Finite Group Databasebeta

← #29663013

Subgroups — #29663013

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456533C211yesmaximal, characteristic, complemented
2#21455890C211yesmaximal, characteristic, complemented
3#21454839C211yesmaximal, characteristic, complemented
4#21445378C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445330C211yesmaximal, characteristic, complemented
7#21445282C211yesmaximal, characteristic, complemented
8#21445261C211yesmaximal, characteristic, complemented
9#21445251C211yesmaximal, characteristic, complemented
10#21445211C211yesmaximal, characteristic, complemented
11#21445165C211yesmaximal, characteristic, complemented
12#21444569C211yesmaximal, characteristic, complemented
13#21444524C211yesmaximal, characteristic, complemented
14#21444517C211yesmaximal, characteristic, complemented
15C61×#3035C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840134C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840103—21no—
22#16840047C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16840006—21no—
25#16840006—21no—
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839924—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31#16839862—21no—
32#16839862—21no—
33C4×#11758202—21no—
34C4×#11758200—21no—
35#16839840—21no—
36#16839840—21no—
37#16839840C2211yescharacteristic
38#16839840C2211yescharacteristic
39#16839819—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42C4×#11758194—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#532C2211yescharacteristic
58C4×#41×C61C2211yescharacteristic
59C61×#464—21no—
60C61×#463—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#460C2211yescharacteristic
64C61×#459C2211yescharacteristic
65C2×C4⋊C8×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519769—21no—
78#14519756—21no—
79#14519756—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82#14519736—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519700—21no—
95#14519700—21no—
96#14519700—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688D411yescharacteristic
107#14519688D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×#41×C61D411yescharacteristic
115#135×C61—44no—
116C2×Q16×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×D8×C61—21no—
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#124×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126C4×Q8×C61D411yescharacteristic
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61D411yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135#117×C61D411yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#103×C61—21no—
143#102×C61—21no—
144#102×C61—21no—
145#41×C61—22no—
146#41×C61—41no—
147#41×C61—42no—
148#41×C61—42no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61#4111yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166Q16×C61—42no—
167SD16×C61—42no—
168SD16×C61—42no—
169D8×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—42no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61#4111yescharacteristic
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758202—42no—
193#11758200—42no—
194#11758200—42no—
195#11758200—42no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—41no—
203#11758194—22no—
204#11758194—41no—
205#11758194—42no—
206#11758194—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C23×C61—21no—
216C23×C61—41no—
217C23×C61#12111yescharacteristic
218Q8×C61—21no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61AGL(1,ℤ/8)12yes—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—42no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230D4×C61—82no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—81no—
244C2×C4×C61—82no—
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61#12111yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×Dic61—41no—
252C2×Dic61—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257C8×C61—41no—
258C8×C61—41no—
259#92418—21no—
260#92418—21no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418#13512yescomplemented
267#19182—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61C2×#12111yescharacteristic
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×#13511yescharacteristic
287C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C4×C61#57511yescharacteristic
289C4×C61#57511yescharacteristic
290Dic61—81no—
291Dic61—81no—
292#3182—611no—
293#3107—611no—
294#3035D6111yescharacteristic
295#2971—611no—
296#1701—611no—
297#1699—611no—
298#1693—611no—
299#1685—611no—
300#1682—611no—
301#1680—611no—
302#1674—611no—
303#1666—611no—
304#1565—611no—
305#1553—611no—
306#1552—611no—
307C2×C61—41no—
308C2×C61—81no—
309C2×C61—161no—
310C2×C61#303511yescharacteristic
311C2×C61#305511yescharacteristic
312C2×C61#306911yescharacteristic
313C2×#135—21no—
314#533—611no—
315#532C2×D6111yescharacteristic
316C4×#41C2×D6111yescharacteristic
317#500—611no—
318#492—611no—
319#487—611no—
320#486—611no—
321#485—1221no—
322#475—611no—
323#474—611no—
324#468—1221no—
325#468—1221no—
326#464—21no—
327#463—21no—
328#462—611no—
329#462C2×D6111yescharacteristic
330#461—611no—
331#461C2×D6111yescharacteristic
332#460C2×D6111yescharacteristic
333#459C2×D6111yescharacteristic
334#452—1221no—
335C8×Q8—611no—
336C8×Q8—611no—
337#440—1221no—
338#440—1221no—
339C4×Q16—1221no—
340C4×SD16—1221no—
341#436—611no—
342#436—611no—
343#436—1221no—
344#436—1221no—
345#432—611no—
346#432—1221no—
347#431—611no—
348C2×C4⋊C8C2×D6111yescharacteristic
349#416—21no—
350C4×M16—611no—
351C4×M16—1221no—
352#335—611no—
353#335—611no—
354#334—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#332—611no—
360#332—611no—
361#332—611no—
362#331—611no—
363#331—611no—
364C61#1918211yescharacteristic, complemented, Sylow
365C2×#41—21no—
366C2×#41C22⋊D6111yescharacteristic
367#135—44no—
368C2×Q16—21no—
369C2×Q16—1221no—
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×D8—21no—
376C2×M16—611no—
377C2×M16—611no—
378C2×M16—1221no—
379C2×M16—1221no—
380C22×C8—21no—
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8C22×D6111yescharacteristic
384#127C22×D6111yescharacteristic
385#125—1221no—
386#124—21no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389#121—21no—
390#121—21no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C22⋊D6111yescharacteristic
394C4×Q8C22×D6111yescharacteristic
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C22⋊D6111yescharacteristic
400C4×D4C22×D6111yescharacteristic
401#117—21no—
402#117—1221no—
403#117C22⋊D6111yescharacteristic
404C2×C42—21no—
405C2×C42C22×D6111yescharacteristic
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—1221no—
410C4⋊C8—21no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—21no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#102—21no—
427#102—21no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435#97—611no—
436#97—611no—
437#97—1221no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—1221no—
441C4×C8—1221no—
442C4×C8—1221no—
443#41—22no—
444#41—41no—
445#41—42no—
446#41—42no—
447C2×Q8—21no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8C2×C22⋊D6111yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4D4×D6111yescharacteristic
462C22×C4#1175822411yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464Q16—42no—
465Q16—2442no—
466SD16—42no—
467SD16—42no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471D8—42no—
472M16—1222no—
473M16—2442no—
474M16—2441no—
475M16—2442no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—42no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—2441no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—2441no—
509C42D4×D6111yescharacteristic
510C42#1175822411yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—41no—
515C23#1451974911yescharacteristic
516Q8—21no—
517Q8—41no—
518Q8—41no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522Q8—2441no—
523Q8#1451980312yes—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—81no—
532D4—82no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451974911yescharacteristic
550C2×C4#1451974911yescharacteristic
551C2×C4#1451974911yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C8—41no—
556C8—41no—
557C8—612no—
558C8—1221no—
559C8—1221no—
560C8—1221no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#1451974911yescharacteristic
575C4—21no—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4#135×D6111yescharacteristic
586C4C2×#1451980311yescharacteristic
587C4#1684055911yescharacteristic
588C4#1684056011yescharacteristic
589C2—41no—
590C2—81no—
591C2—161no—
592C2#2145527011yescharacteristic
593C2#2145543411yescharacteristic
594C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3035C2
socleC22×C61C2×#121
hypercenter#3035D61
hypocenterC61#19182
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #21456533C2semidirect
2. #21455890C2semidirect
3. #21454839C2semidirect
4. #21445378C2semidirect
5. #21445368C2semidirect
6. #21445330C2semidirect
7. #21445282C2semidirect
8. #21445261C2semidirect
9. #21445251C2semidirect
10. #21445211C2semidirect
11. #21445165C2semidirect
12. #21444569C2semidirect
13. #21444524C2semidirect
14. #21444517C2semidirect
266. #92418#135semidirect
364. C61#19182semidirect

Maximal subgroups

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

subgroupindex
#214565332
#214558902
#214548392
#214453782
#214453682
#214453302
#214452822
#214452612
#214452512
#214452112
#214451652
#214445692
#214445242
#214445172
C61×#30352
#1918261