Finite Group Databasebeta

← #10972977

Subgroups — #10972977

Order 768.

2762 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C2×Dic3×#41C211yesmaximal, characteristic, complemented
3C3×#2311C211yesmaximal, characteristic, complemented
4#82269C211yesmaximal, characteristic, complemented
5#82227C211yesmaximal, characteristic, complemented
6#82156C211yesmaximal, characteristic, complemented
7C2×#4988C211yesmaximal, characteristic, complemented
8#76507C214yesmaximal, complemented
9#76317C214yesmaximal, complemented
10#74891C218yesmaximal, complemented
11#74033C214yesmaximal, complemented
12#73999C214yesmaximal, complemented
13#35254—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18#5830—28no—
19Dic3×#41—28no—
20C2×#947C2212yes—
21C2×Q8×Dic3C2212yescomplemented
22C2×#940C2212yes—
23C2×#937C2214yes—
24C2×D4×Dic3C2214yes—
25C2×D4×Dic3C2212yescomplemented
26C2×#930C2214yes—
27C22×(C2×D3)⋊C4C2214yes—
28C2×#926C2211yescharacteristic
29C2×#926C2211yescharacteristic
30C2×#926C2212yescomplemented
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2212yescomplemented
33C2×#886C2218yescomplemented
34C3×#391C2212yescomplemented
35C3×#390C2212yescomplemented
36C3×#386C2214yes—
37C3×#380C2214yescomplemented
38#5212C2214yes—
39#4988C2212yes—
40#4988C2214yes—
41#4988C2214yes—
42#4988C2212yes—
43#4984C2212yescomplemented
44#4984C2214yescomplemented
45#4984C2214yescomplemented
46#4984C2212yescomplemented
47#4961C2218yescomplemented
48#4961C2218yescomplemented
49#4958C2218yescomplemented
50#4955C2214yes—
51#4950C2214yes—
52#4939C2214yes—
53C4×C4⋊Dic3C2214yescomplemented
54#4677C2218yescomplemented
55#4674C2218yescomplemented
56#4670C2218yescomplemented
57#4670C2218yescomplemented
58C2×#535—31no—
59C2×C4×#41—31no—
60#2854—34no—
61#2624—34no—
62#2448—34no—
63#2445—34no—
64#2418—38no—
65#2316—31no—
66#2311D311yescharacteristic, complemented
67#2309—31no—
68#2296—31no—
69C2×#390—31no—
70C24×D3C2312yes—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41D418yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4C2312yes—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2312yes—
78C23×C3×C4C2312yes—
79C23×C3×C4C2311yescharacteristic
80C22×#67C2318yes—
81C23×Dic3C2312yes—
82C23×Dic3C2314yes—
83C22×D12—24no—
84C22×D12—24no—
85C22×D12—24no—
86C22×D12C2312yes—
87C22×D12C2312yes—
88C3×#117—24no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C22⋊C4C2312yes—
94C2×C3×C22⋊C4C2318yes—
95C2×C3×C42C2312yes—
96C2×#217C2312yes—
97C2×#217C2312yes—
98C2×#217C2314yes—
99#947—28no—
100Q8×Dic3—28no—
101#940—28no—
102#937—216no—
103D4×Dic3—216no—
104D4×Dic3—28no—
105#930—216no—
106C2×(C2×D3)⋊C4—48no—
107C2×(C2×D3)⋊C4C2318yes—
108C2×(C2×D3)⋊C4C2318yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×(C2×D3)⋊C4C2318yes—
112#926—24no—
113#926—24no—
114#926—28no—
115C2×C4⋊Dic3C2314yes—
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2318yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4×Dic3—24no—
122C2×C4×Dic3—28no—
123C2×C4×Dic3C2314yes—
124C2×C4×Dic3C2312yes—
125C2×C4×Dic3C2312yes—
126C2×C4×Dic3C2311yescharacteristic
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2312yes—
129#886—416no—
130C3×#95C2318yes—
131C3⋊#95C2314yes—
132C3⋊#95C2318yes—
133C3⋊#95C2314yes—
134C22×#41C2×D311yescharacteristic
135C23×D4—31no—
136#535—68no—
137C2×#127—32no—
138C2×#124—32no—
139C2×#123—38no—
140C2×#121—34no—
141C2×#120—34no—
142C4×#41—68no—
143C2×C4×Q8—32no—
144C2×C4×D4—34no—
145C2×C4×D4—32no—
146C2×#117—31no—
147C2×#117—31no—
148C2×#117—32no—
149C2×#117C2×D311yescharacteristic
150C22×C4⋊C4C2×D311yescharacteristic
151C22×C22⋊C4—34no—
152C22×C42—32no—
153C22×C42—31no—
154#399—38no—
155#397—38no—
156#394—38no—
157#394—38no—
158#391C2×D312yescomplemented
159#390—34no—
160#390—32no—
161#390—32no—
162#390—34no—
163#390—34no—
164#390C2×D312yescomplemented
165#389—34no—
166#389—34no—
167#389—32no—
168#389—32no—
169#388—38no—
170#388—38no—
171#386—38no—
172#386C2×D314yes—
173#385—34no—
174#380—34no—
175#380C2×D314yescomplemented
176#379—34no—
177C4×C4⋊C4—34no—
178C24×C3C2412yes—
179C24×C3C2411yescharacteristic
180C23×D3—24no—
181C23×D3—24no—
182C23×D3—24no—
183C23×D3—44no—
184C23×D3—44no—
185C23×D3C2414yes—
186C3×#41—232no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8C2×D414yes—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4C2×D418yes—
195C2×C3×D4C2×D414yes—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—44no—
200C22×C3×C4C2×D418yes—
201C22×C3×C4C2×D414yes—
202C22×C3×C4#4114yes—
203C22×C3×C4#4114yes—
204C22×C3×C4C2414yes—
205C22×C3×C4C2414yes—
206C22×C3×C4C2412yes—
207C22×C3×C4C2411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—216no—
210C2×#67—48no—
211C2×#67—48no—
212C22×Dic3—28no—
213C22×Dic3—24no—
214C22×Dic3—28no—
215C22×Dic3—24no—
216C22×Dic3—42no—
217C22×Dic3—44no—
218C22×Dic3C2414yes—
219C22×Dic3C2414yes—
220C22×Dic3C2412yes—
221C22×Dic3C2412yes—
222C2×D12—24no—
223C2×D12—24no—
224C2×D12—24no—
225C2×D12—24no—
226C2×D12—24no—
227C2×D12—24no—
228C2×D12—416no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—48no—
234C3×C42—24no—
235#217—28no—
236#217—24no—
237#217—24no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—216no—
240(C2×D3)⋊C4—48no—
241(C2×D3)⋊C4—48no—
242(C2×D3)⋊C4—48no—
243C4⋊Dic3—24no—
244C4⋊Dic3—28no—
245C4⋊Dic3—24no—
246C4⋊Dic3—28no—
247C4⋊Dic3—48no—
248C4⋊Dic3—44no—
249C4×Dic3—22no—
250C4×Dic3—22no—
251C4×Dic3—24no—
252C4×Dic3—28no—
253C4×Dic3—24no—
254C4×Dic3—24no—
255C25—32no—
256C2×#41—24no—
257C2×#41—24no—
258C2×#41D1218yes—
259C22×Q8C22×D311yescharacteristic
260C22×D4—38no—
261C22×D4—32no—
262C22×D4—32no—
263C22×D4—64no—
264C22×D4—64no—
265C22×D4—64no—
266C22×D4C22×D312yes—
267C22×D4C22×D311yescharacteristic
268C23×C4—32no—
269C23×C4—34no—
270C23×C4C22×D312yes—
271C23×C4C22×D312yes—
272C23×C4C22×D311yescharacteristic
273#127—68no—
274#124—68no—
275#123—1216no—
276#121—616no—
277#120—616no—
278C4×Q8—68no—
279C4×D4—616no—
280C4×D4—68no—
281#117—24no—
282#117—68no—
283#117—64no—
284#117—64no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—34no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—34no—
290C2×C4⋊C4—38no—
291C2×C4⋊C4—34no—
292C2×C4⋊C4C22×D312yes—
293C2×C4⋊C4C22×D312yes—
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—32no—
296C2×C22⋊C4—34no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—38no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—38no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—128no—
304C2×C22⋊C4C22×D312yes—
305C2×C22⋊C4C22×D318yes—
306C2×C42—32no—
307C2×C42—31no—
308C2×C42—32no—
309C2×C42—32no—
310C2×C42—34no—
311C2×C42—31no—
312C2×C42—68no—
313C2×C42—64no—
314C2×C42C22×D312yes—
315#95—34no—
316#95—34no—
317#95—38no—
318#95C22×D318yes—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3C22×D414yes—
326C23×C3C22×D412yes—
327C23×C3C2×#4112yes—
328C23×C3C2×#4112yes—
329C23×C3C2511yescharacteristic
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—28no—
334C22×D3—28no—
335C22×D3—416no—
336C22×D3—48no—
337C22×D3—88no—
338C22×D3—84no—
339C22×D3—84no—
340C3×Q8—28no—
341C3×D4—216no—
342C3×D4—28no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—216no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—28no—
350C2×C3×C4—24no—
351C2×C3×C4—84no—
352C2×C3×C4#123116yescomplemented
353C2×C3×C4C22×D414yes—
354C2×C3×C4C22×D412yes—
355C2×C3×C4C22×D412yes—
356C2×C3×C4C2×#4112yes—
357C2×C3×C4C2×#4112yes—
358C2×C3×C4C2×#4112yes—
359C2×C3×C4C2×#4112yes—
360#67—816no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—48no—
373C2×Dic3—416no—
374C2×Dic3H(5,𝔽2)14yes—
375D12—48no—
376D12—48no—
377C24—34no—
378C24—64no—
379C24—64no—
380C24—64no—
381C24—124no—
382C24—124no—
383C24C23×D312yes—
384C24C23×D311yescharacteristic
385#41—232no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C2×D1214yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—616no—
394C2×D4—64no—
395C2×D4—64no—
396C2×D4—64no—
397C2×D4—64no—
398C2×D4—64no—
399C2×D4—64no—
400C2×D4—128no—
401C2×D4—128no—
402C2×D4—1216no—
403C2×D4C2×D1218yes—
404C2×D4C2×D1214yes—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—32no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—34no—
412C22×C4—44no—
413C22×C4—64no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—64no—
417C22×C4—124no—
418C22×C4—122no—
419C22×C4C2×D1218yes—
420C22×C4C2×D1214yes—
421C22×C4#23714yes—
422C22×C4#23714yes—
423C22×C4C23×D314yes—
424C22×C4C23×D314yes—
425C22×C4C23×D312yes—
426C22×C4C23×D311yescharacteristic
427C22×C4C23×D311yescharacteristic
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—68no—
432C4⋊C4—68no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—128no—
436C4⋊C4—124no—
437C22⋊C4—24no—
438C22⋊C4—48no—
439C22⋊C4—616no—
440C22⋊C4—68no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C22⋊C4—616no—
444C22⋊C4—128no—
445C22⋊C4—128no—
446C22⋊C4—128no—
447C42—24no—
448C42—62no—
449C42—64no—
450C42—62no—
451C42—68no—
452C42—64no—
453C42—64no—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—24no—
459C22×C3—28no—
460C22×C3—44no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3C2×#12314yes—
464C22×C3C23×D411yescharacteristic
465C22×C3C22×#4111yescharacteristic
466C22×C3C22×#4111yescharacteristic
467C22×C3C2×H(5,𝔽2)12yes—
468C22×C3#58512yes—
469C2×D3—44no—
470C2×D3—48no—
471C2×D3—48no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—816no—
475C3×C4—24no—
476C3×C4—44no—
477C3×C4—44no—
478C3×C4C2×#12314yes—
479C3×C4C2×#12314yes—
480Dic3—24no—
481Dic3—44no—
482Dic3—42no—
483Dic3—44no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—64no—
491C23—64no—
492C23—64no—
493C23—68no—
494C23—68no—
495C23—1216no—
496C23—128no—
497C23—244no—
498C23—244no—
499C23—248no—
500C23C22×D1214yes—
501C23C22×D1212yes—
502C23C2×#23712yes—
503C23C2×#23712yes—
504C23C24×D311yescharacteristic
505Q8—28no—
506D4—216no—
507D4—28no—
508D4—128no—
509D4—128no—
510D4—2416no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—216no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—34no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—84no—
532C2×C4—128no—
533C2×C4—1216no—
534C2×C4#886116yescomplemented
535C2×C4C22×D1214yes—
536C2×C4C22×D1212yes—
537C2×C4C22×D1212yes—
538C2×C4C2×#23712yes—
539C2×C4C2×#23712yes—
540C2×C4C2×#23712yes—
541C2×C4C2×#23712yes—
542C2×C3—24no—
543C2×C3—44no—
544C2×C3C22×#12311yescharacteristic
545C2×C3#346012yes—
546C2×C3#348812yes—
547C2×C3#350811yescharacteristic
548C2×C3#351011yescharacteristic
549D3—84no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—28no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—124no—
560C22—124no—
561C22—128no—
562C22—128no—
563C22—124no—
564C22—2416no—
565C22C2×#88614yes—
566C22C23×D1211yescharacteristic
567C22C22×#23711yescharacteristic
568C22C22×#23711yescharacteristic
569C22D3×H(5,𝔽2)12yes—
570C22#596812yes—
571C4—24no—
572C4—44no—
573C4—44no—
574C4—64no—
575C4—122no—
576C4—124no—
577C4—124no—
578C4C2×#88614yes—
579C4C2×#88614yes—
580C3#3525411yescharacteristic, complemented, Sylow
581C2—24no—
582C2—44no—
583C2—244no—
584C2C22×#88611yescharacteristic
585C2#8590012yes—
586C2#8631911yescharacteristic
587C2#8634411yescharacteristic
588C2#8636412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2311C2
prime radicalC2×#5830C2
hypercenter#2311D3
hypocenterC3#35254
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×#41C2semidirect
3. C3×#2311C2semidirect
4. #82269C2semidirect
5. #82227C2semidirect
6. #82156C2semidirect
7. C2×#4988C2semidirect
8. #76507C2semidirect
9. #76317C2semidirect
10. #74891C2semidirect
11. #74033C2semidirect
12. #73999C2semidirect
21. C2×Q8×Dic3C22semidirect
25. C2×D4×Dic3C22semidirect
30. C2×#926C22semidirect
32. C22×C4×Dic3C22semidirect
33. C2×#886C22semidirect
34. C3×#391C22semidirect
35. C3×#390C22semidirect
37. C3×#380C22semidirect
43. #4984C22semidirect
44. #4984C22semidirect
45. #4984C22semidirect
46. #4984C22semidirect
47. #4961C22semidirect
48. #4961C22semidirect
49. #4958C22semidirect
53. C4×C4⋊Dic3C22semidirect
54. #4677C22semidirect
55. #4674C22semidirect
56. #4670C22semidirect
57. #4670C22semidirect
66. #2311D3semidirect
158. #391C2×D3semidirect
164. #390C2×D3semidirect
175. #380C2×D3semidirect
352. C2×C3×C4#123semidirect
534. C2×C4#886semidirect
580. C3#35254semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C2×Dic3×#412
C3×#23112
#822692
#822272
#821562
C2×#49882
#765072
#763172
#748912
#740332
#739992
#352543