
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C22×C414yes—
151C22×C4×C61C2414yes—
152C22×C4×C61C2414yes—
153C22×C4×C61C2412yes—
154C22×C4×C61C2412yes—
155C22×C4×C61C2411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61C22×C414yes—
162C2×C22⋊D61C22×C414yes—
163C22×Dic61—24no—
164C22×Dic61—24no—
165C22×Dic61C22×C418yes—
166C22×Dic61C2414yes—
167C22×Dic61C2414yes—
168C22×Dic61C2411yescharacteristic
169C22×Dic61C2412yes—
170C22×Dic61C2411yescharacteristic
171C22×Dic61C2412yes—
172#11758222—232no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244C22×C414yes—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—24no—
179C2×C4×D61—24no—
180C2×C4×D61—24no—
181C2×C4×D61—24no—
182C2×C4×D61—28no—
183C2×C4×D61—44no—
184C2×C4×D61—44no—
185C2×C4×D61C22×C414yes—
186C2×C4×D61C22×C418yes—
187C2×C4×D61C22×C418yes—
188C2×C4×D61C22×C414yes—
189C2×C4×D61#4114yes—
190C2×C4×D61#4114yes—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122C22×C414yes—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—44no—
197C4⋊C4×C61C22×C418yes—
198C22⋊C4×C61—44no—
199C22⋊C4×C61C22×C418yes—
200C42×C61—44no—
201C42×C61C22×C414yes—
202C42×C61C22×C414yes—
203C22⋊Dic61—44no—
204#11758198—28no—
205#11758198—44no—
206#11758198—44no—
207#11758198C22×C418yes—
208#11758198C22×C418yes—
209C4⋊Dic61—24no—
210C4⋊Dic61—24no—
211C4⋊Dic61—44no—
212#11758196—28no—
213#11758196—28no—
214#11758196—28no—
215#11758196—28no—
216#11758196—28no—
217#11758196C22×C418yes—
218#11758196C22×C418yes—
219C4×Dic61—28no—
220C4×Dic61—44no—
221C4×Dic61—44no—
222C4×Dic61—44no—
223C4×Dic61C22×C414yes—
224C4×Dic61C22×C418yes—
225C4×Dic61C22×C418yes—
226C4×Dic61C22×C414yes—
227C4×Dic61C22×C418yes—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61C23×C412yes—
233C23×C61C2511yescharacteristic
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61C23×C412yes—
241C22×D61C23×C412yes—
242C22×D61C2×#4112yes—
243C22×D61C2×#4112yes—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—28no—
251C2×C4×C61—24no—
252C2×C4×C61—24no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—24no—
257C2×C4×C61C23×C412yes—
258C2×C4×C61C23×C414yes—
259C2×C4×C61C23×C414yes—
260C2×C4×C61C23×C412yes—
261C2×C4×C61C2×#4112yes—
262C2×C4×C61C2×#4112yes—
263C22⋊D61—28no—
264C22⋊D61—28no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—216no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61C23×C414yes—
277C2×Dic61C23×C412yes—
278C2×Dic61C23×C412yes—
279C2×Dic61C23×C414yes—
280C2×Dic61C2×#4112yes—
281C2×Dic61C2×#4112yes—
282C2×Dic61H(5,𝔽2)14yes—
283C2×Dic612−1+414yes—
284D244—28no—
285C4×D61—28no—
286C4×D61—216no—
287C4×D61—216no—
288C4×D61—88no—
289C4×D61#117116yescomplemented
290Dic122—28no—
291#34188—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—24no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—42no—
297C22×C61C24×C411yescharacteristic
298C22×C61C22×#4111yescharacteristic
299C22×C61C22×#4111yescharacteristic
300C22×C61C2×H(5,𝔽2)12yes—
301C22×C61C2×2−1+412yes—
302C2×D61—24no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—24no—
306C2×D61—22no—
307C2×D61—24no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—42no—
311C2×D61C2×#11714yes—
312C4×C61—22no—
313C4×C61—24no—
314C4×C61—24no—
315C4×C61—44no—
316C4×C61—44no—
317C4×C61C2×#11714yes—
318Dic61—22no—
319Dic61—24no—
320Dic61—44no—
321Dic61—44no—
322Dic61C2×#11714yes—
323Dic61#52018yes—
324C2×C4×#41—611no—
325#2355—614no—
326#2354—614no—
327#2352—614no—
328#2352—614no—
329#2314—611no—
330#2314—611no—
331#2314—611no—
332#2314D6111yescharacteristic, complemented
333#2311—612no—
334#2285—611no—
335#2285—612no—
336#2285—612no—
337C2×#384—611no—
338#2274—612no—
339C2×C61—42no—
340C2×C61C22×#11711yescharacteristic
341C2×C61C2×#52012yes—
342C2×C61#350811yescharacteristic
343C2×C61#350911yescharacteristic
344C2×C61#351112yes—
345D61—24no—
346D61—42no—
347C22×#41—611no—
348C4×#41—6116no—
349C2×C4×Q8—612no—
350C2×C4×D4—612no—
351C2×C4×D4—612no—
352C2×C4×D4—612no—
353C2×#117—612no—
354C2×#117—614no—
355C2×#117—611no—
356C2×#117—611no—
357C2×#117—614no—
358C2×#117C2×D6111yescharacteristic
359C22×C4⋊C4—612no—
360C22×C4⋊C4—614no—
361C22×C42—611no—
362C22×C42—611no—
363C22×C42—611no—
364#391—614no—
365#390—614no—
366#388—614no—
367#388—614no—
368#388—614no—
369#388C2×D6114yes—
370#387—614no—
371#387—614no—
372#387—614no—
373#387C2×D6114yes—
374#386—614no—
375#386—614no—
376#384—614no—
377#384—612no—
378#384—612no—
379#384—611no—
380#384C2×D6111yescharacteristic
381#382—611no—
382#382—614no—
383#382—614no—
384#382C2×D6111yescharacteristic
385#380—612no—
386#380—612no—
387#380—612no—
388#380—614no—
389#380—614no—
390#380—614no—
391#380C2×D6112yes—
392C4×C4⋊C4—612no—
393C4×C4⋊C4—612no—
394C4×C4⋊C4—612no—
395C4×C4⋊C4—614no—
396C4×C4⋊C4—614no—
397C4×C4⋊C4—614no—
398C4×C4⋊C4—614no—
399C4×C4⋊C4—614no—
400C4×C4⋊C4C2×D6112yes—
401C4×C22⋊C4—614no—
402C4×C22⋊C4—614no—
403C61#3418811yescharacteristic, complemented, Sylow
404C2×#41—618no—
405C2×#41—1224no—
406C2×#41—1224no—
407C22×Q8—611no—
408C22×D4—611no—
409C22×D4—611no—
410C22×D4—611no—
411C23×C4—611no—
412C23×C4—612no—
413C23×C4—612no—
414C23×C4—612no—
415C23×C4—611no—
416C23×C4—614no—
417C23×C4C22×D6111yescharacteristic
418C4×Q8—6116no—
419C4×D4—6116no—
420C4×D4—6116no—
421C4×D4—6116no—
422#117—618no—
423#117—618no—
424#117—1228no—
425#117—2448no—
426#117—2448no—
427#117C4×D6118yescomplemented
428C2×C4⋊C4—612no—
429C2×C4⋊C4—612no—
430C2×C4⋊C4—614no—
431C2×C4⋊C4—614no—
432C2×C4⋊C4—612no—
433C2×C4⋊C4—614no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—614no—
437C2×C4⋊C4—614no—
438C2×C4⋊C4—1228no—
439C2×C4⋊C4—2448no—
440C2×C4⋊C4C22×D6112yes—
441C2×C4⋊C4C22×D6112yes—
442C2×C4⋊C4C22×D6114yes—
443C2×C4⋊C4C22×D6112yes—
444C2×C22⋊C4—612no—
445C2×C22⋊C4—612no—
446C2×C22⋊C4—614no—
447C2×C22⋊C4—614no—
448C2×C22⋊C4—614no—
449C2×C22⋊C4—614no—
450C2×C22⋊C4C22×D6114yes—
451C2×C22⋊C4C22×D6112yes—
452C2×C42—618no—
453C2×C42—611no—
454C2×C42—612no—
455C2×C42—612no—
456C2×C42—612no—
457C2×C42—611no—
458C2×C42—618no—
459C2×C42—618no—
460C2×C42—614no—
461C2×C42—614no—
462C2×C42—614no—
463C2×C42—614no—
464C2×C42C22×D6114yes—
465C2×C42C22×D6111yescharacteristic
466C2×C42C22×D6111yescharacteristic
467#95—614no—
468#95—614no—
469#95—614no—
470#95—614no—
471#95—614no—
472#95—614no—
473#95—614no—
474#95C22×D6114yes—
475#95C22×D6114yes—
476#95C22×D6114yes—
477C24—611no—
478C24—611no—
479C24C23×D6111yescharacteristic
480#41—12232no—
481C2×Q8—614no—
482C2×Q8—1222no—
483C2×Q8—1222no—
484C2×D4—614no—
485C2×D4—614no—
486C2×D4—614no—
487C2×D4—1222no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—614no—
498C22×C4—614no—
499C22×C4—614no—
500C22×C4—618no—
501C22×C4—611no—
502C22×C4—612no—
503C22×C4—614no—
504C22×C4—618no—
505C22×C4—618no—
506C22×C4—614no—
507C22×C4—614no—
508C22×C4—1224no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—1224no—
512C22×C4—1228no—
513C22×C4—1224no—
514C22×C4—1224no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—2444no—
518C22×C4—2444no—
519C22×C4C2×C4×D6114yes—
520C22×C4C23×D6114yes—
521C22×C4C23×D6114yes—
522C22×C4C23×D6111yescharacteristic
523C22×C4C23×D6112yes—
524C22×C4C23×D6112yes—
525C22×C4C23×D6111yescharacteristic
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—618no—
530C4⋊C4—618no—
531C4⋊C4—1228no—
532C4⋊C4—1228no—
533C4⋊C4—1228no—
534C4⋊C4—1228no—
535C4⋊C4—1228no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—2444no—
539C4⋊C4C2×C4×D6118yes—
540C22⋊C4—44no—
541C22⋊C4—618no—
542C22⋊C4—618no—
543C22⋊C4—1228no—
544C22⋊C4—2444no—
545C22⋊C4—2444no—
546C22⋊C4—2444no—
547C22⋊C4C2×C4×D6118yes—
548C42—44no—
549C42—614no—
550C42—614no—
551C42—618no—
552C42—618no—
553C42—618no—
554C42—1228no—
555C42—2444no—
556C42—2444no—
557C42—2444no—
558C42C2×C4×D6114yes—
559C42C2×C4×D6114yes—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—612no—
565C23—612no—
566C23—612no—
567C23—612no—
568C23—1222no—
569C23—1222no—
570C23—1221no—
571C23—1221no—
572C23—1222no—
573C23—1222no—
574C23C22×C4×D6112yes—
575C23C24×D6111yescharacteristic
576Q8—1228no—
577D4—1228no—
578D4—1228no—
579D4—1228no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—614no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—6116no—
600C2×C4—614no—
601C2×C4—612no—
602C2×C4—12216no—
603C2×C4—12216no—
604C2×C4—12216no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1228no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—4888no—
617C2×C4C22×C4×D6112yes—
618C2×C4C22×C4×D6114yes—
619C2×C4C22×C4×D6114yes—
620C2×C4C22×C4×D6112yes—
621C2×C4#41×D6112yes—
622C2×C4#41×D6112yes—
623C22—22no—
624C22—24no—
625C22—42no—
626C22—41no—
627C22—41no—
628C22—614no—
629C22—1224no—
630C22—1224no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1224no—
635C22—2442no—
636C22—2441no—
637C22—2441no—
638C22C23×C4×D6111yescharacteristic
639C22C2×#41×D6111yescharacteristic
640C22C2×#41×D6111yescharacteristic
641C22H(5,𝔽2)×D6112yes—
642C222−1+4×D6112yes—
643C4—22no—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—614no—
649C4—618no—
650C4—1222no—
651C4—1224no—
652C4—2444no—
653C4—2444no—
654C4#117×D6114yes—
655C2—42no—
656C2—1224no—
657C2—2442no—
658C2C2×#117×D6111yescharacteristic
659C2#520×D6112yes—
660C2#2145969111yescharacteristic
661C2#2145970111yescharacteristic
662C2#2145972412yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2314C2
socle, cosocleC23×C61C25
prime radicalC22×#11758222C22
hypercenter#2314D61
hypocenterC61#34188
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #21448773C2semidirect
3. #21448765C2semidirect
4. #21448747C2semidirect
5. #21448743C2semidirect
6. #21448429C2semidirect
7. #21448428C2semidirect
8. #21448427C2semidirect
10. #21448225C2semidirect
11. #21448223C2semidirect
12. #21448221C2semidirect
13. #384×D61C2semidirect
14. #21448173C2semidirect
15. C61×#2314C2semidirect
17. #16840230C4semidirect
85. #14519742C2×C4semidirect
88. #14519737C2×C4semidirect
92. #14519734C2×C4semidirect
112. C2×C4×Dic61C2×C4semidirect
123. C42×D61C2×C4semidirect
132. #117×C61C2×C4semidirect
289. C4×D61#117semidirect
332. #2314D61semidirect
403. C61#34188semidirect
427. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
#214487732
#214487652
#214487472
#214487432
#214484292
#214484282
#214484272
#214484082
#214482252
#214482232
#214482212
#384×D612
#214481732
C61×#23142
#3418861