Finite Group Databasebeta

← #30615905

Subgroups — #30615905

Order 15616.

1226 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
1C2×#16840769C211yesmaximal, characteristic, complemented
2C2×#16840733C211yesmaximal, characteristic
3C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic
4C2×#16840727C211yesmaximal, characteristic, complemented
5#21461894C212yesmaximal, complemented
6#21461893C212yesmaximal, complemented
7#21461890C212yesmaximal, complemented
8#21461889C212yesmaximal, complemented
9#21461486C211yesmaximal, characteristic, complemented
10#21461486C211yesmaximal, characteristic, complemented
11#21448201C211yesmaximal, characteristic
12C22×#11758213C2211yescharacteristic
13C22×#11758213C2211yescharacteristic
14C22×D61⋊C8C2211yescharacteristic
15C22×D61⋊C8C2211yescharacteristic
16#16840769—24no—
17C2×#14519844C2212yescomplemented
18C2×C4⋊#92420C2212yescomplemented
19#16840733—24no—
20C2×#14519834C2211yescharacteristic
21C2×#14519834C2211yescharacteristic
22C2×#14519834C2212yes—
23D61⋊(C22⋊C8)—24no—
24C2×C22⋊#92420C2211yescharacteristic
25C2×C22⋊#92420C2211yescharacteristic
26C2×C22⋊#92420C2212yes—
27#16840727—24no—
28C2×#14519832C2212yescomplemented
29C2×C4×#92420C2212yescomplemented
30#16840628C2212yescomplemented
31#16840628C2212yescomplemented
32#16840628C2212yescomplemented
33#16840628C2212yescomplemented
34C23×C4×D61C2211yescharacteristic
35C2×#14519735C2211yescharacteristic
36C2×#14519735C2211yescharacteristic
37#16839624C412yes—
38#16839622C411yescharacteristic
39#16839619C411yescharacteristic
40#16839619C411yescharacteristic
41#16839617C412yes—
42C22⋊C4×Dic61C2212yes—
43#16839609C2212yes—
44C61×#380C411yescharacteristic
45C2×#11758213—22no—
46C2×#11758213—22no—
47C2×#11758213—22no—
48C2×#11758213—22no—
49C2×#11758213—22no—
50C2×#11758213D414yes—
51C2×D61⋊C8—22no—
52C2×D61⋊C8—22no—
53C2×D61⋊C8—22no—
54C2×D61⋊C8—22no—
55C2×D61⋊C8—22no—
56C2×D61⋊C8D414yes—
57C22×#92420C2312yes—
58C22×#92420C2311yescharacteristic
59C22×#92420C2311yescharacteristic
60C22×#92420C2311yescharacteristic
61C22×#92420C2312yes—
62C22×#92420C2311yescharacteristic
63#14519844—24no—
64C4⋊#92420—24no—
65#14519834—22no—
66#14519834—22no—
67#14519834—24no—
68C22⋊#92420—22no—
69C22⋊#92420—22no—
70C22⋊#92420—24no—
71#14519832—24no—
72C4×#92420—24no—
73C24×D61C2×C411yescharacteristic
74C23×Dic61C2311yescharacteristic
75C22×C4×D61—22no—
76C22×C4×D61—22no—
77C22×C4×D61—22no—
78C22×C4×D61—22no—
79C22×C4×D61—22no—
80C22×C4×D61—22no—
81C22×C4×D61C2×C411yescharacteristic
82C22×C4×D61C2×C411yescharacteristic
83C22×C4×D61C2311yescharacteristic
84C22×C4×D61C2311yescharacteristic
85#14519735—22no—
86#14519735—22no—
87#14519735—22no—
88#14519735—22no—
89C2×C4⋊Dic61C2×C411yescharacteristic
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×C22⋊Dic61C2×C412yes—
92C2×#11758198C2×C412yes—
93C2×#11758198C2×C412yes—
94C2×#11758198C2×C412yes—
95C2×#11758198C2×C412yes—
96C2×C4×Dic61C2312yes—
97C2×C4×Dic61C2312yes—
98#14519684C2×C412yes—
99#14519684C2×C412yes—
100#14519684C2×C412yes—
101C23×C4×C61C2×C411yescharacteristic
102C2×C4⋊C4×C61C2×C411yescharacteristic
103C2×C4⋊C4×C61C2×C411yescharacteristic
104C2×C22⋊C4×C61C2×C412yes—
105#95×C61C2×C412yes—
106C24×C61C22×C411yescharacteristic
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—41no—
113C23×D61—41no—
114C23×D61C22×C411yescharacteristic
115C23×D61C22×C411yescharacteristic
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C412yes—
131C22×Dic61C22×C412yes—
132C22×Dic61C2411yescharacteristic
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C4×D61—44no—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61#4112yes—
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—22no—
160C2×#92420—22no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—22no—
164C2×#92420—22no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—22no—
170C2×#92420C2×D414yes—
171C2×#92420C2×D412yes—
172C2×#92420C2×D412yes—
173#11758213—28no—
174#11758213—44no—
175D61⋊C8—28no—
176D61⋊C8—44no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—42no—
180C22⋊Dic61—24no—
181#11758198—24no—
182#11758198—24no—
183#11758198—24no—
184#11758198—42no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61#13112yes—
198C23×C61C23×C411yescharacteristic
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—28no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—44no—
211C22×D61—44no—
212C22×D61—44no—
213C22×D61—44no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61#11712yes—
217C22×D61C4×D412yes—
218C22×D61C4×D412yes—
219C22×D61#13112yes—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—44no—
230C2×C4×C61#11712yes—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C4×D412yes—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61C22×D411yescharacteristic
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C2×#4111yescharacteristic
249C2×Dic61C2×#4111yescharacteristic
250C2×Dic61C2×#4111yescharacteristic
251C2×Dic61C2×#4111yescharacteristic
252C4×D61—44no—
253C4×D61—44no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61#12114yes—
259C4×D61#12314yes—
260C4×D61#12414yes—
261C4×D61#12614yes—
262#92420—22no—
263#92420—22no—
264#92420—24no—
265#92420—42no—
266#92420—41no—
267#92420—41no—
268#21053—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×#11711yescharacteristic
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C2×C4×D411yescharacteristic
279C22×C61C2×#13111yescharacteristic
280C22×C61C2×#13111yescharacteristic
281C22×C61#56811yescharacteristic
282C22×C61#56811yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—416no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61#38814yes—
298C4×C61—42no—
299C4×C61—42no—
300C4×C61—41no—
301C4×C61#38814yes—
302Dic61—42no—
303Dic61—42no—
304Dic61—41no—
305Dic61C2×#12111yescharacteristic
306Dic61C2×#12311yescharacteristic
307Dic61C2×#12411yescharacteristic
308Dic61C2×#12611yescharacteristic
309C2×#424—611no—
310C2×#407—611no—
311C22×C22⋊C8—611no—
312C2×#405—611no—
313#2280—611no—
314#1924—612no—
315#1924—612no—
316#1923—612no—
317#1923—612no—
318#1729—611no—
319#1729—611no—
320C2×C61—42no—
321C2×C61C2×#38811yescharacteristic
322C2×C61#292411yescharacteristic
323C2×C61#292411yescharacteristic
324C2×C61#292911yescharacteristic
325C2×C61#293011yescharacteristic
326C2×C61#293311yescharacteristic
327C2×C61#293311yescharacteristic
328D61—24no—
329D61—42no—
330C24×C4—611no—
331C22×M16—611no—
332C22×M16—611no—
333C23×C8—611no—
334C23×C8—611no—
335C2×#117—611no—
336C2×#117—611no—
337#424—1224no—
338C2×C4⋊C8—612no—
339C2×C4⋊C8—612no—
340#407—1224no—
341C2×C22⋊C8—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—612no—
344C2×C22⋊C8—611no—
345C2×C22⋊C8—611no—
346C2×C22⋊C8—612no—
347C2×C22⋊C8—1224no—
348#405—1224no—
349C2×#97—612no—
350C2×C4×C8—612no—
351#381—612no—
352#380—611no—
353#380—611no—
354#380—611no—
355#380#692311yescharacteristic
356#379—612no—
357C4×C22⋊C4—612no—
358C4×C22⋊C4—612no—
359#336—612no—
360#336—612no—
361#336—612no—
362#336—612no—
363C61#2105311yescharacteristic, complemented, Sylow
364C25—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—1222no—
371C23×C4—1222no—
372C23×C4—1222no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—1222no—
376C23×C4C2×#692311yescharacteristic
377C2×M16—614no—
378C2×M16—1222no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1222no—
382C2×M16—1222no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—612no—
386C22×C8—614no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—612no—
390C22×C8—1222no—
391C22×C8—1222no—
392C22×C8—1222no—
393C22×C8—1222no—
394C22×C8—1222no—
395#117—1222no—
396#117—1222no—
397#117—1222no—
398#117—1222no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4C2×#692311yescharacteristic
402C2×C4⋊C4C2×#692311yescharacteristic
403C2×C22⋊C4—612no—
404C2×C22⋊C4—612no—
405C2×C22⋊C4—612no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4—612no—
408C2×C22⋊C4C2×#692312yes—
409C2×C42—612no—
410C2×C42—612no—
411C4⋊C8—1224no—
412C4⋊C8—1224no—
413C22⋊C8—1222no—
414C22⋊C8—1222no—
415C22⋊C8—1224no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1224no—
419#97—1224no—
420C4×C8—1224no—
421#95—612no—
422#95—612no—
423#95—612no—
424#95C2×#692312yes—
425C24—611no—
426C24—611no—
427C24—1222no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24—2441no—
433C24—2441no—
434C24C22×#692311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—2444no—
467C22×C4—2444no—
468C22×C4—2444no—
469C22×C4—2444no—
470C22×C4C22×#692311yescharacteristic
471C22×C4C22×#692312yes—
472C22×C4C22×#692312yes—
473C22×C4C22×#692311yescharacteristic
474M16—1228no—
475M16—2444no—
476C2×C8—614no—
477C2×C8—612no—
478C2×C8—612no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1228no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1222no—
495C2×C8—2444no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C22⋊C4—42no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—2442no—
506C42—1222no—
507C42—1222no—
508C42—1222no—
509C42—1222no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—612no—
517C23—612no—
518C23—612no—
519C23—612no—
520C23—1228no—
521C23—1221no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—2444no—
532C23—2444no—
533C23—2444no—
534C23—2444no—
535C23—2444no—
536C23—2444no—
537C23#1451986512yes—
538C23C23×#692311yescharacteristic
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—614no—
556C2×C4—614no—
557C2×C4—614no—
558C2×C4—614no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2444no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2444no—
578C2×C4#1451985612yes—
579C2×C4D4×#692312yes—
580C2×C4D4×#692312yes—
581C8—1222no—
582C8—1222no—
583C8—1224no—
584C8—2441no—
585C8—2441no—
586C8—2442no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—614no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22—1222no—
600C22—1222no—
601C22—1222no—
602C22—1222no—
603C22—1222no—
604C22—24416no—
605C22—2441no—
606C22—2441no—
607C22—2441no—
608C22—2441no—
609C22C2×#1451985611yescharacteristic
610C22C2×D4×#692311yescharacteristic
611C22C2×D4×#692311yescharacteristic
612C22D61⋊#13111yescharacteristic
613C22C2×#1451986511yescharacteristic
614C22#1684084011yescharacteristic
615C22#1684084111yescharacteristic
616C4—41no—
617C4—42no—
618C4—42no—
619C4—611no—
620C4—611no—
621C4—611no—
622C4—611no—
623C4—2441no—
624C4—2442no—
625C4—2442no—
626C4#1684071214yes—
627C2—42no—
628C2—1224no—
629C2—2442no—
630C2C2×#1684071211yescharacteristic
631C2#2146251011yescharacteristic
632C2#2146251211yescharacteristic
633C2#2146253511yescharacteristic
634C2#2146253711yescharacteristic
635C2#2146255011yescharacteristic
636C2#2146255211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#380C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenter#380#6923
hypocenterC61#21053
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#16840769C2semidirect
4. C2×#16840727C2semidirect
5. #21461894C2semidirect
6. #21461893C2semidirect
7. #21461890C2semidirect
8. #21461889C2semidirect
9. #21461486C2semidirect
10. #21461486C2semidirect
17. C2×#14519844C22semidirect
18. C2×C4⋊#92420C22semidirect
28. C2×#14519832C22semidirect
29. C2×C4×#92420C22semidirect
30. #16840628C22semidirect
31. #16840628C22semidirect
32. #16840628C22semidirect
33. #16840628C22semidirect
363. C61#21053semidirect

Maximal subgroups

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

subgroupindex
C2×#168407692
C2×#168407332
C2×D61⋊(C22⋊C8)2
C2×#168407272
#214618942
#214618932
#214618902
#214618892
#214614862
#214482012
#2105361