Finite Group Databasebeta

← #29840333

Subgroups — #29840333

Order 15616.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic
3#21448842C214yesmaximal, complemented
4#21448820C214yesmaximal, complemented
5#21448689C214yesmaximal, complemented
6#21448669C214yesmaximal
7#21448495C214yesmaximal, complemented
8#21448435C211yesmaximal, characteristic, complemented
9#21448405C211yesmaximal, characteristic, complemented
10C4×#14519753C214yesmaximal, complemented
11#21448198C211yesmaximal, characteristic
12#21448175C211yesmaximal, characteristic, complemented
13C2×C61×#519C211yesmaximal, characteristic, complemented
14C23×C4×D61C2211yescharacteristic
15C2×#14519753C4116yescomplemented
16C2×#14519753C2212yes—
17C2×#14519753C2212yes—
18C2×#14519753C2212yes—
19C2×#14519753C2212yes—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22C2×#14519735C2212yes—
23C2×#14519734C2212yes—
24C2×#14519733C2212yes—
25#117×D61—28no—
26C22×C4⋊Dic61C2211yescharacteristic
27C22×#11758196C2212yes—
28C2×C4⋊C4×D61C2212yes—
29C22×#11758198C2212yes—
30C2×C22⋊C4×D61C2212yes—
31C2×C42×D61C2212yes—
32#16839669C2214yescomplemented
33#16839669C2214yescomplemented
34#16839660C2214yescomplemented
35#16839658C2214yescomplemented
36#16839657C2218yescomplemented
37#16839653C2218yes—
38#16839652C2218yes—
39#16839646C2218yescomplemented
40#16839643C2218yescomplemented
41#16839633C2218yes—
42#16839625C2214yes—
43#16839623C2212yes—
44#16839620C2214yes—
45#16839619C2212yes—
46#16839619C2212yes—
47#16839617C2212yes—
48C4⋊C4×Dic61C2214yescomplemented
49C4×#11758196C2218yescomplemented
50C4×C4⋊Dic61C2214yescomplemented
51C22⋊C4×Dic61C2214yes—
52C4×#11758198C2212yes—
53C4×#11758198C2212yes—
54C4×#11758198C2218yes—
55C23×Q8×C61C2211yescharacteristic
56C61×#519—28no—
57C2×C4×Q8×C61C2214yescomplemented
58C2×C4×Q8×C61C2214yescomplemented
59C2×#117×C61C2211yescharacteristic
60C2×#117×C61C2214yes—
61C2×#117×C61C2211yescharacteristic
62C24×D61C2311yescharacteristic
63C23×Dic61C2311yescharacteristic
64C23×Dic61C2312yes—
65C22×C4×D61—24no—
66C22×C4×D61—24no—
67C22×C4×D61C2×C418yes—
68C22×C4×D61C2314yes—
69C22×C4×D61C2311yescharacteristic
70C22×C4×D61C2311yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C22×C4×D61C2311yescharacteristic
73#14519753C2×C4116yes—
74#14519753C2×C4116yes—
75#14519753C2×C4116yescomplemented
76#14519753C2×C4116yescomplemented
77#14519736—24no—
78#14519736—24no—
79#14519735—28no—
80#14519734—28no—
81#14519733—28no—
82C2×C4⋊Dic61C2×C418yescomplemented
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2312yes—
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2312yes—
87C2×#11758196C2×C4116yescomplemented
88C2×#11758196C2314yes—
89C2×#11758196C2318yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C4⋊C4×D61—28no—
93C2×C22⋊Dic61C2312yes—
94C2×C22⋊Dic61C2312yes—
95C2×#11758198C2×C4116yes—
96C2×#11758198C2314yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C22⋊C4×D61—28no—
102C2×C4×Dic61C2314yes—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2312yes—
105C2×C4×Dic61C2318yes—
106C42×D61—28no—
107#14519684C2314yes—
108#14519684C2314yes—
109#14519684C2318yes—
110#14519684C2318yes—
111#14519684C2318yes—
112#14519684C2318yes—
113C22×Q8×C61—24no—
114C22×Q8×C61—24no—
115C22×Q8×C61C2×C418yescomplemented
116C22×Q8×C61C2312yes—
117C22×Q8×C61C2312yes—
118C23×C4×C61C2312yes—
119C23×C4×C61C2311yescharacteristic
120C4×Q8×C61—216no—
121C4×Q8×C61—216no—
122#117×C61—24no—
123#117×C61—216no—
124#117×C61D418yes—
125C2×C4⋊C4×C61C2312yes—
126C2×C4⋊C4×C61C2318yes—
127C2×C4⋊C4×C61C2312yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C22⋊C4×C61C2312yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2318yes—
132C2×C42×C61C2312yes—
133C24×C61C2411yescharacteristic
134C23×D61—22no—
135C23×D61—22no—
136C23×D61—24no—
137C23×D61—42no—
138C23×D61C22×C414yes—
139C23×D61C2411yescharacteristic
140C23×D61C2411yescharacteristic
141C2×Q8×C61—232no—
142C2×Q8×C61—24no—
143C2×Q8×C61—24no—
144C2×Q8×C61—24no—
145C2×Q8×C61—24no—
146C2×Q8×C61C22×C418yes—
147C2×Q8×C61C22×C418yes—
148C22×C4×C61—24no—
149C22×C4×C61—24no—
150C22×C4×C61C22×C418yes—
151C22×C4×C61C22×C414yes—
152C22×C4×C61C2×D414yes—
153C22×C4×C61#4114yes—
154C22×C4×C61C2414yes—
155C22×C4×C61C2414yes—
156C22×C4×C61C2414yes—
157C22×C4×C61C2411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—44no—
162C22×Dic61C22×C414yes—
163C22×Dic61C22×C418yes—
164C22×Dic61C2414yes—
165C22×Dic61C2414yes—
166C22×Dic61C2411yescharacteristic
167C22×Dic61C2411yescharacteristic
168C22×Dic61C2414yes—
169C2×C4×D61—232no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—28no—
179C2×C4×D61—28no—
180C2×C4×D61—44no—
181C2×C4×D61C22×C414yes—
182C2×C4×D61C22×C414yes—
183C2×C4×D61C22×C414yes—
184C2×C4×D61C22×C414yes—
185C4⋊C4×C61—216no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61C2×D418yes—
188C22⋊C4×C61—24no—
189C22⋊C4×C61C2×D418yes—
190C42×C61—216no—
191C42×C61—24no—
192C42×C61C2×D418yes—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195#11758198—28no—
196#11758198—28no—
197#11758198—44no—
198#11758198C22×C4116yes—
199#11758198C22×C4116yes—
200C4⋊Dic61—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61C22×C418yes—
203C4⋊Dic61C22×C418yes—
204#11758196—28no—
205#11758196—48no—
206#11758196C22×C4116yes—
207#11758196C22×C4116yes—
208C4×Dic61—28no—
209C4×Dic61—24no—
210C4×Dic61—24no—
211C4×Dic61—48no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61C23×C412yes—
215C23×C61C22×D412yes—
216C23×C61C2×#4112yes—
217C23×C61C2511yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—22no—
221C22×D61—28no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—28no—
227C22×D61—44no—
228C22×D61—48no—
229C22×D61—48no—
230C22×D61C23×C412yes—
231C22×D61C23×C412yes—
232C22×D612−1+414yes—
233Q8×C61—216no—
234Q8×C61—216no—
235C2×C4×C61—216no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61C4×D4116yes—
244C2×C4×C61C23×C418yes—
245C2×C4×C61C23×C412yes—
246C2×C4×C61C23×C412yes—
247C2×C4×C61C22×D418yes—
248C2×C4×C61C22×D412yes—
249C2×C4×C61C22×D412yes—
250C2×C4×C61C2×#4112yes—
251C2×C4×C61C2×#4112yes—
252C2×Dic61—28no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—28no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—48no—
265C2×Dic61C23×C412yes—
266C2×Dic61C23×C412yes—
267C2×Dic61C23×C418yes—
268C4×D61—28no—
269C4×D61—28no—
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—416no—
273#34203—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—24no—
277C22×C61—24no—
278C22×C61C2×C4×D414yes—
279C22×C61C24×C411yescharacteristic
280C22×C61C23×D411yescharacteristic
281C22×C61C22×#4111yescharacteristic
282C22×C61C2×2−1+412yes—
283C22×C61#58512yes—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—24no—
287C2×D61—24no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—416no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61#51914yescomplemented
295C4×C61—28no—
296C4×C61—28no—
297C4×C61—44no—
298C4×C61C2×C4×D414yes—
299C4×C61C2×C4×D414yes—
300Dic61—22no—
301Dic61—22no—
302Dic61—28no—
303Dic61—44no—
304Dic61—44no—
305C22×C22⋊Q8—611no—
306C2×#519D6111yescharacteristic, complemented
307C22×#117—611no—
308#2362—614no—
309#2360—614no—
310#2344—614no—
311#2342—614no—
312#2323—614no—
313#2316—611no—
314#2310—611no—
315C4×C22⋊Q8—614no—
316#2280—611no—
317#2274—611no—
318C2×C61—24no—
319C2×C61C22×C4×D411yescharacteristic
320C2×C61C2×#51911yescharacteristic
321C2×C61#342811yescharacteristic
322C2×C61#348912yes—
323C2×C61#349512yes—
324D61—24no—
325D61—42no—
326C23×Q8C2×D6111yescharacteristic
327C24×C4—611no—
328C2×C22⋊Q8—6116no—
329C2×C22⋊Q8—612no—
330C2×C22⋊Q8—612no—
331C2×C22⋊Q8—612no—
332C2×C22⋊Q8—612no—
333#519—28no—
334C2×C4×Q8C2×D6114yescomplemented
335C2×C4×Q8C2×D6114yescomplemented
336C2×#117—611no—
337C2×#117—611no—
338C2×#117—612no—
339C2×#117—612no—
340C2×#117—612no—
341C2×#117—1228no—
342C2×#117C2×D6111yescharacteristic
343C2×#117C2×D6114yes—
344C2×#117C2×D6111yescharacteristic
345C22×C4⋊C4—611no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4—612no—
348C22×C22⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C42—612no—
351#391—614no—
352#391—614no—
353#389—614no—
354#389—618no—
355#389—614no—
356#388—618no—
357#388—618no—
358#387—618no—
359#387—618no—
360#385—618no—
361#381—614no—
362#381—612no—
363#380—612no—
364#380—612no—
365#380—614no—
366#379—612no—
367C4×C4⋊C4—614no—
368C4×C4⋊C4—614no—
369C4×C4⋊C4—618no—
370C4×C22⋊C4—614no—
371C4×C22⋊C4—618no—
372C4×C22⋊C4—612no—
373C4×C22⋊C4—612no—
374C61#3420311yescharacteristic, complemented, Sylow
375C25—611no—
376C22×Q8—24no—
377C22×Q8—24no—
378C22×Q8C4×D6118yescomplemented
379C22×Q8C22×D6112yes—
380C22×Q8C22×D6112yes—
381C23×C4—611no—
382C23×C4—612no—
383C23×C4—618no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—614no—
389C23×C4—1224no—
390C23×C4—1224no—
391C23×C4C22×D6112yes—
392C23×C4C22×D6111yescharacteristic
393C22⋊Q8—6116no—
394C22⋊Q8—6116no—
395C22⋊Q8—6116no—
396C22⋊Q8—6116no—
397C4×Q8—216no—
398C4×Q8—216no—
399#117—24no—
400#117—216no—
401#117—1224no—
402#117—1224no—
403#117—1228no—
404#117—1228no—
405#117—1228no—
406#117C22⋊D6118yes—
407C2×C4⋊C4—6116no—
408C2×C4⋊C4—618no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—618no—
417C2×C4⋊C4—1228no—
418C2×C4⋊C4C22×D6112yes—
419C2×C4⋊C4C22×D6118yes—
420C2×C4⋊C4C22×D6112yes—
421C2×C22⋊C4—6116no—
422C2×C22⋊C4—614no—
423C2×C22⋊C4—614no—
424C2×C22⋊C4—612no—
425C2×C22⋊C4—612no—
426C2×C22⋊C4—614no—
427C2×C22⋊C4—614no—
428C2×C22⋊C4—614no—
429C2×C22⋊C4—1228no—
430C2×C22⋊C4C22×D6112yes—
431C2×C22⋊C4C22×D6112yes—
432C2×C42—614no—
433C2×C42—612no—
434C2×C42—612no—
435C2×C42—618no—
436C2×C42—1228no—
437C2×C42C22×D6112yes—
438C2×C42C22×D6118yes—
439C2×C42C22×D6112yes—
440#95—614no—
441#95—618no—
442#95—614no—
443#95—618no—
444#95—618no—
445#95—618no—
446C24—611no—
447C24—611no—
448C24—614no—
449C24—1224no—
450C24—1222no—
451C24—1222no—
452C24—2442no—
453C24C23×D6111yescharacteristic
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—232no—
459C2×Q8C2×C4×D6118yes—
460C2×Q8C2×C4×D6118yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—614no—
464C22×C4—611no—
465C22×C4—614no—
466C22×C4—611no—
467C22×C4—618no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—1228no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—12232no—
486C22×C4—1228no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4C2×C4×D6118yes—
490C22×C4C2×C4×D6114yes—
491C22×C4#1175822214yes—
492C22×C4C2×C22⋊D6114yes—
493C22×C4C23×D6114yes—
494C22×C4C23×D6114yes—
495C22×C4C23×D6114yes—
496C22×C4C23×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—6116no—
501C4⋊C4—6116no—
502C4⋊C4—618no—
503C4⋊C4—618no—
504C4⋊C4—1228no—
505C4⋊C4—1224no—
506C4⋊C4—1224no—
507C4⋊C4—2448no—
508C4⋊C4C2×C22⋊D6118yes—
509C22⋊C4—24no—
510C22⋊C4—6116no—
511C22⋊C4—6116no—
512C22⋊C4—1228no—
513C22⋊C4—1224no—
514C22⋊C4—1224no—
515C22⋊C4—1228no—
516C22⋊C4—2444no—
517C22⋊C4C2×C22⋊D6118yes—
518C42—216no—
519C42—24no—
520C42—1228no—
521C42—1224no—
522C42—1224no—
523C42—2448no—
524C42C2×C22⋊D6118yes—
525C23—22no—
526C23—22no—
527C23—612no—
528C23—614no—
529C23—612no—
530C23—1228no—
531C23—1221no—
532C23—1221no—
533C23—1222no—
534C23—1228no—
535C23—1221no—
536C23—1221no—
537C23—1222no—
538C23—1222no—
539C23—2444no—
540C23—2448no—
541C23—2448no—
542C23C22×C4×D6112yes—
543C23C22×C22⋊D6112yes—
544C23C2×#1175822212yes—
545C23C24×D6111yescharacteristic
546Q8—216no—
547Q8—216no—
548C2×C4—216no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—618no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—1228no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1228no—
565C2×C4—1228no—
566C2×C4—1228no—
567C2×C4—1228no—
568C2×C4—1228no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—24416no—
576C2×C4—2448no—
577C2×C4C4×C22⋊D61116yes—
578C2×C4C22×C4×D6118yes—
579C2×C4C22×C4×D6112yes—
580C2×C4C22×C4×D6112yes—
581C2×C4C22×C22⋊D6118yes—
582C2×C4C22×C22⋊D6112yes—
583C2×C4C22×C22⋊D6112yes—
584C2×C4C2×#1175822212yes—
585C2×C4C2×#1175822212yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—614no—
591C22—1224no—
592C22—1222no—
593C22—1222no—
594C22—1224no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22—24416no—
599C22—2442no—
600C22—2442no—
601C22C2×C4×C22⋊D6114yes—
602C22C23×C4×D6111yescharacteristic
603C22C23×C22⋊D6111yescharacteristic
604C22C22×#1175822211yescharacteristic
605C222−1+4×D6112yes—
606C22#1684059912yes—
607C4—28no—
608C4—28no—
609C4—44no—
610C4—1222no—
611C4—1222no—
612C4—1228no—
613C4—2444no—
614C4—2444no—
615C4C2×C4×C22⋊D6114yes—
616C4C2×C4×C22⋊D6114yes—
617C2—24no—
618C2—1224no—
619C2—2442no—
620C2C22×C4×C22⋊D6111yescharacteristic
621C2#519×D6111yescharacteristic
622C2#2145902111yescharacteristic
623C2#2145949212yes—
624C2#2145956112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC2×C61×#519C2
socle, cosocleC23×C61C25
prime radicalC24×D61C23
hypercenterC2×#519D61
hypocenterC61#34203
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
3. #21448842C2semidirect
4. #21448820C2semidirect
5. #21448689C2semidirect
7. #21448495C2semidirect
8. #21448435C2semidirect
9. #21448405C2semidirect
10. C4×#14519753C2semidirect
12. #21448175C2semidirect
13. C2×C61×#519C2semidirect
15. C2×#14519753C4semidirect
32. #16839669C22semidirect
33. #16839669C22semidirect
34. #16839660C22semidirect
35. #16839658C22semidirect
36. #16839657C22semidirect
39. #16839646C22semidirect
40. #16839643C22semidirect
48. C4⋊C4×Dic61C22semidirect
49. C4×#11758196C22semidirect
50. C4×C4⋊Dic61C22semidirect
57. C2×C4×Q8×C61C22semidirect
58. C2×C4×Q8×C61C22semidirect
75. #14519753C2×C4semidirect
76. #14519753C2×C4semidirect
82. C2×C4⋊Dic61C2×C4semidirect
87. C2×#11758196C2×C4semidirect
115. C22×Q8×C61C2×C4semidirect
294. C2×D61#519semidirect
306. C2×#519D61semidirect
334. C2×C4×Q8C2×D61semidirect
335. C2×C4×Q8C2×D61semidirect
374. C61#34203semidirect
378. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197532
C2×#117×D612
#214488422
#214488202
#214486892
#214486692
#214484952
#214484352
#214484052
C4×#145197532
#214481982
#214481752
C2×C61×#5192
#3420361