Finite Group Databasebeta

← #11724384

Subgroups — #11724384

Order 864.

828 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#89041C211yesmaximal, characteristic, complemented
2C3⋊D3×D12C211yesmaximal, characteristic, complemented
3D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4#89022C211yesmaximal, characteristic, complemented
5C3×D3×D12C211yesmaximal, characteristic, complemented
6C3×#3798C211yesmaximal, characteristic, complemented
7C3×#3795C211yesmaximal, characteristic, complemented
8#88962C212yesmaximal, complemented
9#88954C212yesmaximal, complemented
10#88952C212yesmaximal, complemented
11D3×#626C212yesmaximal, complemented
12#64314—31nomaximal
13#64308—31nomaximal
14#64307—31nomaximal
15C2×D3×C3⋊D3C2212yescomplemented
16C2×D3×C3⋊D3C2212yescomplemented
17C2×C3×D32C2212yescomplemented
18C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
19C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C3×C4×C3⋊D3C2211yescharacteristic, complemented
21C3×#634C2211yescharacteristic, complemented
22C32×D12C2211yescharacteristic, complemented
23C32×C4×D3C2211yescharacteristic, complemented
24C32×C4×D3C2211yescharacteristic, complemented
25#6295C2212yescomplemented
26#6294C2212yescomplemented
27#6292C2212yescomplemented
28#6291C2212yescomplemented
29#6291C2212yescomplemented
30#6289C2212yescomplemented
31C3×#627C2211yescharacteristic, complemented
32C3×#626C2211yescharacteristic, complemented
33C3×#626C2211yescharacteristic, complemented
34C3×#626C2212yescomplemented
35C3×#625C2211yescharacteristic, complemented
36C3×#625C2212yescomplemented
37C3×D3×Dic3C2212yescomplemented
38D4×C3⋊D3—31no—
39#3828—31no—
40C2×C4⋊(C3⋊D3)—31no—
41C3×#237—31no—
42C3×D3×D4—31no—
43C3×#235—31no—
44C3×#235—31no—
45C3×#235—31no—
46C2×C3×D12—31no—
47#3811—32no—
48#3811—32no—
49#3811—32no—
50D3×#67—32no—
51D3×#67—32no—
52D3×#67—32no—
53C2×#626—32no—
54#3804—32no—
55#3802—32no—
56D3×D12—32no—
57D3×D12—31no—
58D3×D12—31no—
59D3×D12—31no—
60D3×D12—31no—
61D3×D12—61no—
62D3×D12—61no—
63D3×D12D311yescharacteristic, complemented
64#3799—31no—
65#3799—31no—
66#3799—31no—
67#3799—32no—
68#3799—61no—
69#3798D311yescharacteristic, complemented
70#3796—31no—
71#3796—32no—
72#3795—31no—
73#3795D311yescharacteristic, complemented
74C2×C3⋊(C3⋊D3)C2312yes—
75C2×C3×C3⋊D3C2311yescharacteristic
76C2×C3×C3⋊D3C2312yes—
77C2×C32×D3C2312yes—
78C2×C32×D3C2311yescharacteristic
79C2×C32×D3C2311yescharacteristic
80D3×C3⋊D3—42no—
81D3×C3⋊D3—42no—
82C3×D32—42no—
83C33×C4C2311yescharacteristic
84C3×C3⋊Dic3C2312yes—
85C3×C3⋊Dic3C2311yescharacteristic
86C32×Dic3C2311yescharacteristic
87C32×Dic3C2311yescharacteristic
88#1009—91no—
89#1009—91no—
90#1004—91no—
91C22×C3⋊D3—32no—
92C22×C3⋊D3—32no—
93C22×C3×D3—32no—
94C22×C3×D3—32no—
95C2×D32—32no—
96C2×D32—32no—
97C2×D32—32no—
98C2×D32—32no—
99C2×D32—62no—
100C2×D32—62no—
101C2×D32C2×D312yescomplemented
102C32×D4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—31no—
105#638—32no—
106#638—32no—
107C4⋊(C3⋊D3)—32no—
108C4⋊(C3⋊D3)—31no—
109C4⋊(C3⋊D3)—31no—
110C4⋊(C3⋊D3)—31no—
111C4⋊(C3⋊D3)—61no—
112C4⋊(C3⋊D3)—61no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—121no—
115C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
116C4×C3⋊D3—32no—
117C4×C3⋊D3C2×D311yescharacteristic, complemented
118#634C2×D311yescharacteristic, complemented
119C3×#67—31no—
120C3×#67—32no—
121C3×#67—31no—
122C3×#67—32no—
123C3×#67—32no—
124C3×#67—31no—
125C3×#67—31no—
126C2×C3×Dic3—32no—
127C3×D12—21no—
128C3×D12—32no—
129C3×D12—31no—
130C3×D12—31no—
131C3×D12—31no—
132C3×D12—31no—
133C3×D12—61no—
134C3×D12C2×D311yescharacteristic, complemented
135C3×D12C2×D311yescharacteristic, complemented
136C3×C4×D3—21no—
137C3×C4×D3—21no—
138C3×C4×D3—32no—
139C3×C4×D3—31no—
140C3×C4×D3—31no—
141C3×C4×D3—61no—
142C3×C4×D3C2×D311yescharacteristic, complemented
143C3×C4×D3C2×D311yescharacteristic, complemented
144C3×C4×D3C2×D311yescharacteristic, complemented
145C3×C4×D3C2×D311yescharacteristic, complemented
146C3×Dic6—31no—
147C3×Dic6—31no—
148C3×Dic6—31no—
149C3×Dic6—31no—
150C3×Dic6—61no—
151#627C2×D311yescharacteristic, complemented
152#626—32no—
153#626—32no—
154#626—32no—
155#626—32no—
156#626—32no—
157#626—32no—
158#626—32no—
159#626—32no—
160#626—62no—
161#626—62no—
162#626—62no—
163#626C2×D311yescharacteristic, complemented
164#626C2×D311yescharacteristic, complemented
165#626C2×D312yescomplemented
166#625—32no—
167#625C2×D312yescomplemented
168#625C2×D311yescharacteristic, complemented
169#624—32no—
170#624—32no—
171#624—32no—
172#624—62no—
173D3×Dic3—32no—
174D3×Dic3—32no—
175D3×Dic3C2×D312yescomplemented
176C2×C33C2411yescharacteristic
177C3⋊(C3⋊D3)—22no—
178C3×C3⋊D3—21no—
179C3×C3⋊D3—22no—
180C32×D3—21no—
181C32×D3—21no—
182C32×D3—22no—
183C3×#41—91no—
184C3×#41—91no—
185C2×C3×D4—91no—
186C2×#67—92no—
187C2×#67—92no—
188#239—92no—
189#239—91no—
190#239—91no—
191#237—31no—
192#237—91no—
193#237—92no—
194D3×D4—31no—
195D3×D4—91no—
196D3×D4—91no—
197D3×D4—92no—
198D3×D4—92no—
199D3×D4—92no—
200D3×D4—92no—
201D3×D4—92no—
202D3×D4—92no—
203D3×D4—91no—
204D3×D4—181no—
205#235—31no—
206#235—31no—
207#235—31no—
208#235—91no—
209#235—92no—
210#235—91no—
211#235—91no—
212#235—181no—
213C2×D12—31no—
214C2×D12—91no—
215C2×D12—91no—
216C2×D12—92no—
217C2×D12—91no—
218C2×D12—181no—
219C22×C32—32no—
220C22×C32—31no—
221C22×C32—31no—
222C2×C3⋊D3—32no—
223C2×C3⋊D3—32no—
224C2×C3⋊D3—32no—
225C2×C3⋊D3—62no—
226C2×C3⋊D3—62no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—122no—
229C2×C3⋊D3—122no—
230C2×C3⋊D3—122no—
231C2×C3⋊D3C22×D312yes—
232C2×C3⋊D3C22×D311yescharacteristic
233C2×C3×D3—21no—
234C2×C3×D3—21no—
235C2×C3×D3—22no—
236C2×C3×D3—32no—
237C2×C3×D3—32no—
238C2×C3×D3—31no—
239C2×C3×D3—31no—
240C2×C3×D3—61no—
241C2×C3×D3—62no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C2×C3×D3C22×D312yes—
245C2×C3×D3C22×D311yescharacteristic
246C2×C3×D3C22×D311yescharacteristic
247C2×C3×D3C22×D311yescharacteristic
248C2×C3×D3C22×D311yescharacteristic
249C2×C3×D3C22×D312yes—
250D32—42no—
251D32—122no—
252D32—122no—
253D32—122no—
254D32—122no—
255D32—124no—
256D32—124no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—31no—
261C32×C4—31no—
262C32×C4—41no—
263C32×C4C22×D311yescharacteristic
264C32×C4C22×D311yescharacteristic
265C32×C4C22×D311yescharacteristic
266C3⋊Dic3C22×D311yescharacteristic
267C3⋊Dic3C22×D312yes—
268C3×Dic3—21no—
269C3×Dic3—21no—
270C3×Dic3—31no—
271C3×Dic3—32no—
272C3×Dic3—32no—
273C3×Dic3—31no—
274C3×Dic3—62no—
275C3×Dic3—61no—
276C3×Dic3C22×D311yescharacteristic
277C3×Dic3C22×D311yescharacteristic
278C3×Dic3C22×D311yescharacteristic
279C3×Dic3C22×D311yescharacteristic
280H(5,𝔽2)—271noSylow
281C33H(5,𝔽2)11yescharacteristic, complemented, Sylow
282C23×C3—92no—
283C22×D3—32no—
284C22×D3—32no—
285C22×D3—92no—
286C22×D3—92no—
287C22×D3—92no—
288C22×D3—92no—
289C22×D3—92no—
290C22×D3—92no—
291C22×D3—182no—
292C22×D3—182no—
293C3×Q8—91no—
294C3×Q8—91no—
295C3×D4—31no—
296C3×D4—31no—
297C3×D4—61no—
298C3×D4—91no—
299C3×D4—91no—
300C3×D4—92no—
301C3×D4—92no—
302C3×D4—91no—
303C3×D4—91no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312#67—32no—
313#67—31no—
314#67—31no—
315#67—32no—
316#67—31no—
317#67—31no—
318#67—32no—
319#67—92no—
320#67—92no—
321#67—92no—
322#67—92no—
323#67—92no—
324#67—92no—
325#67—92no—
326#67—182no—
327#67—182no—
328C2×Dic3—32no—
329C2×Dic3—92no—
330D12—32no—
331D12—31no—
332D12—31no—
333D12—31no—
334D12—31no—
335D12—61no—
336D12—92no—
337D12—92no—
338D12—92no—
339D12—92no—
340D12—92no—
341D12—91no—
342D12—91no—
343D12—91no—
344D12—181no—
345D12—181no—
346D12—181no—
347D12—182no—
348D12—361no—
349D12D3211yescharacteristic, complemented
350C4×D3—32no—
351C4×D3—31no—
352C4×D3—31no—
353C4×D3—61no—
354C4×D3—92no—
355C4×D3—92no—
356C4×D3—92no—
357C4×D3—92no—
358C4×D3—92no—
359C4×D3—182no—
360C4×D3D3211yescharacteristic, complemented
361C4×D3D3211yescharacteristic, complemented
362Dic6—31no—
363Dic6—31no—
364Dic6—31no—
365Dic6—31no—
366Dic6—61no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—61no—
372C2×C32—61no—
373C2×C32—62no—
374C2×C32C23×D311yescharacteristic
375C2×C32C23×D311yescharacteristic
376C2×C32C23×D311yescharacteristic
377C3⋊D3—21no—
378C3⋊D3—22no—
379C3⋊D3—62no—
380C3⋊D3—62no—
381C3⋊D3—62no—
382C3⋊D3—122no—
383C3⋊D3—122no—
384C3⋊D3—122no—
385C3⋊D3—242no—
386C3×D3—22no—
387C3×D3—21no—
388C3×D3—22no—
389C3×D3—21no—
390C3×D3—21no—
391C3×D3—22no—
392C3×D3—21no—
393C3×D3—42no—
394C3×D3—41no—
395C3×D3—62no—
396C3×D3—61no—
397C3×D3—61no—
398C3×D3—62no—
399C3×D3—62no—
400C3×D3—122no—
401#41—91no—
402#41—91no—
403#41—271no—
404#41—272no—
405#41—271no—
406C2×D4—91no—
407C2×D4—271no—
408C2×D4—271no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—272no—
412C22×C3—32no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—32no—
417C22×C3—31no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—62no—
421C22×C3—92no—
422C22×C3—91no—
423C22×C3—362no—
424C2×D3—31no—
425C2×D3—31no—
426C2×D3—32no—
427C2×D3—32no—
428C2×D3—61no—
429C2×D3—62no—
430C2×D3—92no—
431C2×D3—92no—
432C2×D3—92no—
433C2×D3—122no—
434C2×D3—122no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—362no—
439C2×D3—362no—
440C2×D3—362no—
441C2×D3—362no—
442C2×D3—362no—
443C2×D3—362no—
444C2×D3—362no—
445C2×D3—364no—
446C2×D3—364no—
447C2×D3C2×D3212yes—
448C2×D3C2×D3211yescharacteristic
449C2×D3C2×D3211yescharacteristic
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—31no—
454C3×C4—31no—
455C3×C4—31no—
456C3×C4—31no—
457C3×C4—41no—
458C3×C4—61no—
459C3×C4—61no—
460C3×C4—92no—
461C3×C4—91no—
462C3×C4C2×D3211yescharacteristic
463C3×C4C2×D3211yescharacteristic
464C3×C4C2×D3211yescharacteristic
465Dic3—31no—
466Dic3—32no—
467Dic3—31no—
468Dic3—32no—
469Dic3—62no—
470Dic3—61no—
471Dic3C2×D3211yescharacteristic
472Dic3C2×D3211yescharacteristic
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#100411yescharacteristic, complemented
478C32#100911yescharacteristic, complemented
479C32#100911yescharacteristic, complemented
480C23—92no—
481C23—272no—
482C23—272no—
483Q8—91no—
484Q8—91no—
485D4—31no—
486D4—92no—
487D4—91no—
488D4—91no—
489D4—91no—
490D4—92no—
491D4—91no—
492D4—272no—
493D4—272no—
494D4—272no—
495D4—272no—
496D4—271no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—92no—
500C2×C4—91no—
501C2×C4—272no—
502C2×C4—272no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—62no—
508C2×C3—62no—
509C2×C3—61no—
510C2×C3—62no—
511C2×C3—61no—
512C2×C3—61no—
513C2×C3—61no—
514C2×C3—122no—
515C2×C3—121no—
516C2×C3—182no—
517C2×C3—181no—
518C2×C3C22×D3211yescharacteristic
519C2×C3C22×D3211yescharacteristic
520C2×C3C22×D3211yescharacteristic
521D3—21no—
522D3—21no—
523D3—22no—
524D3—62no—
525D3—62no—
526D3—61no—
527D3—61no—
528D3—62no—
529D3—122no—
530D3—182no—
531D3—182no—
532D3—182no—
533D3—362no—
534D3—362no—
535D3—362no—
536D3—722no—
537C22—31no—
538C22—31no—
539C22—32no—
540C22—91no—
541C22—92no—
542C22—272no—
543C22—362no—
544C22—1082no—
545C22—1082no—
546C4—31no—
547C4—31no—
548C4—92no—
549C4—91no—
550C4D3311yescharacteristic
551C3—21no—
552C3—21no—
553C3—21no—
554C3—41no—
555C3#6430711yescharacteristic, complemented
556C3#6430811yescharacteristic, complemented
557C3#6431411yescharacteristic, complemented
558C2—61no—
559C2—62no—
560C2—61no—
561C2—182no—
562C2—181no—
563C2—542no—
564C2C2×D3311yescharacteristic
subgroupG/H
center, FrattiniC2C2×D33
commutator, socle, cosocleC2×C33C24
FittingC33×C4C23
hypercenterC4D33
hypocenterC33H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #89041C2semidirect
2. C3⋊D3×D12C2semidirect
3. D3×C4⋊(C3⋊D3)C2semidirect
4. #89022C2semidirect
5. C3×D3×D12C2semidirect
6. C3×#3798C2semidirect
7. C3×#3795C2semidirect
8. #88962C2semidirect
9. #88954C2semidirect
10. #88952C2semidirect
11. D3×#626C2semidirect
15. C2×D3×C3⋊D3C22semidirect
16. C2×D3×C3⋊D3C22semidirect
17. C2×C3×D32C22semidirect
18. C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C3×C4⋊(C3⋊D3)C22semidirect
20. C3×C4×C3⋊D3C22semidirect
21. C3×#634C22semidirect
22. C32×D12C22semidirect
23. C32×C4×D3C22semidirect
24. C32×C4×D3C22semidirect
25. #6295C22semidirect
26. #6294C22semidirect
27. #6292C22semidirect
28. #6291C22semidirect
29. #6291C22semidirect
30. #6289C22semidirect
31. C3×#627C22semidirect
32. C3×#626C22semidirect
33. C3×#626C22semidirect
34. C3×#626C22semidirect
35. C3×#625C22semidirect
36. C3×#625C22semidirect
37. C3×D3×Dic3C22semidirect
63. D3×D12D3semidirect
69. #3798D3semidirect
73. #3795D3semidirect
101. C2×D32C2×D3semidirect
115. C4⋊(C3⋊D3)C2×D3semidirect
117. C4×C3⋊D3C2×D3semidirect
118. #634C2×D3semidirect
134. C3×D12C2×D3semidirect
135. C3×D12C2×D3semidirect
142. C3×C4×D3C2×D3semidirect
143. C3×C4×D3C2×D3semidirect
144. C3×C4×D3C2×D3semidirect
145. C3×C4×D3C2×D3semidirect
151. #627C2×D3semidirect
163. #626C2×D3semidirect
164. #626C2×D3semidirect
165. #626C2×D3semidirect
167. #625C2×D3semidirect
168. #625C2×D3semidirect
175. D3×Dic3C2×D3semidirect
281. C33H(5,𝔽2)semidirect
349. D12D32semidirect
360. C4×D3D32semidirect
361. C4×D3D32semidirect
477. C32#1004semidirect
478. C32#1009semidirect
479. C32#1009semidirect
555. C3#64307semidirect
556. C3#64308semidirect
557. C3#64314semidirect

Maximal subgroups

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

subgroupindex
#890412
C3⋊D3×D122
D3×C4⋊(C3⋊D3)2
#890222
C3×D3×D122
C3×#37982
C3×#37952
#889622
#889542
#889522
D3×#6262
#643143
#643083
#643073