Finite Group Databasebeta

← #29689588

Subgroups — #29689588

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459222C211yesmaximal, characteristic
2#21454455C211yesmaximal, characteristic
3#21448312C211yesmaximal, characteristic
4#21446798C212yesmaximal, complemented
5#21446794C212yesmaximal, complemented
6#21446764C211yesmaximal, characteristic, complemented
7#21446763C211yesmaximal, characteristic, complemented
8#21446762C211yesmaximal, characteristic, complemented
9#21445917C212yesmaximal
10#21445917C212yesmaximal
11C61×#1986C211yesmaximal, characteristic, complemented
12#16840392C412yescomplemented
13#16840392C412yescomplemented
14#16840391—21no—
15#16840391—21no—
16#16840389C412yescomplemented
17#16840389C412yescomplemented
18#16840387—21no—
19#16840387—21no—
20#16840319C2211yescharacteristic
21#16840316—22no—
22#16840316—22no—
23#16840315C2212yes—
24C2×#14519734C2211yescharacteristic
25#117×D61C2212yes—
26#16839824—22no—
27#16839821—22no—
28#16839785C2212yes—
29#16839725C2212yes—
30C4×#11758194C2211yescharacteristic
31#16839654—21no—
32#16839652—21no—
33#16839649—21no—
34#16839643—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198C2211yescharacteristic
37C4×#11758198C2211yescharacteristic
38#16839608C2211yescharacteristic
39C42×Dic61C2211yescharacteristic
40#16839523C2212yes—
41#16839523C2212yes—
42#16839522C2212yes—
43#16839522C2212yes—
44#16839521C2212yes—
45#16839521C2212yes—
46C61×#533C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C61×#339C2212yes—
50C61×#339C2212yes—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194C2312yes—
53#14519777C2×C411yescharacteristic
54#14519777C2×C411yescharacteristic
55#14519776—22no—
56#14519776—22no—
57#14519776C2×C412yes—
58#14519776C2×C412yes—
59#14519775—22no—
60#14519775—22no—
61#14519769C2×C412yes—
62#14519768C2×C412yes—
63#14519768C2×C412yes—
64#14519767—22no—
65#14519767—22no—
66#14519767—22no—
67#14519767—22no—
68#14519757—22no—
69#14519757—22no—
70#14519756—21no—
71#14519756—21no—
72#14519755—22no—
73#14519755—22no—
74#14519755—22no—
75#14519755—22no—
76#14519754—22no—
77#14519754—22no—
78#14519753—22no—
79#14519753—22no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84#14519736C2312yes—
85#14519735C2×C412yes—
86#14519735C2×C412yes—
87#14519734—22no—
88#14519734C2×C412yes—
89#14519734C2×C412yes—
90#14519734D412yes—
91#14519734D412yes—
92#14519733—22no—
93C2×C4⋊Dic61—21no—
94C2×#11758196D411yescharacteristic
95C2×#11758196D411yescharacteristic
96C4⋊C4×D61C2×C412yes—
97C4⋊C4×D61C2×C412yes—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198D411yescharacteristic
101C2×#11758198D411yescharacteristic
102C22⋊C4×D61—22no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61—22no—
109#14519706—24no—
110#14519689—24no—
111#14519688—21no—
112C4×#92418—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C2×M16×C61C2312yes—
119C4⋊Q8×C61C2×C411yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121#125×C61C2×C412yes—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126#117×C61C2312yes—
127C2×C4⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131#97×C61—21no—
132C4×C8×C61—21no—
133#95×C61—21no—
134C23×D61C2×D411yescharacteristic
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D412yes—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157M16×C61—22no—
158M16×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—22no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61C22×C411yescharacteristic
172C42×C61C22×C411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C22⋊Dic61—22no—
176#11758198—22no—
177#11758198—22no—
178#11758198—42no—
179#11758198—42no—
180#11758198C22⋊C412yes—
181#11758198C22⋊C412yes—
182#11758198C22⋊C412yes—
183#11758198C22⋊C412yes—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—42no—
189C4⋊Dic61C22×C412yes—
190C4⋊Dic61C22×C412yes—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196C22⋊C412yes—
200#11758196C22⋊C412yes—
201#11758196C22⋊C412yes—
202#11758196C22⋊C412yes—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C4×Dic61C22×C412yes—
211C4×Dic61C2×D412yes—
212#11758194—22no—
213#11758194—22no—
214C2×#92418—22no—
215C23×C61C22×D411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61C2×C22⋊C412yes—
221Q8×C61—81no—
222Q8×C61—81no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C612−1+412yes—
240C2×C4×C612−1+412yes—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—82no—
259C4×D61—82no—
260Dic122—82no—
261Dic122—82no—
262Dic122—81no—
263Dic122—81no—
264C8×C61—42no—
265#92418—42no—
266#21846—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61C22×C22⋊C411yescharacteristic
270C22×C61#53611yescharacteristic
271C22×C61#53611yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—81no—
276C2×D61—81no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—81no—
285C4×C61#51912yes—
286C4×C61#53612yes—
287Dic61—22no—
288Dic61—22no—
289Dic61—22no—
290Dic61—42no—
291Dic61—42no—
292Dic61—42no—
293Dic61—81no—
294#3458—611no—
295#2921—611no—
296#2298—611no—
297#1995—612no—
298#1994—612no—
299#1986—611no—
300#1986—611no—
301#1986—611no—
302#1986D6111yescharacteristic, complemented
303#1826—612no—
304#1826—612no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61#231011yescharacteristic
308C2×C61#290111yescharacteristic
309C2×C61#290111yescharacteristic
310D61—42no—
311#544—612no—
312#544—612no—
313#544—612no—
314#544—612no—
315#544—1221no—
316#544—1221no—
317#543—1221no—
318#543—1221no—
319#533—612no—
320#533—611no—
321#533—1222no—
322#533—1222no—
323#533C2×D6111yescharacteristic
324C2×#117—611no—
325C2×#117—612no—
326#433—1222no—
327#433—1222no—
328#423—612no—
329#407—612no—
330C4×M16—611no—
331C4×M16C2×D6111yescharacteristic
332#388—1221no—
333#388—1221no—
334#387—1221no—
335#387—1221no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4C2×D6111yescharacteristic
338C4×C22⋊C4—611no—
339C4×C22⋊C4—611no—
340#376—611no—
341C43—611no—
342#339—612no—
343#339—612no—
344#339—612no—
345#339—612no—
346#339—612no—
347#339—612no—
348#339C2×D6112yes—
349#339C2×D6112yes—
350C61#2184611yescharacteristic, complemented, Sylow
351C23×C4—611no—
352C2×M16—612no—
353C2×M16C22×D6112yes—
354C4⋊Q8—612no—
355C4⋊Q8—612no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1222no—
359C4⋊Q8—1222no—
360C4⋊Q8—1222no—
361C4⋊Q8—1222no—
362C4⋊Q8C4×D6111yescharacteristic
363C4⋊Q8C4×D6111yescharacteristic
364#125—612no—
365#125—612no—
366#125—612no—
367#125—1222no—
368#125—1222no—
369#125—1222no—
370#125—1222no—
371#125C4×D6112yes—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—1222no—
375C22⋊Q8—1222no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392#117—612no—
393#117—612no—
394#117—612no—
395#117—612no—
396#117—612no—
397#117—612no—
398#117—612no—
399#117—1222no—
400#117—1222no—
401#117C22×D6112yes—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1221no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—611no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1222no—
417C2×C42—1221no—
418C2×C42—1221no—
419C2×C42C22×D6111yescharacteristic
420C2×C42C22×D6111yescharacteristic
421C2×C42C22×D6111yescharacteristic
422C4⋊C8—1224no—
423C22⋊C8—1224no—
424#97—21no—
425#97—1221no—
426C4×C8—21no—
427C4×C8—1221no—
428#95—21no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—611no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1222no—
438C2×Q8—1222no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457M16—22no—
458M16—22no—
459M16—1222no—
460M16—1222no—
461C2×C8—22no—
462C2×C8—1222no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—612no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2442no—
487C4⋊C4C2×C4×D6112yes—
488C4⋊C4C2×C4×D6112yes—
489C22⋊C4—22no—
490C22⋊C4—612no—
491C22⋊C4—612no—
492C22⋊C4—612no—
493C22⋊C4—612no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—612no—
503C42—612no—
504C42—1222no—
505C42—1222no—
506C42—1222no—
507C42—1222no—
508C42—1222no—
509C42—2442no—
510C42—2442no—
511C42C2×C4×D6111yescharacteristic
512C42C2×C4×D6111yescharacteristic
513C42D4×D6111yescharacteristic
514C42D4×D6111yescharacteristic
515C23—612no—
516C23—1221no—
517C23—1221no—
518C23—2441no—
519C23—2441no—
520C23C2×D4×D6111yescharacteristic
521Q8—81no—
522Q8—81no—
523Q8—4881no—
524Q8—4881no—
525Q8—4882no—
526Q8—4882no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4882no—
548C2×C4—4882no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4C22⋊C4×D6111yescharacteristic
557C2×C4C22⋊C4×D6111yescharacteristic
558C2×C4C22⋊C4×D6111yescharacteristic
559C2×C4C22⋊C4×D6111yescharacteristic
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4#1451982412yes—
563C2×C4#1451982512yes—
564C8—42no—
565C8—2442no—
566C22—21no—
567C22—21no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—4881no—
572C22—4881no—
573C22C2×C22⋊C4×D6111yescharacteristic
574C22#1684033211yescharacteristic
575C22#1684033711yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—4881no—
588C4—2442no—
589C4—2442no—
590C4—2442no—
591C4#1684023812yes—
592C4#1684033512yes—
593C2—21no—
594C2—21no—
595C2—2442no—
596C2#2144840211yescharacteristic
597C2#2145436311yescharacteristic
598C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61#21846
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #21446798C2semidirect
5. #21446794C2semidirect
6. #21446764C2semidirect
7. #21446763C2semidirect
8. #21446762C2semidirect
11. C61×#1986C2semidirect
12. #16840392C4semidirect
13. #16840392C4semidirect
16. #16840389C4semidirect
17. #16840389C4semidirect
302. #1986D61semidirect
350. C61#21846semidirect

Maximal subgroups

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

subgroupindex
#214592222
#214544552
#214483122
#214467982
#214467942
#214467642
#214467632
#214467622
#214459172
C61×#19862
#2184661