Finite Group Databasebeta

← #30620600

Subgroups — #30620600

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462328C211yesmaximal, characteristic
2Q8×C22⋊#6923C211yesmaximal, characteristic
3#21461910C212yesmaximal, complemented
4#21461898C212yesmaximal, complemented
5#21461785C212yesmaximal, complemented
6#21461723C211yesmaximal, characteristic
7#21461695C211yesmaximal, characteristic
8#21461658C211yesmaximal, characteristic
9#21461592C212yesmaximal, complemented
10#21461555C211yesmaximal, characteristic
11C2×#103×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×#14519845C2212yes—
14C2×#14519840C2212yes—
15C2×#14519840C2212yes—
16D61⋊(C22⋊C8)C2211yescharacteristic
17C2×#14519831C2212yes—
18(C2×#6923)⋊Q8—22no—
19(C2×#6923)⋊Q8C2212yes—
20#16840713C2212yes—
21#16840711—22no—
22#16840704—21no—
23#16840704—21no—
24#16840704C2211yescharacteristic
25#16840704C2211yescharacteristic
26C4⋊C4×#6923C2212yes—
27C4×C22⋊#6923—21no—
28C4×C22⋊#6923—21no—
29C4×C22⋊#6923C2211yescharacteristic
30#16840636C2212yes—
31#16840635C2212yes—
32#16840635C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×Q8×D61C2211yescharacteristic
35C22×C8×D61C2211yescharacteristic
36C2×C4⋊C4×D61C2211yescharacteristic
37C2×#14519702C411yescharacteristic, complemented
38C2×#14519701C411yescharacteristic, complemented
39C2×#14519700C411yescharacteristic, complemented
40#103×D61C411yescharacteristic, complemented
41#103×D61C411yescharacteristic, complemented
42#103×D61C411yescharacteristic, complemented
43#103×D61C411yescharacteristic, complemented
44#103×D61C2211yescharacteristic
45#103×D61C2211yescharacteristic
46#103×D61C2211yescharacteristic
47#103×D61C2211yescharacteristic
48C2×#103×C61C411yescharacteristic, complemented
49C2×D61⋊C8C2312yes—
50Q8×#6923—24no—
51Q8×#6923—42no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2312yes—
56C2×C22⋊#6923—21no—
57C2×C22⋊#6923—21no—
58C2×C22⋊#6923D411yescharacteristic
59C2×C22⋊#6923D411yescharacteristic
60C2×C4×#6923—22no—
61C2×C4×#6923—22no—
62C2×C4×#6923C2312yes—
63#14519845—42no—
64#14519840—24no—
65#14519840—42no—
66#14519834—21no—
67C22⋊#92420—21no—
68#14519831—42no—
69(C2×#6923)⋊C4—22no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—21no—
74(C2×#6923)⋊C4—21no—
75(C2×#6923)⋊C4D412yes—
76C22×Dic122—21no—
77C22×Dic122C2×C411yescharacteristic
78C2×Q8×D61—21no—
79C2×Q8×D61—21no—
80C2×Q8×D61—24no—
81C2×Q8×D61—42no—
82C2×Q8×D61C2×C411yescharacteristic
83C2×Q8×D61C2×C411yescharacteristic
84C2×Q8×D61C2311yescharacteristic
85C2×Q8×D61C2311yescharacteristic
86C22×C4×D61—21no—
87C22×C4×D61—21no—
88C22×C4×D61C2311yescharacteristic
89C22×#92418C2×C411yescharacteristic
90C2×C8×D61—42no—
91C2×C8×D61C2×C411yescharacteristic
92C2×C8×D61C2×C411yescharacteristic
93C2×C8×D61C2311yescharacteristic
94C2×C8×D61C2311yescharacteristic
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×#11758196—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61C2×C411yescharacteristic
100C4⋊C4×D61C2×C411yescharacteristic
101C4⋊C4×D61C2311yescharacteristic
102C4⋊C4×D61C2311yescharacteristic
103#14519702C2×C411yescharacteristic
104#14519702C2×C411yescharacteristic
105#14519702C2×C411yescharacteristic
106#14519702C2×C411yescharacteristic
107#14519701C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519701C2×C411yescharacteristic
111#14519700C2×C411yescharacteristic
112#14519700C2×C411yescharacteristic
113#14519700C2×C411yescharacteristic
114#14519700C2×C411yescharacteristic
115C22×Q8×C61C2×C411yescharacteristic
116C22×C8×C61C2×C411yescharacteristic
117C2×C4⋊C4×C61C2×C411yescharacteristic
118#103×C61C2×C411yescharacteristic
119#103×C61C2×C411yescharacteristic
120#103×C61C2×C411yescharacteristic
121#103×C61C2×C411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C22×#6923—22no—
124C22×#6923—22no—
125C22×#6923C2×D412yes—
126C22×#6923C2×D412yes—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—41no—
130C2×Q8×C61C22×C411yescharacteristic
131C2×Q8×C61C22×C411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61#4111yescharacteristic
138Q8×D61—22no—
139Q8×D61—22no—
140Q8×D61—22no—
141Q8×D61—22no—
142Q8×D61—22no—
143Q8×D61—41no—
144Q8×D61—41no—
145Q8×D61—44no—
146Q8×D61—44no—
147Q8×D61#4114yes—
148C2×C4×D61—21no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—44no—
159C2×C4×D61—82no—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×C4×D61C2411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—41no—
171C2×Dic122—44no—
172C2×Dic122C22×C411yescharacteristic
173C2×Dic122C22×C411yescharacteristic
174C22⋊#6923—22no—
175C22⋊#6923—22no—
176C22⋊#6923—42no—
177C22⋊#6923—42no—
178C2×#92420—22no—
179C4⋊#6923—22no—
180C4⋊#6923—22no—
181C4⋊#6923—22no—
182C4⋊#6923—22no—
183C4⋊#6923—44no—
184C4⋊#6923—82no—
185C4×#6923—22no—
186C4×#6923—22no—
187C4×#6923—44no—
188C4×#6923—82no—
189D61⋊C8—22no—
190D61⋊C8—22no—
191C2×C8×C61—41no—
192C2×C8×C61C22×C411yescharacteristic
193C2×C8×C61C22×C411yescharacteristic
194C4⋊C4×C61—21no—
195C4⋊C4×C61C22×C411yescharacteristic
196C4⋊C4×C61C22×C411yescharacteristic
197C4⋊Dic61—21no—
198C4⋊Dic61C22×C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200#11758196—21no—
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204C2×#92418—41no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C8×D61—22no—
208C8×D61—22no—
209C8×D61—41no—
210C8×D61—41no—
211C23×C61C4×D411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61C4×D411yescharacteristic
215C22×D61#13712yes—
216C22×D61#13712yes—
217C22×D61C22×D411yescharacteristic
218C2×#6923—22no—
219C2×#6923—22no—
220C2×#6923—22no—
221C2×#6923—22no—
222C2×#6923—22no—
223C2×#6923—22no—
224C2×#6923—42no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—42no—
230Q8×C61—21no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—42no—
234Q8×C61#11712yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—42no—
242C2×C4×C61—81no—
243C2×C4×C61C4×D411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—42no—
251C2×Dic61—81no—
252C2×Dic61C4×D411yescharacteristic
253C2×Dic61C2×#4111yescharacteristic
254C4×D61—24no—
255C4×D61—22no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—22no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61—42no—
263C4×D61—42no—
264C4×D61—81no—
265C4×D61—81no—
266C4×D61C2×#4112yes—
267C4×D61H(5,𝔽2)12yes—
268Dic122—21no—
269Dic122—24no—
270Dic122—21no—
271Dic122—21no—
272Dic122—42no—
273Dic122—42no—
274Dic122—42no—
275Dic122—42no—
276Dic122#11712yes—
277#92420—42no—
278C8×C61—21no—
279C8×C61—21no—
280#92418—21no—
281#92418—21no—
282#22567—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61#44112yes—
286C22×C61C2×C4×D411yescharacteristic
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—24no—
290C2×D61—24no—
291C2×D61#44112yes—
292C2×D61#54612yes—
293C2×D61C2×#13711yescharacteristic
294C2×D61C2×#13711yescharacteristic
295C2×D61#57712yes—
296#6923—42no—
297#6923—42no—
298#6923—82no—
299#6923—82no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61C2×#11711yescharacteristic
306C4×C61#51811yescharacteristic
307Dic61—22no—
308Dic61—21no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61#55112yes—
313C22×#103—611no—
314#2345—611no—
315Q8×C22⋊C4—611no—
316#1935—612no—
317#1927—612no—
318#1865—612no—
319#1824—611no—
320#1807—611no—
321#1787—611no—
322#1763—612no—
323#1752—611no—
324C2×C61—22no—
325C2×C61#234211yescharacteristic
326C2×C61C2×#44111yescharacteristic
327C2×C61#294911yescharacteristic
328D61—22no—
329D61#331612yes—
330D61#332012yes—
331C23×Q8—611no—
332C23×C8—611no—
333C2×C4×Q8—612no—
334C22×C4⋊C4—611no—
335C2×#106—612no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—612no—
348C2×#103—612no—
349C2×#103#692311yescharacteristic, complemented
350C2×C22⋊C8—611no—
351C2×#97—612no—
352#391—612no—
353#391—1222no—
354#389—612no—
355#387—1222no—
356#380—611no—
357#380—611no—
358#380—1221no—
359#380—1221no—
360C4×C4⋊C4—612no—
361C4×C22⋊C4—611no—
362C4×C22⋊C4—1221no—
363C4×C22⋊C4—1221no—
364#340—612no—
365#340—612no—
366#340—612no—
367#336—612no—
368C61#2256711yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—611no—
371C22×Q8—611no—
372C22×Q8—611no—
373C22×Q8—611no—
374C22×Q8—1221no—
375C22×Q8—1221no—
376C22×Q8—1221no—
377C22×Q8—1224no—
378C22×Q8—2442no—
379C22×Q8C2×#692311yescharacteristic
380C23×C4—611no—
381C23×C4—1221no—
382C23×C4—1221no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—612no—
389C22×C8—2442no—
390C22×C8C2×#692311yescharacteristic
391C4×Q8—1224no—
392C4×Q8—2442no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4C2×#692311yescharacteristic
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C42—612no—
411C2×C42—1222no—
412C2×C42—1222no—
413#106—2442no—
414#103—611no—
415#103—611no—
416#103—611no—
417#103—611no—
418#103—611no—
419#103—611no—
420#103—611no—
421#103—611no—
422#103—611no—
423#103—611no—
424#103—611no—
425#103—611no—
426#103—1224no—
427#103—2442no—
428#103C2×#692311yescharacteristic
429#103C2×#692311yescharacteristic
430#103C2×#692311yescharacteristic
431#103C2×#692311yescharacteristic
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434#97—2442no—
435#95—612no—
436#95—1222no—
437#95—1222no—
438#95—1222no—
439#95—1221no—
440#95—1221no—
441#95—1222no—
442C24—611no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—41no—
446C2×Q8—614no—
447C2×Q8—611no—
448C2×Q8—611no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1222no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×Q8—1222no—
457C2×Q8—1221no—
458C2×Q8—1222no—
459C2×Q8—1222no—
460C2×Q8—2441no—
461C2×Q8—2444no—
462C2×Q8—2444no—
463C2×Q8—2441no—
464C2×Q8—2441no—
465C2×Q8—2444no—
466C2×Q8C22×#692311yescharacteristic
467C2×Q8C22×#692311yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—4882no—
492C22×C4—2444no—
493C22×C4C22×#692311yescharacteristic
494C2×C8—41no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—2441no—
503C2×C8—2441no—
504C2×C8—2441no—
505C2×C8C22×#692311yescharacteristic
506C2×C8C22×#692311yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—611no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4—4882no—
520C4⋊C4—2444no—
521C4⋊C4C22×#692311yescharacteristic
522C4⋊C4C22×#692311yescharacteristic
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C42—1222no—
528C42—1222no—
529C42—4882no—
530C42—2444no—
531C23—611no—
532C23—612no—
533C23—612no—
534C23—611no—
535C23—1222no—
536C23—1222no—
537C23D4×#692311yescharacteristic
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—42no—
542Q8—612no—
543Q8—1224no—
544Q8—1221no—
545Q8—1221no—
546Q8—1221no—
547Q8—2442no—
548Q8—2442no—
549Q8—2442no—
550Q8—2442no—
551Q8#1451985612yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—81no—
560C2×C4—611no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—611no—
564C2×C4—1222no—
565C2×C4—1224no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—4881no—
581C2×C4—4881no—
582C2×C4—4881no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4D4×#692311yescharacteristic
596C2×C4C23×#692311yescharacteristic
597C8—21no—
598C8—21no—
599C8—1221no—
600C8—1221no—
601C8—2442no—
602C22—21no—
603C22—21no—
604C22—611no—
605C22—611no—
606C22—612no—
607C22—612no—
608C22—612no—
609C22—1224no—
610C22—1224no—
611C22—1221no—
612C22—1221no—
613C22#1684080212yes—
614C22C2×D4×#692311yescharacteristic
615C4—21no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—612no—
621C4—1221no—
622C4—1222no—
623C4—4882no—
624C4—4882no—
625C4—2442no—
626C4—2442no—
627C4—2441no—
628C4—2441no—
629C4—2441no—
630C4C2×#1451985611yescharacteristic
631C4#1684082911yescharacteristic
632C2—22no—
633C2—612no—
634C2—612no—
635C2—1222no—
636C2#2146232511yescharacteristic
637C2C2×#1684080211yescharacteristic
638C2#2146257911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#103×C61C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#103#6923
hypocenterC61#22567
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #21461910C2semidirect
4. #21461898C2semidirect
5. #21461785C2semidirect
9. #21461592C2semidirect
37. C2×#14519702C4semidirect
38. C2×#14519701C4semidirect
39. C2×#14519700C4semidirect
40. #103×D61C4semidirect
41. #103×D61C4semidirect
42. #103×D61C4semidirect
43. #103×D61C4semidirect
48. C2×#103×C61C4semidirect
349. C2×#103#6923semidirect
368. C61#22567semidirect

Maximal subgroups

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

subgroupindex
#214623282
Q8×C22⋊#69232
#214619102
#214618982
#214617852
#214617232
#214616952
#214616582
#214615922
#214615552
C2×#103×D612
#2256761