Finite Group Databasebeta

← #13176720

Subgroups — #13176720

Order 1728.

1900 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
1#11724586C212yesmaximal, complemented
2D8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64121C211yesmaximal, characteristic, complemented
4C32×#912C211yesmaximal, characteristic, complemented
5#11723561C212yesmaximal, complemented
6#11723558C212yesmaximal, complemented
7#11723553C212yesmaximal, complemented
8#11723341C211yesmaximal, characteristic, complemented
9#11723337C211yesmaximal, characteristic, complemented
10#11723330C211yesmaximal, characteristic, complemented
11D3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
12C3⋊D3×#135—31nomaximal
13#10594730—34nomaximal
14D4×C3⋊(C3⋊D3)C2212yes—
15C3×#3830C2212yescomplemented
16C32×#237C2212yescomplemented
17#89039—22no—
18C2×#6289—22no—
19#89031—22no—
20#89028—22no—
21C4×D3×C3⋊D3C2211yescharacteristic
22#89022C2212yescomplemented
23#89019C2212yes—
24#89018C2212yescomplemented
25C33×D8C2211yescharacteristic, complemented
26#88860C2212yescomplemented
27#88852C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic
29C3×#3754C2212yescomplemented
30C3×#3746C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#214C2212yescomplemented
33C32×Dic12C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic, complemented
35#88798C2212yescomplemented
36#88797C2212yescomplemented
37#88795C2212yescomplemented
38#88785C2211yescharacteristic
39C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
40D3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
41#41×C3⋊D3—32no—
42#64312—38no—
43C32×#135—31no—
44#64159—32no—
45#64128—31no—
46Q16×C3⋊D3—31no—
47#64125—32no—
48SD16×C3⋊D3—32no—
49#64121D311yescharacteristic, complemented
50D8×C3⋊D3—31no—
51D8×C3⋊D3—34no—
52D8×C3⋊D3—64no—
53#64113—31no—
54C2×C8×C3⋊D3—31no—
55C3×#912—34no—
56C3×#912D314yescomplemented
57#63935—38no—
58#63935—38no—
59#63930—38no—
60#63808—34no—
61#63808—34no—
62#63802—34no—
63D32×C8—34no—
64C22×C3⋊(C3⋊D3)—22no—
65C2×D3×C3⋊D3D411yescharacteristic
66C33×D4C2312yes—
67#6312—22no—
68C4⋊(C3⋊(C3⋊D3))C2312yes—
69C4×C3⋊(C3⋊D3)C2311yescharacteristic
70C3×#638—22no—
71C2×C3×C3⋊Dic3—22no—
72C3×C4×C3⋊D3C2311yescharacteristic
73C3×#634C2312yes—
74C32×#67—22no—
75C2×C32×Dic3—22no—
76C32×C4×D3C2311yescharacteristic
77C32×Dic6C2312yes—
78#6293—22no—
79#6292—22no—
80#6291—22no—
81#6289—22no—
82#6289—22no—
83#6289—22no—
84#6289D411yescharacteristic
85Dic3×C3⋊D3D411yescharacteristic
86D3×C3⋊Dic3D411yescharacteristic
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91D3×#135—94no—
92D3×#135—91no—
93C32×#41—32no—
94#3832—32no—
95Q8×C3⋊D3—32no—
96#3830—62no—
97#3830C2×D312yescomplemented
98D4×C3⋊D3—32no—
99D4×C3⋊D3—38no—
100D4×C3⋊D3—62no—
101D4×C3⋊D3—68no—
102#3828—32no—
103#3828—62no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—62no—
106C3×#237—38no—
107C3×#237C2×D318yescomplemented
108#3811—68no—
109C2×#624—68no—
110#3804—68no—
111#3804—68no—
112C4×D32—34no—
113#3799—38no—
114#3799—38no—
115#3797—38no—
116C32×Q16—31no—
117C32×SD16—32no—
118C32×D8—24no—
119C32×D8C2×D314yescomplemented
120C32×D8C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3756—32no—
123#3755—32no—
124#3754C2×D312yescomplemented
125#3753—32no—
126#3753—38no—
127#3753—68no—
128C2×C3⋊(C3⋊C8)—31no—
129#3746C2×D311yescharacteristic, complemented
130#3745—31no—
131#3745—34no—
132#3745—64no—
133#3744—32no—
134C8×C3⋊D3—31no—
135C8×C3⋊D3—31no—
136C8×C3⋊D3—34no—
137C8×C3⋊D3—31no—
138C8×C3⋊D3—64no—
139C8×C3⋊D3C2×D311yescharacteristic, complemented
140C3×#214—38no—
141C3×#214C2×D318yescomplemented
142C3×Dic12—34no—
143C3×Dic12C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yescomplemented
146#3718—38no—
147#3717—38no—
148#3717—38no—
149#3710—34no—
150D3×C3⋊C8—34no—
151D3×C3⋊C8—34no—
152C22×C33—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3⋊(C3⋊D3)—44no—
155C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
156C2×C3×C3⋊D3C2×D411yescharacteristic
157C2×C32×D3C2×D411yescharacteristic
158D3×C3⋊D3—22no—
159C33×C4C2411yescharacteristic
160C3⋊(C3⋊Dic3)C2×D411yescharacteristic
161C3×C3⋊Dic3—22no—
162C3×C3⋊Dic3C2×D411yescharacteristic
163C32×Dic3—22no—
164C32×Dic3C2×D411yescharacteristic
165D3×#41—98no—
166D3×#41—92no—
167C3×#135—34no—
168C3×#135—91no—
169#950—98no—
170#950—92no—
171#919—94no—
172#919—91no—
173D3×Q16—94no—
174D3×Q16—91no—
175#916—98no—
176#916—92no—
177D3×SD16—98no—
178D3×SD16—92no—
179#912—34no—
180#912C3⋊D311yescharacteristic, complemented
181D3×D8—94no—
182D3×D8—91no—
183D3×D8—184no—
184#904—94no—
185#904—91no—
186C2×D3×C8—94no—
187C2×D3×C8—91no—
188C22×C3⋊D3—31no—
189C22×C3⋊D3—62no—
190C22×C3⋊D3—68no—
191C22×C3⋊D3—128no—
192C2×D32—34no—
193C32×Q8—32no—
194C32×D4—28no—
195C32×D4—62no—
196C32×D4C22×D318yes—
197C32×D4C22×D312yes—
198C2×C32×C4—31no—
199C2×C32×C4—62no—
200#638—22no—
201#638—62no—
202#638—62no—
203#638—68no—
204#638—128no—
205C2×C3⋊Dic3—22no—
206C2×C3⋊Dic3—31no—
207C4⋊(C3⋊D3)—32no—
208C4⋊(C3⋊D3)—38no—
209C4⋊(C3⋊D3)—68no—
210C4⋊(C3⋊D3)—62no—
211C4×C3⋊D3—31no—
212C4×C3⋊D3—31no—
213C4×C3⋊D3—31no—
214C4×C3⋊D3—34no—
215C4×C3⋊D3—64no—
216C4×C3⋊D3—62no—
217C4×C3⋊D3—62no—
218C4×C3⋊D3—62no—
219C4×C3⋊D3C22×D311yescharacteristic
220#634—62no—
221#634C22×D312yes—
222C3×#67—28no—
223C3×#67—68no—
224C2×C3×Dic3—28no—
225C2×C3×Dic3—68no—
226C3×C4×D3—34no—
227C3×C4×D3C22×D314yes—
228C3×Dic6—38no—
229C3×Dic6C22×D318yes—
230#627—68no—
231#626—68no—
232#626—68no—
233#624—34no—
234#624—68no—
235#624—68no—
236#624—68no—
237D3×Dic3—34no—
238D3×Dic3—34no—
239C32×C8—24no—
240C32×C8—31no—
241C32×C8C22×D314yes—
242C32×C8C22×D311yescharacteristic
243C3⋊(C3⋊C8)—31no—
244C3⋊(C3⋊C8)—34no—
245C3⋊(C3⋊C8)—64no—
246C3⋊(C3⋊C8)C22×D311yescharacteristic
247C3×C3⋊C8—34no—
248C3×C3⋊C8C22×D314yes—
249C2×#135—271noSylow
250C2×C33—42no—
251C2×C33C22×D411yescharacteristic
252C3⋊(C3⋊D3)—42no—
253C3⋊(C3⋊D3)#13512yescomplemented
254C3×C3⋊D3—21no—
255C32×D3—21no—
256C3×#41—38no—
257C3×#41—92no—
258#239—98no—
259#239—92no—
260D3×Q8—98no—
261D3×Q8—92no—
262#237—38no—
263#237—182no—
264#237—188no—
265#237C2×C3⋊D312yescomplemented
266D3×D4—98no—
267D3×D4—92no—
268D3×D4—188no—
269D3×D4—182no—
270D3×D4—188no—
271#235—98no—
272#235—92no—
273#235—182no—
274#235—188no—
275C2×C4×D3—94no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×C4×D3—188no—
279C3×Q16—34no—
280C3×Q16—91no—
281C3×SD16—38no—
282C3×SD16—92no—
283C3×D8—24no—
284C3×D8D3214yescomplemented
285C3×D8C2×C3⋊D311yescharacteristic, complemented
286C2×C3×C8—34no—
287C2×C3×C8—91no—
288#216—98no—
289#216—92no—
290#215—98no—
291#215—92no—
292#214—38no—
293#214C2×C3⋊D312yescomplemented
294#213—98no—
295#213—92no—
296#213—188no—
297C2×C3⋊C8—94no—
298C2×C3⋊C8—91no—
299Dic12—34no—
300Dic12C2×C3⋊D311yescharacteristic, complemented
301D24—94no—
302D24—91no—
303D24—184no—
304#204—98no—
305#204—92no—
306D3×C8—34no—
307D3×C8—94no—
308D3×C8—94no—
309D3×C8—91no—
310D3×C8—94no—
311D3×C8—91no—
312D3×C8—91no—
313D3×C8—184no—
314D3×C8C2×C3⋊D311yescharacteristic, complemented
315C22×C32—22no—
316C22×C32—28no—
317C22×C32—31no—
318C22×C32—48no—
319C2×C3⋊D3—31no—
320C2×C3⋊D3—34no—
321C2×C3⋊D3—64no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—128no—
326C2×C3⋊D3—124no—
327C2×C3⋊D3—1216no—
328C2×C3⋊D3—2416no—
329C2×C3⋊D3D3×D411yescharacteristic
330C2×C3×D3—34no—
331C2×C3×D3D3×D414yes—
332D32—68no—
333C32×C4—24no—
334C32×C4—31no—
335C32×C4—62no—
336C32×C4C23×D314yes—
337C32×C4C23×D311yescharacteristic
338C3⋊Dic3—22no—
339C3⋊Dic3—31no—
340C3⋊Dic3—34no—
341C3⋊Dic3—64no—
342C3⋊Dic3D3×D411yescharacteristic
343C3×Dic3—28no—
344C3×Dic3—34no—
345C3×Dic3—68no—
346C3×Dic3D3×D414yes—
347C2×#41—272no—
348#135—31no—
349#135—91no—
350#135—271no—
351#135—271no—
352#135—272no—
353#135—272no—
354C2×Q16—271no—
355C2×SD16—272no—
356C2×D8—271no—
357C22×C8—271no—
358C33C2×#13511yescharacteristic, complemented, Sylow
359C22×D3—94no—
360C22×D3—91no—
361C22×D3—182no—
362C22×D3—188no—
363C22×D3—368no—
364C3×Q8—38no—
365C3×Q8—92no—
366C3×D4—28no—
367C3×D4—68no—
368C3×D4—182no—
369C3×D4C2×D3218yes—
370C3×D4C22×C3⋊D312yes—
371C2×C3×C4—34no—
372C2×C3×C4—68no—
373C2×C3×C4—91no—
374C2×C3×C4—182no—
375#67—22no—
376#67—68no—
377#67—182no—
378#67—188no—
379#67—182no—
380#67—188no—
381#67—368no—
382C2×Dic3—22no—
383C2×Dic3—68no—
384C2×Dic3—94no—
385C2×Dic3—91no—
386D12—98no—
387D12—92no—
388D12—188no—
389D12—188no—
390D12—182no—
391C4×D3—34no—
392C4×D3—94no—
393C4×D3—94no—
394C4×D3—91no—
395C4×D3—94no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—184no—
399C4×D3—182no—
400C4×D3—182no—
401C4×D3—188no—
402C4×D3—188no—
403C4×D3—188no—
404C4×D3—182no—
405C4×D3C22×C3⋊D311yescharacteristic
406Dic6—38no—
407Dic6—182no—
408Dic6—188no—
409Dic6C22×C3⋊D312yes—
410C3×C8—24no—
411C3×C8—34no—
412C3×C8—91no—
413C3×C8C2×D3214yes—
414C3×C8C22×C3⋊D311yescharacteristic
415C3⋊C8—34no—
416C3⋊C8—94no—
417C3⋊C8—91no—
418C3⋊C8—184no—
419C3⋊C8C22×C3⋊D311yescharacteristic
420C2×C32—24no—
421C2×C32—42no—
422C2×C32—48no—
423C2×C32—61no—
424C2×C32—88no—
425C2×C32C2×D3×D414yes—
426C2×C32C2×D3×D411yescharacteristic
427C3⋊D3—21no—
428C3⋊D3—32no—
429C3⋊D3—38no—
430C3⋊D3—68no—
431C3⋊D3—122no—
432C3⋊D3—128no—
433C3⋊D3—248no—
434C3×D3—24no—
435C3×D3—64no—
436#41—32no—
437#41—92no—
438#41—272no—
439#41—272no—
440#41—542no—
441#41—542no—
442C2×Q8—272no—
443C2×D4—272no—
444C2×D4—542no—
445C22×C4—271no—
446C22×C4—542no—
447Q16—31no—
448Q16—91no—
449Q16—272no—
450SD16—32no—
451SD16—92no—
452SD16—272no—
453SD16—272no—
454D8—272no—
455D8—271no—
456D8D3×C3⋊D311yescharacteristic, complemented
457C2×C8—31no—
458C2×C8—91no—
459C2×C8—271no—
460C2×C8—271no—
461C2×C8—271no—
462C2×C8—271no—
463C22×C3—28no—
464C22×C3—22no—
465C22×C3—34no—
466C22×C3—48no—
467C22×C3—91no—
468C2×D3—34no—
469C2×D3—94no—
470C2×D3—91no—
471C2×D3—184no—
472C2×D3—182no—
473C2×D3—188no—
474C2×D3—188no—
475C2×D3—182no—
476C2×D3—368no—
477C2×D3—364no—
478C2×D3—3616no—
479C2×D3—7216no—
480C2×D3D4×C3⋊D311yescharacteristic
481C3×C4—24no—
482C3×C4—34no—
483C3×C4—68no—
484C3×C4—91no—
485C3×C4—182no—
486C3×C4C22×D3214yes—
487C3×C4C23×C3⋊D311yescharacteristic
488Dic3—22no—
489Dic3—34no—
490Dic3—68no—
491Dic3—94no—
492Dic3—91no—
493Dic3—184no—
494Dic3D4×C3⋊D311yescharacteristic
495C32—24no—
496C32D3×#13514yescomplemented
497C32D3×#13511yescharacteristic, complemented
498C23—271no—
499C23—542no—
500Q8—32no—
501Q8—92no—
502Q8—542no—
503D4—62no—
504D4—182no—
505D4—272no—
506D4—542no—
507D4—542no—
508D4—542no—
509D4C2×D3×C3⋊D312yes—
510C2×C4—31no—
511C2×C4—62no—
512C2×C4—91no—
513C2×C4—182no—
514C2×C4—271no—
515C2×C4—271no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—542no—
519C2×C4—542no—
520C2×C4—542no—
521C8—31no—
522C8—91no—
523C8—271no—
524C8C2×D3×C3⋊D311yescharacteristic
525C2×C3—24no—
526C2×C3—42no—
527C2×C3—48no—
528C2×C3—64no—
529C2×C3—88no—
530C2×C3—181no—
531C2×C3D32×D414yes—
532C2×C3C2×D4×C3⋊D311yescharacteristic
533D3—21no—
534D3—64no—
535D3—92no—
536D3—98no—
537D3—188no—
538D3—362no—
539D3—368no—
540D3—728no—
541C22—22no—
542C22—31no—
543C22—91no—
544C22—271no—
545C22—542no—
546C22—542no—
547C22—1084no—
548C4—31no—
549C4—62no—
550C4—91no—
551C4—182no—
552C4—271no—
553C4C22×D3×C3⋊D311yescharacteristic
554C3—24no—
555C3#1059473014yescomplemented
556C3C3⋊D3×#13511yescharacteristic, complemented
557C2—42no—
558C2—61no—
559C2—181no—
560C2—272no—
561C2—1082no—
562C2D3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2D3×D4×C3⋊D3
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×D8C22
socleC2×C33C22×D4
hypercenterD8D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11724586C2semidirect
2. D8×C3⋊(C3⋊D3)C2semidirect
3. C3×#64121C2semidirect
4. C32×#912C2semidirect
5. #11723561C2semidirect
6. #11723558C2semidirect
7. #11723553C2semidirect
8. #11723341C2semidirect
9. #11723337C2semidirect
10. #11723330C2semidirect
11. D3×C8×C3⋊D3C2semidirect
15. C3×#3830C22semidirect
16. C32×#237C22semidirect
22. #89022C22semidirect
24. #89018C22semidirect
25. C33×D8C22semidirect
26. #88860C22semidirect
27. #88852C22semidirect
29. C3×#3754C22semidirect
30. C3×#3746C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#214C22semidirect
33. C32×Dic12C22semidirect
34. C32×D3×C8C22semidirect
35. #88798C22semidirect
36. #88797C22semidirect
37. #88795C22semidirect
39. C3⋊D3×C3⋊C8C22semidirect
40. D3×C3⋊(C3⋊C8)C22semidirect
49. #64121D3semidirect
56. C3×#912D3semidirect
97. #3830C2×D3semidirect
107. C3×#237C2×D3semidirect
119. C32×D8C2×D3semidirect
120. C32×D8C2×D3semidirect
124. #3754C2×D3semidirect
129. #3746C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
141. C3×#214C2×D3semidirect
143. C3×Dic12C2×D3semidirect
145. C3×D3×C8C2×D3semidirect
180. #912C3⋊D3semidirect
253. C3⋊(C3⋊D3)#135semidirect
265. #237C2×C3⋊D3semidirect
284. C3×D8D32semidirect
285. C3×D8C2×C3⋊D3semidirect
293. #214C2×C3⋊D3semidirect
300. Dic12C2×C3⋊D3semidirect
314. D3×C8C2×C3⋊D3semidirect
358. C33C2×#135semidirect
456. D8D3×C3⋊D3semidirect
496. C32D3×#135semidirect
497. C32D3×#135semidirect
555. C3#10594730semidirect
556. C3C3⋊D3×#135semidirect

Maximal subgroups

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

subgroupindex
#117245862
D8×C3⋊(C3⋊D3)2
C3×#641212
C32×#9122
#117235612
#117235582
#117235532
#117233412
#117233372
#117233302
D3×C8×C3⋊D32
C3⋊D3×#1353
#105947303