Finite Group Databasebeta

← #11724385

Subgroups — #11724385

Order 864.

814 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#1113C211yesmaximal, characteristic, complemented
2#89026C211yesmaximal, characteristic, complemented
3#89020C211yesmaximal, characteristic, complemented
4#89017C211yesmaximal, characteristic, complemented
5C3×D3×D12C211yesmaximal, characteristic, complemented
6C3×#3799C211yesmaximal, characteristic, complemented
7C3×#3796C211yesmaximal, characteristic, complemented
8#88960C212yesmaximal, complemented
9#88955C212yesmaximal, complemented
10#88953C212yesmaximal, complemented
11D3×#626C212yesmaximal, complemented
12#64320—31nomaximal
13#64314—31nomaximal
14#64307—31nomaximal
15C2×#1113C2212yescomplemented
16C2×D3×C3⋊D3C2212yescomplemented
17C2×C3×D32C2212yescomplemented
18C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
19C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
21C3×C4×C3⋊D3C2211yescharacteristic, complemented
22C32×D12C2211yescharacteristic, complemented
23C32×C4×D3C2211yescharacteristic, complemented
24C32×Dic6C2211yescharacteristic, complemented
25#6295C2212yescomplemented
26#6293C2211yescharacteristic, complemented
27#6292C2212yescomplemented
28#6292C2211yescharacteristic, complemented
29#6291C2211yescharacteristic, complemented
30#6290C2211yescharacteristic, complemented
31#6290C2212yescomplemented
32Dic3×C3⋊D3C2212yescomplemented
33C3×#626C2212yescomplemented
34C3×#626C2212yescomplemented
35C3×#626C2212yescomplemented
36C3×#624C2212yescomplemented
37C3×D3×Dic3C2212yescomplemented
38#3832—31no—
39D4×C3⋊D3—31no—
40#3828—31no—
41C3×#239—31no—
42C3×#239—31no—
43C3×#237—31no—
44C3×D3×D4—31no—
45C3×#235—31no—
46C2×C3×D12—31no—
47#3811—32no—
48D3×#67—32no—
49D3×#67—32no—
50D3×#67—32no—
51C2×#626—32no—
52#3804—32no—
53#3802—32no—
54#3802—31no—
55#3802—31no—
56#3802—31no—
57#3802—61no—
58D3×D12—32no—
59D3×D12—31no—
60D3×D12—31no—
61D3×D12—32no—
62D3×D12—32no—
63D3×D12D311yescharacteristic, complemented
64#3799D311yescharacteristic, complemented
65#3798—31no—
66#3798—31no—
67#3798—32no—
68#3798—61no—
69#3796—31no—
70#3796—31no—
71#3796—61no—
72#3796D311yescharacteristic, complemented
73#3795—32no—
74C2×C3⋊(C3⋊D3)C2311yescharacteristic
75C2×C3×C3⋊D3C2312yes—
76C2×C3×C3⋊D3C2311yescharacteristic
77C2×C3×C3⋊D3C2312yes—
78C2×C32×D3C2312yes—
79C2×C32×D3C2311yescharacteristic
80#1113—42no—
81D3×C3⋊D3—42no—
82C3×D32—42no—
83C33×C4C2311yescharacteristic
84C3⋊(C3⋊Dic3)C2311yescharacteristic
85C3×C3⋊Dic3C2311yescharacteristic
86C32×Dic3C2312yes—
87C32×Dic3C2311yescharacteristic
88#1009—91no—
89#1009—91no—
90#1004—91no—
91C22×C3⋊D3—32no—
92C22×C3×D3—32no—
93C22×C3×D3—32no—
94C2×D32—32no—
95C2×D32—32no—
96C2×D32—32no—
97C2×D32—32no—
98C2×D32—32no—
99C2×D32—62no—
100C2×D32C2×D312yescomplemented
101C32×Q8—31no—
102C32×D4—31no—
103C2×C32×C4—31no—
104#638—32no—
105#638—31no—
106#638—31no—
107C4⋊(C3⋊D3)—31no—
108C4⋊(C3⋊D3)—32no—
109C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
110C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
111C4×C3⋊D3—32no—
112C4×C3⋊D3—31no—
113C4×C3⋊D3—31no—
114C4×C3⋊D3—31no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—61no—
118C4×C3⋊D3—121no—
119C4×C3⋊D3C2×D311yescharacteristic, complemented
120#634—31no—
121C3×#67—32no—
122C3×#67—32no—
123C3×#67—32no—
124C2×C3×Dic3—32no—
125C3×D12—21no—
126C3×D12—32no—
127C3×D12—32no—
128C3×D12—32no—
129C3×D12—31no—
130C3×D12—31no—
131C3×D12—31no—
132C3×D12—31no—
133C3×D12—61no—
134C3×D12—61no—
135C3×D12C2×D311yescharacteristic, complemented
136C3×D12C2×D311yescharacteristic, complemented
137C3×C4×D3—21no—
138C3×C4×D3—32no—
139C3×C4×D3—31no—
140C3×C4×D3—31no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×C4×D3—61no—
144C3×C4×D3C2×D311yescharacteristic, complemented
145C3×C4×D3C2×D311yescharacteristic, complemented
146C3×Dic6—21no—
147C3×Dic6C2×D311yescharacteristic, complemented
148C3×Dic6C2×D311yescharacteristic, complemented
149#627—31no—
150#627—31no—
151#627—61no—
152#626—32no—
153#626—32no—
154#626—31no—
155#626—31no—
156#626—32no—
157#626—32no—
158#626—31no—
159#626—31no—
160#626—61no—
161#626—62no—
162#626—61no—
163#626C2×D312yescomplemented
164#626C2×D312yescomplemented
165#626C2×D312yescomplemented
166#625—32no—
167#625—31no—
168#625—32no—
169#625—31no—
170#625—32no—
171#625—61no—
172#625—62no—
173#624C2×D312yescomplemented
174D3×Dic3—32no—
175D3×Dic3—32no—
176D3×Dic3—62no—
177D3×Dic3C2×D312yescomplemented
178C2×C33C2411yescharacteristic
179C3⋊(C3⋊D3)—21no—
180C3×C3⋊D3—22no—
181C3×C3⋊D3—22no—
182C3×C3⋊D3—21no—
183C32×D3—21no—
184C32×D3—22no—
185C3×#41—91no—
186C3×#41—91no—
187C2×C3×D4—91no—
188C2×#67—92no—
189C2×#67—92no—
190#239—31no—
191#239—31no—
192#239—91no—
193#239—91no—
194#239—181no—
195#237—31no—
196#237—92no—
197#237—91no—
198D3×D4—31no—
199D3×D4—91no—
200D3×D4—91no—
201D3×D4—92no—
202D3×D4—92no—
203D3×D4—92no—
204D3×D4—91no—
205D3×D4—91no—
206D3×D4—92no—
207D3×D4—92no—
208D3×D4—91no—
209D3×D4—181no—
210#235—31no—
211#235—92no—
212#235—92no—
213#235—91no—
214#235—91no—
215#235—91no—
216#235—181no—
217C2×D12—31no—
218C2×D12—92no—
219C2×D12—91no—
220C2×D12—92no—
221C22×C32—32no—
222C22×C32—31no—
223C2×C3⋊D3—31no—
224C2×C3⋊D3—31no—
225C2×C3⋊D3—31no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—61no—
229C2×C3⋊D3—121no—
230C2×C3⋊D3—122no—
231C2×C3⋊D3C22×D312yes—
232C2×C3⋊D3C22×D312yes—
233C2×C3⋊D3C22×D311yescharacteristic
234C2×C3×D3—21no—
235C2×C3×D3—22no—
236C2×C3×D3—32no—
237C2×C3×D3—32no—
238C2×C3×D3—32no—
239C2×C3×D3—32no—
240C2×C3×D3—31no—
241C2×C3×D3—31no—
242C2×C3×D3—62no—
243C2×C3×D3—61no—
244C2×C3×D3—62no—
245C2×C3×D3—122no—
246C2×C3×D3—122no—
247C2×C3×D3C22×D312yes—
248C2×C3×D3C22×D311yescharacteristic
249C2×C3×D3C22×D311yescharacteristic
250C2×C3×D3C22×D312yes—
251D32—42no—
252D32—122no—
253D32—122no—
254D32—124no—
255D32—122no—
256D32—122no—
257D32—122no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—31no—
262C32×C4—32no—
263C32×C4—41no—
264C32×C4C22×D311yescharacteristic
265C32×C4C22×D311yescharacteristic
266C32×C4C22×D311yescharacteristic
267C3⋊Dic3—31no—
268C3⋊Dic3—31no—
269C3⋊Dic3—31no—
270C3⋊Dic3—61no—
271C3⋊Dic3—61no—
272C3⋊Dic3—61no—
273C3⋊Dic3—121no—
274C3⋊Dic3C22×D311yescharacteristic
275C3×Dic3—22no—
276C3×Dic3—21no—
277C3×Dic3—31no—
278C3×Dic3—31no—
279C3×Dic3—61no—
280C3×Dic3C22×D312yes—
281C3×Dic3C22×D312yes—
282C3×Dic3C22×D311yescharacteristic
283C3×Dic3C22×D311yescharacteristic
284H(5,𝔽2)—271noSylow
285C33H(5,𝔽2)11yescharacteristic, complemented, Sylow
286C23×C3—92no—
287C22×D3—32no—
288C22×D3—32no—
289C22×D3—92no—
290C22×D3—92no—
291C22×D3—92no—
292C22×D3—92no—
293C22×D3—92no—
294C22×D3—92no—
295C22×D3—182no—
296C3×Q8—31no—
297C3×Q8—31no—
298C3×Q8—61no—
299C3×D4—31no—
300C3×D4—31no—
301C3×D4—61no—
302C3×D4—92no—
303C3×D4—92no—
304C3×D4—92no—
305C3×D4—91no—
306C3×D4—91no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—61no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—91no—
313#67—32no—
314#67—32no—
315#67—32no—
316#67—92no—
317#67—92no—
318#67—91no—
319#67—91no—
320#67—92no—
321#67—91no—
322#67—91no—
323#67—92no—
324#67—92no—
325#67—91no—
326#67—91no—
327#67—92no—
328#67—181no—
329#67—181no—
330#67—182no—
331C2×Dic3—32no—
332C2×Dic3—92no—
333D12—32no—
334D12—32no—
335D12—32no—
336D12—31no—
337D12—31no—
338D12—31no—
339D12—31no—
340D12—61no—
341D12—61no—
342D12—91no—
343D12—92no—
344D12—92no—
345D12—91no—
346D12—92no—
347D12—91no—
348D12—181no—
349D12—182no—
350D12D3211yescharacteristic, complemented
351C4×D3—32no—
352C4×D3—31no—
353C4×D3—31no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—61no—
357C4×D3—92no—
358C4×D3—92no—
359C4×D3—91no—
360C4×D3—91no—
361C4×D3—91no—
362C4×D3—181no—
363C4×D3—181no—
364C4×D3—181no—
365C4×D3—182no—
366C4×D3—361no—
367C4×D3D3211yescharacteristic, complemented
368Dic6—91no—
369Dic6—91no—
370Dic6—91no—
371Dic6—181no—
372Dic6D3211yescharacteristic, complemented
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—41no—
377C2×C32—61no—
378C2×C32—62no—
379C2×C32C23×D311yescharacteristic
380C2×C32C23×D311yescharacteristic
381C2×C32C23×D311yescharacteristic
382C3⋊D3—22no—
383C3⋊D3—21no—
384C3⋊D3—22no—
385C3⋊D3—61no—
386C3⋊D3—62no—
387C3⋊D3—61no—
388C3⋊D3—61no—
389C3⋊D3—121no—
390C3⋊D3—121no—
391C3⋊D3—122no—
392C3×D3—21no—
393C3×D3—22no—
394C3×D3—22no—
395C3×D3—21no—
396C3×D3—42no—
397C3×D3—41no—
398C3×D3—62no—
399C3×D3—61no—
400C3×D3—61no—
401C3×D3—62no—
402C3×D3—62no—
403C3×D3—62no—
404C3×D3—122no—
405C3×D3—121no—
406C3×D3—122no—
407#41—91no—
408#41—91no—
409#41—271no—
410#41—272no—
411#41—271no—
412C2×D4—91no—
413C2×D4—272no—
414C2×D4—271no—
415C2×D4—272no—
416C2×D4—271no—
417C2×D4—272no—
418C22×C3—32no—
419C22×C3—31no—
420C22×C3—32no—
421C22×C3—31no—
422C22×C3—61no—
423C22×C3—62no—
424C22×C3—92no—
425C22×C3—91no—
426C22×C3—92no—
427C22×C3—362no—
428C2×D3—32no—
429C2×D3—31no—
430C2×D3—32no—
431C2×D3—31no—
432C2×D3—32no—
433C2×D3—32no—
434C2×D3—61no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—91no—
438C2×D3—91no—
439C2×D3—91no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—181no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—361no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3—362no—
449C2×D3—362no—
450C2×D3—362no—
451C2×D3—362no—
452C2×D3—364no—
453C2×D3C2×D3212yes—
454C2×D3C2×D3211yescharacteristic
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—31no—
459C3×C4—31no—
460C3×C4—32no—
461C3×C4—32no—
462C3×C4—41no—
463C3×C4—61no—
464C3×C4—62no—
465C3×C4—91no—
466C3×C4C2×D3211yescharacteristic
467C3×C4C2×D3211yescharacteristic
468C3×C4C2×D3211yescharacteristic
469Dic3—31no—
470Dic3—31no—
471Dic3—61no—
472Dic3—91no—
473Dic3—91no—
474Dic3—91no—
475Dic3—181no—
476Dic3—181no—
477Dic3—181no—
478Dic3—361no—
479Dic3C2×D3211yescharacteristic
480Dic3C2×D3212yes—
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#100411yescharacteristic, complemented
486C32#100911yescharacteristic, complemented
487C32#100911yescharacteristic, complemented
488C23—92no—
489C23—272no—
490C23—272no—
491Q8—31no—
492Q8—271no—
493D4—31no—
494D4—92no—
495D4—92no—
496D4—92no—
497D4—91no—
498D4—91no—
499D4—271no—
500D4—272no—
501D4—271no—
502D4—272no—
503D4—271no—
504D4—272no—
505C2×C4—31no—
506C2×C4—92no—
507C2×C4—92no—
508C2×C4—91no—
509C2×C4—272no—
510C2×C4—271no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—62no—
516C2×C3—61no—
517C2×C3—62no—
518C2×C3—61no—
519C2×C3—122no—
520C2×C3—121no—
521C2×C3—181no—
522C2×C3—182no—
523C2×C3—182no—
524C2×C3C22×D3211yescharacteristic
525C2×C3C22×D3211yescharacteristic
526C2×C3C22×D3211yescharacteristic
527D3—21no—
528D3—22no—
529D3—62no—
530D3—62no—
531D3—62no—
532D3—61no—
533D3—62no—
534D3—61no—
535D3—122no—
536D3—122no—
537D3—121no—
538D3—182no—
539D3—181no—
540D3—181no—
541D3—181no—
542D3—361no—
543D3—361no—
544D3—362no—
545C22—31no—
546C22—32no—
547C22—92no—
548C22—91no—
549C22—92no—
550C22—271no—
551C22—362no—
552C22—1082no—
553C22—1082no—
554C4—31no—
555C4—32no—
556C4—91no—
557C4—271no—
558C4D3311yescharacteristic
559C3—21no—
560C3—21no—
561C3—21no—
562C3—41no—
563C3#6430711yescharacteristic, complemented
564C3#6431411yescharacteristic, complemented
565C3#6432011yescharacteristic, complemented
566C2—61no—
567C2—62no—
568C2—182no—
569C2—182no—
570C2—181no—
571C2—541no—
572C2C2×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. C4⋊#1113C2semidirect
2. #89026C2semidirect
3. #89020C2semidirect
4. #89017C2semidirect
5. C3×D3×D12C2semidirect
6. C3×#3799C2semidirect
7. C3×#3796C2semidirect
8. #88960C2semidirect
9. #88955C2semidirect
10. #88953C2semidirect
11. D3×#626C2semidirect
15. C2×#1113C22semidirect
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⋊D3)C22semidirect
21. C3×C4×C3⋊D3C22semidirect
22. C32×D12C22semidirect
23. C32×C4×D3C22semidirect
24. C32×Dic6C22semidirect
25. #6295C22semidirect
26. #6293C22semidirect
27. #6292C22semidirect
28. #6292C22semidirect
29. #6291C22semidirect
30. #6290C22semidirect
31. #6290C22semidirect
32. Dic3×C3⋊D3C22semidirect
33. C3×#626C22semidirect
34. C3×#626C22semidirect
35. C3×#626C22semidirect
36. C3×#624C22semidirect
37. C3×D3×Dic3C22semidirect
63. D3×D12D3semidirect
64. #3799D3semidirect
72. #3796D3semidirect
100. C2×D32C2×D3semidirect
109. C4⋊(C3⋊D3)C2×D3semidirect
110. C4⋊(C3⋊D3)C2×D3semidirect
119. C4×C3⋊D3C2×D3semidirect
135. C3×D12C2×D3semidirect
136. C3×D12C2×D3semidirect
144. C3×C4×D3C2×D3semidirect
145. C3×C4×D3C2×D3semidirect
147. C3×Dic6C2×D3semidirect
148. C3×Dic6C2×D3semidirect
163. #626C2×D3semidirect
164. #626C2×D3semidirect
165. #626C2×D3semidirect
173. #624C2×D3semidirect
177. D3×Dic3C2×D3semidirect
285. C33H(5,𝔽2)semidirect
350. D12D32semidirect
367. C4×D3D32semidirect
372. Dic6D32semidirect
485. C32#1004semidirect
486. C32#1009semidirect
487. C32#1009semidirect
563. C3#64307semidirect
564. C3#64314semidirect
565. C3#64320semidirect

Maximal subgroups

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

subgroupindex
C4⋊#11132
#890262
#890202
#890172
C3×D3×D122
C3×#37992
C3×#37962
#889602
#889552
#889532
D3×#6262
#643203
#643143
#643073