Finite Group Databasebeta

← #10786200

Subgroups — #10786200

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3360C211yesmaximal, characteristic
2#83659C211yesmaximal, characteristic, complemented
3#83626C211yesmaximal, characteristic, complemented
4#79579C211yesmaximal, characteristic
5#71933C211yesmaximal, characteristic, complemented
6#71925C211yesmaximal, characteristic, complemented
7#71241C211yesmaximal, characteristic, complemented
8#71226C211yesmaximal, characteristic, complemented
9#71147C211yesmaximal, characteristic, complemented
10#71132C211yesmaximal, characteristic, complemented
11#71049C211yesmaximal, characteristic, complemented
12#71041C211yesmaximal, characteristic, complemented
13#70942C211yesmaximal, characteristic, complemented
14#70932C211yesmaximal, characteristic, complemented
15#70879C211yesmaximal, characteristic
16#19866—31nomaximal, Sylow
17C3×#556C2211yescharacteristic
18C3×#554C2211yescharacteristic
19#5542C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#488—21no—
22C3×#487C2211yescharacteristic
23C3×#479—21no—
24C3×#476C2211yescharacteristic
25C3×#463—21no—
26C3×#461C2211yescharacteristic
27C3×C8×Q8C2211yescharacteristic
28C3×C4×SD16—21no—
29C3×C4×D8C2211yescharacteristic
30#5155—21no—
31#5094C2211yescharacteristic
32#5091C2211yescharacteristic
33#5080C2211yescharacteristic
34#5079C2211yescharacteristic
35#5078C2211yescharacteristic
36#5078C2211yescharacteristic
37#5052—21no—
38#5051—21no—
39#5045—21no—
40#5045—21no—
41C4×#216C2211yescharacteristic
42#5020C2211yescharacteristic
43#5020C2211yescharacteristic
44#5019C2211yescharacteristic
45#5019C2211yescharacteristic
46#5012C2211yescharacteristic
47D4×C3⋊C8C2211yescharacteristic
48#4998—21no—
49#4996—21no—
50#4971—21no—
51#4964—21no—
52#4873—21no—
53#4767—21no—
54#4767—21no—
55#4699C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4484C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4458C2211yescharacteristic
67#4454C2211yescharacteristic
68#3393—31no—
69#3360D311yescharacteristic
70#3356—31no—
71#3342—31no—
72#1723—31no—
73#1693—31no—
74#1684—31no—
75#1671—31no—
76#1670—31no—
77#1650—31no—
78#1650—31no—
79#1643—31no—
80#1638—31no—
81#1581—31no—
82#1579—31no—
83C2×C3×SD16—21no—
84C2×C3×D8D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#126—41no—
88C3×#125—21no—
89C3×#124—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—41no—
93C3×C4×Q8—21no—
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4C2311yescharacteristic
97C3×C4×D4C2311yescharacteristic
98C3×#117—21no—
99C2×C3×C4⋊C4—21no—
100C2×#216D411yescharacteristic
101#936—21no—
102#936—21no—
103C2×#214—21no—
104C2×#214—21no—
105#926—21no—
106#926—21no—
107#924—21no—
108#924—21no—
109C2×#208—21no—
110C22×C3⋊C8—21no—
111C2×Dic12—21no—
112C2×Dic12—21no—
113#887—21no—
114#887—21no—
115#879—41no—
116#879—41no—
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×#107D411yescharacteristic
121C3×#106—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#102—21no—
128C3×#102D411yescharacteristic
129C3×#102D411yescharacteristic
130C3×C4×C8—21no—
131C3×C4×C8C2311yescharacteristic
132#835—21no—
133#835—21no—
134#835—41no—
135#835D411yescharacteristic
136#830—21no—
137#830—21no—
138Dic3⋊C8—21no—
139C8×Dic3—21no—
140#810—21no—
141#810—21no—
142#810—21no—
143#810—21no—
144#810D411yescharacteristic
145#808—21no—
146#808—21no—
147#808—21no—
148#808—21no—
149#808—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153#807—41no—
154#807D411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#556—31no—
162#556C2×D311yescharacteristic
163#554C2×D311yescharacteristic
164#500—31no—
165#500—31no—
166#498—31no—
167#495—61no—
168#494—31no—
169#488—21no—
170#487—31no—
171#487C2×D311yescharacteristic
172#485—61no—
173#485—61no—
174#485—61no—
175#483—61no—
176#479—21no—
177#477—31no—
178#476C2×D311yescharacteristic
179#475—31no—
180#475—31no—
181#475—31no—
182#470—61no—
183#467—61no—
184#464—61no—
185#464—61no—
186#463—21no—
187#461—31no—
188#461—31no—
189#461C2×D311yescharacteristic
190C8×Q8—31no—
191C8×Q8C2×D311yescharacteristic
192C4×Q16—31no—
193C4×Q16—61no—
194C4×SD16—21no—
195C4×D8C2×D311yescharacteristic
196#436—31no—
197C8×D4—31no—
198#428—61no—
199#427—61no—
200#335—31no—
201#334—31no—
202#333—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4C2×D411yescharacteristic
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—41no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C2×D411yescharacteristic
222C3×SD16—42no—
223C3×D8—41no—
224C2×C3×C8—21no—
225C2×C3×C8C2×D411yescharacteristic
226C2×C3×C8C2×D411yescharacteristic
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4C2×D411yescharacteristic
236C3×C4⋊C4C2×D411yescharacteristic
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—41no—
241C3×C22⋊C4—41no—
242C3×C22⋊C4—41no—
243C3×C42—21no—
244C3×C42—41no—
245C3×C42C2411yescharacteristic
246#217—41no—
247#217—41no—
248#217—41no—
249#217—41no—
250#216—22no—
251#214—81no—
252#214—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264#208—42no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—42no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272Dic12—42no—
273Dic12—81no—
274C2×Q16—31no—
275C2×Q16—61no—
276C2×Q16—61no—
277C2×SD16—21no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×D8#6711yescharacteristic
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8C22×D311yescharacteristic
286C4⋊Q8C22×D311yescharacteristic
287#126—41no—
288#126—121no—
289#126—121no—
290#125—21no—
291#124—21no—
292#124—61no—
293#124—61no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—41no—
297C22⋊Q8—61no—
298C22⋊Q8—61no—
299C4×Q8—21no—
300C4×Q8—31no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×Q8C22×D311yescharacteristic
304C4×D4—21no—
305C4×D4C22×D311yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—21no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—21no—
311#107—31no—
312#107—61no—
313#107—61no—
314#107—61no—
315#107—121no—
316#107#6711yescharacteristic
317#106—21no—
318#106—61no—
319#106—61no—
320#106—61no—
321#106—61no—
322#106—61no—
323C4⋊C8—21no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—31no—
327C4⋊C8—61no—
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—21no—
331#103—21no—
332#103—31no—
333#103—31no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—121no—
341#102—21no—
342#102#6711yescharacteristic
343#102#6711yescharacteristic
344C22⋊C8—61no—
345C22⋊C8—61no—
346C4×C8—21no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C23×C3—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—81no—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—161no—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380Dic6—21no—
381Dic6—21no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3⋊C8—21no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8#13512yes—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—31no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8D3×D411yescharacteristic
401C2×D4C2×#6711yescharacteristic
402C2×D4C2×#6711yescharacteristic
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406Q16—62no—
407Q16—122no—
408Q16—241no—
409SD16—42no—
410SD16—241no—
411SD16—241no—
412D8—41no—
413M16—122no—
414C2×C8—21no—
415C2×C8—31no—
416C2×C8—31no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—122no—
422C2×C8C2×#6711yescharacteristic
423C2×C8C2×#6711yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—31no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4D3×D411yescharacteristic
441C4⋊C4C2×#6711yescharacteristic
442C4⋊C4C2×#6711yescharacteristic
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C42—21no—
453C42—41no—
454C42—61no—
455C42—121no—
456C42—121no—
457C42C23×D311yescharacteristic
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3D4211yescharacteristic
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4C2×#13511yescharacteristic
473C3×C4#57511yescharacteristic
474C3×C4#57711yescharacteristic
475C3×C4#57811yescharacteristic
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—61no—
487Q8—61no—
488Q8—81no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493D4—21no—
494D4—21no—
495D4—21no—
496D4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—161no—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4#100411yescharacteristic
514C2×C4C22×#6711yescharacteristic
515C8—21no—
516C8—21no—
517C8—32no—
518C8—41no—
519C8—61no—
520C8—121no—
521C8—121no—
522C8—121no—
523C8—121no—
524C2×C3—81no—
525C2×C3—81no—
526C2×C3#327111yescharacteristic
527C2×C3#328311yescharacteristic
528C2×C3#328911yescharacteristic
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22D4×#6711yescharacteristic
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—81no—
542C4—81no—
543C4—121no—
544C4—121no—
545C4—241no—
546C4—241no—
547C4D3×#13511yescharacteristic
548C4#577111yescharacteristic
549C4#582211yescharacteristic
550C4#583711yescharacteristic
551C3#1986611yescharacteristic, complemented, Sylow
552C2—81no—
553C2—81no—
554C2#8354711yescharacteristic
555C2#8392611yescharacteristic
556C2#8401311yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3360C2
socleC22×C3D42
prime radicalC2×C3×D8D4
hypercenter#3360D3
hypocenterC3#19866
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
2. #83659C2semidirect
3. #83626C2semidirect
5. #71933C2semidirect
6. #71925C2semidirect
7. #71241C2semidirect
8. #71226C2semidirect
9. #71147C2semidirect
10. #71132C2semidirect
11. #71049C2semidirect
12. #71041C2semidirect
13. #70942C2semidirect
14. #70932C2semidirect
551. C3#19866semidirect

Maximal subgroups

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

subgroupindex
C3×#33602
#836592
#836262
#795792
#719332
#719252
#712412
#712262
#711472
#711322
#710492
#710412
#709422
#709322
#708792
#198663