Finite Group Databasebeta

← #30616749

Subgroups — #30616749

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462266C211yesmaximal, characteristic, complemented
2C4×#14519856C211yesmaximal, characteristic, complemented
3#21462008C212yesmaximal, complemented
4#21461959C212yesmaximal
5#21461843C214yesmaximal, complemented
6C4×#14519836C214yesmaximal, complemented
7C2×#16840299C211yesmaximal, characteristic
8C2×#14519856C2212yes—
9C2×#14519836C2218yes—
10#16840707—21no—
11#16840706—21no—
12#16840705C2212yes—
13#16840704C2212yes—
14C4×C4⋊#6923—21no—
15C4×C4⋊#6923—21no—
16C4×C4⋊#6923—21no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2212yes—
19C4×C22⋊#6923C2212yes—
20#16840698—21no—
21#16840698—21no—
22C42×#6923—21no—
23#16840640C2218yes—
24#16840299—21no—
25#16840299—21no—
26#16840299—21no—
27#16840299—21no—
28#16840299—21no—
29#16840299—21no—
30#16840299—21no—
31#16840299—21no—
32C2×#14519772C412yescomplemented
33C2×#14519768C411yescharacteristic, complemented
34C2×#14519766C411yescharacteristic, complemented
35C2×#14519762C2212yes—
36C2×#14519753C2211yescharacteristic
37C2×#14519752C2211yescharacteristic
38C2×C4×D244C411yescharacteristic, complemented
39C2×C4×D244C2212yes—
40C2×#14519734C412yescomplemented
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic, complemented
43#14519856—28no—
44C2×C4⋊#6923—22no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923C2314yes—
47C2×C22⋊#6923C2314yes—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—22no—
51C2×C4×#6923—22no—
52#14519836—216no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C22×D244C2×C411yescharacteristic
60C22×D244C2312yes—
61C22×C4×D61C2×C412yes—
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519772—22no—
68#14519772—22no—
69#14519768—21no—
70#14519768—21no—
71#14519768—21no—
72#14519768—21no—
73#14519766—21no—
74#14519766—21no—
75#14519766—21no—
76#14519766—21no—
77#14519762—22no—
78#14519762—22no—
79#14519762—22no—
80#14519762—22no—
81#14519753—21no—
82#14519753—21no—
83#14519753—21no—
84#14519753—21no—
85#14519752—21no—
86#14519752—21no—
87#14519752—21no—
88#14519752—21no—
89C4×D244—22no—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—22no—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244—21no—
96C4×D244—21no—
97#14519734—22no—
98#14519734—22no—
99#14519734—22no—
100#14519734—22no—
101C2×C4⋊Dic61C2×C411yescharacteristic
102C2×C4⋊Dic61C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2×C412yes—
107C2×#11758198C2×C412yes—
108C2×#11758198C2×C412yes—
109C2×#11758198C2312yes—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C4⋊C4×C61C2×C412yes—
128C2×C42×C61C2×C412yes—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C22×C412yes—
131C23×D61C2411yescharacteristic
132C22×#6923—24no—
133C22×#6923—24no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C412yes—
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—28no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—42no—
152C2×C4×D61—24no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—44no—
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C22⋊#6923—24no—
171C22⋊#6923—28no—
172C22⋊#6923#4118yes—
173C4⋊#6923—28no—
174C4⋊#6923—44no—
175C4×#6923—28no—
176C4×#6923—44no—
177C4×#6923—44no—
178C4×#6923—44no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61C22⋊C411yescharacteristic
186C42×C61C22⋊C411yescharacteristic
187C42×C61C22⋊C411yescharacteristic
188C42×C61C22⋊C411yescharacteristic
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C23×C61C23×C411yescharacteristic
220C22×D61—22no—
221C22×D61—24no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61C22×D411yescharacteristic
227C22×D61C22×D411yescharacteristic
228C22×D61C2×#4114yes—
229C2×#6923—28no—
230C2×#6923—44no—
231C2×#6923—44no—
232C2×#6923—44no—
233C2×#6923—44no—
234C2×#6923—44no—
235Q8×C61—42no—
236Q8×C61—42no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261D244—42no—
262D244—44no—
263D244—44no—
264D244—42no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—44no—
278C4×D61—44no—
279#21421—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C4×#4112yes—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—84no—
293C2×D61#52512yes—
294C2×D61#53212yes—
295#6923—48no—
296#6923—84no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61#41012yes—
304C4×C61#41012yes—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—21no—
311Dic61—42no—
312C2×#529—611no—
313#2283—611no—
314C4×#117—611no—
315#1996—612no—
316#1964—612no—
317#1899—614no—
318C4×#99—614no—
319C2×C61—22no—
320C2×C61#229711yescharacteristic
321C2×C61C2×#41011yescharacteristic
322C2×C61C2×#41011yescharacteristic
323D61—22no—
324D61—22no—
325D61—82no—
326#529—1221no—
327#529—1221no—
328#529—1221no—
329#529—1221no—
330#529—1221no—
331#529—1221no—
332#529—1221no—
333#529—1221no—
334C2×#126—612no—
335C2×#125—611no—
336C2×#124—611no—
337C2×#123—612no—
338C2×C22⋊Q8—611no—
339C2×#121—611no—
340C2×C4×Q8#692311yescharacteristic, complemented
341C2×C4×D4—612no—
342C2×C4×D4—611no—
343C2×#117—612no—
344C2×#117—612no—
345C22×C42—611no—
346C2×#99—618no—
347#383—1221no—
348#382—1221no—
349#381—612no—
350#380—612no—
351C4×C4⋊C4—1221no—
352C4×C4⋊C4—1221no—
353C4×C4⋊C4—1221no—
354C4×C22⋊C4—612no—
355C4×C22⋊C4—612no—
356C4×C22⋊C4—612no—
357#376—1221no—
358#376—1221no—
359C43—1221no—
360#342—618no—
361C61#2142111yescharacteristic, complemented, Sylow
362C22×Q8C2×#692311yescharacteristic
363C22×D4—612no—
364C22×D4—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—612no—
369#126—1222no—
370#126—1222no—
371#126—1222no—
372#126—1222no—
373#125—1221no—
374#125—1221no—
375#125—1221no—
376#125—1221no—
377#124—1221no—
378#124—1221no—
379#124—1221no—
380#124—1221no—
381#123—1222no—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389#121—1221no—
390#121—1221no—
391#121—1221no—
392#121—1221no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1221no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117—1222no—
409#117—1228no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4C2×#692311yescharacteristic
417C2×C4⋊C4C2×#692312yes—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—612no—
421C2×C22⋊C4—612no—
422C2×C22⋊C4—612no—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—614no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1222no—
437C2×C42—1222no—
438C2×C42—1222no—
439C2×C42—1222no—
440C2×C42C2×#692312yes—
441C2×C42C2×#692311yescharacteristic
442#99—12216no—
443#95—1222no—
444#95—1222no—
445#95—1222no—
446#95—1222no—
447#95—1222no—
448#95—1222no—
449C24—611no—
450C24—612no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—42no—
454C2×D4—1228no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—2442no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1224no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2444no—
484C22×C4C22×#692312yes—
485C22×C4C22×#692311yescharacteristic
486C22×C4C22×#692311yescharacteristic
487C22×C4C22×#692311yescharacteristic
488C22×C4C22×#692312yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1228no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2444no—
503C22⋊C4—618no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1224no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1228no—
516C42—22no—
517C42—22no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—1228no—
531C42—2444no—
532C42—2444no—
533C42—2444no—
534C42C22⋊#692311yescharacteristic
535C42C22⋊#692311yescharacteristic
536C42C22⋊#692311yescharacteristic
537C42C22⋊#692311yescharacteristic
538C23—611no—
539C23—611no—
540C23—614no—
541C23—1224no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2442no—
546C23—2442no—
547C23C23×#692311yescharacteristic
548Q8—42no—
549Q8—42no—
550D4—2442no—
551D4—2444no—
552D4—2444no—
553D4—2442no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1228no—
577C2×C4—1222no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2444no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2444no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2442no—
592C2×C4C2×C22⋊#692311yescharacteristic
593C2×C4C2×C22⋊#692311yescharacteristic
594C2×C4C2×C22⋊#692311yescharacteristic
595C2×C4C2×C22⋊#692311yescharacteristic
596C2×C4C2×C22⋊#692311yescharacteristic
597C2×C4C2×C22⋊#692311yescharacteristic
598C22—21no—
599C22—21no—
600C22—612no—
601C22—612no—
602C22—1221no—
603C22—1221no—
604C22—1221no—
605C22—1221no—
606C22—4884no—
607C22—2444no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22C22×C22⋊#692311yescharacteristic
612C22#41×#692312yes—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—1221no—
620C4—1221no—
621C4—1221no—
622C4—1221no—
623C4—1221no—
624C4—1221no—
625C4—4884no—
626C4—2448no—
627C4—2442no—
628C4#1684074112yes—
629C4#1684074312yes—
630C2—22no—
631C2—1222no—
632C2—1222no—
633C2—4882no—
634C2#2146228011yescharacteristic
635C2C2×#1684074111yescharacteristic
636C2C2×#1684074311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519752C22
cosocleC23×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#21421
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462266C2semidirect
2. C4×#14519856C2semidirect
3. #21462008C2semidirect
5. #21461843C2semidirect
6. C4×#14519836C2semidirect
32. C2×#14519772C4semidirect
33. C2×#14519768C4semidirect
34. C2×#14519766C4semidirect
38. C2×C4×D244C4semidirect
40. C2×#14519734C4semidirect
42. C2×C4×Q8×C61C4semidirect
340. C2×C4×Q8#6923semidirect
361. C61#21421semidirect

Maximal subgroups

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

subgroupindex
#214622662
C4×#145198562
#214620082
#214619592
#214618432
C4×#145198362
C2×#168402992
#2142161