Finite Group Databasebeta

← D3×#64128 #13176770

Subgroups — D3×#64128 #13176770

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3832C212yesmaximal, complemented
2#11723960C211yesmaximal, characteristic, complemented
3C3×#64128C211yesmaximal, characteristic, complemented
4C32×D3×Q16C211yesmaximal, characteristic, complemented
5#11723589C212yesmaximal, complemented
6#11723585C212yesmaximal, complemented
7D3×#3755C212yesmaximal, complemented
8#11723345C211yesmaximal, characteristic, complemented
9#11723341C211yesmaximal, characteristic, complemented
10D3×#3745C211yesmaximal, characteristic, complemented
11D3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
12C2×#64128—31nomaximal
13D3×#919—34nomaximal
14#89080C2212yescomplemented
15C3×#3832C2212yescomplemented
16C32×D3×Q8C2212yes—
17D3×C4⋊(C3⋊D3)—22no—
18D3×C4⋊(C3⋊D3)C2212yes—
19C4×D3×C3⋊D3—22no—
20C4×D3×C3⋊D3C2211yescharacteristic
21#89018—22no—
22#89018C2212yescomplemented
23#89017—22no—
24#89017C2212yescomplemented
25C33×Q16C2211yescharacteristic, complemented
26#88862C2212yescomplemented
27#88852C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
29C3×#3755C2212yescomplemented
30C3×#3745C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#216C2212yescomplemented
33C32×Dic12C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic
35#88795C2212yescomplemented
36#88793C2212yescomplemented
37#88791C2212yescomplemented
38#88785C2211yescharacteristic, complemented
39C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
40D3×C3⋊(C3⋊C8)C2211yescharacteristic
41C2×#3832—32no—
42D3×#239—38no—
43C2×C32×Q16—31no—
44C2×#3755—32no—
45#64128—32no—
46#64128—31no—
47#64128—32no—
48#64128—31no—
49#64128—31no—
50#64128—34no—
51#64128—64no—
52#64128D311yescharacteristic, direct product factor
53C2×#3745—31no—
54C2×C8×C3⋊D3—31no—
55C3×#919—34no—
56C3×D3×Q16D314yescomplemented
57#63950—38no—
58#63949—38no—
59D3×#215—38no—
60#63810—34no—
61#63808—34no—
62D3×D24—34no—
63D32×C8—34no—
64C2×D3×C3⋊D3—22no—
65C2×D3×C3⋊D3D411yescharacteristic
66C33×Q8C2312yes—
67C4⋊(C3⋊(C3⋊D3))—22no—
68C4⋊(C3⋊(C3⋊D3))C2312yes—
69C4×C3⋊(C3⋊D3)—22no—
70C4×C3⋊(C3⋊D3)C2311yescharacteristic
71C3×C4⋊(C3⋊D3)—22no—
72C3×C4⋊(C3⋊D3)C2312yes—
73C3×C4×C3⋊D3—22no—
74C3×C4×C3⋊D3C2311yescharacteristic
75C32×C4×D3—22no—
76C32×C4×D3C2311yescharacteristic
77C32×Dic6—22no—
78C32×Dic6C2312yes—
79#6292—22no—
80#6292—22no—
81#6292—22no—
82#6289—22no—
83#6289D411yescharacteristic
84Dic3×C3⋊D3—22no—
85Dic3×C3⋊D3D411yescharacteristic
86D3×C3⋊Dic3D411yescharacteristic
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91D3×#135—91no—
92C2×#919—94no—
93C2×C32×Q8—32no—
94#3832—32no—
95#3832—38no—
96#3832—32no—
97#3832—32no—
98#3832—62no—
99#3832—62no—
100#3832—68no—
101#3832C2×D312yescomplemented
102C2×C4⋊(C3⋊D3)—32no—
103C2×C4⋊(C3⋊D3)—62no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—62no—
106C3×#239—38no—
107C3×D3×Q8C2×D318yes—
108D3×D12—38no—
109D3×D12—68no—
110C4×D32—34no—
111C4×D32—68no—
112#3799—38no—
113#3799—68no—
114#3796—38no—
115#3796—68no—
116C32×Q16—24no—
117C32×Q16—32no—
118C32×Q16—31no—
119C32×Q16C2×D314yescomplemented
120C32×Q16C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3755—32no—
123#3755—38no—
124#3755—32no—
125#3755—32no—
126#3755—68no—
127#3755C2×D312yescomplemented
128C2×C3⋊(C3⋊C8)—31no—
129#3745—32no—
130#3745—31no—
131#3745—34no—
132#3745—64no—
133#3745C2×D311yescharacteristic, complemented
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×#216C2×D318yescomplemented
141C3×#215—38no—
142C3×Dic12C2×D314yescomplemented
143C3×D24—34no—
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yes—
146#3717—38no—
147#3715—38no—
148#3714—38no—
149#3710—34no—
150D3×C3⋊C8—34no—
151D3×C3⋊C8—34no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3C2×D411yescharacteristic
156C2×C32×D3C2×D411yescharacteristic
157D3×C3⋊D3—22no—
158D3×C3⋊D3—44no—
159C33×C4—22no—
160C33×C4C2411yescharacteristic
161C3⋊(C3⋊Dic3)C2×D411yescharacteristic
162C3×C3⋊Dic3C2×D411yescharacteristic
163C32×Dic3—22no—
164C32×Dic3C2×D411yescharacteristic
165D3×#41—92no—
166C2×#239—98no—
167C3×#135—91no—
168C2×C3×Q16—34no—
169#950—92no—
170C2×#215—98no—
171#919—34no—
172#919—94no—
173#919—94no—
174#919—98no—
175#919—98no—
176#919—94no—
177#919—91no—
178#919—184no—
179D3×Q16C3⋊D311yescharacteristic, complemented
180#916—92no—
181D3×SD16—92no—
182#912—91no—
183D3×D8—91no—
184#904—91no—
185C2×D24—94no—
186C2×D3×C8—94no—
187C2×D3×C8—91no—
188C22×C3⋊D3—31no—
189C22×C3⋊D3—62no—
190C2×D32—34no—
191C2×D32—68no—
192C32×Q8—28no—
193C32×Q8—32no—
194C32×Q8—62no—
195C32×Q8C22×D318yes—
196C32×Q8C22×D312yes—
197C2×C32×C4—31no—
198C2×C32×C4—62no—
199C2×C3⋊Dic3—31no—
200C4⋊(C3⋊D3)—22no—
201C4⋊(C3⋊D3)—32no—
202C4⋊(C3⋊D3)—38no—
203C4⋊(C3⋊D3)—68no—
204C4⋊(C3⋊D3)—62no—
205C4⋊(C3⋊D3)—62no—
206C4⋊(C3⋊D3)—62no—
207C4⋊(C3⋊D3)—62no—
208C4⋊(C3⋊D3)—68no—
209C4⋊(C3⋊D3)—128no—
210C4⋊(C3⋊D3)C22×D312yes—
211C4×C3⋊D3—22no—
212C4×C3⋊D3—31no—
213C4×C3⋊D3—31no—
214C4×C3⋊D3—31no—
215C4×C3⋊D3—34no—
216C4×C3⋊D3—64no—
217C4×C3⋊D3—62no—
218C4×C3⋊D3—62no—
219C4×C3⋊D3—62no—
220C4×C3⋊D3—68no—
221C4×C3⋊D3—128no—
222C4×C3⋊D3C22×D311yescharacteristic
223C3×D12—38no—
224C3×D12—68no—
225C3×C4×D3—28no—
226C3×C4×D3—34no—
227C3×C4×D3—68no—
228C3×C4×D3C22×D314yes—
229C3×Dic6—28no—
230C3×Dic6C22×D318yes—
231#626—68no—
232#626—68no—
233#626—68no—
234#624—34no—
235#624—68no—
236D3×Dic3—34no—
237D3×Dic3—34no—
238D3×Dic3—68no—
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×C33C22×D411yescharacteristic
251C3⋊(C3⋊D3)—21no—
252C3⋊(C3⋊D3)—42no—
253C3×C3⋊D3—21no—
254C3×C3⋊D3—42no—
255C32×D3#13512yescomplemented
256C3×#41—92no—
257C2×C3×Q8—38no—
258#239—38no—
259#239—98no—
260#239—98no—
261#239—98no—
262#239—92no—
263#239—188no—
264#239—188no—
265#239—188no—
266D3×Q8C2×C3⋊D312yes—
267#237—92no—
268#237—182no—
269D3×D4—92no—
270D3×D4—182no—
271#235—92no—
272#235—182no—
273C2×D12—98no—
274C2×D12—188no—
275C2×C4×D3—94no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×C4×D3—188no—
279C3×Q16—24no—
280C3×Q16—38no—
281C3×Q16—34no—
282C3×Q16D3214yescomplemented
283C3×Q16C2×C3⋊D311yescharacteristic, complemented
284C3×SD16—92no—
285C3×D8—91no—
286C2×C3×C8—34no—
287C2×C3×C8—91no—
288#216C2×C3⋊D312yescomplemented
289#215—38no—
290#215—98no—
291#215—98no—
292#215—98no—
293#215—92no—
294#215—188no—
295#214—92no—
296#213—92no—
297C2×C3⋊C8—94no—
298C2×C3⋊C8—91no—
299Dic12C2×C3⋊D311yescharacteristic, complemented
300D24—34no—
301D24—98no—
302D24—94no—
303D24—91no—
304D24—184no—
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
315C22×C32—31no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—31no—
318C2×C3⋊D3—34no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—62no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—128no—
324C2×C3⋊D3—124no—
325C2×C3⋊D3D3×D411yescharacteristic
326C2×C3×D3—34no—
327C2×C3×D3—68no—
328C2×C3×D3D3×D414yes—
329D32—68no—
330D32—1216no—
331C32×C4—22no—
332C32×C4—24no—
333C32×C4—28no—
334C32×C4—31no—
335C32×C4—48no—
336C32×C4—62no—
337C32×C4C23×D314yes—
338C32×C4C23×D311yescharacteristic
339C3⋊Dic3—31no—
340C3⋊Dic3—34no—
341C3⋊Dic3—64no—
342C3⋊Dic3D3×D411yescharacteristic
343C3×Dic3—28no—
344C3×Dic3—34no—
345C3×Dic3D3×D414yes—
346C2×#41—272no—
347#135—91no—
348#135—271no—
349#135—271no—
350#135—272no—
351#135—272no—
352#135—271no—
353C2×Q16—31no—
354C2×SD16—272no—
355C2×D8—271no—
356C22×C8—271no—
357C33C2×#13511yescharacteristic, complemented, Sylow
358C22×D3—94no—
359C22×D3—91no—
360C22×D3—188no—
361C22×D3—182no—
362C3×Q8—28no—
363C3×Q8—38no—
364C3×Q8—68no—
365C3×Q8C2×D3218yes—
366C3×Q8C22×C3⋊D312yes—
367C3×D4—92no—
368C3×D4—182no—
369C2×C3×C4—34no—
370C2×C3×C4—68no—
371C2×C3×C4—91no—
372C2×C3×C4—182no—
373#67—182no—
374#67—182no—
375#67—182no—
376C2×Dic3—94no—
377C2×Dic3—91no—
378C2×Dic3—182no—
379D12—38no—
380D12—68no—
381D12—98no—
382D12—92no—
383D12—188no—
384D12—182no—
385D12—188no—
386D12—188no—
387D12—188no—
388D12—188no—
389D12—368no—
390C4×D3—22no—
391C4×D3—34no—
392C4×D3—68no—
393C4×D3—94no—
394C4×D3—94no—
395C4×D3—91no—
396C4×D3—94no—
397C4×D3—91no—
398C4×D3—91no—
399C4×D3—184no—
400C4×D3—182no—
401C4×D3—188no—
402C4×D3—182no—
403C4×D3—188no—
404C4×D3—188no—
405C4×D3—368no—
406C4×D3C22×C3⋊D311yescharacteristic
407Dic6—22no—
408Dic6C22×C3⋊D312yes—
409C3×C8—24no—
410C3×C8—34no—
411C3×C8—91no—
412C3×C8C2×D3214yes—
413C3×C8C22×C3⋊D311yescharacteristic
414C3⋊C8—34no—
415C3⋊C8—94no—
416C3⋊C8—91no—
417C3⋊C8—184no—
418C3⋊C8C22×C3⋊D311yescharacteristic
419C2×C32—24no—
420C2×C32—32no—
421C2×C32C2×D3×D414yes—
422C2×C32C2×D3×D411yescharacteristic
423C3⋊D3—21no—
424C3⋊D3—42no—
425C3⋊D3—61no—
426C3⋊D3—64no—
427C3⋊D3—124no—
428C3⋊D3—122no—
429C3⋊D3—128no—
430C3⋊D3—248no—
431C3×D3—64no—
432C3×D3—128no—
433C3×D3#91918yescomplemented
434#41—92no—
435#41—272no—
436#41—272no—
437#41—272no—
438#41—542no—
439#41—542no—
440C2×Q8—32no—
441C2×D4—272no—
442C2×D4—542no—
443C22×C4—271no—
444C22×C4—542no—
445Q16—32no—
446Q16—31no—
447Q16D3×C3⋊D311yescharacteristic, complemented
448SD16—92no—
449SD16—272no—
450SD16—272no—
451SD16—272no—
452D8—91no—
453D8—272no—
454D8—271no—
455C2×C8—31no—
456C2×C8—91no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C2×C8—271no—
461C22×C3—34no—
462C22×C3—91no—
463C22×C3—182no—
464C2×D3—34no—
465C2×D3—68no—
466C2×D3—94no—
467C2×D3—91no—
468C2×D3—184no—
469C2×D3—188no—
470C2×D3—182no—
471C2×D3—188no—
472C2×D3—182no—
473C2×D3—368no—
474C2×D3—3616no—
475C2×D3—364no—
476C2×D3D4×C3⋊D311yescharacteristic
477C3×C4—28no—
478C3×C4—22no—
479C3×C4—24no—
480C3×C4—34no—
481C3×C4—48no—
482C3×C4—68no—
483C3×C4—91no—
484C3×C4C22×D3214yes—
485C3×C4C23×C3⋊D311yescharacteristic
486Dic3—22no—
487Dic3—34no—
488Dic3—94no—
489Dic3—91no—
490Dic3—184no—
491Dic3D4×C3⋊D311yescharacteristic
492C32—24no—
493C32C2×#91914yescomplemented
494C32D3×#13511yescharacteristic, complemented
495C23—271no—
496C23—542no—
497Q8—32no—
498Q8—62no—
499Q8C2×D3×C3⋊D312yes—
500D4—92no—
501D4—182no—
502D4—272no—
503D4—542no—
504D4—542no—
505D4—542no—
506D4—542no—
507C2×C4—31no—
508C2×C4—62no—
509C2×C4—91no—
510C2×C4—182no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—542no—
516C2×C4—542no—
517C2×C4—542no—
518C8—31no—
519C8—91no—
520C8—271no—
521C8C2×D3×C3⋊D311yescharacteristic
522C2×C3—24no—
523C2×C3—38no—
524C2×C3—181no—
525C2×C3—362no—
526C2×C3D32×D414yes—
527C2×C3C2×D4×C3⋊D311yescharacteristic
528D3—64no—
529D3—128no—
530D3—181no—
531D3—184no—
532D3—364no—
533D3—362no—
534D3—368no—
535D3—728no—
536D3#6412812yesdirect product factor
537C22—31no—
538C22—91no—
539C22—182no—
540C22—271no—
541C22—542no—
542C22—542no—
543C22—1084no—
544C4—22no—
545C4—31no—
546C4—62no—
547C4—91no—
548C4—271no—
549C4C22×D3×C3⋊D311yescharacteristic
550C3—24no—
551C3D3×#91914yescomplemented
552C3C2×#6412811yescharacteristic, complemented
553C2—32no—
554C2—181no—
555C2—362no—
556C2—541no—
557C2—1082no—
558C2D3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2D3×D4×C3⋊D3
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C22×D4
hypercenterQ16D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. D3×#3832C2semidirect
2. #11723960C2semidirect
3. C3×#64128C2semidirect
4. C32×D3×Q16C2semidirect
5. #11723589C2semidirect
6. #11723585C2semidirect
7. D3×#3755C2semidirect
8. #11723345C2semidirect
9. #11723341C2semidirect
10. D3×#3745C2semidirect
11. D3×C8×C3⋊D3C2semidirect
14. #89080C22semidirect
15. C3×#3832C22semidirect
22. #89018C22semidirect
24. #89017C22semidirect
25. C33×Q16C22semidirect
26. #88862C22semidirect
27. #88852C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3755C22semidirect
30. C3×#3745C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#216C22semidirect
33. C32×Dic12C22semidirect
35. #88795C22semidirect
36. #88793C22semidirect
37. #88791C22semidirect
38. #88785C22semidirect
39. C3⋊D3×C3⋊C8C22semidirect
52. #64128D3direct (N × H)
56. C3×D3×Q16D3semidirect
101. #3832C2×D3semidirect
119. C32×Q16C2×D3semidirect
120. C32×Q16C2×D3semidirect
127. #3755C2×D3semidirect
133. #3745C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
140. C3×#216C2×D3semidirect
142. C3×Dic12C2×D3semidirect
179. D3×Q16C3⋊D3semidirect
255. C32×D3#135semidirect
282. C3×Q16D32semidirect
283. C3×Q16C2×C3⋊D3semidirect
288. #216C2×C3⋊D3semidirect
299. Dic12C2×C3⋊D3semidirect
357. C33C2×#135semidirect
433. C3×D3#919semidirect
447. Q16D3×C3⋊D3semidirect
493. C32C2×#919semidirect
494. C32D3×#135semidirect
536. D3#64128direct (N × H)
551. C3D3×#919semidirect
552. C3C2×#64128semidirect

Maximal subgroups

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

subgroupindex
D3×#38322
#117239602
C3×#641282
C32×D3×Q162
#117235892
#117235852
D3×#37552
#117233452
#117233412
D3×#37452
D3×C8×C3⋊D32
C2×#641283
D3×#9193