Finite Group Databasebeta

← #10945040

Subgroups — #10945040

Order 768.

1432 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5521C211yesmaximal, characteristic
2C3×#2625C211yesmaximal, characteristic, complemented
3C2×#5046C211yesmaximal, characteristic
4C2×#5045C211yesmaximal, characteristic, complemented
5#71584C211yesmaximal, characteristic, complemented
6#69687C212yesmaximal, complemented
7#69687C212yesmaximal, complemented
8#69626C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69521C211yesmaximal, characteristic
11#69503C211yesmaximal, characteristic, complemented
12#33018—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#215C2212yes—
17C2×#926C2211yescharacteristic
18#5521—24no—
19#5521—24no—
20C2×#893—21no—
21C2×#893—21no—
22C2×D3×C22⋊C4—22no—
23C2×C4×D12C2212yes—
24C2×C4×D12C2212yes—
25C3×#400—21no—
26C3×#398—21no—
27C3×#397—22no—
28C3×#391C2212yescomplemented
29C3×#389C2212yescomplemented
30C3×#380C2211yescharacteristic
31C2×#835C2211yescharacteristic, complemented
32C2×#835C2211yescharacteristic, complemented
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5046—24no—
36#5045—24no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic, complemented
41C2×#807C2211yescharacteristic, complemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44#4524C2212yescomplemented
45#4524C2212yescomplemented
46C2×#518—31no—
47C2×#485—31no—
48C2×#450—31no—
49#2625D311yescharacteristic, complemented
50#1877—32no—
51#1877—32no—
52#1864—32no—
53#1862—32no—
54#1825—31no—
55#1818—31no—
56#1786—31no—
57C24×D3—21no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×D12—21no—
61C22×D12—22no—
62C22×D12—24no—
63C22×D12—24no—
64C22×D12D414yes—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4D412yes—
81C2×C3×C42C2312yes—
82C2×#217—21no—
83C2×#215—24no—
84C2×#215—24no—
85C2×#215—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88#926—22no—
89#926D414yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4×Dic3—21no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96#893—28no—
97#893—28no—
98D3×C22⋊C4—216no—
99C4×D12—24no—
100C4×D12—24no—
101C4×D12—24no—
102C4×D12—24no—
103C3×#95—21no—
104C3×#95—22no—
105C3×#95—22no—
106C3×#95—22no—
107#835—22no—
108#835—22no—
109#830—22no—
110#830—22no—
111#809—24no—
112#809—24no—
113#809—24no—
114#808—22no—
115#807—22no—
116C23×D4—31no—
117C22×SD16—32no—
118C2×C22⋊Q8C2×D311yescharacteristic
119#518—64no—
120#518—64no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C2×C4×D4—61no—
125C2×#117—31no—
126C2×#117C2×D311yescharacteristic
127C22×C22⋊C4—62no—
128#485—64no—
129#450—64no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—31no—
139#400—21no—
140#398—21no—
141#397—22no—
142#391C2×D312yescomplemented
143#389C2×D312yescomplemented
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C23×D3—22no—
151C23×D3—28no—
152C23×D3—24no—
153C23×D3—81no—
154C23×D3—82no—
155C23×D3—81no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—21no—
172C22×Dic3—21no—
173C2×D12—28no—
174C2×D12—28no—
175C2×D12—28no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—216no—
181C2×D12C2×D414yes—
182C2×D12#4114yes—
183C2×D12#4114yes—
184C2×C4×D3—28no—
185C2×C4×D3—28no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×C4×D3—82no—
189C2×C4×D3—82no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—82no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—81no—
201C3×C42—22no—
202C3×C42—22no—
203#217—24no—
204#215—48no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—28no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3C2×D412yes—
210C4⋊Dic3C2×D412yes—
211C4×Dic3—24no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C25—61no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—34no—
222C22×D4—32no—
223C22×D4—64no—
224C22×D4—64no—
225C22×D4—62no—
226C22×D4—61no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4C22×D311yescharacteristic
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—64no—
233C22×C8—32no—
234C22×C8—31no—
235C22×C8—31no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—68no—
243C4×D4—68no—
244#117—22no—
245#117—22no—
246#117—34no—
247#117—62no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4#6711yescharacteristic
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—616no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4#6712yes—
263C2×C42—61no—
264C2×C42C22×D312yes—
265#107—62no—
266#106—62no—
267#103—62no—
268#103—62no—
269#102—64no—
270#102—64no—
271#102—64no—
272C22⋊C8—62no—
273C22⋊C8—62no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—22no—
281C23×C3—21no—
282C23×C3#12012yes—
283C23×C3C22×D411yescharacteristic
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—216no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—84no—
293C22×D3—84no—
294C22×D3—88no—
295C22×D3—88no—
296C3×Q8—42no—
297C3×Q8—41no—
298C3×Q8—41no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4112yes—
321C2×C3×C4C2×#4112yes—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327D12—216no—
328D12—24no—
329D12—24no—
330D12—24no—
331D12#12318yes—
332C4×D3—88no—
333C4×D3—88no—
334C3⋊C8—42no—
335C3⋊C8—41no—
336C3⋊C8—41no—
337C24—68no—
338C24—62no—
339C24—64no—
340C24—241no—
341C24—242no—
342C24—241no—
343C24C2×#6711yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×D4—34no—
349C2×D4—34no—
350C2×D4—34no—
351C2×D4—616no—
352C2×D4—68no—
353C2×D4—68no—
354C2×D4—68no—
355C2×D4—64no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—68no—
365C22×C4—68no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—242no—
369C22×C4—242no—
370C22×C4—242no—
371C22×C4—242no—
372C22×C4D3×D412yes—
373C22×C4D3×D412yes—
374C22×C4C2×#6712yes—
375C22×C4C2×#6712yes—
376C22×C4C2×#6711yescharacteristic
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—62no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—42no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4—82no—
398C22⋊C4—24no—
399C22⋊C4—42no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C22⋊C4—64no—
403C22⋊C4—81no—
404C42—22no—
405C42—22no—
406C42—64no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C2×#12011yescharacteristic
413C22×C3#52512yes—
414C22×C3#57611yescharacteristic
415C22×C3#57712yes—
416C22×C3#57711yescharacteristic
417C2×D3—28no—
418C2×D3—44no—
419C2×D3—44no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—44no—
423C2×D3—44no—
424C2×D3—816no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—81no—
429C3×C4—82no—
430C3×C4—81no—
431C3×C4C2×#12312yes—
432C3×C4#52511yescharacteristic
433C3×C4#52511yescharacteristic
434Dic3—42no—
435Dic3—42no—
436C23—22no—
437C23—21no—
438C23—22no—
439C23—21no—
440C23—616no—
441C23—64no—
442C23—64no—
443C23—64no—
444C23—122no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23—244no—
449C23—244no—
450C23—248no—
451C23—248no—
452C23#93712yes—
453C23C22×#6711yescharacteristic
454Q8—42no—
455Q8—41no—
456Q8—41no—
457D4—38no—
458D4—616no—
459D4—64no—
460D4—64no—
461D4—64no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—248no—
486C2×C4—248no—
487C2×C4#93712yes—
488C2×C4C2×D3×D411yescharacteristic
489C2×C4C2×D3×D411yescharacteristic
490C2×C4D3×#4112yes—
491C2×C4D3×#4112yes—
492C8—122no—
493C8—121no—
494C8—121no—
495C2×C3—42no—
496C2×C3#284911yescharacteristic
497C2×C3#300911yescharacteristic
498C2×C3#301211yescharacteristic
499C2×C3#311812yes—
500C2×C3#312312yes—
501D3—48no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—42no—
506C22—42no—
507C22—68no—
508C22—124no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22—124no—
514C22—2416no—
515C22C2×#93711yescharacteristic
516C22#582812yes—
517C22#583511yescharacteristic
518C22#583712yes—
519C22#583711yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—122no—
527C4—122no—
528C4D3×#12312yes—
529C4#565611yescharacteristic
530C4#565611yescharacteristic
531C3#3301811yescharacteristic, complemented, Sylow
532C2—42no—
533C2—128no—
534C2#8291211yescharacteristic
535C2#8340411yescharacteristic
536C2#8342411yescharacteristic
537C2#8344712yes—
538C2#8346012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2625D3
hypocenterC3#33018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
2. C3×#2625C2semidirect
4. C2×#5045C2semidirect
5. #71584C2semidirect
6. #69687C2semidirect
7. #69687C2semidirect
8. #69626C2semidirect
9. #69625C2semidirect
11. #69503C2semidirect
28. C3×#391C22semidirect
29. C3×#389C22semidirect
31. C2×#835C22semidirect
32. C2×#835C22semidirect
40. C2×#808C22semidirect
41. C2×#807C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
44. #4524C22semidirect
45. #4524C22semidirect
49. #2625D3semidirect
142. #391C2×D3semidirect
143. #389C2×D3semidirect
531. C3#33018semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
C3×#26252
C2×#50462
C2×#50452
#715842
#696872
#696262
#696252
#695212
#695032
#330183