Finite Group Databasebeta

← #11235715

Subgroups — #11235715

Order 768.

3346 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2840C211yesmaximal, characteristic, complemented
2#76881C213yesmaximal, complemented
3#76876C216yesmaximal, complemented
4#76873C213yesmaximal, complemented
5#76868C213yesmaximal, complemented
6#76863C213yesmaximal, complemented
7#76790C216yesmaximal, complemented
8#73861C212yesmaximal, complemented
9#73860C211yesmaximal, characteristic, complemented
10#73851C212yesmaximal, complemented
11D3×#379C211yesmaximal, characteristic, complemented
12#45458—31nomaximal, Sylow
13C25×D3C2211yescharacteristic
14C22×#239C2211yescharacteristic, complemented
15C2×#953C2212yes—
16C2×#947C2216yescomplemented
17C2×#946C2213yescomplemented
18C2×#882C2216yes—
19C2×#882C2216yes—
20C2×D3×C22⋊C4C2213yes—
21C2×D3×C22⋊C4C2213yes—
22C3×#393C2213yescomplemented
23C3×#393C2213yescomplemented
24C3×#393C2216yescomplemented
25C3×#379C2211yescharacteristic
26C3×#379C2212yes—
27#5247C2211yescharacteristic
28#5231C2213yescomplemented
29#5231C2213yescomplemented
30#4961C2216yescomplemented
31#4960C2213yescomplemented
32#4957C2216yescomplemented
33#4956C2216yescomplemented
34#4955C2216yes—
35#4955C2213yes—
36#4955C2213yes—
37#4955C2216yes—
38#4953C2213yescomplemented
39#4945C2216yescomplemented
40#4944C2216yescomplemented
41#4675C2216yescomplemented
42#4674C2213yescomplemented
43#4672C2216yescomplemented
44#4671C2213yescomplemented
45#4670C2216yescomplemented
46#4669C2216yescomplemented
47#4666C2213yescomplemented
48#4665C2216yescomplemented
49#4665C2216yescomplemented
50#4664C2213yescomplemented
51#2846—32no—
52#2840D311yescharacteristic, complemented
53#2812—31no—
54#2736—33no—
55#2684—36no—
56#2670—33no—
57#2561—33no—
58#2472—36no—
59#2404—33no—
60#2375—32no—
61C2×#379—31no—
62C25×C3C2311yescharacteristic
63C24×D3—24no—
64C24×D3—46no—
65C24×D3—46no—
66C24×D3C2313yes—
67C24×D3C2313yes—
68C22×C3×Q8C2311yescharacteristic, complemented
69C22×C3×Q8C2313yescomplemented
70C22×#67C2316yes—
71C2×#239—46no—
72C22×D12C2312yes—
73C22×D12C2316yes—
74C22×D12C2311yescharacteristic, complemented
75C22×C4×D3C2311yescharacteristic
76C22×C4×D3C2313yes—
77C22×C4×D3C2312yescomplemented
78C2×C3×C22⋊C4C2316yes—
79C2×C3×C22⋊C4C2316yes—
80C2×C3×C22⋊C4C2313yes—
81C2×C3×C22⋊C4C2313yes—
82#953—28no—
83C2×#217C2316yes—
84C2×#217C2313yes—
85C2×#217C2313yes—
86#947—412no—
87#946—46no—
88C2×(C2×D3)⋊C4C2316yes—
89C2×(C2×D3)⋊C4C2316yes—
90C2×(C2×D3)⋊C4C2316yes—
91C2×(C2×D3)⋊C4C2313yes—
92C2×(C2×D3)⋊C4C2316yes—
93C2×(C2×D3)⋊C4C2313yes—
94C2×(C2×D3)⋊C4C2316yes—
95C2×(C2×D3)⋊C4C2316yescomplemented
96C2×(C2×D3)⋊C4C2316yescomplemented
97C2×C4⋊Dic3C2313yescomplemented
98C2×Dic3⋊C4C2316yescomplemented
99C2×C4×Dic3C2316yescomplemented
100#882—412no—
101#882—412no—
102D3×C22⋊C4—46no—
103D3×C22⋊C4—46no—
104C3×#95C2313yescomplemented
105C3×#95C2316yescomplemented
106C3×#95C2313yescomplemented
107C3⋊#95C2316yescomplemented
108C3⋊#95C2313yescomplemented
109C3⋊#95C2316yescomplemented
110C3⋊#95C2313yescomplemented
111C26—31no—
112C22×#41—31no—
113C2×#124—36no—
114C2×C22⋊Q8—33no—
115C2×#120—32no—
116C2×#120—36no—
117C2×#120—36no—
118C22×C22⋊C4—33no—
119C22×C22⋊C4—33no—
120#397—33no—
121#397—33no—
122#397—36no—
123#394—36no—
124#394—36no—
125#393—33no—
126#393—33no—
127#393C2×D313yescomplemented
128#393C2×D313yescomplemented
129#393C2×D316yescomplemented
130#392—33no—
131#392—33no—
132#390—36no—
133#388—36no—
134#388—36no—
135#388—36no—
136#388—36no—
137#386—36no—
138#386—36no—
139#386—36no—
140#381—33no—
141#379—31no—
142#379—36no—
143#379—36no—
144#379—33no—
145#379—33no—
146#379C2×D312yes—
147#379C2×D311yescharacteristic
148C4×C22⋊C4—33no—
149C24×C3—43no—
150C24×C3—43no—
151C24×C3#4114yescomplemented
152C24×C3C2413yes—
153C23×D3—26no—
154C23×D3—26no—
155C23×D3—26no—
156C23×D3—26no—
157C23×D3—212no—
158C23×D3—212no—
159C23×D3—46no—
160C23×D3—46no—
161C23×D3—43no—
162C23×D3—43no—
163C23×D3—448no—
164C23×D3—448no—
165C23×D3—416no—
166C23×D3C2412yes—
167C23×D3C2413yes—
168C23×D3C2411yescharacteristic
169C2×C3×Q8—43no—
170C2×C3×Q8—43no—
171C2×C3×Q8—46no—
172C22×C3×C4C2411yescharacteristic
173C22×C3×C4C2412yes—
174C22×C3×C4C2416yes—
175C22×C3×C4C2413yes—
176C2×#67—212no—
177C2×#67—46no—
178C2×#67—46no—
179C22×Dic3C2411yescharacteristic
180C22×Dic3C2416yes—
181C22×Dic3C2413yes—
182#239—416no—
183C2×D12—212no—
184C2×D12—212no—
185C2×D12—43no—
186C2×D12—46no—
187C2×D12—46no—
188C2×C4×D3—26no—
189C2×C4×D3—26no—
190C2×C4×D3—46no—
191C2×C4×D3—46no—
192C3×C22⋊C4—43no—
193C3×C22⋊C4—43no—
194C3×C22⋊C4—46no—
195C3×C22⋊C4—46no—
196#217—212no—
197#217—43no—
198#217—43no—
199(C2×D3)⋊C4—212no—
200(C2×D3)⋊C4—212no—
201(C2×D3)⋊C4—212no—
202(C2×D3)⋊C4—43no—
203(C2×D3)⋊C4—43no—
204(C2×D3)⋊C4—46no—
205(C2×D3)⋊C4—46no—
206(C2×D3)⋊C4—46no—
207(C2×D3)⋊C4—46no—
208C4⋊Dic3—43no—
209Dic3⋊C4—46no—
210C4×Dic3—46no—
211C25—33no—
212C25—33no—
213C25—64no—
214C25—126no—
215C25—126no—
216C25C22×D311yescharacteristic
217C2×#41—126no—
218C22×Q8C22×D313yescomplemented
219C22×Q8C22×D311yescharacteristic, complemented
220C22×D4—36no—
221C22×D4—32no—
222C22×D4—36no—
223C22×D4—31no—
224C23×C4—31no—
225C23×C4—33no—
226C23×C4—32no—
227#124—1212no—
228C22⋊Q8—126no—
229#120—68no—
230#120—1212no—
231#120—1212no—
232C2×C4⋊C4—36no—
233C2×C4⋊C4—33no—
234C2×C22⋊C4—36no—
235C2×C22⋊C4—36no—
236C2×C22⋊C4—36no—
237C2×C22⋊C4—36no—
238C2×C22⋊C4—33no—
239C2×C22⋊C4—33no—
240C2×C22⋊C4—33no—
241C2×C22⋊C4—33no—
242C2×C22⋊C4—36no—
243C2×C22⋊C4—36no—
244C2×C22⋊C4—36no—
245C2×C22⋊C4—36no—
246C2×C22⋊C4—126no—
247C2×C22⋊C4—126no—
248C2×C22⋊C4C22×D313yes—
249C2×C22⋊C4C22×D313yes—
250C2×C22⋊C4C22×D316yes—
251C2×C22⋊C4C22×D316yes—
252C2×C42—36no—
253#95—33no—
254#95—36no—
255#95—33no—
256#95—36no—
257#95C22×D313yescomplemented
258#95C22×D313yescomplemented
259#95C22×D316yescomplemented
260C23×C3—23no—
261C23×C3—23no—
262C23×C3—26no—
263C23×C3—26no—
264C23×C3—412no—
265C23×C3—44no—
266C23×C3—412no—
267C23×C3C2×#4116yes—
268C23×C3C2511yescharacteristic
269C22×D3—21no—
270C22×D3—23no—
271C22×D3—224no—
272C22×D3—23no—
273C22×D3—23no—
274C22×D3—26no—
275C22×D3—26no—
276C22×D3—22no—
277C22×D3—26no—
278C22×D3—26no—
279C22×D3—424no—
280C22×D3—412no—
281C22×D3—424no—
282C22×D3—424no—
283C22×D3—412no—
284C22×D3—424no—
285C22×D3—412no—
286C22×D3—4128no—
287C22×D3—412no—
288C22×D3—412no—
289C22×D3H(5,𝔽2)16yes—
290C22×D3H(5,𝔽2)16yes—
291C3×Q8—44no—
292C3×Q8—86no—
293C2×C3×C4—26no—
294C2×C3×C4—26no—
295C2×C3×C4—23no—
296C2×C3×C4—26no—
297C2×C3×C4—26no—
298C2×C3×C4—23no—
299C2×C3×C4—26no—
300#67—812no—
301C2×Dic3—26no—
302C2×Dic3—26no—
303C2×Dic3—26no—
304C2×Dic3—23no—
305C2×Dic3—26no—
306C2×Dic3—23no—
307D12—44no—
308D12—48no—
309D12—812no—
310C4×D3—48no—
311C4×D3—44no—
312C4×D3—86no—
313C24—31no—
314C24—33no—
315C24—32no—
316C24—43no—
317C24—43no—
318C24—66no—
319C24—66no—
320C24—612no—
321C24—612no—
322C24—66no—
323C24—66no—
324C24—123no—
325C24—123no—
326C24—126no—
327C24—126no—
328C24—1248no—
329C24—1216no—
330C24—1248no—
331C24#23914yescomplemented
332C24C23×D313yes—
333#41—1216no—
334C2×Q8—43no—
335C2×Q8—43no—
336C2×Q8—46no—
337C2×D4—612no—
338C2×D4—612no—
339C2×D4—612no—
340C2×D4—123no—
341C2×D4—126no—
342C2×D4—126no—
343C2×D4—126no—
344C2×D4—126no—
345C22×C4—31no—
346C22×C4—36no—
347C22×C4—33no—
348C22×C4—66no—
349C22×C4—66no—
350C22×C4—126no—
351C22×C4—126no—
352C22×C4C23×D311yescharacteristic
353C22×C4C23×D312yes—
354C22×C4C23×D316yes—
355C22×C4C23×D313yes—
356C4⋊C4—126no—
357C4⋊C4—123no—
358C22⋊C4—43no—
359C22⋊C4—43no—
360C22⋊C4—46no—
361C22⋊C4—46no—
362C22⋊C4—612no—
363C22⋊C4—612no—
364C22⋊C4—612no—
365C22⋊C4—612no—
366C22⋊C4—123no—
367C22⋊C4—123no—
368C22⋊C4—126no—
369C22⋊C4—126no—
370C22⋊C4—126no—
371C22⋊C4—126no—
372C22⋊C4—123no—
373C22⋊C4—123no—
374C42—126no—
375C22×C3—212no—
376C22×C3—26no—
377C22×C3—416no—
378C22×C3—43no—
379C22×C3—46no—
380C22×C3—43no—
381C22×C3C22×#4111yescharacteristic
382C22×C3C2×H(5,𝔽2)13yes—
383C22×C3C2×H(5,𝔽2)13yes—
384C2×D3—26no—
385C2×D3—26no—
386C2×D3—26no—
387C2×D3—212no—
388C2×D3—448no—
389C2×D3—448no—
390C2×D3—46no—
391C2×D3—43no—
392C2×D3—43no—
393C2×D3—46no—
394C2×D3—42no—
395C2×D3—46no—
396C2×D3H(3,𝔽4)14yes—
397C3×C4—42no—
398C3×C4—41no—
399C3×C4—43no—
400C3×C4—46no—
401Dic3—46no—
402Dic3—41no—
403Dic3—43no—
404C23—23no—
405C23—23no—
406C23—26no—
407C23—26no—
408C23—36no—
409C23—36no—
410C23—412no—
411C23—44no—
412C23—412no—
413C23—624no—
414C23—61no—
415C23—63no—
416C23—63no—
417C23—63no—
418C23—66no—
419C23—66no—
420C23—62no—
421C23—66no—
422C23—66no—
423C23—1224no—
424C23—1224no—
425C23—1212no—
426C23—1212no—
427C23—1224no—
428C23—1212no—
429C23—1212no—
430C23—1224no—
431C23—12128no—
432C23—1212no—
433C23C2×#23916yes—
434C23C24×D311yescharacteristic
435Q8—44no—
436Q8—86no—
437D4—128no—
438D4—124no—
439D4—2412no—
440D4—2412no—
441C2×C4—26no—
442C2×C4—26no—
443C2×C4—26no—
444C2×C4—26no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—26no—
448C2×C4—66no—
449C2×C4—66no—
450C2×C4—66no—
451C2×C4—66no—
452C2×C4—63no—
453C2×C4—63no—
454C2×C4—124no—
455C2×C4—128no—
456C2×C4—246no—
457C2×C3—46no—
458C2×C3#349013yes—
459C2×C3C2×H(3,𝔽4)11yescharacteristic
460C2×C3#353213yes—
461D3—24no—
462D3—46no—
463D3—82no—
464C22—212no—
465C22—26no—
466C22—34no—
467C22—416no—
468C22—43no—
469C22—43no—
470C22—46no—
471C22—66no—
472C22—66no—
473C22—612no—
474C22—66no—
475C22—1248no—
476C22—1248no—
477C22—123no—
478C22—123no—
479C22—126no—
480C22—122no—
481C22—126no—
482C22—126no—
483C22C22×#23911yescharacteristic
484C22D3×H(5,𝔽2)13yes—
485C22D3×H(5,𝔽2)13yes—
486C4—41no—
487C4—42no—
488C4—43no—
489C4—46no—
490C4—121no—
491C4—123no—
492C4—126no—
493C3#4545811yescharacteristic, complemented, Sylow
494C2—46no—
495C2—64no—
496C2—126no—
497C2—242no—
498C2#8657913yes—
499C2D3×H(3,𝔽4)11yescharacteristic
500C2#8709013yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2840C2
hypercenter#2840D3
hypocenterC3#45458
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
1. C3×#2840C2semidirect
2. #76881C2semidirect
3. #76876C2semidirect
4. #76873C2semidirect
5. #76868C2semidirect
6. #76863C2semidirect
7. #76790C2semidirect
8. #73861C2semidirect
9. #73860C2semidirect
10. #73851C2semidirect
11. D3×#379C2semidirect
14. C22×#239C22semidirect
16. C2×#947C22semidirect
17. C2×#946C22semidirect
22. C3×#393C22semidirect
23. C3×#393C22semidirect
24. C3×#393C22semidirect
28. #5231C22semidirect
29. #5231C22semidirect
30. #4961C22semidirect
31. #4960C22semidirect
32. #4957C22semidirect
33. #4956C22semidirect
38. #4953C22semidirect
39. #4945C22semidirect
40. #4944C22semidirect
41. #4675C22semidirect
42. #4674C22semidirect
43. #4672C22semidirect
44. #4671C22semidirect
45. #4670C22semidirect
46. #4669C22semidirect
47. #4666C22semidirect
48. #4665C22semidirect
49. #4665C22semidirect
50. #4664C22semidirect
52. #2840D3semidirect
68. C22×C3×Q8C23semidirect
69. C22×C3×Q8C23semidirect
74. C22×D12C23semidirect
77. C22×C4×D3C23semidirect
95. C2×(C2×D3)⋊C4C23semidirect
96. C2×(C2×D3)⋊C4C23semidirect
97. C2×C4⋊Dic3C23semidirect
98. C2×Dic3⋊C4C23semidirect
99. C2×C4×Dic3C23semidirect
104. C3×#95C23semidirect
105. C3×#95C23semidirect
106. C3×#95C23semidirect
107. C3⋊#95C23semidirect
108. C3⋊#95C23semidirect
109. C3⋊#95C23semidirect
110. C3⋊#95C23semidirect
127. #393C2×D3semidirect
128. #393C2×D3semidirect
129. #393C2×D3semidirect
151. C24×C3#41semidirect
218. C22×Q8C22×D3semidirect
219. C22×Q8C22×D3semidirect
257. #95C22×D3semidirect
258. #95C22×D3semidirect
259. #95C22×D3semidirect
331. C24#239semidirect
493. C3#45458semidirect

Maximal subgroups

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

subgroupindex
C3×#28402
#768812
#768762
#768732
#768682
#768632
#767902
#738612
#738602
#738512
D3×#3792
#454583