Finite Group Databasebeta

← #29840901

Subgroups — #29840901

Order 15616.

2618 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519755C211yesmaximal, characteristic
2#21448903C214yesmaximal, complemented
3#21448847C214yesmaximal, complemented
4#21448689C212yesmaximal, complemented
5#21448671C212yesmaximal
6#21448669C212yesmaximal
7C22⋊C4×Dic122C212yesmaximal, complemented
8#21448570C214yesmaximal, complemented
9#21448422C211yesmaximal, characteristic, complemented
10#21448419C211yesmaximal, characteristic, complemented
11C4×#14519755C214yesmaximal, complemented
12#21448200C211yesmaximal, characteristic
13#21448174C211yesmaximal, characteristic, complemented
14C4×C22⋊C4×D61C211yesmaximal, characteristic
15C61×#2274C211yesmaximal, characteristic, complemented
16C23×Dic122C2211yescharacteristic
17C23×C4×D61C2211yescharacteristic
18C2×#14519755C4116yescomplemented
19C2×#14519755C2212yes—
20C2×#14519755C2212yes—
21C2×#14519755C2212yes—
22C2×#14519755C2212yes—
23C2×C4×Dic122C2214yescomplemented
24C2×#14519734C2212yes—
25C2×#14519733C2212yes—
26C22×#11758196C2211yescharacteristic
27C22×#11758196C2211yescharacteristic
28C22×#11758198C2211yescharacteristic
29C22×#11758198C2211yescharacteristic
30C2×C22⋊C4×D61C2212yes—
31C2×C42×D61C2212yes—
32#16839668C2214yescomplemented
33#16839666C2214yescomplemented
34#16839666C2214yescomplemented
35#16839657C2214yescomplemented
36#16839654C2214yes—
37#16839653C2214yes—
38#16839653C2214yes—
39#16839652C2214yes—
40#16839648C2214yescomplemented
41#16839648C2214yescomplemented
42#16839646C2214yescomplemented
43#16839646C2214yescomplemented
44#16839644C2214yescomplemented
45#16839643C2214yescomplemented
46#16839636C2214yes—
47#16839633C2214yes—
48#16839625C2212yes—
49#16839624C2212yes—
50#16839623C2212yes—
51#16839622C2212yes—
52#16839621C2212yes—
53#16839620C2212yes—
54#16839620C2212yes—
55#16839617C2212yes—
56C4×#11758196C2214yescomplemented
57C4×#11758196C2214yescomplemented
58#16839613C2214yescomplemented
59C22⋊C4×Dic61C2212yes—
60C22⋊C4×Dic61C2212yes—
61C4×#11758198C2212yes—
62C4×#11758198C2212yes—
63C4×#11758198C2214yes—
64C4×#11758198C2214yes—
65C4×C22⋊Dic61C2211yescharacteristic
66C4×C22⋊Dic61C2211yescharacteristic
67#16839609C2212yes—
68#95×D61C2212yes—
69C2×#117×C61C2212yes—
70C22×C4⋊C4×C61C2211yescharacteristic
71C4×C4⋊C4×C61C2214yescomplemented
72C4×C4⋊C4×C61C2214yescomplemented
73C4×C22⋊C4×C61C2211yescharacteristic
74C4×C22⋊C4×C61C2212yes—
75C4×C22⋊C4×C61C2211yescharacteristic
76C24×D61C2311yescharacteristic
77C22×Dic122—44no—
78C22×Dic122C2×C418yescomplemented
79C22×Dic122C2312yes—
80C22×Dic122C2312yes—
81C23×Dic61C2311yescharacteristic
82C23×Dic61C2311yescharacteristic
83C23×Dic61C2311yescharacteristic
84C22×C4×D61—44no—
85C22×C4×D61C2×C418yes—
86C22×C4×D61C2314yes—
87C22×C4×D61C2311yescharacteristic
88C22×C4×D61C2311yescharacteristic
89C22×C4×D61C2311yescharacteristic
90C22×C4×D61C2311yescharacteristic
91#14519755C2×C4116yes—
92#14519755C2×C4116yes—
93#14519755C2×C4116yescomplemented
94#14519755C2×C4116yescomplemented
95C4×Dic122—48no—
96#14519734—44no—
97#14519733—28no—
98C2×C4⋊Dic61C2314yes—
99C2×#11758196C2×C418yescomplemented
100C2×#11758196C2×C418yescomplemented
101C2×#11758196C2314yes—
102C2×#11758196C2314yes—
103C2×#11758196C2314yes—
104C2×#11758196C2312yes—
105C2×#11758196C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2312yes—
108C2×C22⋊Dic61C2312yes—
109C2×C22⋊Dic61C2312yes—
110C2×#11758198C2×C418yes—
111C2×#11758198C2×C418yes—
112C2×#11758198C2314yes—
113C2×#11758198C2314yes—
114C2×#11758198C2314yes—
115C2×#11758198C2312yes—
116C2×#11758198C2312yes—
117C2×#11758198C2312yes—
118C2×#11758198C2312yes—
119C22⋊C4×D61—28no—
120C2×C4×Dic61C2314yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2312yes—
123C2×C4×Dic61C2312yes—
124C2×C4×Dic61C2314yes—
125C42×D61—44no—
126#14519684C2312yes—
127#14519684C2314yes—
128#14519684C2312yes—
129#14519684C2314yes—
130#14519684C2314yes—
131#14519684C2314yes—
132#14519684C2314yes—
133#14519684C2314yes—
134#14519684C2314yes—
135#14519684C2314yes—
136#14519684C2314yes—
137C23×C4×C61C2311yescharacteristic
138C23×C4×C61C2311yescharacteristic
139C23×C4×C61C2311yescharacteristic
140#117×C61—28no—
141C2×C4⋊C4×C61C2×C418yescomplemented
142C2×C4⋊C4×C61C2314yes—
143C2×C4⋊C4×C61C2312yes—
144C2×C4⋊C4×C61C2312yes—
145C2×C22⋊C4×C61C2312yes—
146C2×C22⋊C4×C61C2312yes—
147C2×C42×C61C2314yes—
148C2×C42×C61C2312yes—
149C2×C42×C61C2312yes—
150C2×C42×C61C2314yes—
151#95×C61C2312yes—
152#95×C61C2312yes—
153#95×C61C2314yes—
154C24×C61C2411yescharacteristic
155C23×D61—24no—
156C23×D61—24no—
157C23×D61—42no—
158C23×D61C22×C414yes—
159C23×D61C2411yescharacteristic
160C23×D61C2411yescharacteristic
161C22×C4×C61—24no—
162C22×C4×C61—42no—
163C22×C4×C61—42no—
164C22×C4×C61C22×C414yes—
165C22×C4×C61C22×C414yes—
166C22×C4×C61C22×C414yes—
167C22×C4×C61C2414yes—
168C22×C4×C61C2411yescharacteristic
169C22×C4×C61C2411yescharacteristic
170C22×C4×C61C2412yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2414yes—
173C22×Dic61—24no—
174C22×Dic61—24no—
175C22×Dic61—42no—
176C22×Dic61C22×C414yes—
177C22×Dic61C22×C414yes—
178C22×Dic61C22×C414yes—
179C22×Dic61C2414yes—
180C22×Dic61C2414yes—
181C22×Dic61C2411yescharacteristic
182C22×Dic61C2412yes—
183C22×Dic61C2412yes—
184C22×Dic61C2411yescharacteristic
185C2×C4×D61—24no—
186C2×C4×D61—24no—
187C2×C4×D61—28no—
188C2×C4×D61—28no—
189C2×C4×D61—416no—
190C2×C4×D61—42no—
191C2×C4×D61—42no—
192C2×C4×D61—44no—
193C2×C4×D61C22×C414yes—
194C2×C4×D61C22×C414yes—
195C2×C4×D61C22×C414yes—
196C2×C4×D61C22×C414yes—
197C2×Dic122—28no—
198C2×Dic122—416no—
199C2×Dic122—44no—
200C2×Dic122C22×C418yes—
201C2×Dic122C22×C418yes—
202C4⋊C4×C61—28no—
203C4⋊C4×C61C22×C418yes—
204C4⋊C4×C61C22×C418yes—
205C22⋊C4×C61—24no—
206C22⋊C4×C61#4118yes—
207C42×C61—28no—
208C42×C61—42no—
209C42×C61—42no—
210C42×C61—44no—
211C22⋊Dic61—24no—
212C22⋊Dic61—24no—
213#11758198—28no—
214#11758198—28no—
215#11758198—44no—
216#11758198C22×C418yes—
217#11758198C22×C418yes—
218#11758198C22×C418yes—
219#11758198C22×C418yes—
220C4⋊Dic61—44no—
221#11758196—28no—
222#11758196—28no—
223#11758196—44no—
224#11758196C22×C418yes—
225#11758196C22×C418yes—
226#11758196C22×C418yes—
227#11758196C22×C418yes—
228C4×Dic61—28no—
229C4×Dic61—28no—
230C4×Dic61—42no—
231C4×Dic61—42no—
232C4×Dic61—44no—
233C23×C61—22no—
234C23×C61—22no—
235C23×C61C23×C412yes—
236C23×C61C2×#4114yes—
237C23×C61C2511yescharacteristic
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—28no—
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—28no—
245C22×D61—44no—
246C22×D61—416no—
247C22×D61C23×C412yes—
248C22×D61C23×C412yes—
249C22×D612−1+414yes—
250C2×C4×C61—28no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—44no—
262C2×C4×C61—44no—
263C2×C4×C61C23×C414yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61C23×C412yes—
266C2×C4×C61C23×C414yes—
267C2×C4×C61C2×#4118yes—
268C2×Dic61—28no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—24no—
272C2×Dic61—22no—
273C2×Dic61—28no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—24no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—24no—
280C2×Dic61—44no—
281C2×Dic61C23×C414yes—
282C2×Dic61C23×C414yes—
283C2×Dic61C23×C412yes—
284C2×Dic61C23×C412yes—
285C2×Dic61H(5,𝔽2)14yes—
286C4×D61—28no—
287C4×D61—28no—
288C4×D61—44no—
289C4×D61—44no—
290C4×D61—416no—
291Dic122—216no—
292Dic122—48no—
293#34245—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—28no—
297C22×C61C4×#4114yes—
298C22×C61C24×C411yescharacteristic
299C22×C61C22×#4112yes—
300C22×C61C2×H(5,𝔽2)12yes—
301C22×C61C2×2−1+412yes—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—28no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—416no—
309C2×D61—44no—
310C2×D61#51914yes—
311C4×C61—24no—
312C4×C61—24no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—28no—
316C4×C61—44no—
317Dic61—24no—
318Dic61—24no—
319Dic61—22no—
320Dic61—44no—
321Dic61—44no—
322Dic61#51814yes—
323C22×C22⋊Q8—611no—
324#2368—614no—
325#2362—614no—
326#2344—612no—
327#2342—612no—
328#2342—612no—
329Q8×C22⋊C4—612no—
330#2332—614no—
331#2313—611no—
332#2313—611no—
333C4×C22⋊Q8—614no—
334#2280—611no—
335#2274—611no—
336#2274D6111yescharacteristic, complemented
337C2×C4×C22⋊C4—611no—
338C2×C61—24no—
339C2×C61C2×C4×#4111yescharacteristic
340C2×C61C2×#51811yescharacteristic
341C2×C61C2×#51911yescharacteristic
342C2×C61#352014yes—
343D61—24no—
344D61—42no—
345C23×Q8—611no—
346C24×C4—611no—
347C2×C22⋊Q8—6116no—
348C2×C22⋊Q8—612no—
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—612no—
351C2×C22⋊Q8—612no—
352C2×C4×Q8—614no—
353C2×#117—612no—
354C2×#117—612no—
355C2×#117C2×D6112yes—
356C22×C4⋊C4—611no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4C2×D6111yescharacteristic
359C22×C22⋊C4—611no—
360C22×C22⋊C4—611no—
361C22×C22⋊C4—612no—
362C22×C42—612no—
363#391—614no—
364#391—614no—
365#391—614no—
366#389—614no—
367#388—614no—
368#388—614no—
369#388—614no—
370#388—614no—
371#387—614no—
372#387—614no—
373#387—614no—
374#387—614no—
375#387—614no—
376#387—614no—
377#385—614no—
378#385—614no—
379#381—612no—
380#381—612no—
381#381—612no—
382#380—612no—
383#380—612no—
384#380—612no—
385#380—612no—
386#379—612no—
387C4×C4⋊C4—614no—
388C4×C4⋊C4—614no—
389C4×C4⋊C4—614no—
390C4×C4⋊C4C2×D6114yescomplemented
391C4×C4⋊C4C2×D6114yescomplemented
392C4×C22⋊C4—611no—
393C4×C22⋊C4—611no—
394C4×C22⋊C4—612no—
395C4×C22⋊C4—612no—
396C4×C22⋊C4—612no—
397C4×C22⋊C4—614no—
398C4×C22⋊C4—614no—
399C4×C22⋊C4—612no—
400C4×C22⋊C4—612no—
401C4×C22⋊C4C2×D6111yescharacteristic
402C4×C22⋊C4C2×D6112yes—
403C4×C22⋊C4C2×D6111yescharacteristic
404C2×#95—612no—
405C61#3424511yescharacteristic, complemented, Sylow
406C25—611no—
407C22×Q8—618no—
408C22×Q8—612no—
409C22×Q8—612no—
410C22×Q8—2444no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—611no—
416C23×C4—618no—
417C23×C4—611no—
418C23×C4—611no—
419C23×C4—614no—
420C23×C4—2444no—
421C23×C4C22×D6111yescharacteristic
422C23×C4C22×D6111yescharacteristic
423C23×C4C22×D6111yescharacteristic
424C22⋊Q8—6116no—
425C22⋊Q8—6116no—
426C22⋊Q8—6116no—
427C22⋊Q8—6116no—
428C4×Q8—2448no—
429#117—28no—
430#117—1228no—
431#117—2444no—
432C2×C4⋊C4—618no—
433C2×C4⋊C4—618no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—614no—
439C2×C4⋊C4—614no—
440C2×C4⋊C4—614no—
441C2×C4⋊C4—614no—
442C2×C4⋊C4C4×D6118yescomplemented
443C2×C4⋊C4C22×D6114yes—
444C2×C4⋊C4C22×D6112yes—
445C2×C4⋊C4C22×D6112yes—
446C2×C22⋊C4—618no—
447C2×C22⋊C4—618no—
448C2×C22⋊C4—612no—
449C2×C22⋊C4—612no—
450C2×C22⋊C4—612no—
451C2×C22⋊C4—612no—
452C2×C22⋊C4—612no—
453C2×C22⋊C4—612no—
454C2×C22⋊C4—614no—
455C2×C22⋊C4—614no—
456C2×C22⋊C4—614no—
457C2×C22⋊C4—1228no—
458C2×C22⋊C4C22×D6112yes—
459C2×C22⋊C4C22×D6112yes—
460C2×C42—614no—
461C2×C42—614no—
462C2×C42—612no—
463C2×C42—614no—
464C2×C42—612no—
465C2×C42—2444no—
466C2×C42C22×D6114yes—
467C2×C42C22×D6112yes—
468C2×C42C22×D6114yes—
469C2×C42C22×D6112yes—
470#95—612no—
471#95—614no—
472#95—612no—
473#95—614no—
474#95—614no—
475#95—614no—
476#95—614no—
477#95—614no—
478#95—614no—
479#95—614no—
480#95—614no—
481#95C22×D6112yes—
482#95C22×D6112yes—
483#95C22×D6114yes—
484C24—611no—
485C24—611no—
486C24—614no—
487C24—1224no—
488C24—1224no—
489C24—2442no—
490C24C23×D6111yescharacteristic
491C2×Q8—618no—
492C2×Q8—618no—
493C2×Q8—1228no—
494C2×Q8—2444no—
495C2×Q8—24416no—
496C22×C4—24no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—611no—
500C22×C4—614no—
501C22×C4—614no—
502C22×C4—611no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—614no—
506C22×C4—614no—
507C22×C4—614no—
508C22×C4—614no—
509C22×C4—614no—
510C22×C4—614no—
511C22×C4—614no—
512C22×C4—1224no—
513C22×C4—1224no—
514C22×C4—1228no—
515C22×C4—1224no—
516C22×C4—1224no—
517C22×C4—1228no—
518C22×C4—2444no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—24416no—
522C22×C4—2442no—
523C22×C4C2×C4×D6114yes—
524C22×C4C2×C4×D6114yes—
525C22×C4C2×C4×D6114yes—
526C22×C4C23×D6114yes—
527C22×C4C23×D6112yes—
528C22×C4C23×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530C22×C4C23×D6112yes—
531C22×C4C23×D6114yes—
532C4⋊C4—28no—
533C4⋊C4—618no—
534C4⋊C4—618no—
535C4⋊C4—618no—
536C4⋊C4—618no—
537C4⋊C4—1228no—
538C4⋊C4—1228no—
539C4⋊C4—2444no—
540C4⋊C4—2444no—
541C4⋊C4C2×C4×D6118yes—
542C4⋊C4C2×C4×D6118yes—
543C22⋊C4—24no—
544C22⋊C4—618no—
545C22⋊C4—618no—
546C22⋊C4—618no—
547C22⋊C4—618no—
548C22⋊C4—1228no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1228no—
552C22⋊C4—2444no—
553C22⋊C4#1175822218yes—
554C42—28no—
555C42—42no—
556C42—42no—
557C42—44no—
558C42—1228no—
559C42—1228no—
560C42—2442no—
561C42—2442no—
562C42—2444no—
563C23—22no—
564C23—22no—
565C23—612no—
566C23—614no—
567C23—612no—
568C23—1228no—
569C23—1222no—
570C23—1222no—
571C23—1228no—
572C23—1222no—
573C23—1222no—
574C23—1222no—
575C23—2444no—
576C23—24416no—
577C23C22×C4×D6112yes—
578C23C2×#1175822214yes—
579C23C24×D6111yescharacteristic
580Q8—12216no—
581Q8—2448no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—612no—
596C2×C4—614no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—612no—
600C2×C4—1228no—
601C2×C4—1228no—
602C2×C4—1228no—
603C2×C4—1228no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1224no—
607C2×C4—1222no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—2444no—
615C2×C4—2444no—
616C2×C4—24416no—
617C2×C4—2444no—
618C2×C4C22×C4×D6114yes—
619C2×C4C22×C4×D6114yes—
620C2×C4C22×C4×D6112yes—
621C2×C4C22×C4×D6112yes—
622C2×C4C2×#1175822218yes—
623C22—22no—
624C22—22no—
625C22—28no—
626C22—614no—
627C22—1222no—
628C22—1222no—
629C22—1228no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—24416no—
634C22—2444no—
635C22C4×#1175822214yes—
636C22C23×C4×D6111yescharacteristic
637C22C22×#1175822212yes—
638C22H(5,𝔽2)×D6112yes—
639C222−1+4×D6112yes—
640C4—24no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—28no—
645C4—44no—
646C4—614no—
647C4—1224no—
648C4—1222no—
649C4—1224no—
650C4—2444no—
651C4—2444no—
652C2—24no—
653C2—1224no—
654C2—2442no—
655C2C2×C4×#1175822211yescharacteristic
656C2#518×D6111yescharacteristic
657C2#519×D6111yescharacteristic
658C2#2145985814yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2274C2
socle, cosocleC23×C61C25
prime radicalC24×D61C23
hypercenter#2274D61
hypocenterC61#34245
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
2. #21448903C2semidirect
3. #21448847C2semidirect
4. #21448689C2semidirect
7. C22⋊C4×Dic122C2semidirect
8. #21448570C2semidirect
9. #21448422C2semidirect
10. #21448419C2semidirect
11. C4×#14519755C2semidirect
13. #21448174C2semidirect
15. C61×#2274C2semidirect
18. C2×#14519755C4semidirect
23. C2×C4×Dic122C22semidirect
32. #16839668C22semidirect
33. #16839666C22semidirect
34. #16839666C22semidirect
35. #16839657C22semidirect
40. #16839648C22semidirect
41. #16839648C22semidirect
42. #16839646C22semidirect
43. #16839646C22semidirect
44. #16839644C22semidirect
45. #16839643C22semidirect
56. C4×#11758196C22semidirect
57. C4×#11758196C22semidirect
58. #16839613C22semidirect
71. C4×C4⋊C4×C61C22semidirect
72. C4×C4⋊C4×C61C22semidirect
78. C22×Dic122C2×C4semidirect
93. #14519755C2×C4semidirect
94. #14519755C2×C4semidirect
99. C2×#11758196C2×C4semidirect
100. C2×#11758196C2×C4semidirect
141. C2×C4⋊C4×C61C2×C4semidirect
336. #2274D61semidirect
390. C4×C4⋊C4C2×D61semidirect
391. C4×C4⋊C4C2×D61semidirect
405. C61#34245semidirect
442. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197552
#214489032
#214488472
#214486892
#214486712
#214486692
C22⋊C4×Dic1222
#214485702
#214484222
#214484192
C4×#145197552
#214482002
#214481742
C4×C22⋊C4×D612
C61×#22742
#3424561