Finite Group Databasebeta

← #29647240

Subgroups — #29647240

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#21459896C211yesmaximal, characteristic
2#21454906C211yesmaximal, characteristic, complemented
3#21448183C211yesmaximal, characteristic, complemented
4#21444342C211yesmaximal, characteristic, complemented
5#21444341C212yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444325C212yesmaximal, complemented
8#21444312C212yesmaximal, complemented
9#21444306C211yesmaximal, characteristic, complemented
10#21444302C212yesmaximal, complemented
11C61×#1502C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic, complemented
13#16840495C412yescomplemented
14#16840492C411yescharacteristic, complemented
15#16840473C411yescharacteristic, complemented
16#16840471C412yescomplemented
17#16840432C411yescharacteristic, complemented
18#16840418—21no—
19#16840412—21no—
20#16840411—22no—
21#16840377—21no—
22#16840376—22no—
23#16840307C2212yes—
24#16840296C2211yescharacteristic
25#16840293C2212yes—
26#16840241—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839892—21no—
30#16839891—22no—
31C8×Dic122—21no—
32#16839819C2211yescharacteristic
33#16839816C2212yes—
34#16839812C2212yes—
35C4⋊C4×Dic61—21no—
36C4×#11758196—22no—
37#16839613C2212yes—
38#16839608C2211yescharacteristic
39#16839607C2212yes—
40#16839500C2211yescharacteristic
41#16839500C2211yescharacteristic
42#16839498C2212yes—
43#16839497C2211yescharacteristic
44#16839497C2211yescharacteristic
45#16839496C2212yes—
46#16839496C2212yes—
47#16839495C2212yes—
48C61×#532C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#327C2212yes—
52C61×#327C2212yes—
53C2×#11758224—22no—
54C2×#11758224—21no—
55C22×Dic122D411yescharacteristic
56#14519776—21no—
57#14519776C2×C412yes—
58#14519775—22no—
59#14519773C2×C412yes—
60#14519771—21no—
61#14519770—22no—
62#14519769C2×C411yescharacteristic
63#14519768C2×C412yes—
64#14519765C2×C411yescharacteristic
65#14519764—21no—
66#14519764C2×C412yes—
67#14519763—22no—
68#14519763—22no—
69#14519760—21no—
70#14519760—22no—
71#14519759—22no—
72#14519758—22no—
73#14519758—22no—
74#14519757—21no—
75#14519756—22no—
76#14519756C2311yescharacteristic
77#14519754—22no—
78#14519754—22no—
79#14519753—22no—
80#14519751C2311yescharacteristic
81#14519749—22no—
82#14519749—22no—
83#14519748—21no—
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122C2×C412yes—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89C4×Dic122D412yes—
90C4×Dic122D411yescharacteristic
91#14519742—28no—
92#14519742—21no—
93#14519742C2×C412yes—
94C4×D244C2×C411yescharacteristic
95D4×Dic61—22no—
96D4×Dic61—21no—
97D4×Dic61C2×C412yes—
98D4×Dic61C2×C412yes—
99#14519739C2×C412yes—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519734—21no—
104#14519733—22no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196—22no—
108C2×#11758196D412yes—
109C4⋊C4×D61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—22no—
113C2×C4×Dic61C2312yes—
114C4⋊#92418—21no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519688—21no—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684C2311yescharacteristic
124#14519684C2311yescharacteristic
125C2×#41×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127#127×C61C2×C411yescharacteristic
128#124×C61—21no—
129#121×C61—21no—
130#121×C61C2312yes—
131C4×D4×C61—21no—
132C4×D4×C61C2×C412yes—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61C2311yescharacteristic
139C4×C8×C61—21no—
140#95×C61C2312yes—
141#41×C61—81no—
142#41×C61—81no—
143C2×Q8×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162#11758224—44no—
163#11758224—82no—
164#11758224—82no—
165C2×D244C22×C411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×Dic122—24no—
169C2×Dic122—22no—
170C2×Dic122C22⋊C412yes—
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22⋊C411yescharacteristic
173C2×Dic122C22×C411yescharacteristic
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—81no—
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C22⋊Dic61—21no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194#11758198—21no—
195#11758198—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22⋊C412yes—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—28no—
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196—22no—
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—21no—
217C4×Dic61—22no—
218C4×Dic61—82no—
219C4×Dic61C22×C412yes—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C2×#92418—22no—
224C23×C61—22no—
225C23×C61C22×D411yescharacteristic
226C22×D61—21no—
227Q8×C61—81no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—81no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—82no—
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—41no—
249C22⋊D61—41no—
250C22⋊D61—44no—
251C22⋊D61—82no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—44no—
265C2×Dic61—82no—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270D244—41no—
271C4×D61—42no—
272C4×D61—82no—
273Dic122—22no—
274Dic122—24no—
275Dic122—41no—
276Dic122—82no—
277Dic122#10414yes—
278C8×C61—41no—
279C8×C61—41no—
280#92418—42no—
281#17550—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57711yescharacteristic
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—44no—
299C4×C61—42no—
300C4×C61C2×#10412yes—
301Dic61—22no—
302Dic61—22no—
303Dic61—22no—
304Dic61—42no—
305Dic61—42no—
306Dic61—41no—
307Dic61—41no—
308Dic61—41no—
309Dic61C2×#10414yes—
310#3522—611no—
311#2984—611no—
312#2276—611no—
313#1509—611no—
314#1509—612no—
315#1505—612no—
316#1505—611no—
317#1502—612no—
318#1502D6111yescharacteristic, complemented
319#1500—611no—
320#1500—612no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—82no—
324C2×C61#288411yescharacteristic
325C2×C61#289711yescharacteristic
326C2×C61C22×#10411yescharacteristic
327D61—81no—
328#559—611no—
329#559—612no—
330Q82—611no—
331#554—611no—
332#554—612no—
333#550—611no—
334#548—1221no—
335#547—1221no—
336#547—1222no—
337#542—1221no—
338#542—1222no—
339#532—612no—
340#532C2×D6111yescharacteristic
341#529—612no—
342#529—611no—
343#520—1221no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346#446—1221no—
347#446—1222no—
348C8×Q8—1221no—
349#432—611no—
350#432—612no—
351#431—612no—
352#431C2×D6111yescharacteristic
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#328—612no—
362#327—612no—
363#327—612no—
364#327—611no—
365#327—611no—
366#327—612no—
367#327C2×D6112yes—
368#327C2×D6112yes—
369C61#1755011yescharacteristic, complemented, Sylow
370C2×#41—21no—
371C2×#41—1222no—
372C2×#41—1221no—
373C22×Q8—611no—
374C4⋊Q8—612no—
375C4⋊Q8—1221no—
376C4⋊Q8—1222no—
377C4⋊Q8C4×D6111yescharacteristic
378#127—612no—
379#127C4×D6111yescharacteristic
380#126—1222no—
381#126—1221no—
382#125—612no—
383#125—611no—
384#124—21no—
385#124—612no—
386#124—611no—
387#124—1222no—
388#124—1222no—
389#124—1221no—
390#123—1222no—
391#123—1222no—
392#123—1222no—
393#123—1222no—
394#123—1221no—
395C22⋊Q8—611no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1221no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401#121—21no—
402#121—611no—
403#121—1221no—
404#121—1222no—
405#121—1222no—
406#121C22×D6112yes—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—612no—
410C4×Q8—611no—
411C4×Q8—612no—
412C4×Q8—611no—
413C4×Q8—612no—
414C4×Q8—611no—
415C4×Q8—1221no—
416C4×Q8—1228no—
417C4×D4—21no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—611no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1221no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4C4×D6112yes—
428#117—1222no—
429#117—1221no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1222no—
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—612no—
447C22⋊C8C22×D6111yescharacteristic
448C22⋊C8C22×D6111yescharacteristic
449#97—1222no—
450#97—1221no—
451C4×C8—21no—
452C4×C8—1222no—
453#95—611no—
454#95—611no—
455#95—1222no—
456#95—1221no—
457#95C22×D6112yes—
458#41—81no—
459#41—81no—
460#41—4882no—
461#41—4882no—
462#41—2444no—
463C2×Q8—21no—
464C2×Q8—611no—
465C2×Q8—612no—
466C2×Q8—611no—
467C2×Q8—611no—
468C2×Q8—1222no—
469C2×Q8—1224no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—611no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—1221no—
477C2×D4C2×C4×D6112yes—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—1222no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—612no—
497C4⋊C4—611no—
498C4⋊C4—612no—
499C4⋊C4—612no—
500C4⋊C4—612no—
501C4⋊C4—611no—
502C4⋊C4—612no—
503C4⋊C4—611no—
504C4⋊C4—611no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1228no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4C2×C4×D6112yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1221no—
526C22⋊C4—1221no—
527C42—81no—
528C42—612no—
529C42—612no—
530C42—612no—
531C42—612no—
532C42—1222no—
533C42—1224no—
534C42—1228no—
535C42—1221no—
536C42—4882no—
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C42D4×D6111yescharacteristic
540C42D4×D6111yescharacteristic
541C23—22no—
542C23—1221no—
543C23C2×D4×D6111yescharacteristic
544Q8—81no—
545Q8—614no—
546Q8—1224no—
547Q8—1222no—
548Q8—4882no—
549Q8—2441no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—81no—
555D4—4882no—
556D4—2441no—
557D4—2444no—
558D4—2441no—
559D4—2441no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1224no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—4882no—
582C2×C4—4882no—
583C2×C4—2444no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2442no—
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—41no—
596C8—41no—
597C8—2442no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22C2×C22⋊C4×D6111yescharacteristic
607C22#1684056311yescharacteristic
608C22#1684056811yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—44no—
615C4—42no—
616C4—614no—
617C4—1222no—
618C4—1222no—
619C4—1222no—
620C4—2442no—
621C4—2442no—
622C4—2441no—
623C4—2441no—
624C4—2441no—
625C4#104×D6112yes—
626C2—21no—
627C2—21no—
628C2—82no—
629C2—4881no—
630C2#2145428111yescharacteristic
631C2#2145434011yescharacteristic
632C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459896C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17550
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454906C2semidirect
3. #21448183C2semidirect
4. #21444342C2semidirect
5. #21444341C2semidirect
6. #21444327C2semidirect
7. #21444325C2semidirect
8. #21444312C2semidirect
9. #21444306C2semidirect
10. #21444302C2semidirect
11. C61×#1502C2semidirect
12. #16840498C4semidirect
13. #16840495C4semidirect
14. #16840492C4semidirect
15. #16840473C4semidirect
16. #16840471C4semidirect
17. #16840432C4semidirect
318. #1502D61semidirect
369. C61#17550semidirect

Maximal subgroups

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

subgroupindex
#214598962
#214549062
#214481832
#214443422
#214443412
#214443272
#214443252
#214443122
#214443062
#214443022
C61×#15022
#1755061