Finite Group Databasebeta

← #13178002

Subgroups — #13178002

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic, complemented
2#11724566C211yesmaximal, characteristic, complemented
3#11723991C211yesmaximal, characteristic, complemented
4C32×#951C211yesmaximal, characteristic, complemented
5#11723585C212yesmaximal, complemented
6#11723576C212yesmaximal, complemented
7#11723562C212yesmaximal, complemented
8C3⋊D3×#214C212yesmaximal, complemented
9C2×#88793C211yesmaximal, characteristic, complemented
10#11723370C211yesmaximal, characteristic, complemented
11#11723354C211yesmaximal, characteristic, complemented
12#10596249—31nomaximal
13#10595464—34nomaximal
14C33×#41C2211yescharacteristic
15#89076C2211yescharacteristic, complemented
16C3×#3832C2212yes—
17C3×D4×C3⋊D3—22no—
18C3×D4×C3⋊D3C2212yes—
19C3×#3828—21no—
20C3×#3828C2211yescharacteristic
21C2×C3×C4⋊(C3⋊D3)—21no—
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C32×Dic6C2211yescharacteristic, complemented
24#89031—22no—
25#89019—21no—
26#89018C2212yescomplemented
27#89017—21no—
28C3⋊D3×Dic6C2212yescomplemented
29#88863C2211yescharacteristic, complemented
30#88862C2211yescharacteristic, complemented
31#88861C2211yescharacteristic, complemented
32#88860C2211yescharacteristic, complemented
33C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
34C32×#216C2212yescomplemented
35C32×#214C2212yescomplemented
36C32×#208C2211yescharacteristic, complemented
37#88796C2212yescomplemented
38#88793C2212yescomplemented
39#88793C2212yescomplemented
40#88791C2212yescomplemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64366D311yescharacteristic, complemented
44#64364—31no—
45C3×#1009—34no—
46#64297—34no—
47C32×#137—31no—
48#64159—31no—
49#64159—34no—
50#64159—64no—
51C2×#3755—31no—
52#64128—32no—
53#64127—32no—
54#64123—32no—
55SD16×C3⋊D3—32no—
56#64117—31no—
57#64116—31no—
58C3×#951D314yescomplemented
59#63949—38no—
60#63943—38no—
61#63934—38no—
62D3×#214—38no—
63#63830—34no—
64C2×#3715—34no—
65#63815—34no—
66C22×C3×C3⋊D3—21no—
67C22×C3×C3⋊D3—22no—
68C33×Q8C2311yescharacteristic
69C33×D4—21no—
70C33×D4C2311yescharacteristic
71C2×C33×C4—21no—
72C2×C33×C4C2311yescharacteristic
73#6312—21no—
74C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—21no—
76#6308C2311yescharacteristic
77C3×#638—22no—
78C3×#638D412yes—
79C3×C4⋊(C3⋊D3)—22no—
80C3×C4⋊(C3⋊D3)—21no—
81C3×C4⋊(C3⋊D3)D411yescharacteristic
82C3×C4⋊(C3⋊D3)C2312yes—
83C3×C4×C3⋊D3—22no—
84C3×C4×C3⋊D3C2312yes—
85C3×#634D411yescharacteristic
86C2×C32×Dic3—21no—
87C32×Dic6—21no—
88C32×Dic6C2312yes—
89#6293—22no—
90#6292—22no—
91#6289—22no—
92Dic3×C3⋊D3—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C32×C3⋊C8C2312yes—
96#5836—91no—
97#5778—94no—
98C32×#41—24no—
99C32×#41C2×D314yes—
100C32×#41C2×D311yescharacteristic
101C2×C32×Q8—31no—
102#3832—32no—
103#3832—61no—
104#3832C2×D312yes—
105Q8×C3⋊D3—32no—
106Q8×C3⋊D3—61no—
107D4×C3⋊D3—22no—
108D4×C3⋊D3C2×D312yes—
109#3828—21no—
110#3828—31no—
111#3828—34no—
112#3828—62no—
113#3828—64no—
114#3828C2×D311yescharacteristic
115C2×C4⋊(C3⋊D3)—21no—
116C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
117C3×#239—38no—
118C3×D3×D4—38no—
119C3×D3×D4—68no—
120C3×#235—34no—
121C3×#235—64no—
122C2×C3×D12—34no—
123C2×C3×D12—64no—
124C2×C3×Dic6C2×D314yescomplemented
125#3804—68no—
126#3799—38no—
127#3797—64no—
128#3796—64no—
129D3×Dic6—38no—
130C32×Q16—32no—
131C32×SD16—32no—
132C32×M16—31no—
133#3756—31no—
134#3756—34no—
135#3756—64no—
136#3755—31no—
137#3755—34no—
138#3755—32no—
139#3755—32no—
140#3755—64no—
141#3754—31no—
142#3754—34no—
143#3754—64no—
144#3753—31no—
145#3753—34no—
146#3753—64no—
147C2×C3⋊(C3⋊C8)—31no—
148C2×C3⋊(C3⋊C8)—34no—
149C2×C3⋊(C3⋊C8)—64no—
150#3745—32no—
151#3744—32no—
152#3743—32no—
153C8×C3⋊D3—32no—
154C3×#216C2×D318yescomplemented
155C3×#214C2×D318yescomplemented
156C3×#208C2×D314yescomplemented
157#3716—38no—
158#3715—38no—
159#3715—38no—
160#3714—38no—
161#3711—38no—
162D3×C3⋊C8—38no—
163C22×C33—21no—
164C22×C33C2×D411yescharacteristic
165C2×C3⋊(C3⋊D3)—21no—
166C2×C3×C3⋊D3—22no—
167C2×C3×C3⋊D3—82no—
168C2×C3×C3⋊D3—81no—
169C2×C3×C3⋊D3C2×D412yes—
170C33×C4—21no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C3×C3⋊Dic3C2×D412yes—
175C32×Dic3—22no—
176C3×H(5,𝔽2)—91no—
177#1010—91no—
178#1009—34no—
179#1007—94no—
180C3×#137—34no—
181#951—92no—
182#951C3⋊D311yescharacteristic, complemented
183#950—94no—
184#950—91no—
185#950—92no—
186#950—184no—
187#948—91no—
188C2×#215—94no—
189C2×#214—91no—
190C2×#214—92no—
191#932—92no—
192#932—91no—
193#919—98no—
194#918—98no—
195#914—98no—
196D3×SD16—98no—
197#908—94no—
198#907—94no—
199C22×C3⋊D3—21no—
200C22×C3⋊D3—22no—
201C22×C3×D3—68no—
202C22×C3×D3—64no—
203C32×Q8—24no—
204C32×Q8—32no—
205C32×Q8—61no—
206C32×Q8C22×D314yes—
207C32×Q8C22×D311yescharacteristic
208C32×D4—21no—
209C32×D4—24no—
210C32×D4—24no—
211C32×D4—44no—
212C32×D4C22×D314yes—
213C32×D4C22×D311yescharacteristic
214C2×C32×C4—21no—
215C2×C32×C4—24no—
216C2×C32×C4—24no—
217C2×C32×C4—44no—
218C2×C32×C4—61no—
219C2×C32×C4C22×D314yes—
220C2×C32×C4C22×D311yescharacteristic
221#638—22no—
222#638—61no—
223#638—64no—
224#638—124no—
225#638#6712yes—
226C4⋊(C3⋊D3)—22no—
227C4⋊(C3⋊D3)—21no—
228C4⋊(C3⋊D3)—31no—
229C4⋊(C3⋊D3)—34no—
230C4⋊(C3⋊D3)—64no—
231C4⋊(C3⋊D3)—62no—
232C4⋊(C3⋊D3)#6711yescharacteristic
233C4⋊(C3⋊D3)C22×D312yes—
234C4×C3⋊D3—22no—
235C4×C3⋊D3—62no—
236C4×C3⋊D3—61no—
237C4×C3⋊D3—62no—
238C4×C3⋊D3—64no—
239C4×C3⋊D3—124no—
240C4×C3⋊D3C22×D312yes—
241#634—31no—
242#634—34no—
243#634—64no—
244#634—62no—
245#634#6711yescharacteristic
246C3×#67—38no—
247C3×#67—68no—
248C2×C3×Dic3—24no—
249C3×D12—34no—
250C3×D12—38no—
251C3×D12—68no—
252C3×D12—64no—
253C3×C4×D3—38no—
254C3×C4×D3—68no—
255C3×Dic6—24no—
256C3×Dic6—34no—
257C3×Dic6C22×D318yes—
258#627—68no—
259#626—68no—
260#624—68no—
261D3×Dic3—68no—
262C32×C8—32no—
263C3⋊(C3⋊C8)—31no—
264C3⋊(C3⋊C8)—31no—
265C3⋊(C3⋊C8)—34no—
266C3⋊(C3⋊C8)—34no—
267C3⋊(C3⋊C8)—64no—
268C3⋊(C3⋊C8)—64no—
269C3×C3⋊C8C22×D318yes—
270#577—271noSylow
271C2×C33—21no—
272C2×C33—41no—
273C2×C33C22×D411yescharacteristic
274C3⋊(C3⋊D3)—41no—
275C3×C3⋊D3—22no—
276C3×C3⋊D3—42no—
277C3×#41—24no—
278C3×#41—91no—
279C3×#41—92no—
280C3×#41—181no—
281C3×#41D3214yes—
282C3×#41C2×C3⋊D311yescharacteristic
283C2×C3×Q8—34no—
284C2×C3×D4—92no—
285C2×C3×D4—91no—
286C2×C3×D4—181no—
287C2×C3×D4—182no—
288#239—38no—
289#239—98no—
290#239—184no—
291D3×Q8—98no—
292D3×Q8—181no—
293D3×Q8—184no—
294#237—181no—
295#237—182no—
296D3×D4—38no—
297D3×D4—68no—
298#235—34no—
299#235—64no—
300#235—94no—
301#235—91no—
302#235—92no—
303#235—184no—
304#235—188no—
305C2×D12—34no—
306C2×D12—64no—
307C2×Dic6—92no—
308C2×Dic6C2×C3⋊D311yescharacteristic, complemented
309C3×Q16—38no—
310C3×SD16—38no—
311C3×M16—34no—
312#216—94no—
313#216—91no—
314#216—184no—
315#216C2×C3⋊D312yescomplemented
316#215—98no—
317#215—98no—
318#215—94no—
319#215—91no—
320#215—184no—
321#214—94no—
322#214—91no—
323#214—92no—
324#214—92no—
325#214—92no—
326#214—184no—
327#214C2×C3⋊D312yescomplemented
328#213—94no—
329#213—91no—
330#213—92no—
331#213—184no—
332#208—92no—
333#208C2×C3⋊D311yescharacteristic, complemented
334C2×C3⋊C8—94no—
335C2×C3⋊C8—91no—
336C2×C3⋊C8—92no—
337C2×C3⋊C8—184no—
338D24—98no—
339#204—98no—
340ΓL(1,𝔽25)—98no—
341D3×C8—98no—
342C22×C32—21no—
343C22×C32—24no—
344C22×C32—24no—
345C22×C32—44no—
346C22×C32D3×D414yes—
347C22×C32C2×#6711yescharacteristic
348C2×C3⋊D3—22no—
349C2×C3⋊D3—61no—
350C2×C3⋊D3—64no—
351C2×C3⋊D3—82no—
352C2×C3⋊D3—81no—
353C2×C3⋊D3—124no—
354C2×C3⋊D3C2×#6712yes—
355C2×C3×D3—38no—
356C2×C3×D3—68no—
357C2×C3×D3—248no—
358C2×C3×D3—244no—
359C32×C4—21no—
360C32×C4—24no—
361C32×C4—24no—
362C32×C4—24no—
363C32×C4—44no—
364C32×C4—62no—
365C32×C4D3×D414yes—
366C32×C4C2×#6711yescharacteristic
367C32×C4C23×D314yes—
368C32×C4C23×D311yescharacteristic
369C3⋊Dic3—61no—
370C3⋊Dic3—64no—
371C3⋊Dic3—124no—
372C3⋊Dic3C2×#6712yes—
373C3×Dic3—28no—
374C3×Dic3—38no—
3752−1+4—271no—
376H(5,𝔽2)—91no—
377#137—31no—
378#137—272no—
379AGL(1,ℤ/8)—271no—
380AGL(1,ℤ/8)—272no—
381#135—272no—
382#135—271no—
383C2×SD16—272no—
384C2×SD16—271no—
385#131—271no—
386C33#57711yescharacteristic, complemented, Sylow
387C23×C3—182no—
388C23×C3—181no—
389C22×D3—68no—
390C22×D3—64no—
391C3×Q8—24no—
392C3×Q8—38no—
393C3×Q8—64no—
394C3×Q8—91no—
395C3×Q8C2×D3214yes—
396C3×Q8C22×C3⋊D311yescharacteristic
397C3×D4—24no—
398C3×D4—21no—
399C3×D4—24no—
400C3×D4—44no—
401C3×D4—92no—
402C3×D4—92no—
403C3×D4—91no—
404C3×D4—182no—
405C3×D4—182no—
406C3×D4—181no—
407C3×D4C2×D3214yes—
408C3×D4C22×C3⋊D311yescharacteristic
409C2×C3×C4—24no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—44no—
413C2×C3×C4—64no—
414C2×C3×C4—92no—
415C2×C3×C4—182no—
416C2×C3×C4C2×D3214yes—
417C2×C3×C4C22×C3⋊D311yescharacteristic
418#67—38no—
419#67—68no—
420#67—182no—
421#67—184no—
422#67—181no—
423#67—364no—
424C2×Dic3—21no—
425C2×Dic3—182no—
426D12—38no—
427D12—34no—
428D12—68no—
429D12—64no—
430D12—94no—
431D12—91no—
432D12—184no—
433D12—188no—
434C4×D3—38no—
435C4×D3—68no—
436C4×D3—182no—
437C4×D3—188no—
438C4×D3—188no—
439C4×D3—184no—
440C4×D3—181no—
441C4×D3—364no—
442Dic6—21no—
443Dic6—34no—
444Dic6—94no—
445Dic6—91no—
446Dic6—184no—
447Dic6—182no—
448Dic6—188no—
449Dic6C22×C3⋊D312yes—
450C3×C8—38no—
451C3⋊C8—94no—
452C3⋊C8—94no—
453C3⋊C8—91no—
454C3⋊C8—91no—
455C3⋊C8—184no—
456C3⋊C8—184no—
457C3⋊C8C22×C3⋊D312yes—
458C2×C32—21no—
459C2×C32—24no—
460C2×C32—24no—
461C2×C32—44no—
462C2×C32—41no—
463C2×C32—44no—
464C2×C32—84no—
465C2×C32C2×D3×D414yes—
466C2×C32C22×#6711yescharacteristic
467C3⋊D3—22no—
468C3⋊D3—42no—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3⋊D3—244no—
472C3×D3—68no—
473C3×D3—128no—
474#41—91no—
475#41—92no—
476#41—181no—
477#41—272no—
478#41—271no—
479#41—542no—
480#41—541no—
481#41D3×C3⋊D311yescharacteristic
482C2×Q8—31no—
483C2×Q8—272no—
484C2×Q8—541no—
485C2×D4—92no—
486C2×D4—91no—
487C2×D4—181no—
488C2×D4—182no—
489Q16—32no—
490Q16—271no—
491SD16—32no—
492SD16—272no—
493SD16—272no—
494SD16—272no—
495SD16—271no—
496SD16—271no—
497D8—272no—
498D8—271no—
499M16—31no—
500M16—272no—
501C2×C8—272no—
502C2×C8—271no—
503C22×C3—24no—
504C22×C3—21no—
505C22×C3—24no—
506C22×C3—44no—
507C22×C3—92no—
508C22×C3—182no—
509C22×C3—721no—
510C22×C3—722no—
511C22×C3D3×#6714yes—
512C22×C3D4×C3⋊D311yescharacteristic
513C2×D3—38no—
514C2×D3—68no—
515C2×D3—184no—
516C2×D3—181no—
517C2×D3—248no—
518C2×D3—244no—
519C2×D3—364no—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—44no—
525C3×C4—68no—
526C3×C4—92no—
527C3×C4D3×#6714yes—
528C3×C4D4×C3⋊D311yescharacteristic
529C3×C4C22×D3214yes—
530C3×C4C23×C3⋊D311yescharacteristic
531Dic3—22no—
532Dic3—38no—
533Dic3—184no—
534Dic3—181no—
535Dic3—364no—
536C32—24no—
537C32#577814yescomplemented
538C32#583611yescharacteristic, complemented
539C23—182no—
540C23—181no—
541Q8—32no—
542Q8—61no—
543Q8—91no—
544Q8—271no—
545Q8—542no—
546Q8C2×D3×C3⋊D311yescharacteristic
547D4—21no—
548D4—92no—
549D4—92no—
550D4—91no—
551D4—182no—
552D4—182no—
553D4—181no—
554D4—271no—
555D4—542no—
556D4—541no—
557D4C2×D3×C3⋊D311yescharacteristic
558C2×C4—21no—
559C2×C4—61no—
560C2×C4—92no—
561C2×C4—182no—
562C2×C4—542no—
563C2×C4—542no—
564C2×C4—541no—
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C8—32no—
567C8—271no—
568C8—271no—
569C2×C3—24no—
570C2×C3—21no—
571C2×C3—24no—
572C2×C3—44no—
573C2×C3—44no—
574C2×C3—41no—
575C2×C3—84no—
576C2×C3—182no—
577C2×C3—362no—
578C2×C3C2×D3×#6714yes—
579C2×C3C2×D4×C3⋊D311yescharacteristic
580D3—68no—
581D3—128no—
582D3—364no—
583D3—361no—
584D3—724no—
585C22—21no—
586C22—92no—
587C22—182no—
588C22—541no—
589C22—721no—
590C22—722no—
591C22C3⋊D3×#6711yescharacteristic
592C4—21no—
593C4—62no—
594C4—92no—
595C4—541no—
596C4C3⋊D3×#6711yescharacteristic
597C4C22×D3×C3⋊D311yescharacteristic
598C3—24no—
599C3#1059546414yescomplemented
600C3#1059624911yescharacteristic, complemented
601C2—21no—
602C2—41no—
603C2—182no—
604C2—362no—
605C2—1081no—
606C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#64366C2semidirect
2. #11724566C2semidirect
3. #11723991C2semidirect
4. C32×#951C2semidirect
5. #11723585C2semidirect
6. #11723576C2semidirect
7. #11723562C2semidirect
8. C3⋊D3×#214C2semidirect
9. C2×#88793C2semidirect
10. #11723370C2semidirect
11. #11723354C2semidirect
15. #89076C22semidirect
23. C2×C32×Dic6C22semidirect
26. #89018C22semidirect
28. C3⋊D3×Dic6C22semidirect
29. #88863C22semidirect
30. #88862C22semidirect
31. #88861C22semidirect
32. #88860C22semidirect
33. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
34. C32×#216C22semidirect
35. C32×#214C22semidirect
36. C32×#208C22semidirect
37. #88796C22semidirect
38. #88793C22semidirect
39. #88793C22semidirect
40. #88791C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
43. #64366D3semidirect
58. C3×#951D3semidirect
124. C2×C3×Dic6C2×D3semidirect
154. C3×#216C2×D3semidirect
155. C3×#214C2×D3semidirect
156. C3×#208C2×D3semidirect
182. #951C3⋊D3semidirect
308. C2×Dic6C2×C3⋊D3semidirect
315. #216C2×C3⋊D3semidirect
327. #214C2×C3⋊D3semidirect
333. #208C2×C3⋊D3semidirect
386. C33#577semidirect
537. C32#5778semidirect
538. C32#5836semidirect
599. C3#10595464semidirect
600. C3#10596249semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
#117245662
#117239912
C32×#9512
#117235852
#117235762
#117235622
C3⋊D3×#2142
C2×#887932
#117233702
#117233542
#105962493
#105954643