Finite Group Databasebeta

← D3×#2819 #11001702

Subgroups — D3×#2819 #11001702

Order 768.

3342 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#124C212yesmaximal, complemented
2C3×#2819C211yesmaximal, characteristic, complemented
3#83007C212yesmaximal, complemented
4D3×#394C214yesmaximal, complemented
5#77087C214yesmaximal, complemented
6#76885C214yesmaximal, complemented
7D3×#393C214yesmaximal, complemented
8D3×#392C214yesmaximal, complemented
9#76745C214yesmaximal, complemented
10#74254C211yesmaximal, characteristic, complemented
11D3×#382C211yesmaximal, characteristic, complemented
12C2×#2819—31nomaximal, Sylow
13C22×D3×Q8C2212yes—
14C22×D3×D4C2212yes—
15C22×D3×D4C2214yes—
16C2×C3×#124C2212yescomplemented
17C2×#947C2212yescomplemented
18C2×#936C2212yescomplemented
19D3×#124—216no—
20C2×D3×C4⋊C4C2212yes—
21C2×#885C2214yescomplemented
22C2×#885C2214yescomplemented
23C2×D3×C22⋊C4C2214yes—
24C2×D3×C22⋊C4C2214yes—
25C2×D3×C22⋊C4C2218yes—
26C2×#875C2212yescomplemented
27C2×C42×D3C2211yescharacteristic
28C3×#394C2214yescomplemented
29C3×#393C2214yescomplemented
30C3×#392C2214yescomplemented
31C3×#382C2211yescharacteristic, complemented
32#5231C2214yescomplemented
33#5223C2218yescomplemented
34#5223C2214yescomplemented
35#5222C2214yescomplemented
36#4991C2214yescomplemented
37#4977C2212yescomplemented
38#4961C2218yescomplemented
39#4960C2218yescomplemented
40#4960C2214yescomplemented
41#4959C2218yescomplemented
42#4947C2218yescomplemented
43#4947C2214yescomplemented
44#4936C2211yescharacteristic, complemented
45#4673C2214yescomplemented
46#4673C2218yescomplemented
47#4672C2214yescomplemented
48#4671C2214yescomplemented
49D3×#95C2214yes—
50D3×#95C2214yes—
51#4650C2214yescomplemented
52C22×#124—32no—
53#2819—34no—
54#2819—34no—
55#2819—34no—
56#2819—31no—
57#2819—32no—
58#2819D311yescharacteristic, direct product factor
59C2×#394—34no—
60C2×#393—34no—
61C2×#392—34no—
62C2×#382—31no—
63C24×D3C2314yes—
64C22×C3×Q8C2312yes—
65C22×C3×D4C2312yes—
66C22×C3×D4C2314yes—
67C22×#67C2314yes—
68C22×#67C2318yes—
69C2×D3×Q8—28no—
70C2×D3×Q8—48no—
71C2×D3×D4—28no—
72C2×D3×D4—48no—
73C2×D3×D4—416no—
74C2×D3×D4—48no—
75C22×D12C2312yes—
76C22×D12C2314yes—
77C22×C4×D3C2314yes—
78C22×C4×D3C2314yes—
79C22×C4×D3C2312yes—
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2312yes—
82C22×Dic6C2314yes—
83C3×#124—28no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4C2314yes—
86C2×C3×C22⋊C4C2314yes—
87C2×C3×C22⋊C4C2318yes—
88C2×C3×C42C2311yescharacteristic
89C2×#217C2314yes—
90C2×#217C2314yes—
91C2×#217C2318yes—
92#947—28no—
93#936—28no—
94C2×(C2×D3)⋊C4C2314yes—
95C2×(C2×D3)⋊C4C2314yes—
96C2×(C2×D3)⋊C4C2314yes—
97C2×(C2×D3)⋊C4C2314yes—
98C2×(C2×D3)⋊C4C2318yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4C2314yes—
102C2×C4×Dic3C2312yes—
103C2×C4×Dic3C2311yescharacteristic
104D3×C4⋊C4—44no—
105#885—216no—
106#885—216no—
107D3×C22⋊C4—216no—
108D3×C22⋊C4—216no—
109D3×C22⋊C4—416no—
110#875—28no—
111C42×D3D418yescomplemented
112C3×#95C2314yes—
113C3×#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C3⋊#95C2314yes—
117C3⋊#95C2318yes—
118C23×Q8—32no—
119C23×D4—32no—
120C23×D4—34no—
121C2×#124—32no—
122C2×#124—32no—
123C2×#124—34no—
124C2×#124—34no—
125C2×#124—32no—
126C2×#124—616no—
127C2×#124C2×D312yescomplemented
128C22×C4⋊C4—32no—
129C22×C22⋊C4—34no—
130C22×C22⋊C4—34no—
131C22×C22⋊C4—38no—
132C22×C42—31no—
133#394—34no—
134#394—34no—
135#394—38no—
136#394—38no—
137#394—34no—
138#394C2×D314yescomplemented
139#393—34no—
140#393—38no—
141#393—34no—
142#393—38no—
143#393—34no—
144#393C2×D314yescomplemented
145#392—38no—
146#392—34no—
147#392—38no—
148#392—34no—
149#392—34no—
150#392C2×D314yescomplemented
151#382—32no—
152#382—31no—
153#382—34no—
154#382C2×D311yescharacteristic, complemented
155C2×#95—34no—
156C2×#95—34no—
157C24×C3C2414yes—
158C23×D3—28no—
159C23×D3—216no—
160C23×D3—216no—
161C23×D3—28no—
162C23×D3—28no—
163C23×D3C2414yes—
164C23×D3C2411yescharacteristic
165C2×C3×Q8—24no—
166C2×C3×Q8—44no—
167C2×C3×D4—24no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—48no—
171C22×C3×C4C2414yes—
172C22×C3×C4C2414yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—48no—
177C2×#67—48no—
178C2×#67—48no—
179C2×#67—48no—
180C22×Dic3C2414yes—
181C22×Dic3C2414yes—
182C22×Dic3C2412yes—
183C22×Dic3C2411yescharacteristic
184D3×Q8—432no—
185D3×D4—432no—
186D3×D4—464no—
187C2×D12—24no—
188C2×D12—44no—
189C2×D12—44no—
190C2×D12—48no—
191C2×C4×D3—22no—
192C2×C4×D3—28no—
193C2×C4×D3—22no—
194C2×C4×D3—28no—
195C2×C4×D3—216no—
196C2×C4×D3—216no—
197C2×C4×D3—28no—
198C2×C4×D3C2×D414yes—
199C2×C4×D3C2×D418yes—
200C2×Dic6—24no—
201C2×Dic6—28no—
202C2×Dic6—48no—
203C2×Dic6—44no—
204C3×C4⋊C4—42no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—48no—
208C3×C42C2×D414yescomplemented
209#217—28no—
210#217—28no—
211#217—48no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—48no—
217(C2×D3)⋊C4—48no—
218C4⋊Dic3—42no—
219Dic3⋊C4—44no—
220C4×Dic3C2×D414yescomplemented
221C4×Dic3C2×D418yescomplemented
222C25—34no—
223C22×Q8—34no—
224C22×Q8—32no—
225C22×Q8—68no—
226C22×Q8—128no—
227C22×Q8C22×D312yes—
228C22×D4—34no—
229C22×D4—38no—
230C22×D4—32no—
231C22×D4—34no—
232C22×D4—68no—
233C22×D4—128no—
234C22×D4—1216no—
235C22×D4—128no—
236C22×D4C22×D312yes—
237C22×D4C22×D314yes—
238C23×C4—31no—
239C23×C4—32no—
240C23×C4—34no—
241C23×C4—34no—
242#124—28no—
243#124—616no—
244#124—616no—
245#124—68no—
246#124—68no—
247#124—68no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—124no—
251C2×C4⋊C4C22×D312yes—
252C2×C22⋊C4—34no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—34no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—34no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—38no—
259C2×C22⋊C4—38no—
260C2×C22⋊C4—38no—
261C2×C22⋊C4—616no—
262C2×C22⋊C4—616no—
263C2×C22⋊C4—1216no—
264C2×C22⋊C4C22×D314yes—
265C2×C22⋊C4C22×D314yes—
266C2×C22⋊C4C22×D318yes—
267C2×C42—31no—
268C2×C42—32no—
269C2×C42—38no—
270C2×C42C22×D311yescharacteristic
271#95—38no—
272#95—34no—
273#95—38no—
274#95—34no—
275#95C22×D314yes—
276#95C22×D314yes—
277C23×C3—24no—
278C23×C3—28no—
279C23×C3—28no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3C2511yescharacteristic
283C22×D3—24no—
284C22×D3—28no—
285C22×D3—28no—
286C22×D3—24no—
287C22×D3—24no—
288C22×D3—416no—
289C22×D3—416no—
290C22×D3—416no—
291C22×D3—432no—
292C22×D3—432no—
293C22×D3C22×D412yes—
294C22×D3H(5,𝔽2)18yes—
295C22×D3H(5,𝔽2)12yes—
296C22×D32−1+412yes—
297C3×Q8—48no—
298C3×D4—48no—
299C3×D4—416no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4C22×D412yes—
308C2×C3×C4C22×D414yes—
309#67—416no—
310#67—432no—
311C2×Dic3—21no—
312C2×Dic3—24no—
313C2×Dic3—21no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—28no—
317C2×Dic3—28no—
318C2×Dic3C22×D412yes—
319C2×Dic3C22×D414yes—
320D12—48no—
321D12—416no—
322C4×D3—216no—
323C4×D3—28no—
324C4×D3—416no—
325C4×D3—416no—
326Dic6—416no—
327Dic6—48no—
328C24—31no—
329C24—34no—
330C24—68no—
331C24—616no—
332C24—68no—
333C24—616no—
334C24—68no—
335C24C23×D314yes—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—68no—
339C2×Q8—64no—
340C2×Q8—124no—
341C2×Q8—128no—
342C2×Q8—1232no—
343C2×D4—24no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—44no—
347C2×D4—68no—
348C2×D4—64no—
349C2×D4—124no—
350C2×D4—128no—
351C2×D4—124no—
352C2×D4—1232no—
353C2×D4—128no—
354C2×D4—128no—
355C2×D4—128no—
356C2×D4—128no—
357C2×D4—1264no—
358C22×C4—32no—
359C22×C4—31no—
360C22×C4—34no—
361C22×C4—38no—
362C22×C4—34no—
363C22×C4—34no—
364C22×C4—62no—
365C22×C4—68no—
366C22×C4—62no—
367C22×C4—68no—
368C22×C4—616no—
369C22×C4—616no—
370C22×C4—68no—
371C22×C4C23×D314yes—
372C22×C4C23×D314yes—
373C22×C4C23×D312yes—
374C22×C4C23×D311yescharacteristic
375C4⋊C4—42no—
376C4⋊C4—124no—
377C4⋊C4—122no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—68no—
382C22⋊C4—68no—
383C22⋊C4—68no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C22⋊C4—128no—
388C22⋊C4—128no—
389C22⋊C4—128no—
390C42—38no—
391C42—34no—
392C42D3×D414yescomplemented
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—48no—
397C22×C3—48no—
398C22×C3C23×D411yescharacteristic
399C22×C3C2×H(5,𝔽2)14yes—
400C22×C3C2×H(5,𝔽2)11yescharacteristic
401C22×C3C2×2−1+411yescharacteristic
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—48no—
406C2×D3—48no—
407C2×D3—832no—
408C2×D3#53518yes—
409C2×D3#53714yes—
410C2×D3H(3,𝔽4)18yes—
411C2×D3#56218yes—
412C3×C4—24no—
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—44no—
416Dic3—24no—
417Dic3—22no—
418Dic3—44no—
419Dic3—44no—
420C23—24no—
421C23—28no—
422C23—24no—
423C23—28no—
424C23—24no—
425C23—32no—
426C23—32no—
427C23—38no—
428C23—32no—
429C23—64no—
430C23—68no—
431C23—64no—
432C23—68no—
433C23—64no—
434C23—1232no—
435C23—1216no—
436C23—1216no—
437C23—1232no—
438C23—1216no—
439C23C24×D311yescharacteristic
440Q8—48no—
441Q8—128no—
442Q8—1216no—
443D4—48no—
444D4—416no—
445D4—128no—
446D4—1216no—
447D4—1216no—
448D4—1232no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—32no—
457C2×C4—34no—
458C2×C4—616no—
459C2×C4—68no—
460C2×C4—61no—
461C2×C4—64no—
462C2×C4—61no—
463C2×C4—64no—
464C2×C4—68no—
465C2×C4—68no—
466C2×C4—64no—
467C2×C4—1216no—
468C2×C4—1216no—
469C2×C4C2×D3×D412yes—
470C2×C4C2×D3×D414yes—
471C2×C3—84no—
472C2×C3C2×#53512yes—
473C2×C3C2×#53711yescharacteristic
474C2×C3C2×H(3,𝔽4)12yes—
475C2×C3C2×#56212yes—
476D3—84no—
477D3#281918yesdirect product factor
478C22—38no—
479C22—38no—
480C22—38no—
481C22—34no—
482C22—48no—
483C22—44no—
484C22—44no—
485C22—48no—
486C22—44no—
487C22—128no—
488C22—124no—
489C22—124no—
490C22—128no—
491C22—124no—
492C22—2432no—
493C22C22×D3×D411yescharacteristic
494C22D3×H(5,𝔽2)11yescharacteristic
495C22D3×H(5,𝔽2)14yes—
496C22D3×2−1+411yescharacteristic
497C4—24no—
498C4—22no—
499C4—44no—
500C4—44no—
501C4—64no—
502C4—62no—
503C4—124no—
504C4—124no—
505C3C2×#281911yescharacteristic, complemented, Sylow
506C2—38no—
507C2—84no—
508C2—244no—
509C2D3×#53512yes—
510C2D3×#53711yescharacteristic
511C2D3×H(3,𝔽4)12yes—
512C2D3×#56212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2819C2
hypercenter#2819D3
hypocenterC3C2×#2819
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. C2×D3×#124C2semidirect
2. C3×#2819C2semidirect
3. #83007C2semidirect
4. D3×#394C2semidirect
5. #77087C2semidirect
6. #76885C2semidirect
7. D3×#393C2semidirect
8. D3×#392C2semidirect
9. #76745C2semidirect
10. #74254C2semidirect
11. D3×#382C2semidirect
16. C2×C3×#124C22semidirect
17. C2×#947C22semidirect
18. C2×#936C22semidirect
21. C2×#885C22semidirect
22. C2×#885C22semidirect
26. C2×#875C22semidirect
28. C3×#394C22semidirect
29. C3×#393C22semidirect
30. C3×#392C22semidirect
31. C3×#382C22semidirect
32. #5231C22semidirect
33. #5223C22semidirect
34. #5223C22semidirect
35. #5222C22semidirect
36. #4991C22semidirect
37. #4977C22semidirect
38. #4961C22semidirect
39. #4960C22semidirect
40. #4960C22semidirect
41. #4959C22semidirect
42. #4947C22semidirect
43. #4947C22semidirect
44. #4936C22semidirect
45. #4673C22semidirect
46. #4673C22semidirect
47. #4672C22semidirect
48. #4671C22semidirect
51. #4650C22semidirect
58. #2819D3direct (N × H)
111. C42×D3D4semidirect
127. C2×#124C2×D3semidirect
138. #394C2×D3semidirect
144. #393C2×D3semidirect
150. #392C2×D3semidirect
154. #382C2×D3semidirect
208. C3×C42C2×D4semidirect
220. C4×Dic3C2×D4semidirect
221. C4×Dic3C2×D4semidirect
392. C42D3×D4semidirect
477. D3#2819direct (N × H)
505. C3C2×#2819semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1242
C3×#28192
#830072
D3×#3942
#770872
#768852
D3×#3932
D3×#3922
#767452
#742542
D3×#3822
C2×#28193