Finite Group Databasebeta

← #10786202

Subgroups — #10786202

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
1#83637C211yesmaximal, characteristic
2#83626C211yesmaximal, characteristic, complemented
3#81029C211yesmaximal, characteristic, complemented
4#80421C211yesmaximal, characteristic
5C3×#1643C211yesmaximal, characteristic, complemented
6#71229C211yesmaximal, characteristic, complemented
7#71227C211yesmaximal, characteristic, complemented
8#71117C211yesmaximal, characteristic, complemented
9#71041C211yesmaximal, characteristic, complemented
10#68511C211yesmaximal, characteristic, complemented
11#68496C211yesmaximal, characteristic, complemented
12#68250C211yesmaximal, characteristic, complemented
13#68215C211yesmaximal, characteristic, complemented
14#68209C211yesmaximal, characteristic, complemented
15#68153C211yesmaximal, characteristic
16#19866—31nomaximal, Sylow
17C3×#554C2211yescharacteristic
18#5732C2211yescharacteristic
19#5584C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×#461C2211yescharacteristic
24#5168—21no—
25#5166—21no—
26#5093C2211yescharacteristic
27#5092C2211yescharacteristic
28#5091C2211yescharacteristic
29#5084C2211yescharacteristic
30#5080C2211yescharacteristic
31#5078C2211yescharacteristic
32#5068—21no—
33#5063—21no—
34#5052—21no—
35#5045—21no—
36C4×#216C2211yescharacteristic
37Q8×C3⋊C8C2211yescharacteristic
38#5020C2211yescharacteristic
39#5019C2211yescharacteristic
40C4×#214—21no—
41#5016—21no—
42C4×#213C2211yescharacteristic
43#5012C2211yescharacteristic
44#4971—21no—
45#4964—21no—
46#4868—21no—
47#4860—21no—
48#4851C2211yescharacteristic
49#4841C2211yescharacteristic
50#4834C2211yescharacteristic
51#4830C2211yescharacteristic
52#4815—21no—
53#4813—21no—
54#4798—21no—
55#4790—21no—
56#4682C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×Q8⋊C8C2211yescharacteristic
59C3×D4⋊C8C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4480C2211yescharacteristic
64#4479C2211yescharacteristic
65#4459C2211yescharacteristic
66#4456C2211yescharacteristic
67#4454C2211yescharacteristic
68#3393—31no—
69#3360—31no—
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#1643D311yescharacteristic, complemented
80#1638—31no—
81#1581—31no—
82#1579—31no—
83C2×C3×SD16—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#125—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—41no—
89C3×C4×Q8C2311yescharacteristic
90C3×C4×D4—21no—
91C3×C4×D4C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93#947—21no—
94#946—21no—
95Q8×Dic3—21no—
96C2×#216—21no—
97C2×#216—21no—
98C2×#216D411yescharacteristic
99#936—21no—
100C2×#214—21no—
101C2×#213D411yescharacteristic
102#926—21no—
103#924—21no—
104C2×#208—21no—
105C2×#204—21no—
106C2×D3×C8—21no—
107#898—41no—
108#898—41no—
109#897—21no—
110#892—21no—
111#892—21no—
112#887—21no—
113#879—41no—
114#875—21no—
115C4×D12C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6—21no—
119C4×Dic6C2311yescharacteristic
120C3×#106—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#103—41no—
126C3×C4×C8C2311yescharacteristic
127#835—21no—
128#835D411yescharacteristic
129#832D411yescharacteristic
130#830—21no—
131(C2×D3)⋊C8—21no—
132#818—41no—
133#817—21no—
134#816—21no—
135#816—21no—
136Dic3⋊C8—21no—
137C8×Dic3—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#810D411yescharacteristic
143#809—21no—
144#809D411yescharacteristic
145#808—21no—
146#808—21no—
147#808—21no—
148#808—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807D411yescharacteristic
153#807D411yescharacteristic
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#556—31no—
162#556—31no—
163#554C2×D311yescharacteristic
164#500—31no—
165#500—31no—
166#498—31no—
167#495—61no—
168#494—31no—
169#488—61no—
170#487—31no—
171#487C2×D311yescharacteristic
172#485—61no—
173#485—61no—
174#485—61no—
175#483—61no—
176#479—61no—
177#477—31no—
178#476—31no—
179#475—31no—
180#475—31no—
181#475C2×D311yescharacteristic
182#470—61no—
183#467—61no—
184#464—61no—
185#464—61no—
186#463—61no—
187#461—31no—
188#461—31no—
189#461C2×D311yescharacteristic
190C8×Q8—31no—
191C8×Q8—31no—
192C4×Q16—31no—
193C4×Q16—61no—
194C4×SD16—61no—
195C4×D8—31no—
196#436—31no—
197C8×D4—31no—
198#428—61no—
199#427—61no—
200#335—31no—
201#334—31no—
202#333C2×D311yescharacteristic
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8C2×D311yescharacteristic
208D4⋊C8—31no—
209D4⋊C8C2×D311yescharacteristic
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4C2×D411yescharacteristic
215C22×C3×C4—21no—
216C22×C3×C4—41no—
217C2×D12C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C2×D411yescharacteristic
222C3×SD16—81no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4C2×D411yescharacteristic
234C3×C4⋊C4C2×D411yescharacteristic
235C3×C22⋊C4—21no—
236C3×C22⋊C4—41no—
237C3×C42—21no—
238C3×C42C2411yescharacteristic
239#217—41no—
240#217—41no—
241#216—22no—
242#216—42no—
243#216—81no—
244#214—42no—
245#213—41no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—41no—
250(C2×D3)⋊C4—41no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3C2×D411yescharacteristic
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264C4×Dic3—41no—
265#208—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272#204—81no—
273D3×C8—42no—
274C2×Q16—31no—
275C2×Q16—61no—
276C2×Q16—61no—
277C2×SD16—21no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×D8—31no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8C22×D311yescharacteristic
286C4⋊Q8C22×D311yescharacteristic
287#126—121no—
288#126—121no—
289#126—121no—
290#125—21no—
291#124—61no—
292#124—61no—
293#124—61no—
294C22⋊Q8—21no—
295C22⋊Q8—41no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298C22⋊Q8—61no—
299C4×Q8—31no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×Q8C22×D311yescharacteristic
304C4×D4—21no—
305C4×D4—31no—
306C4×D4C22×D311yescharacteristic
307#117—61no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—21no—
311#107—31no—
312#107—31no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—121no—
317#106—21no—
318#106—61no—
319#106—61no—
320#106—61no—
321#106—61no—
322#106—61no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—21no—
331#103—31no—
332#103—31no—
333#103—41no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#102—31no—
342#102—31no—
343#102—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C22×D3—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—81no—
358C3×D4—21no—
359C3×D4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—161no—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376D12—21no—
377D12—21no—
378C4×D3—41no—
379C4×D3—41no—
380Dic6—21no—
381Dic6—21no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C3⋊C8—21no—
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×D4—31no—
402C2×D4D3×D411yescharacteristic
403C22×C4—21no—
404C22×C4—41no—
405C22×C4—61no—
406Q16—62no—
407Q16—122no—
408Q16—241no—
409SD16—81no—
410SD16—122no—
411SD16—241no—
412D8—121no—
413M16—122no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—122no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—31no—
430C4⋊C4—31no—
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⋊C4—121no—
441C4⋊C4D3×D411yescharacteristic
442C4⋊C4D3×D411yescharacteristic
443C22⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—61no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C42—21no—
453C42—61no—
454C42—121no—
455C42—121no—
456C42—121no—
457C42C23×D311yescharacteristic
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3D4211yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4C2×#13511yescharacteristic
472C3×C4#57511yescharacteristic
473C3×C4#57711yescharacteristic
474C3×C4#57811yescharacteristic
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—61no—
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—61no—
496D4—61no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—161no—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4#100411yescharacteristic
515C8—32no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—61no—
520C8—61no—
521C8—61no—
522C8—121no—
523C8—121no—
524C2×C3—81no—
525C2×C3#327111yescharacteristic
526C2×C3#328311yescharacteristic
527C2×C3#328911yescharacteristic
528D3—81no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—121no—
533C22—121no—
534C22—121no—
535C22#559011yescharacteristic
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—81no—
541C4—81no—
542C4—121no—
543C4—121no—
544C4—121no—
545C4—241no—
546C4—241no—
547C4#576211yescharacteristic
548C4D3×#13511yescharacteristic
549C4#577711yescharacteristic
550C4#577911yescharacteristic
551C3#1986611yescharacteristic, complemented, Sylow
552C2—81no—
553C2—241no—
554C2#8006511yescharacteristic
555C2#8026911yescharacteristic
556C2#8029511yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1643C2
socleC22×C3D42
prime radicalC2×#213D4
hypercenter#1643D3
hypocenterC3#19866
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1643

Complements

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

NH ≅ G/Ntype
2. #83626C2semidirect
3. #81029C2semidirect
5. C3×#1643C2semidirect
6. #71229C2semidirect
7. #71227C2semidirect
8. #71117C2semidirect
9. #71041C2semidirect
10. #68511C2semidirect
11. #68496C2semidirect
12. #68250C2semidirect
13. #68215C2semidirect
14. #68209C2semidirect
79. #1643D3semidirect
551. C3#19866semidirect

Maximal subgroups

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

subgroupindex
#836372
#836262
#810292
#804212
C3×#16432
#712292
#712272
#711172
#710412
#685112
#684962
#682502
#682152
#682092
#681532
#198663