Finite Group Databasebeta

← #29839486

Subgroups — #29839486

Order 15616.

2446 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2#21448785C212yesmaximal, complemented
3#21448784C211yesmaximal, characteristic, complemented
4#21448779C212yesmaximal, complemented
5#21448739C211yesmaximal, characteristic, complemented
6#21448736C211yesmaximal, characteristic, complemented
7#21448685C214yesmaximal
8#21448684C212yesmaximal
9#21448653C212yesmaximal
10#21448294C212yesmaximal, complemented
11#21448292C214yesmaximal, complemented
12#21448216C212yesmaximal
13#21448212C211yesmaximal, characteristic
14C4×#14519742C212yesmaximal, complemented
15C4×#14519735C211yesmaximal, characteristic
16#21448158C212yesmaximal
17C61×#2356C211yesmaximal, characteristic, complemented
18C22×Q8×D61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122C2212yes—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2212yes—
23C2×#14519742C2212yes—
24C2×#14519742C2212yes—
25C2×#14519742C2211yescharacteristic
26C4×Q8×D61C4116yescomplemented
27C2×#14519735C2212yes—
28C2×#14519734C2214yes—
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C4⋊C4×D61C2212yes—
31C2×C42×D61C2211yescharacteristic
32C2×C42×D61C2212yes—
33#16839669C2212yes—
34#16839666C2214yes—
35#16839666C2214yes—
36#16839660C2212yes—
37#16839659C2214yes—
38#16839658C2212yes—
39#16839657C2214yes—
40#16839648C2214yes—
41#16839647C2214yes—
42#16839646C2214yes—
43#16839645C2214yes—
44#16839644C2214yes—
45#16839630C2212yes—
46#16839629C2212yes—
47#16839628C2211yescharacteristic
48#16839626C2212yes—
49#16839626C2212yes—
50#16839624C2212yes—
51#16839624C2214yes—
52#16839621C2214yes—
53#16839619C2212yes—
54C4⋊C4×Dic61C2212yes—
55C4⋊C4×Dic61C2212yes—
56C4⋊C4×Dic61C2212yes—
57C4×#11758196C2214yes—
58C4×C4⋊Dic61C2211yescharacteristic
59#16839613C2214yes—
60#16839613C2212yes—
61#16839613C2212yes—
62#16839613C2214yes—
63#16839613C2214yes—
64C4×#11758198C2212yes—
65#16839609C2212yes—
66#16839609C2211yescharacteristic
67#16839609C2214yes—
68#16839609C2211yescharacteristic
69#16839608C2212yes—
70#16839607C2212yes—
71#16839607C2214yes—
72#16839607C2211yescharacteristic
73#16839607C2211yescharacteristic
74C42×Dic61C2214yes—
75C42×Dic61C2212yes—
76C2×C4×Q8×C61C2211yescharacteristic
77C61×#391C2212yes—
78C61×#387C2214yes—
79C61×#383C2212yes—
80C61×#382C2211yescharacteristic
81C4×C4⋊C4×C61C2211yescharacteristic
82C4×C4⋊C4×C61C2212yes—
83C61×#376C2212yes—
84C22×Dic122C2312yes—
85C22×Dic122C2311yescharacteristic
86C2×Q8×D61—24no—
87C2×Q8×D61—24no—
88C2×Q8×D61C2×C418yes—
89C22×C4×D61C2311yescharacteristic
90C22×C4×D61C2312yes—
91C22×C4×D61C2311yescharacteristic
92C22×C4×D61C2312yes—
93C22×C4×D61C2311yescharacteristic
94Q8×Dic61C2×C418yes—
95C4×Dic122—28no—
96C4×Dic122C2×C4116yes—
97C4×Dic122C2×C418yes—
98#14519742—28no—
99#14519742C2×C4116yes—
100#14519742C2×C418yes—
101#14519735—28no—
102#14519734—216no—
103C2×C4⋊Dic61C2312yes—
104C2×C4⋊Dic61C2312yes—
105C2×C4⋊Dic61C2312yes—
106C2×C4⋊Dic61C2311yescharacteristic
107C2×#11758196C2314yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2314yes—
111C2×#11758196C2312yes—
112C2×#11758196C2312yes—
113C2×#11758196C2312yes—
114C4⋊C4×D61C2×C418yes—
115C4⋊C4×D61C2×C4116yes—
116C2×#11758198C2314yes—
117C2×#11758198C2312yes—
118C2×#11758198C2312yes—
119C2×C4×Dic61C2314yes—
120C2×C4×Dic61C2312yes—
121C2×C4×Dic61C2312yes—
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2312yes—
124C2×C4×Dic61C2314yes—
125C2×C4×Dic61C2312yes—
126C2×C4×Dic61C2314yes—
127C2×C4×Dic61C2312yes—
128C2×C4×Dic61C2312yes—
129C2×C4×Dic61C2311yescharacteristic
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2312yes—
132C2×C4×Dic61C2311yescharacteristic
133C2×C4×Dic61C2312yes—
134C42×D61C2×C418yes—
135C42×D61C2×C4116yes—
136#14519684C2311yescharacteristic
137#14519684C2311yescharacteristic
138#14519684C2312yes—
139#14519684C2312yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144#14519684C2312yes—
145#14519684C2312yes—
146#14519684C2314yes—
147#14519684C2314yes—
148#14519684C2314yes—
149C22×Q8×C61C2311yescharacteristic
150C4×Q8×C61C2×C418yes—
151C2×C4⋊C4×C61C2312yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2311yescharacteristic
154C2×C4⋊C4×C61C2312yes—
155C2×C42×C61C2314yes—
156C2×C42×C61C2312yes—
157C2×C42×C61C2312yes—
158C2×C42×C61C2311yescharacteristic
159#95×C61C2311yescharacteristic
160#95×C61C2312yes—
161#95×C61C2314yes—
162#95×C61C2311yescharacteristic
163#95×C61C2314yes—
164#95×C61C2312yes—
165#95×C61C2314yes—
166C23×D61C2411yescharacteristic
167C2×Q8×C61—22no—
168C2×Q8×C61—22no—
169C2×Q8×C61C22×C414yes—
170C22×C4×C61C2414yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2412yes—
173C22×C4×C61C2411yescharacteristic
174C22×C4×C61C2412yes—
175C22×C4×C61C2411yescharacteristic
176C22×C4×C61C2412yes—
177C22×C4×C61C2411yescharacteristic
178C22×Dic61C2412yes—
179C22×Dic61C2414yes—
180C22×Dic61C2411yescharacteristic
181C22×Dic61C2411yescharacteristic
182C22×Dic61C2412yes—
183C22×Dic61C2412yes—
184C22×Dic61C2412yes—
185C22×Dic61C2411yescharacteristic
186Q8×D61—232no—
187C2×C4×D61—28no—
188C2×C4×D61—24no—
189C2×C4×D61—22no—
190C2×C4×D61—22no—
191C2×C4×D61—24no—
192C2×C4×D61—24no—
193C2×C4×D61—24no—
194C2×C4×D61C22×C414yes—
195C2×C4×D61C22×C418yes—
196C2×C4×D61C22×C414yes—
197C2×C4×D61C22×C418yes—
198C2×C4×D61C22×C414yes—
199C2×Dic122—24no—
200C2×Dic122—24no—
201C2×Dic122C22×C418yes—
202C2×Dic122C22×C414yes—
203C2×Dic122#4114yes—
204C2×Dic122#4114yes—
205C4⋊C4×C61—24no—
206C4⋊C4×C61—24no—
207C4⋊C4×C61C22×C418yes—
208C4⋊C4×C61C22×C414yes—
209C42×C61—28no—
210C42×C61—24no—
211C42×C61C22×C418yes—
212C42×C61C22×C414yes—
213#11758198—28no—
214#11758198—24no—
215#11758198#4118yes—
216C4⋊Dic61—24no—
217C4⋊Dic61—24no—
218C4⋊Dic61C22×C418yes—
219C4⋊Dic61C22×C414yes—
220#11758196—28no—
221#11758196—28no—
222#11758196—28no—
223#11758196—28no—
224#11758196C22×C418yes—
225#11758196C22×C418yes—
226#11758196C22×C418yes—
227C4×Dic61—24no—
228C4×Dic61—24no—
229C4×Dic61—28no—
230C4×Dic61—24no—
231C4×Dic61—28no—
232C4×Dic61—28no—
233C4×Dic61—24no—
234C4×Dic61—28no—
235C4×Dic61—24no—
236C4×Dic61C22×C414yes—
237C4×Dic61C22×C418yes—
238C4×Dic61C22×C418yes—
239C4×Dic61C22×C414yes—
240C4×Dic61C22×C414yes—
241C4×Dic61C22×C418yes—
242C23×C61C2511yescharacteristic
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—22no—
246C22×D61C23×C412yes—
247C22×D61C2×#4114yes—
248Q8×C61—28no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—24no—
262C2×C4×C61C23×C412yes—
263C2×C4×C61C23×C414yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61C23×C414yes—
266C2×C4×C61C23×C412yes—
267C2×C4×C61C2×#4114yes—
268C2×C4×C61C2×#4112yes—
269C2×C4×C61C2×#4112yes—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—22no—
280C2×Dic61—24no—
281C2×Dic61—22no—
282C2×Dic61C23×C412yes—
283C2×Dic61C23×C414yes—
284C2×Dic61C23×C412yes—
285C2×Dic61C23×C414yes—
286C2×Dic61C23×C412yes—
287C2×Dic61C2×#4114yes—
288C2×Dic61C2×#4114yes—
289C2×Dic61C2×#4114yes—
290C2×Dic612−1+414yes—
291C4×D61—216no—
292C4×D61—28no—
293C4×D61—28no—
294C4×D61—216no—
295C4×D61—28no—
296Dic122—216no—
297Dic122#117116yescomplemented
298#34128—611nomaximal, Sylow
299C22×C61C24×C411yescharacteristic
300C22×C61C22×#4111yescharacteristic
301C22×C61C22×#4111yescharacteristic
302C22×C61C22×#4112yes—
303C22×C61C2×2−1+412yes—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—24no—
307C2×D61—24no—
308C2×D61C4×#4114yes—
309C4×C61—24no—
310C4×C61—24no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—24no—
314C4×C61—42no—
315C4×C61—44no—
316C4×C61C2×#11714yes—
317Dic61—22no—
318Dic61—22no—
319Dic61—24no—
320Dic61—24no—
321Dic61—44no—
322Dic61—42no—
323Dic61C2×#11718yes—
324Dic61#51914yes—
325C22×C4×Q8—611no—
326#2356—612no—
327#2356—612no—
328#2356—611no—
329#2356D6111yescharacteristic, complemented
330#2351—611no—
331#2351—611no—
332#2344—614no—
333#2344—612no—
334Q8×C22⋊C4—612no—
335#2295—612no—
336#2295—614no—
337#2283—612no—
338#2283—611no—
339C42×Q8—612no—
340C4×#117—611no—
341C4×#117—612no—
342C2×C61C22×#11711yescharacteristic
343C2×C61C2×C4×#4111yescharacteristic
344C2×C61C2×#51911yescharacteristic
345C2×C61#348212yes—
346C2×C61#348212yes—
347D61—24no—
348C23×Q8—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—6116no—
351C2×C4×Q8—611no—
352C2×C4×Q8—612no—
353C2×C4×Q8—612no—
354C2×C4×Q8—611no—
355C2×C4×Q8—612no—
356C2×C4×Q8—612no—
357C2×C4×Q8C2×D6111yescharacteristic
358C2×#117—612no—
359C2×#117—614no—
360C22×C4⋊C4—612no—
361C22×C4⋊C4—611no—
362C22×C42—611no—
363C22×C42—612no—
364#391—612no—
365#391—614no—
366#391—614no—
367#391C2×D6112yes—
368#389—614no—
369#389—612no—
370#389—612no—
371#389—614no—
372#387—614no—
373#387—614no—
374#387—614no—
375#387—614no—
376#387—614no—
377#387C2×D6114yes—
378#383—612no—
379#383—612no—
380#383C2×D6112yes—
381#382—611no—
382#382—612no—
383#382—612no—
384#382C2×D6111yescharacteristic
385#381—614no—
386#381—612no—
387#380—612no—
388#380—614no—
389C4×C4⋊C4—612no—
390C4×C4⋊C4—612no—
391C4×C4⋊C4—612no—
392C4×C4⋊C4—614no—
393C4×C4⋊C4—611no—
394C4×C4⋊C4—614no—
395C4×C4⋊C4—612no—
396C4×C4⋊C4—612no—
397C4×C4⋊C4—614no—
398C4×C4⋊C4—614no—
399C4×C4⋊C4C2×D6112yes—
400C4×C4⋊C4C2×D6111yescharacteristic
401C4×C22⋊C4—612no—
402C4×C22⋊C4—614no—
403C4×C22⋊C4—611no—
404C4×C22⋊C4—611no—
405C4×C22⋊C4—612no—
406#376—612no—
407#376—611no—
408#376—611no—
409#376—612no—
410#376—614no—
411#376C2×D6112yes—
412C43—612no—
413C43—614no—
414C61#3412811yescharacteristic, complemented, Sylow
415C22×Q8—618no—
416C22×Q8—611no—
417C22×Q8—612no—
418C22×Q8—1224no—
419C22×Q8—1224no—
420C22×Q8C22×D6111yescharacteristic
421C23×C4—611no—
422C23×C4—612no—
423C23×C4—612no—
424C23×C4—611no—
425C23×C4—611no—
426C4×Q8—618no—
427C4×Q8—618no—
428C4×Q8—6116no—
429C4×Q8—618no—
430C4×Q8—6116no—
431C4×Q8—1228no—
432C4×Q8—1228no—
433C4×Q8C4×D6118yes—
434#117—12216no—
435#117—1228no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—6116no—
442C2×C4⋊C4—618no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—614no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—614no—
447C2×C4⋊C4—614no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4C22×D6112yes—
450C2×C4⋊C4C22×D6112yes—
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6112yes—
453C2×C22⋊C4—612no—
454C2×C22⋊C4—612no—
455C2×C22⋊C4—614no—
456C2×C42—612no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—612no—
460C2×C42—611no—
461C2×C42—612no—
462C2×C42—6116no—
463C2×C42—618no—
464C2×C42—612no—
465C2×C42—614no—
466C2×C42—612no—
467C2×C42—614no—
468C2×C42—614no—
469C2×C42—612no—
470C2×C42—612no—
471C2×C42—612no—
472C2×C42—614no—
473C2×C42C22×D6114yes—
474C2×C42C22×D6112yes—
475C2×C42C22×D6112yes—
476C2×C42C22×D6111yescharacteristic
477#95—611no—
478#95—612no—
479#95—611no—
480#95—612no—
481#95—612no—
482#95—614no—
483#95—614no—
484#95—614no—
485#95—614no—
486#95—614no—
487#95—612no—
488#95—614no—
489#95—614no—
490#95C22×D6111yescharacteristic
491#95C22×D6111yescharacteristic
492#95C22×D6114yes—
493#95C22×D6112yes—
494#95C22×D6112yes—
495#95C22×D6114yes—
496#95C22×D6114yes—
497C24—611no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—614no—
501C2×Q8—618no—
502C2×Q8—614no—
503C2×Q8—614no—
504C2×Q8—1224no—
505C2×Q8—1224no—
506C2×Q8—12232no—
507C2×Q8C2×C4×D6114yes—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—611no—
511C22×C4—612no—
512C22×C4—611no—
513C22×C4—612no—
514C22×C4—614no—
515C22×C4—614no—
516C22×C4—618no—
517C22×C4—618no—
518C22×C4—614no—
519C22×C4—614no—
520C22×C4—612no—
521C22×C4—1224no—
522C22×C4—1224no—
523C22×C4—1224no—
524C22×C4—1228no—
525C22×C4—1224no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4C23×D6114yes—
529C22×C4C23×D6112yes—
530C22×C4C23×D6111yescharacteristic
531C22×C4C23×D6112yes—
532C22×C4C23×D6112yes—
533C22×C4C23×D6111yescharacteristic
534C22×C4C23×D6112yes—
535C22×C4C23×D6111yescharacteristic
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—618no—
539C4⋊C4—618no—
540C4⋊C4—618no—
541C4⋊C4—618no—
542C4⋊C4—614no—
543C4⋊C4—1228no—
544C4⋊C4—1228no—
545C4⋊C4—1228no—
546C4⋊C4—1224no—
547C4⋊C4—1228no—
548C4⋊C4—1224no—
549C4⋊C4C2×C4×D6118yes—
550C4⋊C4C2×C4×D6114yes—
551C22⋊C4—618no—
552C22⋊C4—1228no—
553C22⋊C4—1224no—
554C42—28no—
555C42—24no—
556C42—618no—
557C42—618no—
558C42—614no—
559C42—614no—
560C42—618no—
561C42—614no—
562C42—1224no—
563C42—1224no—
564C42—1228no—
565C42—1228no—
566C42—1224no—
567C42—1228no—
568C42—1228no—
569C42—1224no—
570C42—1224no—
571C42C2×C4×D6118yes—
572C42C2×C4×D6114yes—
573C23—612no—
574C23—614no—
575C23—1221no—
576C23—1221no—
577C23—1222no—
578C23C24×D6111yescharacteristic
579Q8—28no—
580Q8—6116no—
581Q8—12216no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—612no—
596C2×C4—614no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—614no—
600C2×C4—612no—
601C2×C4—614no—
602C2×C4—612no—
603C2×C4—614no—
604C2×C4—1228no—
605C2×C4—12216no—
606C2×C4—12216no—
607C2×C4—1228no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1224no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1228no—
615C2×C4—1224no—
616C2×C4—1222no—
617C2×C4—1224no—
618C2×C4—1222no—
619C2×C4—1224no—
620C2×C4—1222no—
621C2×C4C22×C4×D6112yes—
622C2×C4C22×C4×D6114yes—
623C2×C4C22×C4×D6112yes—
624C2×C4C22×C4×D6114yes—
625C2×C4C22×C4×D6112yes—
626C2×C4#41×D6114yes—
627C2×C4#41×D6112yes—
628C2×C4#41×D6112yes—
629C22—614no—
630C22—1222no—
631C22—1222no—
632C22—1224no—
633C22—1224no—
634C22C23×C4×D6111yescharacteristic
635C22C2×#41×D6111yescharacteristic
636C22C2×#41×D6111yescharacteristic
637C22C2×#41×D6112yes—
638C222−1+4×D6112yes—
639C4—24no—
640C4—24no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—42no—
645C4—44no—
646C4—618no—
647C4—614no—
648C4—1222no—
649C4—1222no—
650C4—1224no—
651C4—1224no—
652C4—2444no—
653C4—2442no—
654C4#117×D6114yes—
655C2—1224no—
656C2C2×#117×D6111yescharacteristic
657C2C4×#41×D6111yescharacteristic
658C2#519×D6111yescharacteristic
659C2#2145943312yes—
660C2#2145943812yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2356C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2356D61
hypocenterC61#34128
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. #21448785C2semidirect
3. #21448784C2semidirect
4. #21448779C2semidirect
5. #21448739C2semidirect
6. #21448736C2semidirect
10. #21448294C2semidirect
11. #21448292C2semidirect
14. C4×#14519742C2semidirect
17. C61×#2356C2semidirect
26. C4×Q8×D61C4semidirect
297. Dic122#117semidirect
329. #2356D61semidirect
414. C61#34128semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
#214487852
#214487842
#214487792
#214487392
#214487362
#214486852
#214486842
#214486532
#214482942
#214482922
#214482162
#214482122
C4×#145197422
C4×#145197352
#214481582
C61×#23562
#3412861