Finite Group Databasebeta

← #29690027

Subgroups — #29690027

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459207C211yesmaximal, characteristic
2#21454454C211yesmaximal, characteristic, complemented
3#21448302C211yesmaximal, characteristic, complemented
4#21446761C212yesmaximal, complemented
5#21446736C212yesmaximal, complemented
6#21446268C211yesmaximal, characteristic, complemented
7#21446265C211yesmaximal, characteristic, complemented
8#21446262C211yesmaximal, characteristic, complemented
9#21445838C212yesmaximal, complemented
10#21445837C212yesmaximal, complemented
11C61×#1896C211yesmaximal, characteristic, complemented
12#16840385C412yescomplemented
13#16840379C412yescomplemented
14#16840372C412yescomplemented
15#16840365C412yescomplemented
16#16840356—22no—
17#16840347—22no—
18#16840315C2211yescharacteristic
19#16840308C2211yescharacteristic
20#16840294—22no—
21#16840292—22no—
22#16840273C2211yescharacteristic
23#41×Dic61C2211yescharacteristic
24#16840230C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×#14519736C2211yescharacteristic
27#16839820—22no—
28#16839820—22no—
29#16839785C2212yes—
30C2×#14519705C2212yes—
31C2×#14519704C2212yes—
32C2×#14519703C2212yes—
33#16839726C2211yescharacteristic
34M16×Dic61C2212yes—
35#16839660—22no—
36#16839635—22no—
37C4×#11758196C2212yes—
38C4×#11758196C2212yes—
39#16839523C2212yes—
40#16839522C2212yes—
41#16839521C2212yes—
42C61×#525C2211yescharacteristic
43C22×C42×C61C2211yescharacteristic
44C2×#104×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#339C2212yes—
47C2×#11758222C2311yescharacteristic
48C2×#11758194C2312yes—
49#14519776C2×C412yescomplemented
50#14519775C2×C411yescharacteristic, complemented
51#14519773C2×C411yescharacteristic, complemented
52#14519772—22no—
53#14519770—22no—
54#14519770—22no—
55#14519770—22no—
56#14519768—22no—
57#14519768C2×C412yescomplemented
58#14519767—22no—
59#14519766C2×C411yescharacteristic, complemented
60#14519764—22no—
61#14519764C2×C411yescharacteristic, complemented
62#14519763—22no—
63#14519761—22no—
64#14519760—21no—
65#14519759—22no—
66#14519759—22no—
67#14519758—22no—
68#14519757—22no—
69#14519756C2×C412yes—
70#14519754—22no—
71#14519754—22no—
72#14519751C2×C412yes—
73#14519749—22no—
74#14519749—22no—
75#14519748—22no—
76#14519746—21no—
77Q8×Dic61C2×C412yescomplemented
78C4×Dic122—22no—
79#14519742C2×C412yescomplemented
80D4×Dic61—22no—
81D4×Dic61C2×C412yescomplemented
82#14519739C2×C412yescomplemented
83C4×C22⋊D61—22no—
84C4×C22⋊D61—22no—
85#14519737—22no—
86#14519736—21no—
87#14519736—42no—
88#14519736C2×C412yes—
89#14519736C2×C412yes—
90#14519736D412yes—
91#14519736D412yes—
92#14519736C2311yescharacteristic
93#14519734—21no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61D412yes—
96C2×#11758196—22no—
97C2×#11758196—22no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61D412yes—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2312yes—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—21no—
105#14519706—22no—
106#14519706—22no—
107#14519705—42no—
108#14519704—42no—
109#14519703—42no—
110C22⋊#92418—22no—
111#14519687—22no—
112C8×Dic61—22no—
113#14519684—22no—
114#14519684—22no—
115C2×#41×C61C2311yescharacteristic
116C23×C4×C61—21no—
117C23×C4×C61C2311yescharacteristic
118C2×M16×C61C2312yes—
119#123×C61—21no—
120C22⋊Q8×C61C2×C412yes—
121#121×C61C2×C412yes—
122#120×C61—21no—
123C4×D4×C61—22no—
124#117×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61—42no—
127C2×C42×C61C2312yes—
128#104×C61—42no—
129C22⋊C8×C61—22no—
130C24×C61C2×D411yescharacteristic
131#41×C61—22no—
132#41×C61—22no—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61C22×C411yescharacteristic
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—44no—
142C22×C4×C61—42no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—21no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154#11758222—22no—
155#11758222—22no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×Dic122—22no—
160C2×Dic122C22×C411yescharacteristic
161M16×C61—22no—
162M16×C61—22no—
163C2×C8×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C22×C412yes—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—21no—
168C22⋊C4×C61—22no—
169C42×C61—21no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—42no—
173C22⋊Dic61—22no—
174C22⋊Dic61—21no—
175C22⋊Dic61—22no—
176C22⋊Dic61—22no—
177C22⋊Dic61—42no—
178C22⋊Dic61C22⋊C412yes—
179C22⋊Dic61C22⋊C412yes—
180C22⋊Dic61C22⋊C412yes—
181C22⋊Dic61C22⋊C412yes—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—42no—
188C4⋊Dic61C22⋊C412yes—
189C4⋊Dic61C22⋊C412yes—
190C4⋊Dic61C22⋊C412yes—
191C4⋊Dic61C22⋊C412yes—
192C4⋊Dic61C22×C412yes—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—44no—
199#11758196—44no—
200C4×Dic61—21no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61C22×C412yes—
206C4×Dic61C22×C411yescharacteristic
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C2×D412yes—
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211#11758194—22no—
212#11758194—22no—
213C2×#92418—22no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—42no—
218C23×C61C2×C22⋊C412yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221Q8×C61—22no—
222D4×C61—22no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—82no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—42no—
234C2×C4×C61—48no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)12yes—
241C2×C4×C612−1+412yes—
242C22⋊D61—41no—
243C22⋊D61—41no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—22no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—82no—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258D244—22no—
259C4×D61—42no—
260C4×D61—82no—
261Dic122—22no—
262Dic122—82no—
263C8×C61—42no—
264#92418—42no—
265#21901—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—42no—
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61#53511yescharacteristic
276C22×C61#53611yescharacteristic
277C2×D61—41no—
278C2×D61—42no—
279C4×C61—22no—
280C4×C61—44no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61#52012yes—
284C4×C61#53712yes—
285Dic61—22no—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—41no—
290Dic61—42no—
291Dic61—82no—
292#3455—611no—
293#2921—611no—
294#2296—611no—
295#1986—612no—
296#1980—612no—
297#1897—611no—
298#1897—611no—
299#1896—611no—
300#1896D6111yescharacteristic, complemented
301#1806—612no—
302#1806—612no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—42no—
306C2×C61—81no—
307C2×C61#231111yescharacteristic
308C2×C61#290112yes—
309D61—81no—
310#542—612no—
311#542—612no—
312#541—612no—
313#540—612no—
314#539—1222no—
315#538—1222no—
316#533—611no—
317#532—611no—
318#529—1222no—
319#529—1222no—
320#525—611no—
321#525C2×D6111yescharacteristic
322C4×#41—611no—
323C4×#41—611no—
324C2×#117—611no—
325C2×#117—611no—
326C22×C42C2×D6111yescharacteristic
327#433—1222no—
328#433—1222no—
329#423—612no—
330C2×#104—612no—
331C2×#104—612no—
332C2×#104—612no—
333C2×#104C2×D6112yes—
334#407—611no—
335#407C2×D6111yescharacteristic
336C4×M16—612no—
337#389—1222no—
338#385—1222no—
339C4×C4⋊C4—612no—
340C4×C4⋊C4—612no—
341#339—612no—
342#339—612no—
343#339—612no—
344#339C2×D6112yes—
345C61#2190111yescharacteristic, complemented, Sylow
346C2×#41—611no—
347C2×#41C22×D6111yescharacteristic
348C23×C4—21no—
349C23×C4C22×D6111yescharacteristic
350C2×M16—612no—
351C2×M16C22×D6112yes—
352C4⋊Q8—612no—
353C4⋊Q8—611no—
354#127—611no—
355#126—1222no—
356#126—1222no—
357#126—1222no—
358#126—1222no—
359#125—612no—
360#125—1222no—
361#125—1222no—
362#124—611no—
363#124—611no—
364#124—1222no—
365#124—1222no—
366#123—21no—
367#123—1222no—
368#123—1222no—
369#123—1222no—
370#123—1222no—
371#123—1221no—
372C22⋊Q8—612no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C22⋊Q8—1222no—
376C22⋊Q8C4×D6112yes—
377#121—612no—
378#121—1222no—
379#121—1222no—
380#121—1222no—
381#121C4×D6112yes—
382#120—21no—
383#120—1221no—
384C4×Q8—612no—
385C4×Q8—612no—
386C4×Q8—1222no—
387C4×D4—22no—
388C4×D4—612no—
389C4×D4—612no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394#117—612no—
395#117—612no—
396#117—612no—
397#117—612no—
398#117—611no—
399#117—1221no—
400#117—1221no—
401#117—2442no—
402#117C22×D6111yescharacteristic
403C2×C4⋊C4—612no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C22⋊C4—612no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—42no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—612no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6112yes—
417C4⋊C8—1222no—
418C4⋊C8—1222no—
419#104—42no—
420#104—2442no—
421#104—2442no—
422#104—2442no—
423C22⋊C8—22no—
424C22⋊C8—1222no—
425#97—1222no—
426C4×C8—1222no—
427#95—1222no—
428#95—1222no—
429C24D4×D6111yescharacteristic
430#41—22no—
431#41—22no—
432#41—1222no—
433#41—1222no—
434C2×Q8—611no—
435C2×Q8—1222no—
436C2×Q8C2×C4×D6111yescharacteristic
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—611no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1221no—
443C2×D4C2×C4×D6111yescharacteristic
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4C2×C4×D6112yes—
457C22×C4D4×D6112yes—
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—22no—
461M16—22no—
462M16—1222no—
463M16—1222no—
464C2×C8—22no—
465C2×C8—1222no—
466C4⋊C4—22no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C4⋊C4—2442no—
482C4⋊C4C2×C4×D6112yes—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—612no—
487C22⋊C4—612no—
488C22⋊C4—612no—
489C22⋊C4—612no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1221no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—2442no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—612no—
503C42—611no—
504C42—611no—
505C42—612no—
506C42—611no—
507C42—611no—
508C42—1222no—
509C42—1222no—
510C42—1221no—
511C42—1222no—
512C42—1222no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—1221no—
518C23C22⋊C4×D6112yes—
519C23C2×D4×D6111yescharacteristic
520Q8—22no—
521Q8—1222no—
522Q8—4882no—
523D4—22no—
524D4—41no—
525D4—41no—
526D4—82no—
527D4—1222no—
528D4—4882no—
529D4—4882no—
530D4—2441no—
531D4—2441no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C22⋊C4×D6112yes—
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4#1451982212yes—
561C2×C4#1451982412yes—
562C8—42no—
563C8—2442no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—2442no—
573C22—2441no—
574C22C2×C22⋊C4×D6111yescharacteristic
575C22#1684032711yescharacteristic
576C22#1684033411yescharacteristic
577C4—22no—
578C4—44no—
579C4—42no—
580C4—41no—
581C4—1222no—
582C4—1222no—
583C4—1222no—
584C4—1222no—
585C4—4882no—
586C4—2442no—
587C4—2441no—
588C4#1684024112yes—
589C4#1684033912yes—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—81no—
594C2—4881no—
595C2#2144840611yescharacteristic
596C2#2145436312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1896C2
socleC22×C61C22×C22⋊C4
prime radical#21459207C2
cosocleC22×C4×C61C24
hypercenter#1896D61
hypocenterC61#21901
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1896

Complements

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

NH ≅ G/Ntype
2. #21454454C2semidirect
3. #21448302C2semidirect
4. #21446761C2semidirect
5. #21446736C2semidirect
6. #21446268C2semidirect
7. #21446265C2semidirect
8. #21446262C2semidirect
9. #21445838C2semidirect
10. #21445837C2semidirect
11. C61×#1896C2semidirect
12. #16840385C4semidirect
13. #16840379C4semidirect
14. #16840372C4semidirect
15. #16840365C4semidirect
49. #14519776C2×C4semidirect
50. #14519775C2×C4semidirect
51. #14519773C2×C4semidirect
57. #14519768C2×C4semidirect
59. #14519766C2×C4semidirect
61. #14519764C2×C4semidirect
77. Q8×Dic61C2×C4semidirect
79. #14519742C2×C4semidirect
81. D4×Dic61C2×C4semidirect
82. #14519739C2×C4semidirect
300. #1896D61semidirect
345. C61#21901semidirect

Maximal subgroups

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

subgroupindex
#214592072
#214544542
#214483022
#214467612
#214467362
#214462682
#214462652
#214462622
#214458382
#214458372
C61×#18962
#2190161