Finite Group Databasebeta

← #30620604

Subgroups — #30620604

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462327C211yesmaximal, characteristic
2Q8×C22⋊#6923C211yesmaximal, characteristic
3#21461908C212yesmaximal, complemented
4#21461898C212yesmaximal, complemented
5#21461785C212yesmaximal, complemented
6#21461722C211yesmaximal, characteristic
7#21461695C211yesmaximal, characteristic
8#21461658C211yesmaximal, characteristic
9#103×#6923C212yesmaximal, complemented
10C8×C22⋊#6923C211yesmaximal, characteristic
11C2×#103×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×#14519856—21no—
14C2×#14519845C2212yes—
15C2×#14519840C2212yes—
16C2×#14519840C2212yes—
17D61⋊(C22⋊C8)C2211yescharacteristic
18C2×C8×#6923C2212yes—
19(C2×#6923)⋊Q8—22no—
20(C2×#6923)⋊Q8C2212yes—
21#16840713C2212yes—
22#16840711—22no—
23#16840705—21no—
24#16840704—21no—
25#16840704C2211yescharacteristic
26#16840704C2211yescharacteristic
27C4⋊C4×#6923C2212yes—
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—
52#14519856—44no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923C2312yes—
56C2×C4⋊#6923C2312yes—
57C2×C22⋊#6923—21no—
58C2×C22⋊#6923—21no—
59C2×C22⋊#6923D411yescharacteristic
60C2×C22⋊#6923D411yescharacteristic
61C2×C4×#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923C2312yes—
64#14519845—42no—
65#14519840—24no—
66#14519840—42no—
67#14519834—21no—
68C22⋊#92420—21no—
69C8×#6923—42no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—21no—
75(C2×#6923)⋊C4—21no—
76(C2×#6923)⋊C4D412yes—
77C22×Dic122—21no—
78C22×Dic122C2×C411yescharacteristic
79C2×Q8×D61—21no—
80C2×Q8×D61—21no—
81C2×Q8×D61—24no—
82C2×Q8×D61—42no—
83C2×Q8×D61C2×C411yescharacteristic
84C2×Q8×D61C2×C411yescharacteristic
85C2×Q8×D61C2311yescharacteristic
86C2×Q8×D61C2311yescharacteristic
87C22×C4×D61—21no—
88C22×C4×D61—21no—
89C22×C4×D61C2311yescharacteristic
90C22×#92418C2×C411yescharacteristic
91C2×C8×D61—42no—
92C2×C8×D61C2×C411yescharacteristic
93C2×C8×D61C2×C411yescharacteristic
94C2×C8×D61C2311yescharacteristic
95C2×C8×D61C2311yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61C2×C411yescharacteristic
101C4⋊C4×D61C2×C411yescharacteristic
102C4⋊C4×D61C2311yescharacteristic
103C4⋊C4×D61C2311yescharacteristic
104#14519702C2×C411yescharacteristic
105#14519702C2×C411yescharacteristic
106#14519702C2×C411yescharacteristic
107#14519702C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519701C2×C411yescharacteristic
111#14519701C2×C411yescharacteristic
112#14519700C2×C411yescharacteristic
113#14519700C2×C411yescharacteristic
114#14519700C2×C411yescharacteristic
115#14519700C2×C411yescharacteristic
116C22×Q8×C61C2×C411yescharacteristic
117C22×C8×C61C2×C411yescharacteristic
118C2×C4⋊C4×C61C2×C411yescharacteristic
119#103×C61C2×C411yescharacteristic
120#103×C61C2×C411yescharacteristic
121#103×C61C2×C411yescharacteristic
122#103×C61C2×C411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C22×#6923—22no—
125C22×#6923—22no—
126C22×#6923C2×D412yes—
127C22×#6923C2×D412yes—
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61—41no—
131C2×Q8×C61C22×C411yescharacteristic
132C2×Q8×C61C22×C411yescharacteristic
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61#4111yescharacteristic
139Q8×D61—22no—
140Q8×D61—22no—
141Q8×D61—22no—
142Q8×D61—22no—
143Q8×D61—22no—
144Q8×D61—41no—
145Q8×D61—41no—
146Q8×D61—44no—
147Q8×D61—44no—
148Q8×D61#4114yes—
149C2×C4×D61—21no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
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—21no—
159C2×C4×D61—44no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×C4×D61C2411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—41no—
173C2×Dic122—44no—
174C2×Dic122C22×C411yescharacteristic
175C2×Dic122C22×C411yescharacteristic
176C22⋊#6923—22no—
177C22⋊#6923—42no—
178C22⋊#6923—42no—
179C22⋊#6923SD1614yes—
180C2×#92420—22no—
181C4⋊#6923—22no—
182C4⋊#6923—22no—
183C4⋊#6923—22no—
184C4⋊#6923—22no—
185C4⋊#6923—44no—
186C4⋊#6923—44no—
187C4×#6923—22no—
188C4×#6923—22no—
189C4×#6923—44no—
190C4×#6923—44no—
191D61⋊C8—22no—
192D61⋊C8—22no—
193C2×C8×C61—41no—
194C2×C8×C61C22×C411yescharacteristic
195C2×C8×C61C22×C411yescharacteristic
196C4⋊C4×C61—21no—
197C4⋊C4×C61C22×C411yescharacteristic
198C4⋊C4×C61C22×C411yescharacteristic
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206C2×#92418—41no—
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C8×D61—22no—
210C8×D61—22no—
211C8×D61—41no—
212C8×D61—41no—
213C23×C61C4×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61C4×D411yescharacteristic
217C22×D61C2×SD1612yes—
218C22×D61#13512yes—
219C22×D61C22×D411yescharacteristic
220C2×#6923—22no—
221C2×#6923—22no—
222C2×#6923—22no—
223C2×#6923—22no—
224C2×#6923—22no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—42no—
230C2×#6923—42no—
231C2×#6923C2×SD1614yes—
232Q8×C61—21no—
233Q8×C61—21no—
234Q8×C61—21no—
235Q8×C61—42no—
236Q8×C61#11712yescomplemented
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61C4×D411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C4×D411yescharacteristic
255C2×Dic61C2×#4111yescharacteristic
256C4×D61—24no—
257C4×D61—22no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—22no—
261C4×D61—42no—
262C4×D61—42no—
263C4×D61—42no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61C2×#4112yes—
271C4×D61H(5,𝔽2)12yes—
272Dic122—21no—
273Dic122—24no—
274Dic122—21no—
275Dic122—21no—
276Dic122—42no—
277Dic122—42no—
278Dic122—42no—
279Dic122—42no—
280Dic122#11712yescomplemented
281#92420—42no—
282C8×C61—21no—
283C8×C61—21no—
284#92418—21no—
285#92418—21no—
286#22571—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C4×SD1612yes—
290C22×C61C2×C4×D411yescharacteristic
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—24no—
294C2×D61—24no—
295C2×D61C4×SD1612yes—
296C2×D61#54612yes—
297C2×D61C22×SD1611yescharacteristic
298C2×D61C2×#13511yescharacteristic
299C2×D61#57812yes—
300#6923—24no—
301#6923—42no—
302#6923—82no—
303#6923—82no—
304C4×C61—22no—
305C4×C61—21no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61C2×#11711yescharacteristic
310C4×C61#51811yescharacteristic
311Dic61—22no—
312Dic61—21no—
313Dic61—41no—
314Dic61—41no—
315Dic61—41no—
316Dic61#55112yes—
317C22×#103—611no—
318#2344—611no—
319Q8×C22⋊C4—611no—
320#1933—612no—
321#1927—612no—
322#1865—612no—
323#1823—611no—
324#1807—611no—
325#1787—611no—
326C4×#103—612no—
327C8×C22⋊C4—611no—
328C2×C61—22no—
329C2×C61#234211yescharacteristic
330C2×C61C2×C4×SD1611yescharacteristic
331C2×C61#295011yescharacteristic
332D61—22no—
333D61#329812yes—
334D61#330012yes—
335C23×Q8—611no—
336C23×C8—611no—
337C2×C4×Q8—612no—
338C2×#117—1221no—
339C22×C4⋊C4—611no—
340C2×#106—612no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#103#692311yescharacteristic, complemented
355C2×C22⋊C8—611no—
356C2×C4×C8—612no—
357#391—612no—
358#391—1222no—
359#389—612no—
360#387—1222no—
361#381—1221no—
362#380—611no—
363#380—611no—
364#380—1221no—
365C4×C4⋊C4—612no—
366C4×C22⋊C4—611no—
367C4×C22⋊C4—1221no—
368#340—612no—
369#340—612no—
370#340—612no—
371#336—612no—
372C61#2257111yescharacteristic, complemented, Sylow
373C22×Q8—611no—
374C22×Q8—611no—
375C22×Q8—611no—
376C22×Q8—611no—
377C22×Q8—611no—
378C22×Q8—1221no—
379C22×Q8—1221no—
380C22×Q8—1221no—
381C22×Q8—1224no—
382C22×Q8—2442no—
383C22×Q8C2×#692311yescharacteristic
384C23×C4—611no—
385C23×C4—1221no—
386C23×C4—1221no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—611no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—612no—
393C22×C8—2442no—
394C22×C8C2×#692311yescharacteristic
395C4×Q8—1224no—
396C4×Q8—2442no—
397#117—2444no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4C2×#692311yescharacteristic
411C2×C22⋊C4—611no—
412C2×C22⋊C4—611no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—612no—
416C2×C42—1222no—
417C2×C42—1222no—
418#106—2442no—
419#103—611no—
420#103—611no—
421#103—611no—
422#103—611no—
423#103—611no—
424#103—611no—
425#103—611no—
426#103—611no—
427#103—611no—
428#103—611no—
429#103—611no—
430#103—611no—
431#103—1224no—
432#103—2442no—
433#103C2×#692311yescharacteristic
434#103C2×#692311yescharacteristic
435#103C2×#692311yescharacteristic
436#103C2×#692311yescharacteristic
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C4×C8—2442no—
440#95—612no—
441#95—1222no—
442#95—1222no—
443#95—1221no—
444#95—1221no—
445#95—1222no—
446#95—1222no—
447C24—611no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—614no—
452C2×Q8—611no—
453C2×Q8—611no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1222no—
459C2×Q8—1222no—
460C2×Q8—1222no—
461C2×Q8—1222no—
462C2×Q8—1221no—
463C2×Q8—1222no—
464C2×Q8—1222no—
465C2×Q8—2441no—
466C2×Q8—2444no—
467C2×Q8—2444no—
468C2×Q8—2441no—
469C2×Q8—2441no—
470C2×Q8—2444no—
471C2×Q8C22×#692311yescharacteristic
472C2×Q8C22×#692311yescharacteristic
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—611no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—2444no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4C22×#692311yescharacteristic
500C2×C8—41no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—2441no—
509C2×C8—2441no—
510C2×C8—2441no—
511C2×C8C22×#692311yescharacteristic
512C2×C8C22×#692311yescharacteristic
513C4⋊C4—21no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—2444no—
526C4⋊C4—2444no—
527C4⋊C4C22×#692311yescharacteristic
528C4⋊C4C22×#692311yescharacteristic
529C22⋊C4—614no—
530C22⋊C4—1222no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—1222no—
534C42—1222no—
535C42—2444no—
536C42—2444no—
537C23—611no—
538C23—612no—
539C23—612no—
540C23—611no—
541C23—1222no—
542C23—1222no—
543C23D4×#692311yescharacteristic
544Q8—21no—
545Q8—21no—
546Q8—21no—
547Q8—42no—
548Q8—612no—
549Q8—1224no—
550Q8—1221no—
551Q8—1221no—
552Q8—1221no—
553Q8—2442no—
554Q8—2442no—
555Q8—2442no—
556Q8—2442no—
557Q8#1451985612yescomplemented
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—611no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—614no—
570C2×C4—611no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4D4×#692311yescharacteristic
604C2×C4C23×#692311yescharacteristic
605C8—21no—
606C8—21no—
607C8—1221no—
608C8—1221no—
609C8—2442no—
610C22—21no—
611C22—21no—
612C22—611no—
613C22—611no—
614C22—612no—
615C22—612no—
616C22—612no—
617C22—1224no—
618C22—1224no—
619C22—1221no—
620C22—1221no—
621C22SD16×#692312yes—
622C22C2×D4×#692311yescharacteristic
623C4—21no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—612no—
629C4—1224no—
630C4—1221no—
631C4—1222no—
632C4—4882no—
633C4—4882no—
634C4—2442no—
635C4—2441no—
636C4—2441no—
637C4—2441no—
638C4C2×#1451985611yescharacteristic
639C4#1684082911yescharacteristic
640C2—22no—
641C2—612no—
642C2—612no—
643C2—1222no—
644C2#2146232511yescharacteristic
645C2C2×SD16×#692311yescharacteristic
646C2#2146258011yescharacteristic
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#22571
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. #21461908C2semidirect
4. #21461898C2semidirect
5. #21461785C2semidirect
9. #103×#6923C2semidirect
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
236. Q8×C61#117semidirect
280. Dic122#117semidirect
354. C2×#103#6923semidirect
372. C61#22571semidirect
557. Q8#14519856semidirect

Maximal subgroups

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

subgroupindex
#214623272
Q8×C22⋊#69232
#214619082
#214618982
#214617852
#214617222
#214616952
#214616582
#103×#69232
C8×C22⋊#69232
C2×#103×D612
#2257161