Finite Group Databasebeta

← D3×#64113 #13176635

Subgroups — D3×#64113 #13176635

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×#3828C212yesmaximal, complemented
2#11723945C211yesmaximal, characteristic, complemented
3C3×#64113C211yesmaximal, characteristic, complemented
4C2×C32×D3×C8C211yesmaximal, characteristic, complemented
5#11723369C212yesmaximal, complemented
6#11723337C211yesmaximal, characteristic, complemented
7#11723336C211yesmaximal, characteristic, complemented
8#11723335C212yesmaximal, complemented
9D3×#3746C211yesmaximal, characteristic, complemented
10D3×#3745C211yesmaximal, characteristic, complemented
11D3×#3744C212yesmaximal, complemented
12C2×#64113—31nomaximal
13D3×#904—34nomaximal
14#89076C2212yescomplemented
15C3×#3828C2212yescomplemented
16C2×C32×C4×D3C2211yescharacteristic
17D3×#638—22no—
18#89031—22no—
19D3×C4⋊(C3⋊D3)C2212yes—
20C4×D3×C3⋊D3—22no—
21#89022C2212yescomplemented
22#89021C2212yescomplemented
23#89020—22no—
24D3×#634C2212yes—
25C2×C33×C8C2211yescharacteristic, complemented
26#88853C2211yescharacteristic, complemented
27#88852C2211yescharacteristic, complemented
28#88851C2212yescomplemented
29C3×#3746C2211yescharacteristic, complemented
30C3×#3745C2211yescharacteristic, complemented
31C3×#3744C2212yescomplemented
32C2×C32×C3⋊C8C2211yescharacteristic, complemented
33C32×D3×C8C2211yescharacteristic
34C32×D3×C8C2211yescharacteristic
35C32×D3×C8C2211yescharacteristic, complemented
36C32×D3×C8C2211yescharacteristic, complemented
37#88800C2212yescomplemented
38#88797C2212yescomplemented
39#88796C2212yescomplemented
40#88791C2212yescomplemented
41C2×#3828—32no—
42D3×#235—38no—
43C22×C32×C8—31no—
44C2×#3746—31no—
45#64113—32no—
46#64113—31no—
47#64113—31no—
48#64113—31no—
49#64113—32no—
50#64113—34no—
51#64113—64no—
52#64113D311yescharacteristic, direct product factor
53C2×#3745—31no—
54C2×#3744—32no—
55C3×#904—34no—
56C2×C3×D3×C8D314yescomplemented
57#63829—38no—
58#63808—34no—
59#63807—34no—
60#63806—38no—
61D3×Dic12—34no—
62D3×D24—34no—
63D3×#204—38no—
64C22×C32×D3D411yescharacteristic
65C2×D3×C3⋊D3—22no—
66C2×C33×C4C2311yescharacteristic
67#6312—22no—
68C4⋊(C3⋊(C3⋊D3))C2312yes—
69C4×C3⋊(C3⋊D3)—22no—
70#6308C2312yes—
71C3×#638—22no—
72C3×C4⋊(C3⋊D3)C2312yes—
73C3×C4×C3⋊D3—22no—
74C3×#634C2312yes—
75C2×C32×Dic3D411yescharacteristic
76C32×C4×D3D411yescharacteristic
77C32×C4×D3D411yescharacteristic
78C32×C4×D3C2311yescharacteristic
79C32×C4×D3C2311yescharacteristic
80#6293—22no—
81#6292—22no—
82#6291—22no—
83#6290—22no—
84#6289—22no—
85Dic3×C3⋊D3—22no—
86D3×C3⋊Dic3—22no—
87C33×C8C2311yescharacteristic
88C33×C8C2311yescharacteristic
89C32×C3⋊C8C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91D3×#135—91no—
92C2×#904—94no—
93C22×C32×C4—31no—
94C2×#638—62no—
95#3828—32no—
96#3828—38no—
97#3828—32no—
98#3828—32no—
99#3828—62no—
100#3828—62no—
101#3828—68no—
102#3828C2×D312yescomplemented
103C2×C4⋊(C3⋊D3)—32no—
104C2×C4×C3⋊D3—62no—
105C2×#634—32no—
106C3×#235—38no—
107C2×C3×C4×D3C2×D314yes—
108D3×#67—68no—
109#3804—68no—
110D3×D12—38no—
111C4×D32—68no—
112#3799—38no—
113#3798—68no—
114#3795—38no—
115D3×Dic6—38no—
116C2×C32×C8—24no—
117C2×C32×C8—31no—
118C2×C32×C8—31no—
119C2×C32×C8—31no—
120C2×C32×C8—31no—
121C2×C32×C8—31no—
122C2×C32×C8C2×D314yescomplemented
123C2×C32×C8C2×D311yescharacteristic, complemented
124#3746—32no—
125#3746—31no—
126#3746—34no—
127#3746—64no—
128#3746C2×D311yescharacteristic, complemented
129#3745—32no—
130#3745—31no—
131#3745—34no—
132#3745—64no—
133#3745C2×D311yescharacteristic, complemented
134#3744—32no—
135#3744—32no—
136#3744—32no—
137#3744—38no—
138#3744—68no—
139#3744C2×D312yescomplemented
140C2×C3×C3⋊C8C2×D314yescomplemented
141C3×Dic12—34no—
142C3×D24—34no—
143C3×#204—38no—
144C3×D3×C8C2×D314yes—
145C3×D3×C8C2×D314yes—
146C3×D3×C8C2×D314yescomplemented
147C3×D3×C8C2×D314yescomplemented
148#3719—38no—
149#3717—38no—
150#3716—38no—
151#3714—38no—
152C22×C33C2×D411yescharacteristic
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3×C3⋊D3—22no—
155C2×C32×D3—22no—
156C2×C32×D3C2×D411yescharacteristic
157C2×C32×D3C2×D411yescharacteristic
158D3×C3⋊D3—44no—
159C33×C4C2×D411yescharacteristic
160C33×C4C2411yescharacteristic
161C3⋊(C3⋊Dic3)—22no—
162C3×C3⋊Dic3—22no—
163C32×Dic3C2×D411yescharacteristic
164C32×Dic3C2×D411yescharacteristic
165D3×#41—92no—
166C2×#235—98no—
167C3×#135—91no—
168C22×C3×C8—34no—
169#950—92no—
170#919—91no—
171D3×Q16—91no—
172#916—92no—
173D3×SD16—92no—
174#912—91no—
175D3×D8—91no—
176C2×Dic12—94no—
177#904—34no—
178#904—98no—
179#904—94no—
180#904—94no—
181#904—98no—
182#904—94no—
183#904—91no—
184#904—184no—
185C2×D24—94no—
186C2×#204—98no—
187C2×D3×C8C3⋊D311yescharacteristic, complemented
188C23×C32—31no—
189C22×C3⋊D3—62no—
190C22×C3×D3D1214yes—
191C2×D32—68no—
192C2×C32×C4—24no—
193C2×C32×C4—31no—
194C2×C32×C4—31no—
195C2×C32×C4—31no—
196C2×C32×C4—31no—
197C2×C32×C4—31no—
198C2×C32×C4C22×D314yes—
199C2×C32×C4C22×D311yescharacteristic
200#638—22no—
201#638—62no—
202#638—62no—
203#638—62no—
204#638—68no—
205#638—128no—
206C2×C3⋊Dic3—62no—
207C4⋊(C3⋊D3)—32no—
208C4⋊(C3⋊D3)—38no—
209C4⋊(C3⋊D3)—68no—
210C4⋊(C3⋊D3)—62no—
211C4⋊(C3⋊D3)C22×D312yes—
212C4×C3⋊D3—22no—
213C4×C3⋊D3—62no—
214C4×C3⋊D3—62no—
215C4×C3⋊D3—62no—
216C4×C3⋊D3—68no—
217C4×C3⋊D3—128no—
218#634—32no—
219#634—38no—
220#634—68no—
221#634—62no—
222#634C22×D312yes—
223C3×#67—68no—
224C2×C3×Dic3D1214yes—
225C3×D12—38no—
226C3×C4×D3—68no—
227C3×C4×D3D1214yes—
228C3×C4×D3D1214yes—
229C3×C4×D3C22×D314yes—
230C3×C4×D3C22×D314yes—
231C3×Dic6—38no—
232#627—68no—
233#626—68no—
234#626—68no—
235#625—68no—
236#624—68no—
237D3×Dic3—68no—
238D3×Dic3—68no—
239C32×C8—24no—
240C32×C8—24no—
241C32×C8—31no—
242C32×C8—31no—
243C32×C8C22×D314yes—
244C32×C8C22×D314yes—
245C32×C8C22×D311yescharacteristic
246C32×C8C22×D311yescharacteristic
247C3×C3⋊C8C22×D314yes—
248C3×C3⋊C8C22×D314yes—
249C2×#135—271noSylow
250C2×C33—21no—
251C2×C33C22×D411yescharacteristic
252C3⋊(C3⋊D3)—42no—
253C3×C3⋊D3—42no—
254C32×D3—21no—
255C32×D3#13512yescomplemented
256C3×#41—92no—
257C22×C3×C4—34no—
258C2×#67—188no—
259#239—92no—
260D3×Q8—92no—
261#237—92no—
262#237—182no—
263D3×D4—92no—
264D3×D4—182no—
265#235—38no—
266#235—98no—
267#235—98no—
268#235—98no—
269#235—92no—
270#235—188no—
271#235—188no—
272#235—188no—
273#235—182no—
274C2×D12—98no—
275C2×C4×D3—188no—
276C2×C4×D3—182no—
277C2×C4×D3C2×C3⋊D311yescharacteristic
278C2×Dic6—98no—
279C3×Q16—91no—
280C3×SD16—92no—
281C3×D8—91no—
282C2×C3×C8—24no—
283C2×C3×C8—34no—
284C2×C3×C8—34no—
285C2×C3×C8—34no—
286C2×C3×C8—34no—
287C2×C3×C8—34no—
288C2×C3×C8D3214yescomplemented
289C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
290#216—92no—
291#215—92no—
292#214—92no—
293#213—92no—
294C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
295Dic12—34no—
296Dic12—98no—
297Dic12—94no—
298Dic12—91no—
299Dic12—184no—
300D24—34no—
301D24—98no—
302D24—94no—
303D24—91no—
304D24—184no—
305#204—38no—
306#204—98no—
307#204—98no—
308#204—98no—
309#204—92no—
310#204—188no—
311D3×C8C2×C3⋊D311yescharacteristic
312D3×C8C2×C3⋊D311yescharacteristic
313D3×C8C2×C3⋊D311yescharacteristic, complemented
314D3×C8C2×C3⋊D311yescharacteristic, complemented
315C22×C32—24no—
316C22×C32—31no—
317C22×C32—31no—
318C22×C32—62no—
319C22×C32C2×D1214yes—
320C22×C32D3×D411yescharacteristic
321C2×C3⋊D3—22no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—128no—
325C2×C3⋊D3—124no—
326C2×C3×D3—28no—
327C2×C3×D3—68no—
328C2×C3×D3C2×D1214yes—
329C2×C3×D3C2×D1214yes—
330D32—1216no—
331C32×C4—24no—
332C32×C4—24no—
333C32×C4—31no—
334C32×C4—31no—
335C32×C4C2×D1214yes—
336C32×C4D3×D411yescharacteristic
337C32×C4C23×D314yes—
338C32×C4C23×D311yescharacteristic
339C3⋊Dic3—22no—
340C3⋊Dic3—62no—
341C3⋊Dic3—68no—
342C3⋊Dic3—128no—
343C3×Dic3—68no—
344C3×Dic3C2×D1214yes—
345C3×Dic3C2×D1214yes—
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—271no—
356C22×C8—31no—
357C33C2×#13511yescharacteristic, complemented, Sylow
358C23×C3—34no—
359C22×D3—188no—
360C22×D3—182no—
361C22×D3C4⋊(C3⋊D3)11yescharacteristic
362C3×Q8—92no—
363C3×D4—92no—
364C3×D4—182no—
365C2×C3×C4—24no—
366C2×C3×C4—34no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—34no—
370C2×C3×C4—34no—
371C2×C3×C4—182no—
372C2×C3×C4C2×D3214yes—
373C2×C3×C4C22×C3⋊D311yescharacteristic
374#67—68no—
375#67—188no—
376#67—188no—
377#67—188no—
378#67—182no—
379#67—182no—
380#67—182no—
381#67—368no—
382C2×Dic3—188no—
383C2×Dic3—182no—
384C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
385D12—38no—
386D12—98no—
387D12—92no—
388D12—188no—
389D12—188no—
390D12—182no—
391C4×D3—68no—
392C4×D3—188no—
393C4×D3—188no—
394C4×D3—182no—
395C4×D3—188no—
396C4×D3—182no—
397C4×D3—182no—
398C4×D3—368no—
399C4×D3C4⋊(C3⋊D3)11yescharacteristic
400C4×D3C4⋊(C3⋊D3)11yescharacteristic
401C4×D3C22×C3⋊D311yescharacteristic
402C4×D3C22×C3⋊D311yescharacteristic
403Dic6—38no—
404Dic6—98no—
405Dic6—92no—
406Dic6—188no—
407Dic6—188no—
408Dic6—182no—
409C3×C8—24no—
410C3×C8—24no—
411C3×C8—34no—
412C3×C8—34no—
413C3×C8C2×D3214yes—
414C3×C8C2×D3214yes—
415C3×C8C22×C3⋊D311yescharacteristic
416C3×C8C22×C3⋊D311yescharacteristic
417C3⋊C8C22×C3⋊D311yescharacteristic
418C3⋊C8C22×C3⋊D311yescharacteristic
419C2×C32—21no—
420C2×C32—24no—
421C2×C32—24no—
422C2×C32—32no—
423C2×C32—44no—
424C2×C32—61no—
425C2×C32C22×D1214yes—
426C2×C32C2×D3×D411yescharacteristic
427C3⋊D3—42no—
428C3⋊D3—122no—
429C3⋊D3—128no—
430C3⋊D3—248no—
431C3×D3—24no—
432C3×D3—128no—
433C3×D3#90418yescomplemented
434#41—92no—
435#41—272no—
436#41—272no—
437#41—272no—
438#41—542no—
439#41—542no—
440C2×Q8—272no—
441C2×D4—272no—
442C2×D4—542no—
443C22×C4—31no—
444C22×C4—542no—
445Q16—91no—
446Q16—272no—
447Q16—271no—
448SD16—92no—
449SD16—272no—
450SD16—272no—
451SD16—272no—
452D8—91no—
453D8—272no—
454D8—271no—
455C2×C8—31no—
456C2×C8—31no—
457C2×C8—31no—
458C2×C8—31no—
459C2×C8—31no—
460C2×C8D3×C3⋊D311yescharacteristic, complemented
461C22×C3—24no—
462C22×C3—34no—
463C22×C3—34no—
464C22×C3—68no—
465C22×C3—182no—
466C22×C3D3×D1214yes—
467C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
468C2×D3—22no—
469C2×D3—68no—
470C2×D3—188no—
471C2×D3—182no—
472C2×D3—368no—
473C2×D3—3616no—
474C2×D3—364no—
475C2×D3C2×C4⋊(C3⋊D3)11yescharacteristic
476C2×D3C2×C4⋊(C3⋊D3)11yescharacteristic
477C3×C4—24no—
478C3×C4—24no—
479C3×C4—34no—
480C3×C4—34no—
481C3×C4—182no—
482C3×C4D3×D1214yes—
483C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
484C3×C4C22×D3214yes—
485C3×C4C23×C3⋊D311yescharacteristic
486Dic3—68no—
487Dic3—188no—
488Dic3—182no—
489Dic3—368no—
490Dic3C2×C4⋊(C3⋊D3)11yescharacteristic
491Dic3C2×C4⋊(C3⋊D3)11yescharacteristic
492C32—24no—
493C32C2×#90414yescomplemented
494C32D3×#13511yescharacteristic, complemented
495C23—31no—
496C23—542no—
497Q8—92no—
498Q8—272no—
499Q8—542no—
500D4—92no—
501D4—182no—
502D4—272no—
503D4—542no—
504D4—542no—
505D4—542no—
506D4—542no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—182no—
513C2×C4—542no—
514C2×C4—542no—
515C2×C4—542no—
516C2×C4—542no—
517C2×C4C2×D3×C3⋊D311yescharacteristic
518C8—31no—
519C8—31no—
520C8C2×D3×C3⋊D311yescharacteristic
521C8C2×D3×C3⋊D311yescharacteristic
522C2×C3—24no—
523C2×C3—21no—
524C2×C3—24no—
525C2×C3—38no—
526C2×C3—44no—
527C2×C3—64no—
528C2×C3—362no—
529C2×C3C2×D3×D1214yes—
530C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
531D3—21no—
532D3—128no—
533D3—368no—
534D3—362no—
535D3—728no—
536D3#6411312yesdirect product factor
537C22—31no—
538C22—31no—
539C22—62no—
540C22—182no—
541C22—542no—
542C22—1084no—
543C22D3×C4⋊(C3⋊D3)11yescharacteristic
544C4—31no—
545C4—31no—
546C4—182no—
547C4—542no—
548C4D3×C4⋊(C3⋊D3)11yescharacteristic
549C4C22×D3×C3⋊D311yescharacteristic
550C3—24no—
551C3D3×#90414yescomplemented
552C3C2×#6411311yescharacteristic, complemented
553C2—21no—
554C2—32no—
555C2—61no—
556C2—362no—
557C2—1082no—
558C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC4D3×C4⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C22×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. D3×#3828C2semidirect
2. #11723945C2semidirect
3. C3×#64113C2semidirect
4. C2×C32×D3×C8C2semidirect
5. #11723369C2semidirect
6. #11723337C2semidirect
7. #11723336C2semidirect
8. #11723335C2semidirect
9. D3×#3746C2semidirect
10. D3×#3745C2semidirect
11. D3×#3744C2semidirect
14. #89076C22semidirect
15. C3×#3828C22semidirect
21. #89022C22semidirect
22. #89021C22semidirect
25. C2×C33×C8C22semidirect
26. #88853C22semidirect
27. #88852C22semidirect
28. #88851C22semidirect
29. C3×#3746C22semidirect
30. C3×#3745C22semidirect
31. C3×#3744C22semidirect
32. C2×C32×C3⋊C8C22semidirect
35. C32×D3×C8C22semidirect
36. C32×D3×C8C22semidirect
37. #88800C22semidirect
38. #88797C22semidirect
39. #88796C22semidirect
40. #88791C22semidirect
52. #64113D3direct (N × H)
56. C2×C3×D3×C8D3semidirect
102. #3828C2×D3semidirect
122. C2×C32×C8C2×D3semidirect
123. C2×C32×C8C2×D3semidirect
128. #3746C2×D3semidirect
133. #3745C2×D3semidirect
139. #3744C2×D3semidirect
140. C2×C3×C3⋊C8C2×D3semidirect
146. C3×D3×C8C2×D3semidirect
147. C3×D3×C8C2×D3semidirect
187. C2×D3×C8C3⋊D3semidirect
255. C32×D3#135semidirect
288. C2×C3×C8D32semidirect
289. C2×C3×C8C2×C3⋊D3semidirect
294. C2×C3⋊C8C2×C3⋊D3semidirect
313. D3×C8C2×C3⋊D3semidirect
314. D3×C8C2×C3⋊D3semidirect
357. C33C2×#135semidirect
433. C3×D3#904semidirect
460. C2×C8D3×C3⋊D3semidirect
493. C32C2×#904semidirect
494. C32D3×#135semidirect
536. D3#64113direct (N × H)
551. C3D3×#904semidirect
552. C3C2×#64113semidirect

Maximal subgroups

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

subgroupindex
D3×#38282
#117239452
C3×#641132
C2×C32×D3×C82
#117233692
#117233372
#117233362
#117233352
D3×#37462
D3×#37452
D3×#37442
C2×#641133
D3×#9043