Finite Group Databasebeta

← #11633948

Subgroups — #11633948

Order 768.

3322 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5717C211yesmaximal, characteristic, complemented
2C2×#5716C211yesmaximal, characteristic, complemented
3C3×#2366C211yesmaximal, characteristic, complemented
4#82416C214yesmaximal, complemented
5D4×(C2×D3)⋊C4C214yesmaximal, complemented
6#76440C214yesmaximal, complemented
7#76436C214yesmaximal, complemented
8#74918C214yesmaximal, complemented
9#74885C214yesmaximal, complemented
10#73807C212yesmaximal, complemented
11C4×#895C212yesmaximal, complemented
12#59041—31nomaximal, Sylow
13C22×#237C2212yescomplemented
14C22×D3×D4C2214yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×D4C2214yescomplemented
17C2×#938C2214yes—
18C2×#938C2214yescomplemented
19C2×#937C2214yes—
20C2×#936C2212yescomplemented
21C2×D4×Dic3C2214yescomplemented
22C22×(C2×D3)⋊C4C2214yescomplemented
23#5717—28no—
24#5716—28no—
25C2×#895C2212yes—
26C2×#882C2214yescomplemented
27C2×C4×D12C2212yes—
28C2×C4×D12C2214yescomplemented
29C2×C42×D3C2211yescharacteristic, complemented
30C2×#869C2211yescharacteristic, complemented
31C3×#390C2214yescomplemented
32C3×#386C2214yescomplemented
33C3×C4×C4⋊C4C2212yescomplemented
34#5213C2214yescomplemented
35#5213C2218yescomplemented
36#4988C2212yes—
37#4987C2212yescomplemented
38#4987C2214yescomplemented
39#4987C2214yescomplemented
40#4984C2212yescomplemented
41#4961C2218yescomplemented
42#4960C2218yescomplemented
43#4958C2218yescomplemented
44#4955C2218yescomplemented
45#4953C2218yescomplemented
46#4943C2214yescomplemented
47#4939C2214yes—
48C4×(C2×D3)⋊C4C2214yescomplemented
49#4936C2214yescomplemented
50#4673C2218yescomplemented
51#4670C2214yescomplemented
52#4665C2214yescomplemented
53C2×D42—31no—
54C2×#532—31no—
55#2684—34no—
56#2563—32no—
57#2535—34no—
58#2420—34no—
59#2408—34no—
60#2404—34no—
61#2366D311yescharacteristic, complemented
62D4×C22⋊C4—34no—
63C4×#121—32no—
64C24×D3C2314yes—
65C22×C3×D4C2314yes—
66C22×C3×D4C2312yes—
67C23×C3×C4C2314yes—
68C22×#67C2318yes—
69C22×#67C2314yes—
70C23×Dic3C2314yes—
71C2×#237—28no—
72C2×#237—44no—
73C2×#237—44no—
74C2×D3×D4—216no—
75C2×D3×D4—216no—
76C2×D3×D4—48no—
77C22×D12C2314yes—
78C22×D12C2312yes—
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2314yes—
81C22×C4×D3C2312yes—
82C22×Dic6C2312yes—
83C3×#127D418yescomplemented
84C3×C4×D4—48no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4C2314yes—
88C2×C3×C22⋊C4C2318yes—
89C2×C3×C42C2314yes—
90C2×C3×C42C2311yescharacteristic
91C2×#217C2314yes—
92C2×#217C2318yes—
93#938—216no—
94#938—216no—
95#937—216no—
96#936—28no—
97D4×Dic3—216no—
98C2×(C2×D3)⋊C4—48no—
99C2×(C2×D3)⋊C4C2314yes—
100C2×(C2×D3)⋊C4C2314yes—
101C2×(C2×D3)⋊C4C2314yes—
102C2×(C2×D3)⋊C4C2314yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2314yes—
108C2×C4⋊Dic3C2314yes—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2312yes—
111#895—28no—
112#882—48no—
113C4×D12—28no—
114C4×D12—48no—
115C42×D3—24no—
116#869—24no—
117C3×#95C2314yes—
118C3⋊#95C2314yes—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C22×#41—32no—
123C23×D4—34no—
124D42—68no—
125#532—68no—
126C2×C4⋊Q8—31no—
127C2×#127C2×D311yescharacteristic
128C2×#124—32no—
129C2×#121—34no—
130C2×#121—34no—
131C2×#121—32no—
132C2×#120—34no—
133C2×#120—34no—
134C2×C4×D4—34no—
135C2×C4×D4—32no—
136C2×C4×D4—34no—
137C2×C4×D4C2×D314yescomplemented
138C22×C22⋊C4—34no—
139C22×C42—31no—
140#397—38no—
141#394—38no—
142#393—38no—
143#392—38no—
144#390—32no—
145#390—34no—
146#390C2×D314yescomplemented
147#389—32no—
148#388—34no—
149#386—38no—
150#386—34no—
151#386—38no—
152#386—34no—
153#386C2×D314yescomplemented
154#382—34no—
155#381—34no—
156#380—34no—
157#380—34no—
158#380—32no—
159#379—38no—
160C4×C4⋊C4C2×D312yescomplemented
161C4×C22⋊C4—34no—
162C24×C3C2414yes—
163C23×D3—28no—
164C23×D3—216no—
165C23×D3—44no—
166C23×D3—48no—
167C23×D3—44no—
168C23×D3C2414yes—
169C23×D3C2411yescharacteristic
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4C2×D4116yes—
175C2×C3×D4C2×D418yes—
176C22×C3×C4—44no—
177C22×C3×C4—48no—
178C22×C3×C4C2414yes—
179C22×C3×C4C2414yes—
180C22×C3×C4C2411yescharacteristic
181C22×C3×C4C2412yes—
182C2×#67—216no—
183C2×#67—28no—
184C2×#67—216no—
185C2×#67—28no—
186C2×#67—48no—
187C2×#67—44no—
188C22×Dic3—28no—
189C22×Dic3—28no—
190C22×Dic3—44no—
191C22×Dic3C2414yes—
192C22×Dic3C2414yes—
193C22×Dic3C2412yes—
194C22×Dic3C2411yescharacteristic
195#237—432no—
196D3×D4—464no—
197C2×D12—24no—
198C2×D12—24no—
199C2×D12—28no—
200C2×D12—28no—
201C2×D12—44no—
202C2×D12#4118yescomplemented
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—24no—
207C2×C4×D3—28no—
208C2×C4×D3—24no—
209C2×C4×D3—28no—
210C2×C4×D3—24no—
211C2×C4×D3—28no—
212C2×Dic6—24no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—48no—
219C3×C42—28no—
220C3×C42C2×D414yes—
221#217—216no—
222#217—28no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—28no—
226(C2×D3)⋊C4—44no—
227(C2×D3)⋊C4—44no—
228(C2×D3)⋊C4—48no—
229(C2×D3)⋊C4—48no—
230C4⋊Dic3—28no—
231C4⋊Dic3—24no—
232C4⋊Dic3—44no—
233C4×Dic3—24no—
234C4×Dic3—22no—
235C25—34no—
236C2×#41—68no—
237C2×#41—124no—
238C2×#41—124no—
239C22×Q8—32no—
240C22×D4—34no—
241C22×D4—38no—
242C22×D4—32no—
243C22×D4—34no—
244C22×D4—616no—
245C22×D4—616no—
246C22×D4—128no—
247C22×D4C22×D314yes—
248C22×D4C22×D312yes—
249C23×C4—34no—
250C23×C4—32no—
251C23×C4—34no—
252C23×C4—31no—
253C23×C4C22×D314yes—
254C4⋊Q8—64no—
255#127D1218yescomplemented
256#124—68no—
257#121—68no—
258#121—616no—
259#121—616no—
260#120—616no—
261#120—128no—
262C4×D4—48no—
263C4×D4—616no—
264C4×D4—68no—
265C4×D4—128no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—38no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—34no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—128no—
281C2×C22⋊C4C22×D318yes—
282C2×C22⋊C4C22×D314yes—
283C2×C42—32no—
284C2×C42—31no—
285C2×C42—64no—
286C2×C42C22×D314yes—
287C2×C42C22×D311yescharacteristic
288#95—34no—
289#95—34no—
290#95—38no—
291#95—38no—
292#95C22×D314yes—
293C23×C3—28no—
294C23×C3—24no—
295C23×C3—28no—
296C23×C3—24no—
297C23×C3C22×D418yes—
298C23×C3C2511yescharacteristic
299C22×D3—24no—
300C22×D3—21no—
301C22×D3—22no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—22no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—21no—
308C22×D3—24no—
309C22×D3—416no—
310C22×D3—432no—
311C22×D3—816no—
312C22×D3—88no—
313C22×D3—88no—
314C22×D3C2×#4118yes—
315C22×D3H(5,𝔽2)12yes—
316C3×D4—232no—
317C3×D4—48no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—88no—
327C2×C3×C4C22×D412yes—
328C2×C3×C4C22×D414yes—
329C2×C3×C4C2×#4114yes—
330C2×C3×C4H(5,𝔽2)14yes—
331#67—432no—
332#67—416no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—416no—
345C2×Dic3H(5,𝔽2)12yes—
346D12—216no—
347D12—416no—
348C4×D3—28no—
349C4×D3—48no—
350C4×D3—416no—
351Dic6—48no—
352C24—34no—
353C24—31no—
354C24—616no—
355C24—68no—
356C24—124no—
357C24—128no—
358C24—124no—
359C24C23×D314yes—
360#41—1232no—
361C2×Q8—64no—
362C2×Q8—122no—
363C2×Q8—122no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—38no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—68no—
370C2×D4—68no—
371C2×D4—68no—
372C2×D4—616no—
373C2×D4—616no—
374C2×D4—68no—
375C2×D4—64no—
376C2×D4—64no—
377C2×D4—124no—
378C2×D4—128no—
379C2×D4—124no—
380C2×D4—1264no—
381C2×D4C2×D12116yes—
382C2×D4C2×D1218yes—
383C22×C4—31no—
384C22×C4—34no—
385C22×C4—34no—
386C22×C4—32no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—68no—
392C22×C4—62no—
393C22×C4—68no—
394C22×C4—64no—
395C22×C4—68no—
396C22×C4—64no—
397C22×C4—68no—
398C22×C4—64no—
399C22×C4—68no—
400C22×C4—124no—
401C22×C4C23×D314yes—
402C22×C4C23×D314yes—
403C22×C4C23×D311yescharacteristic
404C22×C4C23×D312yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—64no—
408C4⋊C4—68no—
409C4⋊C4—124no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—616no—
415C22⋊C4—68no—
416C22⋊C4—68no—
417C22⋊C4—128no—
418C22⋊C4—124no—
419C22⋊C4—124no—
420C22⋊C4—128no—
421C42—28no—
422C42—64no—
423C42—62no—
424C42C2×D1214yes—
425C22×C3—216no—
426C22×C3—28no—
427C22×C3—44no—
428C22×C3—48no—
429C22×C3—44no—
430C22×C3C23×D411yescharacteristic
431C22×C3C22×#4112yes—
432C22×C3C2×H(5,𝔽2)12yes—
433C22×C3C2×H(5,𝔽2)11yescharacteristic
434C22×C3C2×H(5,𝔽2)11yescharacteristic
435C2×D3—28no—
436C2×D3—28no—
437C2×D3—22no—
438C2×D3—24no—
439C2×D3—28no—
440C2×D3—42no—
441C2×D3—44no—
442C2×D3—41no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—41no—
446C2×D3—44no—
447C2×D3—832no—
448C3×C4—22no—
449C3×C4—44no—
450C3×C4—44no—
451C3×C4#54618yes—
452Dic3—22no—
453Dic3—42no—
454Dic3—44no—
455Dic3—44no—
456C23—28no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—38no—
461C23—32no—
462C23—64no—
463C23—61no—
464C23—62no—
465C23—64no—
466C23—64no—
467C23—62no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—61no—
472C23—1232no—
473C23—1216no—
474C23—248no—
475C23—2416no—
476C23—248no—
477C23C22×D1218yes—
478C23C24×D311yescharacteristic
479Q8—128no—
480D4—232no—
481D4—48no—
482D4—616no—
483D4—1216no—
484D4—1232no—
485D4—1216no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—32no—
495C2×C4—68no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—88no—
508C2×C4—1216no—
509C2×C4—1216no—
510C2×C4—128no—
511C2×C4C22×D1212yes—
512C2×C4C22×D1214yes—
513C2×C4#100414yes—
514C2×C4D3×#4114yes—
515C2×C3—48no—
516C2×C3C2×#54612yes—
517C2×C3#348111yescharacteristic
518C2×C3#348411yescharacteristic
519C2×C3#351912yes—
520C2×C3#354911yescharacteristic
521D3—42no—
522D3—48no—
523C22—216no—
524C22—28no—
525C22—44no—
526C22—44no—
527C22—48no—
528C22—68no—
529C22—68no—
530C22—62no—
531C22—64no—
532C22—68no—
533C22—121no—
534C22—124no—
535C22—122no—
536C22—124no—
537C22—121no—
538C22—124no—
539C22—124no—
540C22—2432no—
541C22C23×D1211yescharacteristic
542C22C2×#100412yes—
543C22C2×D3×#4112yes—
544C22D3×H(5,𝔽2)11yescharacteristic
545C22D3×H(5,𝔽2)11yescharacteristic
546C4—22no—
547C4—44no—
548C4—44no—
549C4—62no—
550C4—124no—
551C4—122no—
552C4—124no—
553C4#555418yes—
554C3#5904111yescharacteristic, complemented, Sylow
555C2—48no—
556C2—128no—
557C2—122no—
558C2C2×#555412yes—
559C2#8586711yescharacteristic
560C2#8661711yescharacteristic
561C2#8677512yes—
562C2#8705011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2366C2
hypercenter#2366D3
hypocenterC3#59041
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. C2×#5717C2semidirect
2. C2×#5716C2semidirect
3. C3×#2366C2semidirect
4. #82416C2semidirect
5. D4×(C2×D3)⋊C4C2semidirect
6. #76440C2semidirect
7. #76436C2semidirect
8. #74918C2semidirect
9. #74885C2semidirect
10. #73807C2semidirect
11. C4×#895C2semidirect
13. C22×#237C22semidirect
16. C2×C3×C4×D4C22semidirect
18. C2×#938C22semidirect
20. C2×#936C22semidirect
21. C2×D4×Dic3C22semidirect
22. C22×(C2×D3)⋊C4C22semidirect
26. C2×#882C22semidirect
28. C2×C4×D12C22semidirect
29. C2×C42×D3C22semidirect
30. C2×#869C22semidirect
31. C3×#390C22semidirect
32. C3×#386C22semidirect
33. C3×C4×C4⋊C4C22semidirect
34. #5213C22semidirect
35. #5213C22semidirect
37. #4987C22semidirect
38. #4987C22semidirect
39. #4987C22semidirect
40. #4984C22semidirect
41. #4961C22semidirect
42. #4960C22semidirect
43. #4958C22semidirect
44. #4955C22semidirect
45. #4953C22semidirect
46. #4943C22semidirect
48. C4×(C2×D3)⋊C4C22semidirect
49. #4936C22semidirect
50. #4673C22semidirect
51. #4670C22semidirect
52. #4665C22semidirect
61. #2366D3semidirect
83. C3×#127D4semidirect
137. C2×C4×D4C2×D3semidirect
146. #390C2×D3semidirect
153. #386C2×D3semidirect
160. C4×C4⋊C4C2×D3semidirect
202. C2×D12#41semidirect
255. #127D12semidirect
554. C3#59041semidirect

Maximal subgroups

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

subgroupindex
C2×#57172
C2×#57162
C3×#23662
#824162
D4×(C2×D3)⋊C42
#764402
#764362
#749182
#748852
#738072
C4×#8952
#590413