Finite Group Databasebeta

← #30611356

Subgroups — #30611356

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462667C211yesmaximal, characteristic
2#21462332C211yesmaximal, characteristic
3#21461259C212yesmaximal, complemented
4#21461257C212yesmaximal, complemented
5#21461253C211yesmaximal, characteristic, complemented
6#21461252C211yesmaximal, characteristic, complemented
7#21461203C211yesmaximal, characteristic, complemented
8#21461201C211yesmaximal, characteristic, complemented
9C2×#16840616C212yesmaximal, complemented
10D61⋊#328C212yesmaximal, complemented
11#21459614C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13#16840790—21no—
14#16840789—22no—
15D4×#92420—21no—
16#16840781C2211yescharacteristic
17#16840780C2211yescharacteristic
18C2×#14519834C2212yes—
19D61⋊(C22⋊C8)C2212yes—
20#16840713—21no—
21#16840709—22no—
22#16840706C2211yescharacteristic
23#16840704C2212yes—
24C4×C4⋊#6923C2211yescharacteristic
25C2×(C2×#6923)⋊C4C2212yes—
26#16840616C2212yes—
27#16840616C2212yes—
28#16840616C2212yes—
29#16840616C2212yes—
30#16840616C2211yescharacteristic
31#16840616C2212yes—
32#16840616C2211yescharacteristic
33#16840615C2212yes—
34#16840615C2211yescharacteristic
35#16840615C2211yescharacteristic
36#16840473—21no—
37#16840472C411yescharacteristic, complemented
38#16840471—21no—
39#16840470C412yescomplemented
40#16840469C411yescharacteristic, complemented
41#16840466—22no—
42#16840427C411yescharacteristic, complemented
43Q8×C22⋊D61C412yescomplemented
44#16840389—21no—
45#16840387—22no—
46#16840317C2211yescharacteristic
47#16840316C2211yescharacteristic
48C2×#14519753C2212yes—
49C22⋊Q8×D61C2212yes—
50#16840234—21no—
51C4×D4×D61C2211yescharacteristic
52D4×Q8×C61C411yescharacteristic, complemented
53C23×#6923—22no—
54C2×D61⋊C8—22no—
55C22×#92420—22no—
56D4×#6923—44no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C4×#6923—21no—
62#14519844—21no—
63C4⋊#92420—21no—
64#14519834—22no—
65#14519834C2311yescharacteristic
66#14519834C2312yes—
67#14519834C2311yescharacteristic
68C22⋊#92420—22no—
69C4×#92420—21no—
70C4×#92420—21no—
71(C2×#6923)⋊C4—21no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2312yes—
75(C2×#6923)⋊C4C2311yescharacteristic
76C2×Q8×D61C2×C412yescomplemented
77C2×D4×D61D411yescharacteristic
78C22×C4×D61C2312yes—
79#14519777C2×C411yescharacteristic, complemented
80#14519776—21no—
81#14519776C2×C411yescharacteristic, complemented
82#14519775—22no—
83#14519775C2×C411yescharacteristic, complemented
84#14519769—21no—
85#14519768—21no—
86#14519768—21no—
87#14519767—22no—
88#14519767—21no—
89#14519757—21no—
90#14519757—22no—
91#14519757C2311yescharacteristic
92#14519756—22no—
93#14519756C2×C412yescomplemented
94#14519755—22no—
95#14519755—21no—
96#14519755—22no—
97#14519755—22no—
98#14519754—22no—
99#14519754—22no—
100#14519754—22no—
101#14519753—22no—
102#14519753—22no—
103#14519753—22no—
104#14519753C2×C412yescomplemented
105#14519753C2×C412yescomplemented
106#14519753C2312yes—
107#14519753C2311yescharacteristic
108Q8×Dic61C2×C412yescomplemented
109C4×Dic122—21no—
110C4×Dic122C2×C411yescharacteristic, complemented
111#14519742—21no—
112C4×D244—21no—
113C4×D244C2×C411yescharacteristic
114D4×Dic61—21no—
115D4×Dic61D411yescharacteristic
116#14519739—21no—
117#14519739D411yescharacteristic
118C4×C22⋊D61—22no—
119C4×C22⋊D61C2×C412yes—
120#14519737—22no—
121#14519737D412yes—
122#14519735—21no—
123#14519734—21no—
124C2×C4⋊Dic61C2×C412yescomplemented
125C2×#11758196—22no—
126C4⋊C4×D61—22no—
127C4⋊C4×D61C2×C412yescomplemented
128C4⋊C4×D61D411yescharacteristic
129C2×#11758198—22no—
130C22⋊C4×D61—22no—
131C22⋊C4×D61D412yes—
132C42×D61C2311yescharacteristic
133C22×Q8×C61C2×C412yescomplemented
134C4⋊Q8×C61—21no—
135C4⋊Q8×C61C2×C411yescharacteristic, complemented
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61C2×C412yescomplemented
138C4×Q8×C61C2×C411yescharacteristic, complemented
139C4×D4×C61—21no—
140C4×D4×C61C2×C411yescharacteristic
141C23×D61C2×D412yes—
142C22×#6923—21no—
143C22×#6923—22no—
144C22×#6923—21no—
145C22×#6923—44no—
146C22×#6923—44no—
147C22×#6923—82no—
148C2×Q8×C61—22no—
149C2×Q8×C61—82no—
150C2×Q8×C61C22×C412yes—
151C2×Q8×C61C22×C411yescharacteristic
152C2×D4×C61C22⋊C411yescharacteristic
153C22×C4×C61—22no—
154C22×C4×C61C22×C412yes—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157C22×Dic61C2×D412yes—
158Q8×D61—82no—
159D4×D61SD1614yes—
160D4×D61Q1614yes—
161C2×D244C22⋊C411yescharacteristic
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—82no—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—21no—
174C2×Dic122C22×C411yescharacteristic
175C22⋊#6923—44no—
176C2×#92420—21no—
177C2×#92420—22no—
178C2×#92420—21no—
179C2×#92420—82no—
180C4⋊#6923—42no—
181C4⋊#6923—42no—
182C4⋊#6923—82no—
183C4×#6923—42no—
184D61⋊C8—82no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61C22⋊C411yescharacteristic
190C4⋊C4×C61C22×C411yescharacteristic
191C22⋊C4×C61—22no—
192C22⋊C4×C61C22⋊C412yes—
193C42×C61—21no—
194C42×C61C22×C411yescharacteristic
195C22⋊Dic61—22no—
196C22⋊Dic61C22⋊C412yes—
197#11758198—22no—
198#11758198—22no—
199#11758198—21no—
200#11758198—22no—
201#11758198—21no—
202#11758198C22⋊C412yes—
203#11758198C22⋊C412yes—
204C4⋊Dic61—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22×C412yes—
210C4⋊Dic61C22×C411yescharacteristic
211#11758196—22no—
212#11758196—21no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218#11758196—21no—
219#11758196—22no—
220#11758196—21no—
221#11758196C22⋊C412yes—
222C4×Dic61—21no—
223C4×Dic61—22no—
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C23×C61C2×C22⋊C412yes—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61C2×C22⋊C412yes—
231C22×D61C2×SD1614yes—
232C22×D61C2×Q1614yes—
233C22×D61C22×D411yescharacteristic
234C2×#6923—22no—
235C2×#6923—41no—
236C2×#6923—41no—
237C2×#6923—41no—
238C2×#6923—41no—
239C2×#6923—42no—
240C2×#6923—42no—
241C2×#6923—41no—
242C2×#6923—42no—
243C2×#6923—88no—
244Q8×C61—42no—
245Q8×C61—41no—
246Q8×C61—82no—
247D4×C61#10314yes—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—42no—
252C2×C4×C61—82no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C23×C411yescharacteristic
257C22⋊D61#10318yes—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—42no—
262C2×Dic61—82no—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C22×D411yescharacteristic
267D244#10314yes—
268C4×D61—42no—
269C4×D61—41no—
270C4×D61—42no—
271C4×D61—41no—
272C4×D61—42no—
273C4×D61—82no—
274C4×D61—82no—
275C4×D61C2×SD1612yes—
276C4×D61C2×Q1612yes—
277Dic122—41no—
278Dic122—81no—
279Dic122—82no—
280Dic122—81no—
281#92420—41no—
282#92420—42no—
283#92420—41no—
284#17614—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61C2×#10314yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—21no—
290C2×D61—22no—
291C2×D61—21no—
292C2×D61—22no—
293C2×D61—44no—
294C2×D61—44no—
295C2×D61C2×#10314yes—
296C2×D61C22×SD1611yescharacteristic
297C2×D61C22×Q1611yescharacteristic
298#6923—42no—
299#6923—81no—
300#6923—82no—
301C4×C61—21no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61C2×#10312yes—
308Dic61—21no—
309Dic61—22no—
310Dic61—21no—
311Dic61—41no—
312Dic61—42no—
313Dic61—41no—
314Dic61C2×#10312yes—
315#3501—611no—
316#2972—611no—
317#2349—611no—
318#1520—612no—
319#1520—612no—
320#1518—611no—
321#1518—611no—
322#1503—611no—
323#1503—611no—
324C2×#328—612no—
325C2×#328—612no—
326C2×C61—24no—
327C2×C61#288311yescharacteristic
328C2×C61C22×#10311yescharacteristic
329C2×C61#290111yescharacteristic
330D61—21no—
331D61—21no—
332D61—24no—
333#554—612no—
334#554—611no—
335#554—611no—
336#554—1221no—
337#554—1222no—
338#554—1221no—
339D4×Q8—612no—
340D4×Q8—611no—
341D4×Q8#692311yescharacteristic, complemented
342#544—1221no—
343#543—1222no—
344#533—611no—
345#533—611no—
346C2×C22⋊Q8—612no—
347C2×C22⋊Q8—612no—
348#518—1221no—
349C2×C4×D4—611no—
350C2×C4×D4—1221no—
351C8×D4—1221no—
352C8×D4—1222no—
353C8×D4—1221no—
354#431—611no—
355#431—611no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—612no—
358#389—1221no—
359#385—1222no—
360#382—611no—
361#380—612no—
362C4×C4⋊C4—611no—
363C2×#95—612no—
364#328—611no—
365#328—612no—
366#328—611no—
367#328—612no—
368#328—611no—
369#328—612no—
370#328—612no—
371#328—612no—
372#328—612no—
373#328—611no—
374C61#1761411yescharacteristic, complemented, Sylow
375C22×Q8—612no—
376C22×Q8C2×#692312yescomplemented
377C22×D4—611no—
378C23×C4—612no—
379C23×C4—1222no—
380C22×C8—1222no—
381C22×C8—1222no—
382C4⋊Q8—21no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386C4⋊Q8—1221no—
387C4⋊Q8—1222no—
388C4⋊Q8C2×#692311yescharacteristic, complemented
389#125—1221no—
390#125—1221no—
391#125—1221no—
392#125—1222no—
393#125—1221no—
394C22⋊Q8—22no—
395C22⋊Q8—612no—
396C22⋊Q8—611no—
397C22⋊Q8—612no—
398C22⋊Q8—612no—
399C22⋊Q8—612no—
400C22⋊Q8—611no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1221no—
410C22⋊Q8—1221no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8C2×#692312yescomplemented
415C4×Q8—612no—
416C4×Q8—611no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8C2×#692311yescharacteristic, complemented
420C4×D4—21no—
421C4×D4—611no—
422C4×D4—612no—
423C4×D4—611no—
424C4×D4—612no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—2444no—
432C4×D4C2×#692311yescharacteristic
433#117—1221no—
434#117—1221no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C22⋊C4—612no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4—1222no—
447C2×C42—611no—
448C2×C42—1221no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C22⋊C8—611no—
452C22⋊C8—612no—
453C22⋊C8—611no—
454C22⋊C8—1222no—
455C22⋊C8—1222no—
456C4×C8—1221no—
457C4×C8—1221no—
458#95—611no—
459#95—612no—
460#95—611no—
461#95—1222no—
462#95—1221no—
463C24—612no—
464C2×Q8—22no—
465C2×Q8—82no—
466C2×Q8—611no—
467C2×Q8—1221no—
468C2×Q8—1222no—
469C2×Q8—1221no—
470C2×Q8—4882no—
471C2×Q8C22×#692312yes—
472C2×Q8C22×#692311yescharacteristic
473C2×D4—612no—
474C2×D4—614no—
475C2×D4—614no—
476C2×D4—611no—
477C2×D4C22⋊#692311yescharacteristic
478C22×C4—22no—
479C22×C4—611no—
480C22×C4—612no—
481C22×C4—611no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—4882no—
493C22×C4—4882no—
494C22×C4—2444no—
495C22×C4—2444no—
496C22×C4C22×#692312yes—
497C2×C8—1221no—
498C2×C8—1222no—
499C2×C8—1221no—
500C2×C8—4882no—
501C2×C8—4882no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—612no—
507C4⋊C4—611no—
508C4⋊C4—611no—
509C4⋊C4—612no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—4882no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4C22⋊#692311yescharacteristic
528C4⋊C4C22×#692311yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—2444no—
540C22⋊C4C22⋊#692312yes—
541C42—21no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—1222no—
546C42—1221no—
547C42—2442no—
548C42C22×#692311yescharacteristic
549C23—612no—
550C23—611no—
551C23—614no—
552C23—614no—
553C23—2442no—
554C23—2442no—
555C23C2×C22⋊#692312yes—
556Q8—41no—
557Q8—42no—
558Q8—82no—
559Q8—4881no—
560Q8—4881no—
561Q8—4882no—
562Q8—2441no—
563D4—618no—
564D4—614no—
565D4#1451984014yes—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—82no—
571C2×C4—611no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—611no—
575C2×C4—612no—
576C2×C4—611no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—4882no—
582C2×C4—4882no—
583C2×C4—4888no—
584C2×C4—4882no—
585C2×C4—2442no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C2×C22⋊#692311yescharacteristic
600C2×C4C2×C22⋊#692312yes—
601C2×C4C2×C22⋊#692311yescharacteristic
602C2×C4C23×#692311yescharacteristic
603C8—2441no—
604C8—2442no—
605C8—2441no—
606C22—22no—
607C22—22no—
608C22—611no—
609C22—611no—
610C22—614no—
611C22—1222no—
612C22—1221no—
613C22—1221no—
614C22—1222no—
615C22—2444no—
616C22—2444no—
617C22C2×#1451984014yes—
618C22C22×C22⋊#692311yescharacteristic
619C4—21no—
620C4—22no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—612no—
626C4—1221no—
627C4—1222no—
628C4—1221no—
629C4—4882no—
630C4—4881no—
631C4—2442no—
632C4—2441no—
633C4—2442no—
634C4—2441no—
635C4C2×#1451984012yes—
636C2—24no—
637C2—1221no—
638C2—1221no—
639C2—1224no—
640C2#2146241811yescharacteristic
641C2C22×#1451984011yescharacteristic
642C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingD4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×D4×D61D4
cosocleC2×C4×D61C24
hypercenterD4×Q8#6923
hypocenterC61#17614
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
3. #21461259C2semidirect
4. #21461257C2semidirect
5. #21461253C2semidirect
6. #21461252C2semidirect
7. #21461203C2semidirect
8. #21461201C2semidirect
9. C2×#16840616C2semidirect
10. D61⋊#328C2semidirect
37. #16840472C4semidirect
39. #16840470C4semidirect
40. #16840469C4semidirect
42. #16840427C4semidirect
43. Q8×C22⋊D61C4semidirect
52. D4×Q8×C61C4semidirect
76. C2×Q8×D61C2×C4semidirect
79. #14519777C2×C4semidirect
81. #14519776C2×C4semidirect
83. #14519775C2×C4semidirect
93. #14519756C2×C4semidirect
104. #14519753C2×C4semidirect
105. #14519753C2×C4semidirect
108. Q8×Dic61C2×C4semidirect
110. C4×Dic122C2×C4semidirect
124. C2×C4⋊Dic61C2×C4semidirect
127. C4⋊C4×D61C2×C4semidirect
133. C22×Q8×C61C2×C4semidirect
135. C4⋊Q8×C61C2×C4semidirect
137. C22⋊Q8×C61C2×C4semidirect
138. C4×Q8×C61C2×C4semidirect
341. D4×Q8#6923semidirect
374. C61#17614semidirect
376. C22×Q8C2×#6923semidirect
388. C4⋊Q8C2×#6923semidirect
414. C22⋊Q8C2×#6923semidirect
419. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626672
#214623322
#214612592
#214612572
#214612532
#214612522
#214612032
#214612012
C2×#168406162
D61⋊#3282
#214596142
#1761461