Finite Group Databasebeta

← #10972975

Subgroups — #10972975

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×#5805C211yesmaximal, characteristic
2C2×#5522C211yesmaximal, characteristic, complemented
3C3×#2309C211yesmaximal, characteristic, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5#82270C211yesmaximal, characteristic, complemented
6#82232C211yesmaximal, characteristic, complemented
7#82227C211yesmaximal, characteristic, complemented
8#76475C214yesmaximal, complemented
9#76458C214yesmaximal, complemented
10#76321C214yesmaximal, complemented
11#76318C214yesmaximal, complemented
12#73826C218yesmaximal, complemented
13#35254—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×#953C2214yes—
19C22×#217C2214yes—
20#5805—28no—
21C2×#940C2212yes—
22C2×#938C2214yes—
23C2×#936C2212yes—
24C2×#926C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27#5522—28no—
28C2×#893C2212yescomplemented
29C2×#887C2212yescomplemented
30C2×#883C2218yescomplemented
31C2×#67⋊C4C2214yes—
32C2×#872C2212yescomplemented
33C2×C42×D3C2212yescomplemented
34C3×#390C2214yes—
35C3×#390C2214yes—
36C3×#379C2214yes—
37#5221C2214yes—
38#5221C2212yes—
39#5221C2212yes—
40#4987C2214yescomplemented
41#4984C2212yescomplemented
42#4984C2212yescomplemented
43#4972C2214yescomplemented
44#4961C2218yescomplemented
45#4959C2218yescomplemented
46#4958C2214yes—
47#4956C2218yescomplemented
48#4952C2218yescomplemented
49#4951C2218yescomplemented
50#4950C2214yes—
51#4949C2218yescomplemented
52#4945C2218yescomplemented
53#4942C2214yes—
54#4939C2212yescomplemented
55C4×C4⋊Dic3C2214yescomplemented
56#4929C2212yescomplemented
57#4928C2214yescomplemented
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#2311—31no—
67#2309D311yescharacteristic, complemented
68#2296—31no—
69C2×#390—31no—
70C25×C3C2312yes—
71C22×C3×D4—24no—
72C22×C3×D4—24no—
73C22×C3×D4—24no—
74C22×C3×D4C2312yes—
75C22×C3×D4C2312yes—
76C23×C3×C4C2311yescharacteristic
77C22×#67C2312yes—
78C22×#67C2318yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C2×#235—24no—
82C2×#235—24no—
83C2×#235D418yes—
84C22×D12C2311yescharacteristic
85C22×C4×D3C2314yes—
86C22×C4×D3C2312yes—
87C22×Dic6C2311yescharacteristic
88C3×#117—24no—
89C3×#117—24no—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4C2312yes—
93C2×C3×C22⋊C4C2312yes—
94C2×C3×C22⋊C4C2318yes—
95C2×C3×C22⋊C4C2318yes—
96C2×C3×C42C2312yes—
97C2×C3×C42C2312yes—
98#953—216no—
99C2×#217—48no—
100C2×#217C2312yes—
101C2×#217C2318yes—
102C2×#217C2318yes—
103C2×#217C2318yes—
104#940—28no—
105#938—216no—
106#936—28no—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2318yes—
109#926—24no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2312yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2314yes—
116C2×Dic3⋊C4C2314yes—
117C2×Dic3⋊C4C2318yes—
118C2×C4×Dic3—24no—
119C2×C4×Dic3C2312yes—
120C2×C4×Dic3C2312yes—
121C2×C4×Dic3C2314yes—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124#893—28no—
125#887—28no—
126#883—416no—
127#67⋊C4—216no—
128#872—28no—
129C42×D3—28no—
130C3⋊#95C2318yes—
131C3⋊#95C2318yes—
132C3⋊#95C2314yes—
133C3⋊#95C2314yes—
134C22×#41—31no—
135C23×D4C2×D311yescharacteristic
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—32no—
145C2×C4×D4—34no—
146C2×#117—31no—
147C2×#117—32no—
148C2×#117C2×D311yescharacteristic
149C2×#117C2×D311yescharacteristic
150C22×C4⋊C4—31no—
151C22×C22⋊C4—34no—
152C22×C42—31no—
153C22×C42—32no—
154#399—38no—
155#397—38no—
156#394—38no—
157#394—38no—
158#391—32no—
159#390—32no—
160#390—32no—
161#390—34no—
162#390—32no—
163#390C2×D314yes—
164#390C2×D314yes—
165#389—34no—
166#389—32no—
167#389—32no—
168#389—34no—
169#388—38no—
170#388—38no—
171#386—34no—
172#386—38no—
173#385—34no—
174#380—34no—
175#380—34no—
176#379C2×D314yes—
177C4×C4⋊C4—34no—
178C24×C3—24no—
179C24×C3—24no—
180C24×C3—24no—
181C24×C3—44no—
182C24×C3—44no—
183C24×C3C2414yes—
184C24×C3C2411yescharacteristic
185C23×D3C2412yes—
186C2×C3×D4—24no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—416no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4#4114yes—
195C22×C3×C4#4114yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—24no—
202C2×#67—216no—
203C2×#67—48no—
204C2×#67—48no—
205C2×#67C2×D418yes—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—42no—
210C22×Dic3—44no—
211C22×Dic3C2414yes—
212C22×Dic3C2414yes—
213C22×Dic3C2412yes—
214C22×Dic3C2412yes—
215C22×Dic3C2412yes—
216#235—232no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12C2×D414yes—
220C2×C4×D3—24no—
221C2×C4×D3—24no—
222C2×C4×D3—28no—
223C2×C4×D3—28no—
224C2×C4×D3—44no—
225C2×C4×D3C2×D418yes—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6C2×D414yes—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—48no—
234C3×C22⋊C4—48no—
235C3×C42—24no—
236C3×C42—24no—
237#217—24no—
238#217—216no—
239#217—216no—
240#217—48no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—48no—
243C4⋊Dic3—24no—
244C4⋊Dic3—24no—
245C4⋊Dic3—24no—
246C4⋊Dic3—28no—
247C4⋊Dic3—44no—
248Dic3⋊C4—28no—
249Dic3⋊C4—48no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252C4×Dic3—24no—
253C4×Dic3—24no—
254C4×Dic3—28no—
255C25C22×D312yes—
256C2×#41—38no—
257C2×#41—64no—
258C2×#41—64no—
259C22×Q8—31no—
260C22×D4—24no—
261C22×D4—24no—
262C22×D4—24no—
263C22×D4—32no—
264C22×D4—38no—
265C22×D4—31no—
266C22×D4C22×D312yes—
267C22×D4C22×D312yes—
268C23×C4—32no—
269C23×C4—32no—
270C23×C4—32no—
271C23×C4—34no—
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—24no—
283#117—68no—
284#117—64no—
285C2×C4⋊C4—32no—
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⋊C4—64no—
293C2×C4⋊C4C22×D312yes—
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—38no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—38no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—128no—
302C2×C22⋊C4C22×D312yes—
303C2×C22⋊C4C22×D312yes—
304C2×C22⋊C4C22×D318yes—
305C2×C22⋊C4C22×D318yes—
306C2×C42—31no—
307C2×C42—32no—
308C2×C42—32no—
309C2×C42—34no—
310C2×C42—31no—
311C2×C42—68no—
312C2×C42—64no—
313C2×C42C22×D312yes—
314C2×C42C22×D312yes—
315#95—34no—
316#95—34no—
317#95—38no—
318#95—38no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—28no—
325C23×C3—28no—
326C23×C3—416no—
327C23×C3—48no—
328C23×C3—88no—
329C23×C3—84no—
330C23×C3—84no—
331C23×C3C22×D412yes—
332C23×C3C2×#4112yes—
333C23×C3C2×#4112yes—
334C23×C3C2511yescharacteristic
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3C22×D414yes—
340C3×D4—48no—
341C3×D4—48no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4#123116yescomplemented
349C2×C3×C4C22×D412yes—
350C2×C3×C4C22×D412yes—
351C2×C3×C4C2×#4112yes—
352C2×C3×C4C2×#4112yes—
353C2×C3×C4C2×#4112yes—
354C2×C3×C4C2×#4112yes—
355#67—216no—
356#67—816no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—28no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—48no—
370C2×Dic3—84no—
371C2×Dic3C22×D414yes—
372C2×Dic3H(5,𝔽2)14yes—
373D12—28no—
374C4×D3—216no—
375C4×D3—416no—
376Dic6—28no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—32no—
381C24—44no—
382C24—44no—
383C24C23×D314yes—
384C24C23×D311yescharacteristic
385#41—632no—
386C2×Q8—34no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—38no—
396C2×D4—34no—
397C2×D4—416no—
398C2×D4—64no—
399C2×D4—64no—
400C2×D4—616no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—128no—
404C2×D4—128no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—38no—
408C22×C4—34no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—68no—
412C22×C4—68no—
413C22×C4—64no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—64no—
417C22×C4—64no—
418C22×C4—124no—
419C22×C4—122no—
420C22×C4—124no—
421C22×C4#23514yes—
422C22×C4D3×D414yes—
423C22×C4#23714yes—
424C22×C4C23×D314yes—
425C22×C4C23×D314yes—
426C22×C4C23×D311yescharacteristic
427C22×C4C23×D311yescharacteristic
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—68no—
431C4⋊C4—68no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—128no—
436C4⋊C4—124no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—68no—
442C22⋊C4—616no—
443C22⋊C4—616no—
444C22⋊C4—64no—
445C22⋊C4—128no—
446C22⋊C4—128no—
447C42—24no—
448C42—24no—
449C42—68no—
450C42—62no—
451C42—62no—
452C42—64no—
453C42—64no—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—44no—
459C22×C3—48no—
460C22×C3—48no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—816no—
464C22×C3C2×#12314yes—
465C22×C3C23×D411yescharacteristic
466C22×C3C22×#4111yescharacteristic
467C22×C3C22×#4111yescharacteristic
468C22×C3C2×H(5,𝔽2)12yes—
469C22×C3#58512yes—
470C2×D3—24no—
471C2×D3—28no—
472C2×D3—44no—
473C2×D3—42no—
474C2×D3—42no—
475C3×C4—44no—
476C3×C4—44no—
477C3×C4C2×#12314yes—
478C3×C4C2×#12314yes—
479Dic3—24no—
480Dic3—24no—
481Dic3—42no—
482Dic3—44no—
483Dic3—44no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—34no—
492C23—416no—
493C23—48no—
494C23—62no—
495C23—62no—
496C23—64no—
497C23—64no—
498C23—88no—
499C23—84no—
500C23—84no—
501C23C2×#23512yes—
502C23C2×D3×D412yes—
503C23C2×#23712yes—
504C23C24×D311yescharacteristic
505Q8—68no—
506D4—48no—
507D4—48no—
508D4—616no—
509D4—68no—
510D4—2416no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—34no—
518C2×C4—34no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—616no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—68no—
528C2×C4—62no—
529C2×C4—64no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—1216no—
533C2×C4—128no—
534C2×C4—244no—
535C2×C4#883116yescomplemented
536C2×C4C2×#23512yes—
537C2×C4C2×#23512yes—
538C2×C4C2×D3×D412yes—
539C2×C4C2×D3×D412yes—
540C2×C4C2×#23712yes—
541C2×C4C2×#23712yes—
542C2×C3—24no—
543C2×C3—84no—
544C2×C3C22×#12311yescharacteristic
545C2×C3#346012yes—
546C2×C3#348812yes—
547C2×C3#350811yescharacteristic
548C2×C3#351011yescharacteristic
549D3—44no—
550C22—24no—
551C22—24no—
552C22—22no—
553C22—22no—
554C22—48no—
555C22—48no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—68no—
560C22—64no—
561C22—816no—
562C22—124no—
563C22—122no—
564C22—122no—
565C22C2×#88314yes—
566C22C22×#23511yescharacteristic
567C22C22×D3×D411yescharacteristic
568C22C22×#23711yescharacteristic
569C22D3×H(5,𝔽2)12yes—
570C22#596612yes—
571C4—44no—
572C4—44no—
573C4—64no—
574C4—64no—
575C4—122no—
576C4—124no—
577C4—124no—
578C4C2×#88314yes—
579C4C2×#88314yes—
580C3#3525411yescharacteristic, complemented, Sylow
581C2—24no—
582C2—84no—
583C2—124no—
584C2C22×#88311yescharacteristic
585C2#8586211yescharacteristic
586C2#8586311yescharacteristic
587C2#8631412yes—
588C2#8633112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2309C2
prime radicalC2×#5805C2
hypercenter#2309D3
hypocenterC3#35254
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C2×#5522C2semidirect
3. C3×#2309C2semidirect
4. C2×#5221C2semidirect
5. #82270C2semidirect
6. #82232C2semidirect
7. #82227C2semidirect
8. #76475C2semidirect
9. #76458C2semidirect
10. #76321C2semidirect
11. #76318C2semidirect
12. #73826C2semidirect
28. C2×#893C22semidirect
29. C2×#887C22semidirect
30. C2×#883C22semidirect
32. C2×#872C22semidirect
33. C2×C42×D3C22semidirect
40. #4987C22semidirect
41. #4984C22semidirect
42. #4984C22semidirect
43. #4972C22semidirect
44. #4961C22semidirect
45. #4959C22semidirect
47. #4956C22semidirect
48. #4952C22semidirect
49. #4951C22semidirect
51. #4949C22semidirect
52. #4945C22semidirect
54. #4939C22semidirect
55. C4×C4⋊Dic3C22semidirect
56. #4929C22semidirect
57. #4928C22semidirect
67. #2309D3semidirect
348. C2×C3×C4#123semidirect
535. C2×C4#883semidirect
580. C3#35254semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×#55222
C3×#23092
C2×#52212
#822702
#822322
#822272
#764752
#764582
#763212
#763182
#738262
#352543