Finite Group Databasebeta

← #29680844

Subgroups — #29680844

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459017C211yesmaximal, characteristic
2#428×D61C212yesmaximal, complemented
3C2×#16839743C211yesmaximal, characteristic
4#21448409C211yesmaximal, characteristic
5#21446035C212yesmaximal, complemented
6#21446032C212yesmaximal, complemented
7#21446031C211yesmaximal, characteristic, complemented
8#21446029C211yesmaximal, characteristic, complemented
9#21446028C211yesmaximal, characteristic, complemented
10#21445527C212yesmaximal, complemented
11C61×#1856C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840241—22no—
15#16840241—22no—
16#16840239C2212yes—
17#16840237C2211yescharacteristic
18#16840235—22no—
19C4×#11758224—22no—
20C4×#11758224—22no—
21C2×#14519734C2211yescharacteristic
22C2×#14519733—22no—
23#117×D61C2212yes—
24C2×C4⋊C4×D61C2211yescharacteristic
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#107×D61—22no—
31#106×D61—22no—
32#16839743C418yescomplemented
33C2×#14519695C2212yes—
34C2×#14519694C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839638—22no—
39#16839620—22no—
40#16839523C2212yes—
41#16839522C2212yes—
42#16839521C2212yes—
43C61×#518C2211yescharacteristic
44C61×#428C2212yes—
45C2×#100×C61C2211yescharacteristic
46C61×#390C2211yescharacteristic
47C61×#339C2212yes—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—24no—
51C2×#11758224C2×C414yes—
52C2×#11758224Q814yes—
53C22×Dic122C2311yescharacteristic
54C22×C22⋊D61D412yes—
55C23×Dic61D412yes—
56C22×C4×D61—21no—
57C22×C4×D61C2311yescharacteristic
58C2×#11758194C2312yes—
59C2×#11758189—22no—
60M16×D61—24no—
61M16×D61C2×C418yescomplemented
62C2×C8×D61—22no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65#14519742—22no—
66#14519742—22no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69C4×C22⋊D61—24no—
70C4×C22⋊D61—24no—
71#14519737—24no—
72#14519737—24no—
73#14519736C2312yes—
74#14519735D412yes—
75#14519735D412yes—
76#14519734—22no—
77#14519734—24no—
78#14519733—24no—
79#14519733—24no—
80#14519733—22no—
81#14519733—24no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—22no—
84C2×#11758196—22no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61D412yes—
90C4⋊C4×D61D412yes—
91C2×C22⋊Dic61—22no—
92C2×C22⋊Dic61—22no—
93C2×#11758198—21no—
94C22⋊C4×D61—22no—
95C2×C4×Dic61—22no—
96C2×C4×Dic61—22no—
97C2×C4×Dic61C2311yescharacteristic
98C42×D61—22no—
99#14519711—22no—
100#14519710—22no—
101#14519710—22no—
102#14519709—22no—
103#14519708—22no—
104#14519708—22no—
105#14519695C2×C418yescomplemented
106#14519694C2×C414yescomplemented
107#14519684—22no—
108#14519684—22no—
109C22×D4×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113#117×C61C2312yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61—22no—
116C2×C22⋊C4×C61—22no—
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#106×C61—22no—
120#100×C61C2×C414yescomplemented
121C24×C61C2×D412yes—
122C23×D61C2×D411yescharacteristic
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C22×C412yes—
126C2×D4×C61C2×Q812yes—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61C4⋊C418yes—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C4⋊C418yes—
143C22×Dic61C2×D412yes—
144C22×Dic61C2×D411yescharacteristic
145#11758224—28no—
146#11758224—28no—
147#11758224—28no—
148#11758224—28no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—42no—
152C2×C4×D61—82no—
153C2×C4×D61C22×C412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×Q812yes—
157C2×C4×D61#4114yes—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122C22×C412yes—
161C2×Dic122C2×Q812yes—
162M16×C61—22no—
163M16×C61C22×C414yes—
164C2×C8×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—82no—
171C42×C61—22no—
172C42×C61—22no—
173C22⋊Dic61—24no—
174C22⋊Dic61—22no—
175C22⋊Dic61—82no—
176#11758198—22no—
177#11758198—22no—
178#11758198—42no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61C2×D412yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—22no—
185#11758196—22no—
186#11758196—41no—
187#11758196—41no—
188#11758196—42no—
189C4×Dic61—24no—
190C4×Dic61—24no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—22no—
196#11758194C22×C414yes—
197C2×#92418—22no—
198#11758189—28no—
199C8×D61—28no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61C2×C4⋊C414yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—41no—
207C22×D61—41no—
208C22×D61C2×C4⋊C412yes—
209C22×D61#12012yes—
210C22×D61#12312yes—
211D4×C61—24no—
212D4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—82no—
221C2×C4×C61—82no—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D412yes—
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61C2×#4112yes—
226C22⋊D61—216no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—28no—
230C2×Dic61—28no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—82no—
239C2×Dic61—82no—
240C2×Dic61C2×C4⋊C414yes—
241C2×Dic61C2×C4⋊C412yes—
242C2×Dic61#12011yescharacteristic
243C2×Dic61#12011yescharacteristic
244C2×Dic61#12312yes—
245C4×D61—82no—
246C4×D61—82no—
247C4×D61—84no—
248C4×D61C4×D414yes—
249C4×D61C4×D414yes—
250C4×D61C22⋊Q814yes—
251C4×D61C22⋊Q814yes—
252Dic122—24no—
253Dic122—24no—
254C8×C61—24no—
255#92418—24no—
256#20858—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—24no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—82no—
264C22×C61C22×C4⋊C411yescharacteristic
265C22×C61C2×#12011yescharacteristic
266C22×C61C2×#12311yescharacteristic
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61#38514yes—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×C4×D412yes—
277C4×C61C2×C22⋊Q812yes—
278Dic61—24no—
279Dic61—42no—
280Dic61—42no—
281Dic61—81no—
282Dic61—81no—
283Dic61#38512yes—
284Dic61#38512yes—
285#3428—611no—
286C2×#428—612no—
287C2×#413—611no—
288#2311—611no—
289#1857—612no—
290#1857—612no—
291#1856—611no—
292#1856—611no—
293#1856—611no—
294#1856D6111yescharacteristic, complemented
295#1733—612no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—44no—
299C2×C61C2×#38511yescharacteristic
300C2×C61#302111yescharacteristic
301C2×C61#302211yescharacteristic
302D61—24no—
303C22×#41—611no—
304C22×M16—612no—
305#520—1222no—
306#520—1222no—
307#519—612no—
308#518—611no—
309#518—1222no—
310#518C2×D6111yescharacteristic
311C4×#41—1222no—
312C4×#41—1222no—
313C2×#117—611no—
314C2×#117—612no—
315C2×#117—1222no—
316C22×C4⋊C4—611no—
317#428—612no—
318#428—612no—
319#428—612no—
320#428—1222no—
321#428—1222no—
322#428C2×D6112yes—
323C2×#107—1222no—
324C2×#106—1222no—
325#413—618no—
326C2×#101—612no—
327C2×#100—611no—
328C2×#100C2×D6111yescharacteristic
329#391—611no—
330#391—611no—
331#390—611no—
332#390C2×D6111yescharacteristic
333#386—1222no—
334#380—1222no—
335#339—612no—
336#339—612no—
337#339—612no—
338#339C2×D6112yes—
339C61#2085811yescharacteristic, complemented, Sylow
340C2×#41—614no—
341C2×#41—614no—
342C2×#41—1222no—
343C2×#41—1222no—
344C2×#41—1224no—
345C22×Q8—611no—
346C22×D4—612no—
347C22×D4C22×D6111yescharacteristic
348C23×C4—611no—
349C23×C4—612no—
350C23×C4—1221no—
351C2×M16—612no—
352C2×M16—618no—
353C2×M16—1222no—
354C2×M16—1224no—
355C2×M16C22×D6112yes—
356C22×C8—1222no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—1224no—
364C4×D4—1224no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1224no—
368C4×D4—1224no—
369#117—612no—
370#117—612no—
371#117—612no—
372#117—1222no—
373#117—1224no—
374#117—1222no—
375#117—1224no—
376#117—1224no—
377#117—1224no—
378#117C22×D6112yes—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1221no—
395C2×C42—611no—
396C2×C42—1222no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42C22×D6111yescharacteristic
400#107—22no—
401#107—1222no—
402#107—1222no—
403#107—1222no—
404#106—22no—
405#106—1222no—
406#106—1222no—
407#106—1222no—
408#101—618no—
409#100—614no—
410#100C4×D6114yescomplemented
411#95—1222no—
412#95—1222no—
413C24—611no—
414C24D4×D6112yes—
415#41—1228no—
416#41—1228no—
417#41—1228no—
418#41—1228no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—618no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4C2×C4×D6112yes—
429C2×D4Q8×D6112yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—611no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—614no—
440C22×C4—612no—
441C22×C4—618no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—4882no—
452C22×C4—2442no—
453C22×C4C23×D6111yescharacteristic
454M16—22no—
455M16—614no—
456M16—1228no—
457M16—1222no—
458M16C2×C4×D6114yes—
459C2×C8—22no—
460C2×C8—1228no—
461C2×C8—1222no—
462C4⋊C4—22no—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1222no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2442no—
473C4⋊C4D4×D6112yes—
474C4⋊C4D4×D6112yes—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—82no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C22⋊C4—1224no—
481C22⋊C4—1222no—
482C22⋊C4—4882no—
483C22⋊C4—2442no—
484C42—22no—
485C42—22no—
486C42—1222no—
487C42—1222no—
488C42—1224no—
489C42—1224no—
490C42—1222no—
491C42—1222no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—612no—
497C23—612no—
498C23—612no—
499C23—2441no—
500C23—2441no—
501C23C4⋊C4×D6114yes—
502C23C2×D4×D6111yescharacteristic
503Q8—1224no—
504Q8—1224no—
505D4—24no—
506D4—24no—
507D4—12216no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—612no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—612no—
521C2×C4—614no—
522C2×C4—614no—
523C2×C4—614no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—1228no—
527C2×C4—1228no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—4884no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4—2442no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4C22×C4×D6111yescharacteristic
543C2×C4C2×D4×D6112yes—
544C2×C4C2×Q8×D6111yescharacteristic
545C2×C4#41×D6112yes—
546C8—24no—
547C8—1224no—
548C22—21no—
549C22—21no—
550C22—24no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—614no—
556C22—1222no—
557C22—1222no—
558C22—2442no—
559C22—2442no—
560C22C2×C4⋊C4×D6111yescharacteristic
561C22#120×D6111yescharacteristic
562C22#123×D6111yescharacteristic
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—612no—
568C4—612no—
569C4—1224no—
570C4—4881no—
571C4—4881no—
572C4—2442no—
573C4—2442no—
574C4C4×D4×D6112yes—
575C4C22⋊Q8×D6112yes—
576C2—21no—
577C2—21no—
578C2—44no—
579C2—1224no—
580C2#385×D6111yescharacteristic
581C2#2145518711yescharacteristic
582C2#2145519411yescharacteristic
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1856C2
socleC22×C61C2×#120
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1856D61
hypocenterC61#20858
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1856

Complements

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

NH ≅ G/Ntype
2. #428×D61C2semidirect
5. #21446035C2semidirect
6. #21446032C2semidirect
7. #21446031C2semidirect
8. #21446029C2semidirect
9. #21446028C2semidirect
10. #21445527C2semidirect
11. C61×#1856C2semidirect
32. #16839743C4semidirect
61. M16×D61C2×C4semidirect
105. #14519695C2×C4semidirect
106. #14519694C2×C4semidirect
120. #100×C61C2×C4semidirect
294. #1856D61semidirect
339. C61#20858semidirect
410. #100C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590172
#428×D612
C2×#168397432
#214484092
#214460352
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C61×#18562
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