Finite Group Databasebeta

← #13176772

Subgroups — #13176772

Order 1728.

1836 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724596C212yesmaximal, complemented
2Q16×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64128C211yesmaximal, characteristic, complemented
4C32×#919C211yesmaximal, characteristic, complemented
5#11723583C212yesmaximal, complemented
6#11723580C212yesmaximal, complemented
7#11723571C212yesmaximal, complemented
8#11723339C211yesmaximal, characteristic, complemented
9#11723336C211yesmaximal, characteristic, complemented
10#11723322C211yesmaximal, characteristic, complemented
11D3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
12C3⋊D3×#135—31nomaximal
13#10594764—34nomaximal
14Q8×C3⋊(C3⋊D3)C2212yes—
15C3×#3832C2212yescomplemented
16C32×#239C2212yescomplemented
17#89026—22no—
18#89026C2212yes—
19C4×D3×C3⋊D3—22no—
20C4×D3×C3⋊D3C2211yescharacteristic
21#89021—22no—
22#89021C2212yescomplemented
23#89014—22no—
24#89014C2212yescomplemented
25C33×Q16C2211yescharacteristic, complemented
26#88863C2212yescomplemented
27#88853C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic
29C3×#3755C2212yescomplemented
30C3×#3745C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#215C2212yescomplemented
33C32×D24C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic, complemented
35#88796C2212yescomplemented
36#88794C2212yescomplemented
37#88789C2212yescomplemented
38#88785C2211yescharacteristic
39C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
40D3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
41#41×C3⋊D3—32no—
42#64319—38no—
43C32×#135—31no—
44#64159—32no—
45#64128D311yescharacteristic, complemented
46Q16×C3⋊D3—31no—
47Q16×C3⋊D3—34no—
48Q16×C3⋊D3—64no—
49#64125—32no—
50SD16×C3⋊D3—32no—
51#64121—31no—
52D8×C3⋊D3—31no—
53#64113—31no—
54C2×C8×C3⋊D3—31no—
55C3×#919—34no—
56C3×#919D314yescomplemented
57#63947—38no—
58#63947—38no—
59#63940—38no—
60#63807—34no—
61#63807—34no—
62#63797—34no—
63D32×C8—34no—
64C2×D3×C3⋊D3—22no—
65C2×D3×C3⋊D3D411yescharacteristic
66C33×Q8C2312yes—
67C4×C3⋊(C3⋊D3)—22no—
68C4×C3⋊(C3⋊D3)C2311yescharacteristic
69#6308—22no—
70#6308C2312yes—
71C3×C4⋊(C3⋊D3)—22no—
72C3×C4⋊(C3⋊D3)C2312yes—
73C3×C4×C3⋊D3—22no—
74C3×C4×C3⋊D3C2311yescharacteristic
75C32×D12—22no—
76C32×D12C2312yes—
77C32×C4×D3—22no—
78C32×C4×D3C2311yescharacteristic
79#6290—22no—
80#6290—22no—
81#6290—22no—
82#6289D411yescharacteristic
83Dic3×C3⋊D3—22no—
84Dic3×C3⋊D3D411yescharacteristic
85D3×C3⋊Dic3—22no—
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#3832C2×D312yescomplemented
95Q8×C3⋊D3—32no—
96Q8×C3⋊D3—38no—
97Q8×C3⋊D3—68no—
98#3830—32no—
99#3830—62no—
100D4×C3⋊D3—32no—
101D4×C3⋊D3—62no—
102#3828—32no—
103#3828—62no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—62no—
106C3×#239—38no—
107C3×#239C2×D318yescomplemented
108#3802—38no—
109#3802—68no—
110C4×D32—34no—
111C4×D32—68no—
112#3795—38no—
113#3795—38no—
114#3795—68no—
115#3795—68no—
116C32×Q16—24no—
117C32×Q16C2×D314yescomplemented
118C32×Q16C2×D311yescharacteristic, complemented
119C32×SD16—32no—
120C32×D8—31no—
121C2×C32×C8—31no—
122#3756—32no—
123#3756—38no—
124#3756—68no—
125#3755C2×D312yescomplemented
126#3754—32no—
127#3753—32no—
128C2×C3⋊(C3⋊C8)—31no—
129#3746—31no—
130#3746—34no—
131#3746—64no—
132#3745C2×D311yescharacteristic, complemented
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×#215—38no—
141C3×#215C2×D318yescomplemented
142C3×D24—34no—
143C3×D24C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yescomplemented
146#3716—38no—
147#3716—38no—
148#3713—38no—
149#3710—34no—
150D3×C3⋊C8—34no—
151D3×C3⋊C8—34no—
152C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3C2×D411yescharacteristic
155C2×C32×D3—22no—
156C2×C32×D3C2×D411yescharacteristic
157D3×C3⋊D3—22no—
158D3×C3⋊D3—44no—
159C33×C4—22no—
160C33×C4C2411yescharacteristic
161C3⋊(C3⋊Dic3)—22no—
162C3⋊(C3⋊Dic3)C2×D411yescharacteristic
163C3×C3⋊Dic3C2×D411yescharacteristic
164C32×Dic3C2×D411yescharacteristic
165D3×#41—98no—
166D3×#41—92no—
167C3×#135—34no—
168C3×#135—91no—
169#950—98no—
170#950—92no—
171#919—34no—
172#919C3⋊D311yescharacteristic, complemented
173D3×Q16—94no—
174D3×Q16—91no—
175D3×Q16—184no—
176#916—98no—
177#916—92no—
178D3×SD16—98no—
179D3×SD16—92no—
180#912—94no—
181#912—91no—
182D3×D8—94no—
183D3×D8—91no—
184#904—94no—
185#904—91no—
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×Q8C22×D318yes—
194C32×Q8C22×D312yes—
195C32×D4—32no—
196C32×D4—62no—
197C2×C32×C4—31no—
198C2×C32×C4—62no—
199#638—62no—
200#638—62no—
201#638—62no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—62no—
204C4⋊(C3⋊D3)—22no—
205C4⋊(C3⋊D3)C22×D312yes—
206C4×C3⋊D3—22no—
207C4×C3⋊D3—31no—
208C4×C3⋊D3—31no—
209C4×C3⋊D3—31no—
210C4×C3⋊D3—34no—
211C4×C3⋊D3—64no—
212C4×C3⋊D3—62no—
213C4×C3⋊D3—62no—
214C4×C3⋊D3—68no—
215C4×C3⋊D3—128no—
216C4×C3⋊D3C22×D311yescharacteristic
217#634—32no—
218#634—38no—
219#634—68no—
220#634—62no—
221#634—68no—
222#634—128no—
223C3×D12—28no—
224C3×D12—38no—
225C3×D12—68no—
226C3×D12C22×D318yes—
227C3×C4×D3—28no—
228C3×C4×D3—34no—
229C3×C4×D3—68no—
230C3×C4×D3C22×D314yes—
231#625—68no—
232#625—68no—
233#625—68no—
234#624—34no—
235D3×Dic3—34no—
236D3×Dic3—34no—
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×C33C22×D411yescharacteristic
251C3⋊(C3⋊D3)#13512yescomplemented
252C3×C3⋊D3—21no—
253C3×C3⋊D3—42no—
254C32×D3—21no—
255C32×D3—42no—
256C3×#41—38no—
257C3×#41—92no—
258#239—38no—
259#239C2×C3⋊D312yescomplemented
260D3×Q8—98no—
261D3×Q8—92no—
262D3×Q8—188no—
263#237—98no—
264#237—92no—
265#237—182no—
266#237—188no—
267D3×D4—98no—
268D3×D4—92no—
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—24no—
280C3×Q16D3214yescomplemented
281C3×Q16C2×C3⋊D311yescharacteristic, complemented
282C3×SD16—38no—
283C3×SD16—92no—
284C3×D8—34no—
285C3×D8—91no—
286C2×C3×C8—34no—
287C2×C3×C8—91no—
288#216—98no—
289#216—92no—
290#216—188no—
291#215—38no—
292#215C2×C3⋊D312yescomplemented
293#214—98no—
294#214—92no—
295#213—98no—
296#213—92no—
297C2×C3⋊C8—94no—
298C2×C3⋊C8—91no—
299Dic12—94no—
300Dic12—91no—
301Dic12—184no—
302D24—34no—
303D24C2×C3⋊D311yescharacteristic, complemented
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—31no—
316C22×C32—62no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—31no—
319C2×C3⋊D3—34no—
320C2×C3⋊D3—64no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3D3×D411yescharacteristic
324C2×C3×D3—28no—
325C2×C3×D3—34no—
326C2×C3×D3—68no—
327C2×C3×D3D3×D414yes—
328D32—68no—
329D32—1216no—
330C32×C4—22no—
331C32×C4—24no—
332C32×C4—28no—
333C32×C4—31no—
334C32×C4—48no—
335C32×C4C23×D314yes—
336C32×C4C23×D311yescharacteristic
337C3⋊Dic3—31no—
338C3⋊Dic3—34no—
339C3⋊Dic3—64no—
340C3⋊Dic3—62no—
341C3⋊Dic3—68no—
342C3⋊Dic3—128no—
343C3⋊Dic3D3×D411yescharacteristic
344C3×Dic3—34no—
345C3×Dic3D3×D414yes—
346C2×#41—272no—
347#135—31no—
348#135—91no—
349#135—271no—
350#135—271no—
351#135—272no—
352#135—272no—
353C2×Q16—271no—
354C2×SD16—272no—
355C2×D8—271no—
356C22×C8—271no—
357C33C2×#13511yescharacteristic, complemented, Sylow
358C22×D3—94no—
359C22×D3—91no—
360C22×D3—182no—
361C22×D3—188no—
362C3×Q8—28no—
363C3×Q8C2×D3218yes—
364C3×Q8C22×C3⋊D312yes—
365C3×D4—38no—
366C3×D4—68no—
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—188no—
375#67—182no—
376#67—188no—
377#67—188no—
378#67—182no—
379C2×Dic3—94no—
380C2×Dic3—91no—
381C2×Dic3—188no—
382C2×Dic3—182no—
383D12—22no—
384D12—38no—
385D12—68no—
386D12C22×C3⋊D312yes—
387C4×D3—22no—
388C4×D3—34no—
389C4×D3—68no—
390C4×D3—94no—
391C4×D3—94no—
392C4×D3—91no—
393C4×D3—94no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—184no—
397C4×D3—182no—
398C4×D3—188no—
399C4×D3—188no—
400C4×D3—182no—
401C4×D3—368no—
402C4×D3C22×C3⋊D311yescharacteristic
403Dic6—98no—
404Dic6—92no—
405Dic6—188no—
406Dic6—182no—
407Dic6—188no—
408Dic6—368no—
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—61no—
421C2×C32—122no—
422C2×C32C2×D3×D414yes—
423C2×C32C2×D3×D411yescharacteristic
424C3⋊D3—21no—
425C3⋊D3—32no—
426C3⋊D3—38no—
427C3⋊D3—42no—
428C3⋊D3—68no—
429C3×D3—24no—
430C3×D3—48no—
431C3×D3—64no—
432C3×D3—128no—
433#41—32no—
434#41—92no—
435#41—272no—
436#41—272no—
437#41—542no—
438#41—542no—
439C2×Q8—272no—
440C2×D4—272no—
441C2×D4—542no—
442C22×C4—271no—
443C22×C4—542no—
444Q16—272no—
445Q16—271no—
446Q16D3×C3⋊D311yescharacteristic, complemented
447SD16—32no—
448SD16—92no—
449SD16—272no—
450SD16—272no—
451D8—31no—
452D8—91no—
453D8—272no—
454C2×C8—31no—
455C2×C8—91no—
456C2×C8—271no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C22×C3—34no—
461C22×C3—68no—
462C22×C3—91no—
463C22×C3—182no—
464C2×D3—22no—
465C2×D3—34no—
466C2×D3—68no—
467C2×D3—94no—
468C2×D3—91no—
469C2×D3—184no—
470C2×D3—182no—
471C2×D3—188no—
472C2×D3—364no—
473C2×D3—3616no—
474C2×D3D4×C3⋊D311yescharacteristic
475C3×C4—28no—
476C3×C4—22no—
477C3×C4—24no—
478C3×C4—34no—
479C3×C4—48no—
480C3×C4—91no—
481C3×C4C22×D3214yes—
482C3×C4C23×C3⋊D311yescharacteristic
483Dic3—34no—
484Dic3—94no—
485Dic3—91no—
486Dic3—184no—
487Dic3—188no—
488Dic3—182no—
489Dic3—368no—
490Dic3D4×C3⋊D311yescharacteristic
491C32—24no—
492C32D3×#13514yescomplemented
493C32D3×#13511yescharacteristic, complemented
494C23—271no—
495C23—542no—
496Q8—272no—
497Q8—542no—
498Q8C2×D3×C3⋊D312yes—
499D4—32no—
500D4—62no—
501D4—92no—
502D4—182no—
503D4—542no—
504D4—542no—
505D4—542no—
506C2×C4—31no—
507C2×C4—62no—
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—
517C8—31no—
518C8—91no—
519C8—271no—
520C8C2×D3×C3⋊D311yescharacteristic
521C2×C3—24no—
522C2×C3—64no—
523C2×C3—128no—
524C2×C3—181no—
525C2×C3—362no—
526C2×C3D32×D414yes—
527C2×C3C2×D4×C3⋊D311yescharacteristic
528D3—21no—
529D3—42no—
530D3—64no—
531D3—92no—
532D3—98no—
533D3—128no—
534D3—188no—
535C22—31no—
536C22—62no—
537C22—91no—
538C22—182no—
539C22—271no—
540C22—542no—
541C22—1084no—
542C4—22no—
543C4—31no—
544C4—91no—
545C4—271no—
546C4—542no—
547C4C22×D3×C3⋊D311yescharacteristic
548C3—24no—
549C3#1059476414yescomplemented
550C3C3⋊D3×#13511yescharacteristic, complemented
551C2—61no—
552C2—122no—
553C2—181no—
554C2—272no—
555C2—362no—
556C2D3×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. #11724596C2semidirect
2. Q16×C3⋊(C3⋊D3)C2semidirect
3. C3×#64128C2semidirect
4. C32×#919C2semidirect
5. #11723583C2semidirect
6. #11723580C2semidirect
7. #11723571C2semidirect
8. #11723339C2semidirect
9. #11723336C2semidirect
10. #11723322C2semidirect
11. D3×C8×C3⋊D3C2semidirect
15. C3×#3832C22semidirect
16. C32×#239C22semidirect
22. #89021C22semidirect
24. #89014C22semidirect
25. C33×Q16C22semidirect
26. #88863C22semidirect
27. #88853C22semidirect
29. C3×#3755C22semidirect
30. C3×#3745C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#215C22semidirect
33. C32×D24C22semidirect
34. C32×D3×C8C22semidirect
35. #88796C22semidirect
36. #88794C22semidirect
37. #88789C22semidirect
39. C3⋊D3×C3⋊C8C22semidirect
40. D3×C3⋊(C3⋊C8)C22semidirect
45. #64128D3semidirect
56. C3×#919D3semidirect
94. #3832C2×D3semidirect
107. C3×#239C2×D3semidirect
117. C32×Q16C2×D3semidirect
118. C32×Q16C2×D3semidirect
125. #3755C2×D3semidirect
132. #3745C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
141. C3×#215C2×D3semidirect
143. C3×D24C2×D3semidirect
145. C3×D3×C8C2×D3semidirect
172. #919C3⋊D3semidirect
251. C3⋊(C3⋊D3)#135semidirect
259. #239C2×C3⋊D3semidirect
280. C3×Q16D32semidirect
281. C3×Q16C2×C3⋊D3semidirect
292. #215C2×C3⋊D3semidirect
303. D24C2×C3⋊D3semidirect
314. D3×C8C2×C3⋊D3semidirect
357. C33C2×#135semidirect
446. Q16D3×C3⋊D3semidirect
492. C32D3×#135semidirect
493. C32D3×#135semidirect
549. C3#10594764semidirect
550. C3C3⋊D3×#135semidirect

Maximal subgroups

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

subgroupindex
#117245962
Q16×C3⋊(C3⋊D3)2
C3×#641282
C32×#9192
#117235832
#117235802
#117235712
#117233392
#117233362
#117233222
D3×C8×C3⋊D32
C3⋊D3×#1353
#105947643