Finite Group Databasebeta

← #13176645

Subgroups — #13176645

Order 1728.

1900 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724578C212yesmaximal, complemented
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64113C211yesmaximal, characteristic, complemented
4C32×#904C211yesmaximal, characteristic, complemented
5#11723363C212yesmaximal, complemented
6#11723345C211yesmaximal, characteristic, complemented
7#11723343C211yesmaximal, characteristic, complemented
8#11723342C212yesmaximal, complemented
9#11723330C211yesmaximal, characteristic, complemented
10#11723322C211yesmaximal, characteristic, complemented
11#11723321C212yesmaximal, complemented
12C3⋊D3×#135—31nomaximal
13#10594684—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C3×#3828C2212yescomplemented
16C32×#235C2212yescomplemented
17#89039—22no—
18#89029—22no—
19#89026C2212yes—
20C4×D3×C3⋊D3—22no—
21#89020—22no—
22#89019C2212yes—
23#89017C2212yescomplemented
24#89015C2212yescomplemented
25C2×C33×C8C2211yescharacteristic, complemented
26C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
27C8×C3⋊(C3⋊D3)C2211yescharacteristic
28C8×C3⋊(C3⋊D3)C2211yescharacteristic
29C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
30C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
31C3×#3746C2211yescharacteristic, complemented
32C3×#3745C2211yescharacteristic, complemented
33C3×#3744C2212yescomplemented
34C32×Dic12C2211yescharacteristic, complemented
35C32×D24C2211yescharacteristic, complemented
36C32×#204C2212yescomplemented
37#88798C2212yescomplemented
38#88793C2212yescomplemented
39#88792C2212yescomplemented
40#88789C2212yescomplemented
41#41×C3⋊D3—32no—
42#64306—38no—
43C32×#135—31no—
44#64159—32no—
45#64128—31no—
46Q16×C3⋊D3—31no—
47#64125—32no—
48SD16×C3⋊D3—32no—
49#64121—31no—
50D8×C3⋊D3—31no—
51#64113D311yescharacteristic, complemented
52C2×C8×C3⋊D3—31no—
53C2×C8×C3⋊D3—34no—
54C2×C8×C3⋊D3—64no—
55C3×#904—34no—
56C3×#904D314yescomplemented
57#63822—38no—
58#63810—34no—
59#63810—34no—
60#63809—38no—
61#63802—34no—
62#63797—34no—
63#63796—38no—
64C22×C3⋊(C3⋊D3)D411yescharacteristic
65C2×D3×C3⋊D3—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)D411yescharacteristic
68C4×C3⋊(C3⋊D3)D411yescharacteristic
69C4×C3⋊(C3⋊D3)D411yescharacteristic
70C4×C3⋊(C3⋊D3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)C2311yescharacteristic
72C3×#638—22no—
73C3×C4⋊(C3⋊D3)C2312yes—
74C3×C4×C3⋊D3—22no—
75C3×#634C2312yes—
76C32×#67—22no—
77C32×D12C2312yes—
78C32×C4×D3—22no—
79C32×Dic6C2312yes—
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
89C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
90C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
91D3×#135—94no—
92D3×#135—91no—
93C32×#41—32no—
94#3832—32no—
95Q8×C3⋊D3—32no—
96#3830—32no—
97#3830—62no—
98D4×C3⋊D3—32no—
99D4×C3⋊D3—62no—
100#3828—62no—
101#3828C2×D312yescomplemented
102C2×C4×C3⋊D3—31no—
103C2×C4×C3⋊D3—34no—
104C2×C4×C3⋊D3—62no—
105C2×C4×C3⋊D3—64no—
106C3×#235—38no—
107C3×#235C2×D318yescomplemented
108#3811—68no—
109#3805—68no—
110#3802—38no—
111C4×D32—68no—
112#3798—68no—
113#3797—38no—
114#3796—38no—
115#3796—38no—
116C32×Q16—31no—
117C32×SD16—32no—
118C32×D8—31no—
119C2×C32×C8—24no—
120C2×C32×C8C2×D314yescomplemented
121C2×C32×C8C2×D311yescharacteristic, complemented
122#3756—32no—
123#3755—32no—
124#3754—32no—
125#3753—32no—
126C2×C3⋊(C3⋊C8)—31no—
127C2×C3⋊(C3⋊C8)—34no—
128C2×C3⋊(C3⋊C8)—64no—
129#3746C2×D311yescharacteristic, complemented
130#3745C2×D311yescharacteristic, complemented
131#3744C2×D312yescomplemented
132C8×C3⋊D3—31no—
133C8×C3⋊D3—31no—
134C8×C3⋊D3—31no—
135C8×C3⋊D3—31no—
136C8×C3⋊D3—34no—
137C8×C3⋊D3—34no—
138C8×C3⋊D3—34no—
139C8×C3⋊D3—34no—
140C8×C3⋊D3—64no—
141C8×C3⋊D3—64no—
142C8×C3⋊D3—64no—
143C8×C3⋊D3—64no—
144C3×Dic12—34no—
145C3×Dic12C2×D314yescomplemented
146C3×D24—34no—
147C3×D24C2×D314yescomplemented
148C3×#204—38no—
149C3×#204C2×D318yescomplemented
150#3718—38no—
151#3715—38no—
152#3715—38no—
153#3713—38no—
154C22×C33C2×D411yescharacteristic
155C2×C3⋊(C3⋊D3)—22no—
156C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
157C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
158C2×C3×C3⋊D3—22no—
159C2×C32×D3—22no—
160D3×C3⋊D3—44no—
161C33×C4C2×D411yescharacteristic
162C33×C4C2411yescharacteristic
163C3⋊(C3⋊Dic3)C2×D411yescharacteristic
164C3⋊(C3⋊Dic3)C2×D411yescharacteristic
165C3×C3⋊Dic3—22no—
166C32×Dic3—22no—
167D3×#41—98no—
168D3×#41—92no—
169C3×#135—34no—
170C3×#135—91no—
171#950—98no—
172#950—92no—
173#919—94no—
174#919—91no—
175D3×Q16—94no—
176D3×Q16—91no—
177#916—98no—
178#916—92no—
179D3×SD16—98no—
180D3×SD16—92no—
181#912—94no—
182#912—91no—
183D3×D8—94no—
184D3×D8—91no—
185#904—34no—
186#904C3⋊D311yescharacteristic, complemented
187C2×D3×C8—94no—
188C2×D3×C8—91no—
189C2×D3×C8—184no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—62no—
194C2×D32—68no—
195C32×Q8—32no—
196C32×D4—32no—
197C32×D4—62no—
198C2×C32×C4—24no—
199C2×C32×C4—62no—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D311yescharacteristic
202#638—22no—
203#638—62no—
204#638—62no—
205C2×C3⋊Dic3—31no—
206C2×C3⋊Dic3—34no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—62no—
209C4⋊(C3⋊D3)—62no—
210C4⋊(C3⋊D3)C22×D312yes—
211C4×C3⋊D3—22no—
212C4×C3⋊D3—31no—
213C4×C3⋊D3—31no—
214C4×C3⋊D3—31no—
215C4×C3⋊D3—31no—
216C4×C3⋊D3—34no—
217C4×C3⋊D3—34no—
218C4×C3⋊D3—34no—
219C4×C3⋊D3—34no—
220C4×C3⋊D3—64no—
221C4×C3⋊D3—64no—
222C4×C3⋊D3—64no—
223C4×C3⋊D3—64no—
224C4×C3⋊D3—62no—
225C4×C3⋊D3—62no—
226#634—62no—
227#634C22×D312yes—
228C3×#67—28no—
229C3×#67—68no—
230C3×D12—38no—
231C3×D12C22×D318yes—
232C3×C4×D3—28no—
233C3×C4×D3—68no—
234C3×Dic6—38no—
235C3×Dic6C22×D318yes—
236#627—68no—
237#626—68no—
238#626—68no—
239#625—68no—
240#624—68no—
241D3×Dic3—68no—
242D3×Dic3—68no—
243C32×C8—24no—
244C32×C8—24no—
245C32×C8C22×D314yes—
246C32×C8C22×D314yes—
247C32×C8C22×D311yescharacteristic
248C32×C8C22×D311yescharacteristic
249C3⋊(C3⋊C8)—31no—
250C3⋊(C3⋊C8)—31no—
251C3⋊(C3⋊C8)—34no—
252C3⋊(C3⋊C8)—34no—
253C3⋊(C3⋊C8)—64no—
254C3⋊(C3⋊C8)—64no—
255C2×#135—271noSylow
256C2×C33—21no—
257C2×C33C22×D411yescharacteristic
258C3⋊(C3⋊D3)—21no—
259C3⋊(C3⋊D3)#13512yescomplemented
260C3×C3⋊D3—42no—
261C32×D3—42no—
262C3×#41—38no—
263C3×#41—92no—
264#239—98no—
265#239—92no—
266D3×Q8—98no—
267D3×Q8—92no—
268#237—98no—
269#237—92no—
270#237—182no—
271#237—188no—
272D3×D4—98no—
273D3×D4—92no—
274D3×D4—182no—
275D3×D4—188no—
276#235—38no—
277#235—182no—
278#235—188no—
279#235C2×C3⋊D312yescomplemented
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×C4×D3—184no—
283C2×C4×D3—182no—
284C2×C4×D3—188no—
285C3×Q16—34no—
286C3×Q16—91no—
287C3×SD16—38no—
288C3×SD16—92no—
289C3×D8—34no—
290C3×D8—91no—
291C2×C3×C8—24no—
292C2×C3×C8D3214yescomplemented
293C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
294#216—98no—
295#216—92no—
296#215—98no—
297#215—92no—
298#214—98no—
299#214—92no—
300#213—98no—
301#213—92no—
302C2×C3⋊C8—94no—
303C2×C3⋊C8—91no—
304C2×C3⋊C8—184no—
305Dic12—34no—
306Dic12C2×C3⋊D311yescharacteristic, complemented
307D24—34no—
308D24C2×C3⋊D311yescharacteristic, complemented
309#204—38no—
310#204C2×C3⋊D312yescomplemented
311D3×C8—94no—
312D3×C8—94no—
313D3×C8—94no—
314D3×C8—94no—
315D3×C8—91no—
316D3×C8—91no—
317D3×C8—91no—
318D3×C8—91no—
319D3×C8—184no—
320D3×C8—184no—
321D3×C8—184no—
322D3×C8—184no—
323C22×C32—24no—
324C22×C32—62no—
325C22×C32D3×D414yes—
326C22×C32D3×D411yescharacteristic
327C2×C3⋊D3—22no—
328C2×C3⋊D3—31no—
329C2×C3⋊D3—31no—
330C2×C3⋊D3—34no—
331C2×C3⋊D3—34no—
332C2×C3⋊D3—64no—
333C2×C3⋊D3—64no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—68no—
336C2×C3⋊D3—616no—
337C2×C3⋊D3—124no—
338C2×C3×D3—28no—
339C2×C3×D3—68no—
340D32—1216no—
341C32×C4—24no—
342C32×C4—24no—
343C32×C4—62no—
344C32×C4D3×D414yes—
345C32×C4D3×D411yescharacteristic
346C32×C4C23×D314yes—
347C32×C4C23×D311yescharacteristic
348C3⋊Dic3—22no—
349C3⋊Dic3—31no—
350C3⋊Dic3—31no—
351C3⋊Dic3—34no—
352C3⋊Dic3—34no—
353C3⋊Dic3—64no—
354C3⋊Dic3—64no—
355C3×Dic3—28no—
356C3×Dic3—68no—
357C2×#41—272no—
358#135—31no—
359#135—91no—
360#135—271no—
361#135—271no—
362#135—272no—
363#135—272no—
364C2×Q16—271no—
365C2×SD16—272no—
366C2×D8—271no—
367C22×C8—271no—
368C33C2×#13511yescharacteristic, complemented, Sylow
369C22×D3—94no—
370C22×D3—91no—
371C22×D3—184no—
372C22×D3—182no—
373C22×D3—188no—
374C3×Q8—38no—
375C3×Q8—92no—
376C3×D4—38no—
377C3×D4—68no—
378C3×D4—92no—
379C3×D4—182no—
380C2×C3×C4—24no—
381C2×C3×C4—68no—
382C2×C3×C4—182no—
383C2×C3×C4C2×D3214yes—
384C2×C3×C4C22×C3⋊D311yescharacteristic
385#67—22no—
386#67—68no—
387#67—182no—
388#67—188no—
389#67—182no—
390#67—188no—
391C2×Dic3—94no—
392C2×Dic3—91no—
393C2×Dic3—184no—
394C2×Dic3—182no—
395C2×Dic3—188no—
396D12—38no—
397D12—188no—
398D12—182no—
399D12C22×C3⋊D312yes—
400C4×D3—22no—
401C4×D3—68no—
402C4×D3—94no—
403C4×D3—94no—
404C4×D3—91no—
405C4×D3—91no—
406C4×D3—94no—
407C4×D3—94no—
408C4×D3—91no—
409C4×D3—91no—
410C4×D3—184no—
411C4×D3—184no—
412C4×D3—184no—
413C4×D3—184no—
414C4×D3—182no—
415C4×D3—188no—
416C4×D3—188no—
417C4×D3—182no—
418Dic6—38no—
419Dic6—182no—
420Dic6—188no—
421Dic6C22×C3⋊D312yes—
422C3×C8—24no—
423C3×C8—24no—
424C3×C8C2×D3214yes—
425C3×C8C2×D3214yes—
426C3×C8C22×C3⋊D311yescharacteristic
427C3×C8C22×C3⋊D311yescharacteristic
428C3⋊C8—94no—
429C3⋊C8—94no—
430C3⋊C8—91no—
431C3⋊C8—91no—
432C3⋊C8—184no—
433C3⋊C8—184no—
434C2×C32—28no—
435C2×C32—24no—
436C2×C32—21no—
437C2×C32—24no—
438C2×C32—122no—
439C2×C32C2×D3×D414yes—
440C2×C32C2×D3×D411yescharacteristic
441C3⋊D3—32no—
442C3⋊D3—38no—
443C3⋊D3—42no—
444C3⋊D3—68no—
445C3⋊D3—68no—
446C3⋊D3—64no—
447C3⋊D3—61no—
448C3×D3—48no—
449C3×D3—128no—
450#41—32no—
451#41—92no—
452#41—272no—
453#41—272no—
454#41—542no—
455#41—542no—
456C2×Q8—272no—
457C2×D4—272no—
458C2×D4—542no—
459C22×C4—271no—
460C22×C4—542no—
461Q16—31no—
462Q16—91no—
463Q16—272no—
464SD16—32no—
465SD16—92no—
466SD16—272no—
467SD16—272no—
468D8—31no—
469D8—91no—
470D8—272no—
471C2×C8—271no—
472C2×C8—271no—
473C2×C8—271no—
474C2×C8—271no—
475C2×C8—271no—
476C2×C8D3×C3⋊D311yescharacteristic, complemented
477C22×C3—24no—
478C22×C3—68no—
479C22×C3—182no—
480C22×C3#380214yes—
481C22×C3D4×C3⋊D311yescharacteristic
482C2×D3—22no—
483C2×D3—68no—
484C2×D3—94no—
485C2×D3—91no—
486C2×D3—94no—
487C2×D3—91no—
488C2×D3—184no—
489C2×D3—184no—
490C2×D3—188no—
491C2×D3—182no—
492C2×D3—1816no—
493C2×D3—364no—
494C2×D3—3616no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—68no—
498C3×C4—182no—
499C3×C4#380214yes—
500C3×C4D4×C3⋊D311yescharacteristic
501C3×C4C22×D3214yes—
502C3×C4C23×C3⋊D311yescharacteristic
503Dic3—22no—
504Dic3—68no—
505Dic3—94no—
506Dic3—94no—
507Dic3—91no—
508Dic3—91no—
509Dic3—184no—
510Dic3—184no—
511C32—24no—
512C32D3×#13514yescomplemented
513C32D3×#13511yescharacteristic, complemented
514C23—271no—
515C23—542no—
516Q8—32no—
517Q8—92no—
518Q8—542no—
519D4—32no—
520D4—62no—
521D4—92no—
522D4—182no—
523D4—542no—
524D4—542no—
525D4—542no—
526C2×C4—62no—
527C2×C4—182no—
528C2×C4—271no—
529C2×C4—271no—
530C2×C4—271no—
531C2×C4—271no—
532C2×C4—271no—
533C2×C4—542no—
534C2×C4—542no—
535C2×C4—542no—
536C2×C4C2×D3×C3⋊D311yescharacteristic
537C8—271no—
538C8—271no—
539C8C2×D3×C3⋊D311yescharacteristic
540C8C2×D3×C3⋊D311yescharacteristic
541C2×C3—28no—
542C2×C3—24no—
543C2×C3—21no—
544C2×C3—24no—
545C2×C3—128no—
546C2×C3—362no—
547C2×C3C2×#380214yes—
548C2×C3C2×D4×C3⋊D311yescharacteristic
549D3—42no—
550D3—98no—
551D3—92no—
552D3—128no—
553D3—188no—
554D3—188no—
555D3—184no—
556D3—181no—
557C22—62no—
558C22—182no—
559C22—271no—
560C22—271no—
561C22—542no—
562C22—1084no—
563C22#8902611yescharacteristic
564C4—62no—
565C4—182no—
566C4—271no—
567C4—271no—
568C4#8902611yescharacteristic
569C4C22×D3×C3⋊D311yescharacteristic
570C3—24no—
571C3#1059468414yescomplemented
572C3C3⋊D3×#13511yescharacteristic, complemented
573C2—21no—
574C2—122no—
575C2—272no—
576C2—362no—
577C2—541no—
578C2C2×#8902611yescharacteristic
subgroupG/H
centerC4#89026
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. #11724578C2semidirect
2. C2×C8×C3⋊(C3⋊D3)C2semidirect
3. C3×#64113C2semidirect
4. C32×#904C2semidirect
5. #11723363C2semidirect
6. #11723345C2semidirect
7. #11723343C2semidirect
8. #11723342C2semidirect
9. #11723330C2semidirect
10. #11723322C2semidirect
11. #11723321C2semidirect
15. C3×#3828C22semidirect
16. C32×#235C22semidirect
23. #89017C22semidirect
24. #89015C22semidirect
25. C2×C33×C8C22semidirect
26. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
29. C8×C3⋊(C3⋊D3)C22semidirect
30. C8×C3⋊(C3⋊D3)C22semidirect
31. C3×#3746C22semidirect
32. C3×#3745C22semidirect
33. C3×#3744C22semidirect
34. C32×Dic12C22semidirect
35. C32×D24C22semidirect
36. C32×#204C22semidirect
37. #88798C22semidirect
38. #88793C22semidirect
39. #88792C22semidirect
40. #88789C22semidirect
51. #64113D3semidirect
56. C3×#904D3semidirect
101. #3828C2×D3semidirect
107. C3×#235C2×D3semidirect
120. C2×C32×C8C2×D3semidirect
121. C2×C32×C8C2×D3semidirect
129. #3746C2×D3semidirect
130. #3745C2×D3semidirect
131. #3744C2×D3semidirect
145. C3×Dic12C2×D3semidirect
147. C3×D24C2×D3semidirect
149. C3×#204C2×D3semidirect
186. #904C3⋊D3semidirect
259. C3⋊(C3⋊D3)#135semidirect
279. #235C2×C3⋊D3semidirect
292. C2×C3×C8D32semidirect
293. C2×C3×C8C2×C3⋊D3semidirect
306. Dic12C2×C3⋊D3semidirect
308. D24C2×C3⋊D3semidirect
310. #204C2×C3⋊D3semidirect
368. C33C2×#135semidirect
476. C2×C8D3×C3⋊D3semidirect
512. C32D3×#135semidirect
513. C32D3×#135semidirect
571. C3#10594684semidirect
572. C3C3⋊D3×#135semidirect

Maximal subgroups

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

subgroupindex
#117245782
C2×C8×C3⋊(C3⋊D3)2
C3×#641132
C32×#9042
#117233632
#117233452
#117233432
#117233422
#117233302
#117233222
#117233212
C3⋊D3×#1353
#105946843