Finite Group Databasebeta

← #30611395

Subgroups — #30611395

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462682C211yesmaximal, characteristic
2#21462340C211yesmaximal, characteristic
3#21461252C211yesmaximal, characteristic, complemented
4#21461251C211yesmaximal, characteristic, complemented
5#21461228C212yesmaximal, complemented
6#21461227C212yesmaximal, complemented
7#21461218C212yesmaximal, complemented
8#21461217C212yesmaximal, complemented
9#21461202C211yesmaximal, characteristic, complemented
10#21461201C211yesmaximal, characteristic, complemented
11#21459616C211yesmaximal, characteristic
12#16840797—21no—
13#16840796—21no—
14#16840795—22no—
15#16840781C2211yescharacteristic
16#16840780C2211yescharacteristic
17#16840734C2212yes—
18#16840733C2212yes—
19#16840714—21no—
20#16840712—22no—
21#16840706C2211yescharacteristic
22#16840705C2212yes—
23C4⋊C4×#6923—21no—
24C4×C4⋊#6923C2211yescharacteristic
25C4×C22⋊#6923C2212yes—
26#16840616C2212yes—
27#16840616C2211yescharacteristic
28#16840616C2211yescharacteristic
29#16840615C2212yes—
30#16840615C2212yes—
31#16840615C2212yes—
32#16840615C2211yescharacteristic
33#16840615C2211yescharacteristic
34#16840615C2212yes—
35#16840615C2212yes—
36#16840473C411yescharacteristic, complemented
37#16840470—22no—
38#16840469—21no—
39#16840468—21no—
40#16840467C412yescomplemented
41Q8×D244C411yescharacteristic, complemented
42D4×Dic122C411yescharacteristic, complemented
43#16840425C412yescomplemented
44#16840392—21no—
45#16840387—22no—
46#16840317C2211yescharacteristic
47#16840316C2211yescharacteristic
48C2×#14519757C2212yes—
49C22⋊Q8×D61C2212yes—
50#16840234—21no—
51C4×D4×D61C2211yescharacteristic
52C61×#554C411yescharacteristic, complemented
53C2×#11758213—22no—
54C2×#11758217—22no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—21no—
57C2×C22⋊#6923—22no—
58C2×C22⋊#6923—22no—
59C2×C4×#6923—21no—
60C2×C4×#6923—22no—
61#14519844—21no—
62C4⋊#92420—21no—
63#14519834—22no—
64#14519834C2311yescharacteristic
65#14519834C2312yes—
66#14519834C2311yescharacteristic
67C22⋊#92420—22no—
68C4×#92420—21no—
69C4×#92420—21no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—21no—
72(C2×#6923)⋊C4C2311yescharacteristic
73(C2×#6923)⋊C4C2312yes—
74(C2×#6923)⋊C4C2311yescharacteristic
75C22×Dic122C2×C412yes—
76C2×Q8×D61C2×C412yes—
77C2×D4×D61D411yescharacteristic
78C22×C4×D61C2312yes—
79#14519777—21no—
80#14519777C2×C411yescharacteristic
81#14519776—21no—
82#14519776—22no—
83#14519776C2×C411yescharacteristic
84#14519775C2×C411yescharacteristic
85#14519768—21no—
86#14519768—21no—
87#14519767—22no—
88#14519767—21no—
89#14519757—22no—
90#14519757—22no—
91#14519757—22no—
92#14519757C2×C412yes—
93#14519757C2×C412yes—
94#14519757C2312yes—
95#14519757C2311yescharacteristic
96#14519756—22no—
97#14519756C2×C412yes—
98#14519755—22no—
99#14519755—21no—
100#14519755—22no—
101#14519755—22no—
102#14519754—22no—
103#14519754—22no—
104#14519754—22no—
105#14519753—21no—
106#14519753—22no—
107#14519753C2311yescharacteristic
108Q8×Dic61—21no—
109C4×Dic122—21no—
110C4×Dic122C2×C411yescharacteristic
111#14519742C2×C412yes—
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⋊Dic61—22no—
125C4⋊C4×D61—22no—
126C4⋊C4×D61C2×C412yes—
127C4⋊C4×D61D411yescharacteristic
128C2×#11758198—22no—
129C22⋊C4×D61—22no—
130C22⋊C4×D61D412yes—
131C42×D61C2311yescharacteristic
132C4⋊Q8×C61C2×C411yescharacteristic
133#125×C61—21no—
134C22⋊Q8×C61—22no—
135C22⋊Q8×C61C2×C412yes—
136C4×Q8×C61C2×C411yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C412yes—
140C23×D61C2×D412yes—
141C22×#6923—21no—
142C22×#6923—22no—
143C22×#6923—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22⋊C411yescharacteristic
147C22×C4×C61—22no—
148C22×C4×C61C22×C412yes—
149C2×C22⋊D61C22⋊C412yes—
150C22×Dic61—22no—
151C22×Dic61C2×D412yes—
152Q8×D61—82no—
153D4×D61—22no—
154D4×D61—22no—
155C2×D244C22⋊C411yescharacteristic
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—82no—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2411yescharacteristic
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—82no—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C411yescharacteristic
171C22⋊#6923—44no—
172C22⋊#6923—82no—
173#11758217—82no—
174C2×#92420—21no—
175C2×#92420—22no—
176C2×#92420—21no—
177C4⋊#6923—42no—
178C4⋊#6923—42no—
179C4×#6923—42no—
180C4×#6923—82no—
181#11758213—82no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22⋊C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61C22⋊C412yes—
191C42×C61—21no—
192C42×C61C22×C411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—21no—
198#11758198—22no—
199#11758198—21no—
200#11758198C22⋊C412yes—
201#11758198C22⋊C412yes—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—22no—
211#11758196—21no—
212#11758196—22no—
213#11758196—21no—
214#11758196—21no—
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218#11758196C22⋊C412yes—
219C4×Dic61—21no—
220C4×Dic61—22no—
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C23×C61C2×C22⋊C412yes—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61C2×C22⋊C412yes—
230C22×D61C22×D411yescharacteristic
231C2×#6923—22no—
232C2×#6923—41no—
233C2×#6923—41no—
234C2×#6923—42no—
235C2×#6923—41no—
236C2×#6923—42no—
237C2×#6923—41no—
238C2×#6923—41no—
239C2×#6923—42no—
240Q8×C61—41no—
241Q8×C61—81no—
242D4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—42no—
247C2×C4×C61—82no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C22⋊D61—24no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—42no—
257C2×Dic61—82no—
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C22×D411yescharacteristic
262D244—22no—
263C4×D61—42no—
264C4×D61—42no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—42no—
268C4×D61—82no—
269C4×D61—82no—
270C4×D61#13712yes—
271C4×D61#13712yes—
272Dic122—41no—
273Dic122—42no—
274Dic122—81no—
275Dic122—82no—
276Dic122—82no—
277#92420—41no—
278#92420—42no—
279#92420—41no—
280#17627—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61C22×C22⋊C411yescharacteristic
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—82no—
291C2×D61—82no—
292C2×D61C2×#13711yescharacteristic
293C2×D61C2×#13711yescharacteristic
294#6923—42no—
295#6923—81no—
296#6923—82no—
297C4×C61—21no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—42no—
303C4×C61#41912yes—
304Dic61—21no—
305Dic61—22no—
306Dic61—21no—
307Dic61—42no—
308Dic61—41no—
309Dic61—41no—
310Dic61#41912yes—
311#3501—611no—
312#2975—611no—
313#2357—611no—
314#1518—611no—
315#1518—611no—
316#1511—612no—
317#1511—612no—
318#1508—612no—
319#1508—612no—
320#1503—611no—
321#1503—611no—
322C2×C61—42no—
323C2×C61#288311yescharacteristic
324C2×C61C2×#41911yescharacteristic
325C2×C61#290111yescharacteristic
326D61—21no—
327D61—21no—
328D61—42no—
329#554—611no—
330#554—612no—
331#554—1221no—
332#554—1222no—
333#554—1221no—
334#554#692311yescharacteristic, complemented
335D4×Q8—612no—
336D4×Q8—611no—
337D4×Q8—611no—
338#544—1221no—
339#543—1222no—
340#533—611no—
341#533—611no—
342C2×C22⋊Q8—612no—
343C2×C22⋊Q8—612no—
344#518—1221no—
345C2×C4×D4—611no—
346#436—1222no—
347#436—1221no—
348#436—1221no—
349#431—611no—
350#431—611no—
351#407—612no—
352#407—612no—
353#390—1221no—
354#388—1222no—
355#382—611no—
356#381—612no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—612no—
360#328—611no—
361#328—612no—
362#328—611no—
363#328—612no—
364#328—612no—
365#328—612no—
366#328—612no—
367#328—611no—
368#328—611no—
369#328—612no—
370C61#1762711yescharacteristic, complemented, Sylow
371C22×Q8—612no—
372C22×Q8—612no—
373C22×D4—611no—
374C23×C4—612no—
375C2×M16—1222no—
376C2×M16—1222no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8—611no—
380C4⋊Q8—1221no—
381C4⋊Q8—1222no—
382C4⋊Q8—1221no—
383C4⋊Q8C2×#692311yescharacteristic
384#125—21no—
385#125—1221no—
386#125—1221no—
387#125—1222no—
388#125—1221no—
389C22⋊Q8—22no—
390C22⋊Q8—611no—
391C22⋊Q8—612no—
392C22⋊Q8—612no—
393C22⋊Q8—611no—
394C22⋊Q8—612no—
395C22⋊Q8—612no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1221no—
407C22⋊Q8—1221no—
408C22⋊Q8—1222no—
409C22⋊Q8C2×#692312yes—
410C4×Q8—612no—
411C4×Q8—611no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8C2×#692311yescharacteristic
415C4×D4—21no—
416C4×D4—611no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—612no—
420C4×D4—611no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4C2×#692311yescharacteristic
427#117—1221no—
428#117—1221no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C2×#692312yes—
436C2×C22⋊C4—612no—
437C2×C22⋊C4—1222no—
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1222no—
440C2×C22⋊C4—1222no—
441C2×C42—611no—
442C2×C42—1222no—
443C2×C42—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C22⋊C8—611no—
447C22⋊C8—612no—
448C22⋊C8—611no—
449C22⋊C8—1222no—
450C22⋊C8—1222no—
451C4×C8—1221no—
452C4×C8—1221no—
453#95—611no—
454#95—612no—
455#95—611no—
456#95—1222no—
457#95—1221no—
458C24—612no—
459C2×Q8—21no—
460C2×Q8—612no—
461C2×Q8—611no—
462C2×Q8—1221no—
463C2×Q8—1222no—
464C2×Q8—1222no—
465C2×Q8—4882no—
466C2×Q8—4882no—
467C2×Q8C22×#692311yescharacteristic
468C2×D4—612no—
469C2×D4—611no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4C22⋊#692311yescharacteristic
473C22×C4—22no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—611no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—611no—
480C22×C4—612no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—4882no—
488C22×C4C22×#692312yes—
489M16—4882no—
490M16—4882no—
491C2×C8—1221no—
492C2×C8—1222no—
493C2×C8—1221no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—612no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4C22⋊#692311yescharacteristic
518C4⋊C4C22×#692312yes—
519C4⋊C4C22×#692311yescharacteristic
520C22⋊C4—22no—
521C22⋊C4—612no—
522C22⋊C4—612no—
523C22⋊C4—612no—
524C22⋊C4—1221no—
525C22⋊C4—1221no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—4882no—
531C22⋊C4—2444no—
532C22⋊C4C22⋊#692312yes—
533C42—21no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—1222no—
538C42—1221no—
539C42—4882no—
540C42—2442no—
541C42C22×#692311yescharacteristic
542C23—611no—
543C23—612no—
544C23—1222no—
545C23—1222no—
546C23—2442no—
547C23—2442no—
548C23C2×C22⋊#692312yes—
549Q8—41no—
550Q8—81no—
551Q8—4882no—
552Q8—4881no—
553Q8—4882no—
554Q8—2441no—
555Q8—2442no—
556D4—22no—
557D4—1224no—
558D4—1222no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—611no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—611no—
568C2×C4—612no—
569C2×C4—611no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1222no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—4882no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2441no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×C22⋊#692311yescharacteristic
592C2×C4C2×C22⋊#692312yes—
593C2×C4C2×C22⋊#692311yescharacteristic
594C2×C4C23×#692311yescharacteristic
595C8—2441no—
596C8—2442no—
597C8—2441no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—611no—
602C22—611no—
603C22—1221no—
604C22—1221no—
605C22—1222no—
606C22—1222no—
607C22—1222no—
608C22—4882no—
609C22—4882no—
610C22C22×C22⋊#692311yescharacteristic
611C4—21no—
612C4—22no—
613C4—21no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—612no—
618C4—1221no—
619C4—1222no—
620C4—1221no—
621C4—4882no—
622C4—4881no—
623C4—2442no—
624C4—2442no—
625C4—2441no—
626C4—2441no—
627C4#1684075512yes—
628C2—42no—
629C2—1221no—
630C2—1221no—
631C2—2442no—
632C2#2146241911yescharacteristic
633C2C2×#1684075511yescharacteristic
634C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radicalC2×D4×D61D4
cosocleC2×C4×D61C24
hypercenter#554#6923
hypocenterC61#17627
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #21461252C2semidirect
4. #21461251C2semidirect
5. #21461228C2semidirect
6. #21461227C2semidirect
7. #21461218C2semidirect
8. #21461217C2semidirect
9. #21461202C2semidirect
10. #21461201C2semidirect
36. #16840473C4semidirect
40. #16840467C4semidirect
41. Q8×D244C4semidirect
42. D4×Dic122C4semidirect
43. #16840425C4semidirect
52. C61×#554C4semidirect
334. #554#6923semidirect
370. C61#17627semidirect

Maximal subgroups

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

subgroupindex
#214626822
#214623402
#214612522
#214612512
#214612282
#214612272
#214612182
#214612172
#214612022
#214612012
#214596162
#1762761