Finite Group Databasebeta

← #29647238

Subgroups — #29647238

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459902C211yesmaximal, characteristic
2#21454909C211yesmaximal, characteristic, complemented
3#21448183C211yesmaximal, characteristic, complemented
4#21444343C212yesmaximal, complemented
5#21444328C211yesmaximal, characteristic, complemented
6#21444325C212yesmaximal, complemented
7#21444313C211yesmaximal, characteristic, complemented
8#21444312C212yesmaximal, complemented
9#21444305C211yesmaximal, characteristic, complemented
10#21444304C212yesmaximal, complemented
11C61×#1509C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic, complemented
13#16840494C412yescomplemented
14Q8×Dic122C411yescharacteristic, complemented
15#16840469C411yescharacteristic, complemented
16#16840468C412yescomplemented
17#16840433C411yescharacteristic, complemented
18#16840421—21no—
19#16840416—21no—
20#16840410—22no—
21#16840383—21no—
22#16840378—22no—
23#16840312C2211yescharacteristic
24#16840307C2212yes—
25#16840298C2212yes—
26#16840242—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839890—22no—
30#16839889—21no—
31#16839886—21no—
32#16839816C2212yes—
33#16839815C2211yescharacteristic
34#16839812C2212yes—
35C4⋊C4×Dic61—21no—
36C4×#11758196—22no—
37#16839613C2212yes—
38#16839608C2211yescharacteristic
39#16839607C2212yes—
40#16839499C2211yescharacteristic
41#16839499C2211yescharacteristic
42#16839497C2212yes—
43#16839497C2212yes—
44#16839496C2211yescharacteristic
45#16839496C2211yescharacteristic
46#16839495C2212yes—
47#16839495C2212yes—
48C61×#529C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#328C2212yes—
52C61×#327C2212yes—
53C2×#11758222—21no—
54C2×#11758222—21no—
55C2×#11758224—22no—
56C22×Dic122D411yescharacteristic
57#14519777—21no—
58#14519777C2×C411yescharacteristic
59#14519776—22no—
60#14519776C2×C412yes—
61#14519774C2×C411yescharacteristic
62#14519773C2×C412yes—
63#14519772—22no—
64#14519768C2×C412yes—
65#14519766C2×C412yes—
66#14519765—21no—
67#14519765—21no—
68#14519763—22no—
69#14519763—22no—
70#14519762—22no—
71#14519761—22no—
72#14519761—22no—
73#14519759—22no—
74#14519757—22no—
75#14519756—21no—
76#14519755—22no—
77#14519755—22no—
78#14519754—22no—
79#14519751—21no—
80#14519751—21no—
81#14519751C2312yes—
82#14519750—22no—
83#14519749—22no—
84Q8×Dic61C2×C412yes—
85C4×Dic122—21no—
86C4×Dic122C2×C412yes—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×Dic122D412yes—
91C4×Dic122D411yescharacteristic
92#14519742—28no—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244C2×C412yes—
96D4×Dic61C2×C412yes—
97#14519739C2×C412yes—
98#14519739C2×C412yes—
99#14519737—22no—
100#14519737—22no—
101#14519737—22no—
102#14519735—22no—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—22no—
106C2×#11758196D412yes—
107C4⋊C4×D61—22no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61C2312yes—
112C4⋊#92418—21no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418C2311yescharacteristic
116C22⋊#92418C2311yescharacteristic
117#14519687—22no—
118C4×#92418—21no—
119C8×Dic61—22no—
120#14519684—21no—
121#14519684—22no—
122#14519684C2311yescharacteristic
123#14519684C2311yescharacteristic
124#126×C61—21no—
125#125×C61C2×C411yescharacteristic
126#124×C61C2×C411yescharacteristic
127#123×C61—21no—
128C22⋊Q8×C61C2311yescharacteristic
129#121×C61C2311yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133#117×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139#97×C61—21no—
140#95×C61C2312yes—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61C2×D412yes—
156C22×Dic61C2×D412yes—
157#11758224—82no—
158#11758224—82no—
159#11758222—44no—
160#11758222—81no—
161#11758222—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×Dic122—24no—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122C22⋊C412yes—
172C2×Dic122C22⋊C411yescharacteristic
173C2×Dic122C22⋊C411yescharacteristic
174C2×C8×C61—22no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61—21no—
188C42×C61—81no—
189C42×C61C22×C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—28no—
208#11758196—22no—
209#11758196—22no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—28no—
215C4×Dic61—24no—
216C4×Dic61—22no—
217C4×Dic61—82no—
218C4×Dic61C22×C412yes—
219C4×Dic61C22×C412yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—21no—
223C2×#92418—21no—
224C23×C61—21no—
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227Q8×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—42no—
249C22⋊D61—42no—
250C22⋊D61—44no—
251C22⋊D61—82no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—82no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267D244—42no—
268D244—42no—
269D244—81no—
270C4×D61—44no—
271C4×D61—82no—
272C4×D61—82no—
273Dic122—22no—
274Dic122—24no—
275Dic122—82no—
276Dic122—81no—
277Dic122#10414yes—
278C8×C61—42no—
279#92418—41no—
280#92418—41no—
281#17550—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57711yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—44no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61C2×#10412yes—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—42no—
307Dic61—42no—
308Dic61—42no—
309Dic61C2×#10414yes—
310#3522—611no—
311#2984—611no—
312#2276—611no—
313#1509—612no—
314#1509D6111yescharacteristic, complemented
315#1505—611no—
316#1505—612no—
317#1502—611no—
318#1502—612no—
319#1500—612no—
320#1500—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—81no—
324C2×C61#288411yescharacteristic
325C2×C61#289711yescharacteristic
326C2×C61C22×#10411yescharacteristic
327D61—82no—
328#559—612no—
329#559—611no—
330Q82—611no—
331#554—612no—
332#554—611no—
333#550—611no—
334#548—1221no—
335#547—1221no—
336#547—1222no—
337#542—1221no—
338#542—1222no—
339#532—612no—
340#532—611no—
341#529—612no—
342#529C2×D6111yescharacteristic
343#520—1221no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346#446—1222no—
347#446—1221no—
348C8×Q8—1221no—
349#432—612no—
350#432C2×D6111yescharacteristic
351#431—611no—
352#431—612no—
353C4×C4⋊C4—612no—
354C4×C4⋊C4—1222no—
355C4×C4⋊C4—1221no—
356C4×C4⋊C4C2×D6111yescharacteristic
357#376—611no—
358#376—612no—
359#328—611no—
360#328—611no—
361#328C2×D6112yes—
362#327—611no—
363#327—611no—
364#327—612no—
365#327—612no—
366#327—612no—
367#327—612no—
368#327C2×D6112yes—
369C61#1755011yescharacteristic, complemented, Sylow
370C2×#41—1221no—
371C2×#41—1222no—
372C2×#41—1221no—
373C22×Q8—611no—
374C4⋊Q8—612no—
375C4⋊Q8—611no—
376C4⋊Q8—1221no—
377C4⋊Q8—1222no—
378#127—612no—
379#127—611no—
380#126—21no—
381#126—1222no—
382#125—612no—
383#125C4×D6111yescharacteristic
384#124—612no—
385#124—1221no—
386#124—1222no—
387#124—1222no—
388#124—1221no—
389#124C4×D6111yescharacteristic
390#123—21no—
391#123—1222no—
392#123—1222no—
393#123—1222no—
394#123—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1221no—
400C22⋊Q8C22×D6111yescharacteristic
401#121—612no—
402#121—1222no—
403#121—1221no—
404#121—1222no—
405#121—1221no—
406#121C22×D6111yescharacteristic
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—612no—
410C4×Q8—611no—
411C4×Q8—612no—
412C4×Q8—611no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×Q8—1228no—
416C4×Q8C4×D6111yescharacteristic
417C4×D4—21no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—612no—
422C4×D4—1221no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4C4×D6111yescharacteristic
428#117—21no—
429#117—1222no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1222no—
436C2×C42—21no—
437C2×C42—612no—
438C2×C42—1222no—
439C2×C42—1221no—
440C2×C42—1222no—
441C2×C42C22×D6111yescharacteristic
442C4⋊C8—21no—
443C4⋊C8—1222no—
444C4⋊C8—1222no—
445C4⋊C8—1221no—
446C22⋊C8—611no—
447C22⋊C8—611no—
448C22⋊C8C22×D6112yes—
449#97—21no—
450#97—1222no—
451C4×C8—1222no—
452C4×C8—1221no—
453#95—611no—
454#95—611no—
455#95—1222no—
456#95—1221no—
457#95C22×D6112yes—
458#41—4881no—
459#41—4881no—
460#41—4882no—
461#41—4882no—
462#41—2444no—
463C2×Q8—611no—
464C2×Q8—612no—
465C2×Q8—611no—
466C2×Q8—1222no—
467C2×Q8—1224no—
468C2×Q8—1221no—
469C2×Q8C2×C4×D6111yescharacteristic
470C2×D4—21no—
471C2×D4—612no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4C2×C4×D6111yescharacteristic
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491C2×C8—22no—
492C2×C8—1221no—
493C2×C8—1221no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—612no—
500C4⋊C4—611no—
501C4⋊C4—612no—
502C4⋊C4—612no—
503C4⋊C4—612no—
504C4⋊C4—611no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1228no—
512C4⋊C4—1221no—
513C4⋊C4C2×C4×D6111yescharacteristic
514C4⋊C4C2×C4×D6111yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C42—21no—
528C42—81no—
529C42—612no—
530C42—612no—
531C42—612no—
532C42—612no—
533C42—1222no—
534C42—1224no—
535C42—1228no—
536C42—4882no—
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C42D4×D6111yescharacteristic
540C42D4×D6111yescharacteristic
541C23—21no—
542C23—1222no—
543C23C2×D4×D6111yescharacteristic
544Q8—41no—
545Q8—614no—
546Q8—1224no—
547Q8—1222no—
548Q8—4881no—
549Q8—4882no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—4881no—
554D4—4882no—
555D4—2442no—
556D4—2444no—
557D4—2442no—
558D4—2442no—
559D4—2442no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—4882no—
582C2×C4—4882no—
583C2×C4—4882no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4C22⋊C4×D6111yescharacteristic
590C2×C4C22⋊C4×D6111yescharacteristic
591C2×C4C22⋊C4×D6111yescharacteristic
592C2×C4C22⋊C4×D6111yescharacteristic
593C2×C4C22×C4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C8—42no—
596C8—2441no—
597C8—2441no—
598C22—21no—
599C22—21no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22C2×C22⋊C4×D6111yescharacteristic
607C22#1684056611yescharacteristic
608C22#1684057111yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—44no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—614no—
619C4—1222no—
620C4—1222no—
621C4—1222no—
622C4—2442no—
623C4—2442no—
624C4—2442no—
625C4#104×D6112yes—
626C2—21no—
627C2—21no—
628C2—81no—
629C2—4882no—
630C2#2145428111yescharacteristic
631C2#2145434311yescharacteristic
632C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1509C2
socleC22×C61C22×C22⋊C4
prime radical#21459902C2
cosocleC22×C4×C61C24
hypercenter#1509D61
hypocenterC61#17550
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. #21454909C2semidirect
3. #21448183C2semidirect
4. #21444343C2semidirect
5. #21444328C2semidirect
6. #21444325C2semidirect
7. #21444313C2semidirect
8. #21444312C2semidirect
9. #21444305C2semidirect
10. #21444304C2semidirect
11. C61×#1509C2semidirect
12. #16840497C4semidirect
13. #16840494C4semidirect
14. Q8×Dic122C4semidirect
15. #16840469C4semidirect
16. #16840468C4semidirect
17. #16840433C4semidirect
314. #1509D61semidirect
369. C61#17550semidirect

Maximal subgroups

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

subgroupindex
#214599022
#214549092
#214481832
#214443432
#214443282
#214443252
#214443132
#214443122
#214443052
#214443042
C61×#15092
#1755061