Finite Group Databasebeta

← D3×#64121 #13176718

Subgroups — D3×#64121 #13176718

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×#3830C212yesmaximal, complemented
2#11723953C211yesmaximal, characteristic, complemented
3C3×#64121C211yesmaximal, characteristic, complemented
4C32×D3×D8C211yesmaximal, characteristic, complemented
5#11723563C212yesmaximal, complemented
6#11723559C212yesmaximal, complemented
7D3×#3754C212yesmaximal, complemented
8#11723343C211yesmaximal, characteristic, complemented
9#11723339C211yesmaximal, characteristic, complemented
10D3×#3746C211yesmaximal, characteristic, complemented
11D3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
12C2×#64121—31nomaximal
13D3×#912—34nomaximal
14#89078C2212yescomplemented
15C3×#3830C2212yescomplemented
16C32×D3×D4C2212yes—
17D3×#638—22no—
18#89029—22no—
19#89028—22no—
20C2×D3×C3⋊Dic3—22no—
21C4×D3×C3⋊D3C2211yescharacteristic
22#89015C2212yescomplemented
23#89014C2212yescomplemented
24D3×#634C2212yes—
25C33×D8C2211yescharacteristic, complemented
26#88861C2212yescomplemented
27#88853C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
29C3×#3754C2212yescomplemented
30C3×#3746C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#213C2212yescomplemented
33C32×D24C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic
35#88800C2212yescomplemented
36#88794C2212yescomplemented
37#88792C2212yescomplemented
38#88785C2211yescharacteristic, complemented
39C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
40D3×C3⋊(C3⋊C8)C2211yescharacteristic
41C2×#3830—32no—
42D3×#237—38no—
43C2×C32×D8—31no—
44C2×#3754—32no—
45#64121—32no—
46#64121—31no—
47#64121—32no—
48#64121—31no—
49#64121—31no—
50#64121—34no—
51#64121—64no—
52#64121D311yescharacteristic, direct product factor
53C2×#3746—31no—
54C2×C8×C3⋊D3—31no—
55C3×#912—34no—
56C3×D3×D8D314yescomplemented
57#63936—38no—
58#63933—38no—
59D3×#214—38no—
60#63810—34no—
61#63807—34no—
62D3×Dic12—34no—
63D32×C8—34no—
64C22×C32×D3—22no—
65C2×D3×C3⋊D3D411yescharacteristic
66C33×D4C2312yes—
67#6312—22no—
68C2×C3⋊(C3⋊Dic3)—22no—
69C4×C3⋊(C3⋊D3)C2311yescharacteristic
70#6308C2312yes—
71C3×#638—22no—
72C2×C3×C3⋊Dic3—22no—
73C3×C4×C3⋊D3C2311yescharacteristic
74C3×#634C2312yes—
75C32×#67—22no—
76C32×D12C2312yes—
77C32×C4×D3C2311yescharacteristic
78#6293—22no—
79#6291—22no—
80#6290—22no—
81#6289D411yescharacteristic
82Dic3×C3⋊D3D411yescharacteristic
83D3×C3⋊Dic3—22no—
84D3×C3⋊Dic3—22no—
85D3×C3⋊Dic3—22no—
86D3×C3⋊Dic3D411yescharacteristic
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91D3×#135—91no—
92C2×#912—94no—
93C2×C32×D4—32no—
94C2×#638—62no—
95C22×C3⋊Dic3—62no—
96#3830—32no—
97#3830—38no—
98#3830—32no—
99#3830—32no—
100#3830—62no—
101#3830—62no—
102#3830—68no—
103#3830C2×D312yescomplemented
104C2×C4×C3⋊D3—31no—
105C2×#634—32no—
106C3×#237—38no—
107C3×D3×D4C2×D318yes—
108D3×#67—68no—
109#3805—68no—
110#3804—68no—
111C2×D3×Dic3—68no—
112C4×D32—34no—
113#3796—38no—
114#3795—38no—
115D3×Dic6—38no—
116C32×D8—24no—
117C32×D8—32no—
118C32×D8—31no—
119C32×D8C2×D314yescomplemented
120C32×D8C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3754—32no—
123#3754—38no—
124#3754—32no—
125#3754—32no—
126#3754—68no—
127#3754C2×D312yescomplemented
128C2×C3⋊(C3⋊C8)—31no—
129#3746—32no—
130#3746—31no—
131#3746—34no—
132#3746—64no—
133#3746C2×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×#214—38no—
141C3×#213C2×D318yescomplemented
142C3×Dic12—34no—
143C3×D24C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yes—
146#3719—38no—
147#3716—38no—
148#3715—38no—
149#3710—34no—
150D3×C3⋊C8—34no—
151D3×C3⋊C8—34no—
152C22×C33—22no—
153C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
154C2×C3×C3⋊D3C2×D411yescharacteristic
155C2×C32×D3—22no—
156C2×C32×D3—44no—
157C2×C32×D3C2×D411yescharacteristic
158D3×C3⋊D3—22no—
159C33×C4C2411yescharacteristic
160C3⋊(C3⋊Dic3)—22no—
161C3⋊(C3⋊Dic3)C2×D411yescharacteristic
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3C2×D411yescharacteristic
164C32×Dic3C2×D411yescharacteristic
165D3×#41—92no—
166C2×#237—98no—
167C3×#135—91no—
168C2×C3×D8—34no—
169#950—92no—
170C2×#214—98no—
171#919—91no—
172D3×Q16—91no—
173#916—92no—
174D3×SD16—92no—
175#912—34no—
176#912—94no—
177#912—94no—
178#912—98no—
179#912—98no—
180#912—94no—
181#912—91no—
182#912—184no—
183D3×D8C3⋊D311yescharacteristic, complemented
184C2×Dic12—94no—
185#904—91no—
186C2×D3×C8—94no—
187C2×D3×C8—91no—
188C23×C32—62no—
189C22×C3⋊D3—31no—
190C22×C3×D3—28no—
191C2×D32—34no—
192C32×D4—28no—
193C32×D4—32no—
194C32×D4—62no—
195C32×D4C22×D318yes—
196C32×D4C22×D312yes—
197C2×C32×C4—31no—
198#638—22no—
199#638—62no—
200#638—62no—
201#638—62no—
202#638—68no—
203#638—128no—
204C2×C3⋊Dic3—22no—
205C2×C3⋊Dic3—31no—
206C2×C3⋊Dic3—62no—
207C2×C3⋊Dic3—62no—
208C2×C3⋊Dic3—62no—
209C2×C3⋊Dic3—62no—
210C2×C3⋊Dic3—68no—
211C2×C3⋊Dic3—128no—
212C4×C3⋊D3—31no—
213C4×C3⋊D3—31no—
214C4×C3⋊D3—31no—
215C4×C3⋊D3—34no—
216C4×C3⋊D3—64no—
217C4×C3⋊D3C22×D311yescharacteristic
218#634—32no—
219#634—38no—
220#634—68no—
221#634—62no—
222#634C22×D312yes—
223C3×#67—28no—
224C3×#67—68no—
225C2×C3×Dic3—68no—
226C3×D12C22×D318yes—
227C3×C4×D3—34no—
228C3×C4×D3C22×D314yes—
229C3×Dic6—38no—
230#627—68no—
231#626—68no—
232#625—68no—
233#624—34no—
234D3×Dic3—34no—
235D3×Dic3—34no—
236D3×Dic3—68no—
237D3×Dic3—68no—
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×C33—42no—
251C2×C33C22×D411yescharacteristic
252C3⋊(C3⋊D3)—21no—
253C3×C3⋊D3—21no—
254C32×D3—42no—
255C32×D3#13512yescomplemented
256C3×#41—92no—
257C2×C3×D4—38no—
258C2×#67—188no—
259C22×Dic3—188no—
260#239—92no—
261D3×Q8—92no—
262#237—38no—
263#237—98no—
264#237—98no—
265#237—98no—
266#237—92no—
267#237—188no—
268#237—188no—
269#237—188no—
270#237—182no—
271D3×D4—182no—
272D3×D4C2×C3⋊D312yes—
273#235—92no—
274#235—182no—
275C2×C4×D3—94no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×Dic6—98no—
279C3×Q16—91no—
280C3×SD16—92no—
281C3×D8—24no—
282C3×D8—38no—
283C3×D8—34no—
284C3×D8D3214yescomplemented
285C3×D8C2×C3⋊D311yescharacteristic, complemented
286C2×C3×C8—34no—
287C2×C3×C8—91no—
288#216—92no—
289#215—92no—
290#214—38no—
291#214—98no—
292#214—98no—
293#214—98no—
294#214—92no—
295#214—188no—
296#213C2×C3⋊D312yescomplemented
297C2×C3⋊C8—94no—
298C2×C3⋊C8—91no—
299Dic12—34no—
300Dic12—98no—
301Dic12—94no—
302Dic12—91no—
303Dic12—184no—
304D24C2×C3⋊D311yescharacteristic, complemented
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—22no—
316C22×C32—28no—
317C22×C32—31no—
318C22×C32—48no—
319C22×C32—62no—
320C22×C32—124no—
321C2×C3⋊D3—31no—
322C2×C3⋊D3—34no—
323C2×C3⋊D3—64no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3D3×D411yescharacteristic
326C2×C3×D3—28no—
327C2×C3×D3—34no—
328C2×C3×D3—416no—
329C2×C3×D3D3×D414yes—
330D32—68no—
331C32×C4—24no—
332C32×C4—31no—
333C32×C4C23×D314yes—
334C32×C4C23×D311yescharacteristic
335C3⋊Dic3—22no—
336C3⋊Dic3—31no—
337C3⋊Dic3—34no—
338C3⋊Dic3—64no—
339C3⋊Dic3—62no—
340C3⋊Dic3—68no—
341C3⋊Dic3—128no—
342C3⋊Dic3D3×D411yescharacteristic
343C3×Dic3—34no—
344C3×Dic3—68no—
345C3×Dic3D3×D414yes—
346C2×#41—272no—
347#135—91no—
348#135—271no—
349#135—272no—
350#135—271no—
351#135—272no—
352#135—271no—
353C2×Q16—271no—
354C2×SD16—272no—
355C2×D8—31no—
356C22×C8—271no—
357C33C2×#13511yescharacteristic, complemented, Sylow
358C23×C3—68no—
359C22×D3—22no—
360C22×D3—94no—
361C22×D3—91no—
362C3×Q8—92no—
363C3×D4—28no—
364C3×D4—38no—
365C3×D4—68no—
366C3×D4—182no—
367C3×D4C2×D3218yes—
368C3×D4C22×C3⋊D312yes—
369C2×C3×C4—34no—
370C2×C3×C4—91no—
371C2×C3×C4—182no—
372#67—22no—
373#67—68no—
374#67—188no—
375#67—182no—
376#67—188no—
377#67—182no—
378#67—188no—
379#67—368no—
380C2×Dic3—68no—
381C2×Dic3—94no—
382C2×Dic3—91no—
383C2×Dic3—188no—
384C2×Dic3—188no—
385C2×Dic3—188no—
386C2×Dic3—182no—
387C2×Dic3—188no—
388C2×Dic3—368no—
389D12—182no—
390D12C22×C3⋊D312yes—
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—182no—
402C4×D3C22×C3⋊D311yescharacteristic
403Dic6—38no—
404Dic6—98no—
405Dic6—92no—
406Dic6—188no—
407Dic6—188no—
408Dic6—182no—
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×C32—42no—
422C2×C32—48no—
423C2×C32—88no—
424C2×C32—122no—
425C2×C32C2×D3×D414yes—
426C2×C32C2×D3×D411yescharacteristic
427C3⋊D3—21no—
428C3⋊D3—61no—
429C3⋊D3—64no—
430C3⋊D3—124no—
431C3×D3—48no—
432C3×D3—64no—
433C3×D3#91218yescomplemented
434#41—92no—
435#41—272no—
436#41—272no—
437#41—272no—
438#41—542no—
439#41—542no—
440C2×Q8—272no—
441C2×D4—32no—
442C2×D4—542no—
443C22×C4—271no—
444C22×C4—542no—
445Q16—91no—
446Q16—272no—
447Q16—271no—
448SD16—92no—
449SD16—272no—
450SD16—272no—
451SD16—272no—
452D8—32no—
453D8—31no—
454D8D3×C3⋊D311yescharacteristic, complemented
455C2×C8—31no—
456C2×C8—91no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C2×C8—271no—
461C22×C3—28no—
462C22×C3—22no—
463C22×C3—34no—
464C22×C3—48no—
465C22×C3—68no—
466C22×C3—91no—
467C22×C3—1216no—
468C2×D3—22no—
469C2×D3—34no—
470C2×D3—44no—
471C2×D3—94no—
472C2×D3—91no—
473C2×D3—184no—
474C2×D3—188no—
475C2×D3—182no—
476C2×D3D4×C3⋊D311yescharacteristic
477C3×C4—24no—
478C3×C4—34no—
479C3×C4—91no—
480C3×C4—182no—
481C3×C4C22×D3214yes—
482C3×C4C23×C3⋊D311yescharacteristic
483Dic3—34no—
484Dic3—68no—
485Dic3—94no—
486Dic3—91no—
487Dic3—184no—
488Dic3—188no—
489Dic3—182no—
490Dic3—368no—
491Dic3D4×C3⋊D311yescharacteristic
492C32—24no—
493C32C2×#91214yescomplemented
494C32D3×#13511yescharacteristic, complemented
495C23—62no—
496C23—271no—
497Q8—92no—
498Q8—272no—
499Q8—542no—
500D4—32no—
501D4—62no—
502D4—182no—
503D4—542no—
504D4—542no—
505D4—542no—
506D4C2×D3×C3⋊D312yes—
507C2×C4—31no—
508C2×C4—91no—
509C2×C4—182no—
510C2×C4—271no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—542no—
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—42no—
525C2×C3—48no—
526C2×C3—88no—
527C2×C3—128no—
528C2×C3—181no—
529C2×C3D32×D414yes—
530C2×C3C2×D4×C3⋊D311yescharacteristic
531D3—42no—
532D3—64no—
533D3—181no—
534D3—184no—
535D3—364no—
536D3#6412112yesdirect product factor
537C22—22no—
538C22—31no—
539C22—62no—
540C22—91no—
541C22—124no—
542C22—271no—
543C22—542no—
544C4—31no—
545C4—91no—
546C4—182no—
547C4—271no—
548C4—542no—
549C4C22×D3×C3⋊D311yescharacteristic
550C3—24no—
551C3D3×#91214yescomplemented
552C3C2×#6412111yescharacteristic, complemented
553C2—32no—
554C2—42no—
555C2—122no—
556C2—181no—
557C2—541no—
558C2D3×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. D3×#3830C2semidirect
2. #11723953C2semidirect
3. C3×#64121C2semidirect
4. C32×D3×D8C2semidirect
5. #11723563C2semidirect
6. #11723559C2semidirect
7. D3×#3754C2semidirect
8. #11723343C2semidirect
9. #11723339C2semidirect
10. D3×#3746C2semidirect
11. D3×C8×C3⋊D3C2semidirect
14. #89078C22semidirect
15. C3×#3830C22semidirect
22. #89015C22semidirect
23. #89014C22semidirect
25. C33×D8C22semidirect
26. #88861C22semidirect
27. #88853C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3754C22semidirect
30. C3×#3746C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#213C22semidirect
33. C32×D24C22semidirect
35. #88800C22semidirect
36. #88794C22semidirect
37. #88792C22semidirect
38. #88785C22semidirect
39. C3⋊D3×C3⋊C8C22semidirect
52. #64121D3direct (N × H)
56. C3×D3×D8D3semidirect
103. #3830C2×D3semidirect
119. C32×D8C2×D3semidirect
120. C32×D8C2×D3semidirect
127. #3754C2×D3semidirect
133. #3746C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
141. C3×#213C2×D3semidirect
143. C3×D24C2×D3semidirect
183. D3×D8C3⋊D3semidirect
255. C32×D3#135semidirect
284. C3×D8D32semidirect
285. C3×D8C2×C3⋊D3semidirect
296. #213C2×C3⋊D3semidirect
304. D24C2×C3⋊D3semidirect
357. C33C2×#135semidirect
433. C3×D3#912semidirect
454. D8D3×C3⋊D3semidirect
493. C32C2×#912semidirect
494. C32D3×#135semidirect
536. D3#64121direct (N × H)
551. C3D3×#912semidirect
552. C3C2×#64121semidirect

Maximal subgroups

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

subgroupindex
D3×#38302
#117239532
C3×#641212
C32×D3×D82
#117235632
#117235592
D3×#37542
#117233432
#117233392
D3×#37462
D3×C8×C3⋊D32
C2×#641213
D3×#9123