Finite Group Databasebeta

← #13177992

Subgroups — #13177992

Order 1728.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1009C211yesmaximal, characteristic, complemented
2#11724581C211yesmaximal, characteristic, complemented
3#11723990C211yesmaximal, characteristic, complemented
4C3×#64159C211yesmaximal, characteristic, complemented
5#11723582C212yesmaximal, complemented
6#11723568C212yesmaximal, complemented
7#11723557C212yesmaximal, complemented
8D3×#3753C212yesmaximal, complemented
9C2×#88790C211yesmaximal, characteristic, complemented
10#11723364C211yesmaximal, characteristic, complemented
11#11723349C211yesmaximal, characteristic, complemented
12#10596306—31nomaximal
13#10595456—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C3×#3828C2211yescharacteristic, complemented
17C32×#239C2212yescomplemented
18C32×D3×D4—22no—
19C32×D3×D4C2212yescomplemented
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×D12—21no—
23C2×C32×D12C2211yescharacteristic, complemented
24#89039—22no—
25C3⋊D3×D12—21no—
26D3×C4⋊(C3⋊D3)C2212yescomplemented
27#89022C2212yescomplemented
28#89018—21no—
29#88862C2212yescomplemented
30#88860C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C3×#3756C2211yescharacteristic, complemented
33C3×#3755C2211yescharacteristic, complemented
34C3×#3754C2211yescharacteristic, complemented
35C3×#3753C2211yescharacteristic, complemented
36C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
37#88792C2212yescomplemented
38#88790C2212yescomplemented
39#88790C2212yescomplemented
40#88789C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×H(5,𝔽2)—31no—
44#64366—31no—
45C3×#1009D314yescomplemented
46#64308—34no—
47#64159—32no—
48#64159D311yescharacteristic, complemented
49#64158—31no—
50#64158—34no—
51#64158—32no—
52#64158—64no—
53#64157—31no—
54#64141—31no—
55#64141—32no—
56C2×#3753—32no—
57C2×#3753—31no—
58C3×#950—34no—
59#63949—38no—
60#63939—38no—
61#63934—38no—
62D3×#213—38no—
63C2×#3714—34no—
64#63823—34no—
65#63814—34no—
66C22×C3⋊(C3⋊D3)—21no—
67C22×C32×D3—21no—
68C22×C32×D3—22no—
69C2×D3×C3⋊D3—22no—
70C33×Q8C2311yescharacteristic
71C33×D4—21no—
72C33×D4C2311yescharacteristic
73C2×C33×C4—21no—
74C2×C33×C4C2311yescharacteristic
75C4⋊(C3⋊(C3⋊D3))—21no—
76C4⋊(C3⋊(C3⋊D3))C2312yes—
77C3×#638—21no—
78C3×C4⋊(C3⋊D3)C2311yescharacteristic
79C3×C4×C3⋊D3—21no—
80C3×#634C2311yescharacteristic
81C32×#67—22no—
82C32×#67D412yes—
83C32×D12—22no—
84C32×D12—21no—
85C32×D12D411yescharacteristic
86C32×D12C2312yes—
87C32×C4×D3—22no—
88C32×C4×D3C2312yes—
89C32×Dic6D411yescharacteristic
90#6292—22no—
91#6291—22no—
92#6289—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C3×C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2311yescharacteristic
96#5835—94no—
97#5769—91no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41—32no—
101C32×#41—61no—
102C32×#41C2×D314yescomplemented
103C32×#41C2×D311yescharacteristic, complemented
104C2×C32×D4—32no—
105C2×C32×D4—31no—
106C2×C32×D4—61no—
107C2×C32×D4—62no—
108#3832—61no—
109D4×C3⋊D3—61no—
110D4×C3⋊D3—62no—
111#3828—32no—
112#3828C2×D311yescharacteristic, complemented
113C2×C4⋊(C3⋊D3)—31no—
114C2×C4⋊(C3⋊D3)—34no—
115C2×C4⋊(C3⋊D3)—32no—
116C2×C4⋊(C3⋊D3)—64no—
117C3×#239C2×D318yescomplemented
118C3×D3×D4—28no—
119C3×D3×D4C2×D318yescomplemented
120C3×#235—24no—
121C3×#235—34no—
122C3×#235C2×D314yes—
123C2×C3×D12—24no—
124C2×C3×D12C2×D314yescomplemented
125#3811—68no—
126D3×D12—38no—
127D3×D12—64no—
128#3799—38no—
129#3799—64no—
130#3756C2×D311yescharacteristic, complemented
131#3755—32no—
132#3755—38no—
133#3755—68no—
134#3755C2×D311yescharacteristic, complemented
135#3754—32no—
136#3754C2×D311yescharacteristic, complemented
137#3753—32no—
138#3753—38no—
139#3753—32no—
140#3753—32no—
141#3753—32no—
142#3753—68no—
143#3753C2×D311yescharacteristic, complemented
144#3748—31no—
145#3748—34no—
146#3748—32no—
147#3748—64no—
148C2×C3⋊(C3⋊C8)—32no—
149C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
150C3×#216—34no—
151C3×#215—34no—
152C3×#214—34no—
153C3×#213—34no—
154C2×C3×C3⋊C8—34no—
155#3715—38no—
156#3714—38no—
157#3714—38no—
158#3713—38no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—21no—
162C22×C33C2×D411yescharacteristic
163C2×C3⋊(C3⋊D3)—22no—
164C2×C3⋊(C3⋊D3)—81no—
165C2×C3×C3⋊D3—21no—
166C2×C32×D3—22no—
167C2×C32×D3—82no—
168C2×C32×D3—81no—
169C2×C32×D3C2×D412yes—
170D3×C3⋊D3—82no—
171C33×C4—21no—
172C33×C4C2×D411yescharacteristic
173C33×C4C2411yescharacteristic
174C3×C3⋊Dic3—21no—
175C32×Dic3C2×D412yes—
176C3×H(5,𝔽2)—34no—
177#1009—94no—
178#1009—91no—
179#1009C3⋊D311yescharacteristic, complemented
180C3×#135—91no—
181#950—34no—
182#950—98no—
183#949—98no—
184#949—94no—
185#949—91no—
186#949—91no—
187#949—184no—
188#948—94no—
189#932—98no—
190#932—94no—
191C2×#213—94no—
192C2×#213—98no—
193#919—92no—
194#914—92no—
195#914—92no—
196D3×D8—92no—
197C2×D24—91no—
198#901—91no—
199C23×C32—62no—
200C23×C32—61no—
201C22×C3⋊D3—62no—
202C22×C3⋊D3—61no—
203C22×C3⋊D3—64no—
204C22×C3⋊D3—124no—
205C22×C3×D3—24no—
206C22×C3×D3—28no—
207C2×D32—68no—
208C32×Q8—24no—
209C32×Q8—31no—
210C32×Q8C22×D314yes—
211C32×Q8C22×D311yescharacteristic
212C32×D4—21no—
213C32×D4—24no—
214C32×D4—24no—
215C32×D4—32no—
216C32×D4—31no—
217C32×D4—32no—
218C32×D4—44no—
219C32×D4—62no—
220C32×D4—62no—
221C32×D4—61no—
222C32×D4C22×D314yes—
223C32×D4C22×D311yescharacteristic
224C2×C32×C4—21no—
225C2×C32×C4—24no—
226C2×C32×C4—24no—
227C2×C32×C4—32no—
228C2×C32×C4—44no—
229C2×C32×C4—62no—
230C2×C32×C4C22×D314yes—
231C2×C32×C4C22×D311yescharacteristic
232#638—21no—
233#638—62no—
234C4⋊(C3⋊D3)—32no—
235C4⋊(C3⋊D3)—38no—
236C4⋊(C3⋊D3)—68no—
237C4⋊(C3⋊D3)—62no—
238C4⋊(C3⋊D3)—61no—
239C4⋊(C3⋊D3)—64no—
240C4⋊(C3⋊D3)—124no—
241C4⋊(C3⋊D3)C22×D311yescharacteristic
242C4×C3⋊D3—21no—
243C4×C3⋊D3—62no—
244#634C22×D311yescharacteristic
245C3×#67—28no—
246C3×#67—64no—
247C3×#67#6718yes—
248C3×D12—28no—
249C3×D12—24no—
250C3×D12—34no—
251C3×D12#6714yes—
252C3×D12C22×D318yes—
253C3×C4×D3—28no—
254C3×C4×D3—64no—
255C3×C4×D3C22×D318yes—
256C3×Dic6—34no—
257C3×Dic6#6714yes—
258#626—68no—
259#626—68no—
260#624—68no—
261C3⋊(C3⋊C8)—32no—
262C3⋊(C3⋊C8)—38no—
263C3⋊(C3⋊C8)—68no—
264C3⋊(C3⋊C8)C22×D311yescharacteristic
265C3⋊(C3⋊C8)C22×D311yescharacteristic
266C3×C3⋊C8—34no—
267C3×C3⋊C8—34no—
268#576—271noSylow
269C2×C33—21no—
270C2×C33—41no—
271C2×C33C22×D411yescharacteristic
272C3⋊(C3⋊D3)—42no—
273C3×C3⋊D3—41no—
274C32×D3—22no—
275C32×D3—42no—
276C3×#41—24no—
277C3×#41—34no—
278C3×#41—38no—
279C3×#41—64no—
280C3×#41—91no—
281C3×#41D3214yescomplemented
282C3×#41C2×C3⋊D311yescharacteristic, complemented
283C2×C3×D4—38no—
284C2×C3×D4—34no—
285C2×C3×D4—64no—
286C2×C3×D4—68no—
287#239—92no—
288#239—184no—
289#239C2×C3⋊D312yescomplemented
290D3×D4—22no—
291D3×D4—92no—
292D3×D4—184no—
293D3×D4—188no—
294D3×D4—182no—
295D3×D4C2×C3⋊D312yescomplemented
296#235—21no—
297#235—34no—
298#235—98no—
299#235—181no—
300#235C2×C3⋊D311yescharacteristic
301C2×D12—21no—
302C2×D12—98no—
303C2×D12—94no—
304C2×D12—91no—
305C2×D12—184no—
306C2×D12—181no—
307C2×D12C2×C3⋊D311yescharacteristic, complemented
308C3×Q16—91no—
309C3×SD16—91no—
310C3×SD16—91no—
311C3×D8—91no—
312C2×C3×C8—91no—
313#216—34no—
314#215—34no—
315#215—98no—
316#215—92no—
317#215—92no—
318#215—188no—
319#214—34no—
320#214—98no—
321#213—34no—
322#213—98no—
323#213—98no—
324#213—98no—
325#213—98no—
326#213—92no—
327#213—92no—
328#213—188no—
329#208—98no—
330#208—94no—
331#208—91no—
332#208—184no—
333C2×C3⋊C8—34no—
334C2×C3⋊C8—98no—
335D24—92no—
336D24—92no—
337ΓL(1,𝔽25)—92no—
338D3×C8—92no—
339C22×C32—21no—
340C22×C32—24no—
341C22×C32—24no—
342C22×C32—32no—
343C22×C32—44no—
344C22×C32—62no—
345C22×C32—241no—
346C22×C32—242no—
347C22×C32D3×D411yescharacteristic
348C22×C32C2×#6714yes—
349C2×C3⋊D3—21no—
350C2×C3⋊D3—62no—
351C2×C3⋊D3—68no—
352C2×C3⋊D3—128no—
353C2×C3⋊D3—242no—
354C2×C3⋊D3—241no—
355C2×C3⋊D3—244no—
356C2×C3⋊D3—248no—
357C2×C3×D3—28no—
358C2×C3×D3—64no—
359C2×C3×D3—88no—
360C2×C3×D3—84no—
361C2×C3×D3C2×#6718yes—
362D32—248no—
363C32×C4—21no—
364C32×C4—24no—
365C32×C4—24no—
366C32×C4—24no—
367C32×C4—32no—
368C32×C4—44no—
369C32×C4D3×D411yescharacteristic
370C32×C4C2×#6714yes—
371C32×C4C23×D314yes—
372C32×C4C23×D311yescharacteristic
373C3⋊Dic3—21no—
374C3×Dic3—64no—
375C3×Dic3C2×#6718yes—
376H(5,𝔽2)—31no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—272no—
379AGL(1,ℤ/8)—272no—
380AGL(1,ℤ/8)—271no—
381AGL(1,ℤ/8)—271no—
382#135—91no—
383#135—272no—
384C2×D8—271no—
385C2×D8—272no—
386#131—271no—
387C33#57611yescharacteristic, complemented, Sylow
388C23×C3—68no—
389C23×C3—64no—
390C22×D3—21no—
391C22×D3—22no—
392C22×D3—188no—
393C22×D3—184no—
394C22×D3—181no—
395C22×D3—182no—
396C22×D3—364no—
397C3×Q8—24no—
398C3×Q8—34no—
399C3×Q8—91no—
400C3×Q8C2×D3214yes—
401C3×Q8C22×C3⋊D311yescharacteristic
402C3×D4—24no—
403C3×D4—21no—
404C3×D4—24no—
405C3×D4—38no—
406C3×D4—38no—
407C3×D4—34no—
408C3×D4—44no—
409C3×D4—68no—
410C3×D4—68no—
411C3×D4—64no—
412C3×D4—91no—
413C3×D4—181no—
414C3×D4C2×D3214yes—
415C3×D4C22×C3⋊D311yescharacteristic
416C2×C3×C4—24no—
417C2×C3×C4—21no—
418C2×C3×C4—24no—
419C2×C3×C4—38no—
420C2×C3×C4—44no—
421C2×C3×C4—68no—
422C2×C3×C4—181no—
423C2×C3×C4C2×D3214yes—
424C2×C3×C4C22×C3⋊D311yescharacteristic
425#67—22no—
426#67—64no—
427#67—188no—
428#67—182no—
429#67#63812yes—
430D12—22no—
431D12—21no—
432D12—34no—
433D12—98no—
434D12—92no—
435D12—188no—
436D12—188no—
437D12—182no—
438D12—184no—
439D12—181no—
440D12—364no—
441D12#63811yescharacteristic
442D12C22×C3⋊D312yes—
443C4×D3—22no—
444C4×D3—64no—
445C4×D3—188no—
446C4×D3—182no—
447C4×D3C22×C3⋊D312yes—
448Dic6—34no—
449Dic6#63811yescharacteristic
450C3×C8—91no—
451C3×C8—91no—
452C3⋊C8—34no—
453C3⋊C8—34no—
454C3⋊C8—98no—
455C3⋊C8—92no—
456C3⋊C8—188no—
457C2×C32—21no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—44no—
461C2×C32—41no—
462C2×C32—44no—
463C2×C32—62no—
464C2×C32—84no—
465C2×C32—122no—
466C2×C32C2×D3×D411yescharacteristic
467C2×C32C22×#6714yes—
468C3⋊D3—41no—
469C3⋊D3—122no—
470C3⋊D3—128no—
471C3⋊D3—248no—
472C3×D3—28no—
473C3×D3—48no—
474C3×D3—124no—
475#41—31no—
476#41—32no—
477#41—61no—
478#41—91no—
479#41—272no—
480#41—541no—
481#41D3×C3⋊D311yescharacteristic, complemented
482C2×D4—32no—
483C2×D4—31no—
484C2×D4—61no—
485C2×D4—62no—
486C2×D4—272no—
487C2×D4—271no—
488C2×D4—541no—
489C2×D4—542no—
490Q16—91no—
491SD16—91no—
492SD16—91no—
493SD16—272no—
494SD16—272no—
495D8—91no—
496D8—272no—
497D8—272no—
498D8—272no—
499D8—272no—
500M16—272no—
501M16—271no—
502C2×C8—91no—
503C2×C8—272no—
504C22×C3—24no—
505C22×C3—21no—
506C22×C3—24no—
507C22×C3—38no—
508C22×C3—44no—
509C22×C3—68no—
510C22×C3—181no—
511C22×C3—244no—
512C22×C3—248no—
513C22×C3D3×#6714yes—
514C22×C3C2×#63811yescharacteristic
515C2×D3—22no—
516C2×D3—64no—
517C2×D3—82no—
518C2×D3—81no—
519C2×D3—188no—
520C2×D3—182no—
521C2×D3—368no—
522C2×D3—722no—
523C2×D3—728no—
524C2×D3—724no—
525C2×D3—721no—
526C2×D3—728no—
527C2×D3C2×#63812yes—
528C3×C4—24no—
529C3×C4—21no—
530C3×C4—24no—
531C3×C4—24no—
532C3×C4—38no—
533C3×C4—44no—
534C3×C4—181no—
535C3×C4D3×#6714yes—
536C3×C4C2×#63811yescharacteristic
537C3×C4C22×D3214yes—
538C3×C4C23×C3⋊D311yescharacteristic
539Dic3—64no—
540Dic3C2×#63812yes—
541C32—24no—
542C32#576911yescharacteristic, complemented
543C32#583514yescomplemented
544C23—62no—
545C23—61no—
546C23—542no—
547C23—541no—
548Q8—31no—
549Q8—91no—
550Q8C2×D3×C3⋊D311yescharacteristic
551D4—21no—
552D4—32no—
553D4—32no—
554D4—31no—
555D4—62no—
556D4—62no—
557D4—61no—
558D4—91no—
559D4—181no—
560D4—272no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4C2×D3×C3⋊D311yescharacteristic
565C2×C4—21no—
566C2×C4—32no—
567C2×C4—62no—
568C2×C4—181no—
569C2×C4—542no—
570C2×C4C2×D3×C3⋊D311yescharacteristic
571C8—91no—
572C8—91no—
573C8—272no—
574C2×C3—24no—
575C2×C3—21no—
576C2×C3—24no—
577C2×C3—44no—
578C2×C3—44no—
579C2×C3—41no—
580C2×C3—68no—
581C2×C3—84no—
582C2×C3—128no—
583C2×C3—361no—
584C2×C3C2×D3×#6714yes—
585C2×C3C22×#63811yescharacteristic
586D3—22no—
587D3—42no—
588D3—124no—
589D3—368no—
590D3—362no—
591D3—728no—
592C22—21no—
593C22—32no—
594C22—62no—
595C22—181no—
596C22—241no—
597C22—242no—
598C22—542no—
599C22—2161no—
600C22—2162no—
601C22D3×#63811yescharacteristic
602C4—21no—
603C4—32no—
604C4—181no—
605C4D3×#63811yescharacteristic
606C4C22×D3×C3⋊D311yescharacteristic
607C3—24no—
608C3#1059545614yescomplemented
609C3#1059630611yescharacteristic, complemented
610C2—21no—
611C2—41no—
612C2—62no—
613C2—122no—
614C2—361no—
615C2—1082no—
616C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#576
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. C32×#1009C2semidirect
2. #11724581C2semidirect
3. #11723990C2semidirect
4. C3×#64159C2semidirect
5. #11723582C2semidirect
6. #11723568C2semidirect
7. #11723557C2semidirect
8. D3×#3753C2semidirect
9. C2×#88790C2semidirect
10. #11723364C2semidirect
11. #11723349C2semidirect
14. C33×#41C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C3×#3828C22semidirect
17. C32×#239C22semidirect
19. C32×D3×D4C22semidirect
23. C2×C32×D12C22semidirect
26. D3×C4⋊(C3⋊D3)C22semidirect
27. #89022C22semidirect
29. #88862C22semidirect
30. #88860C22semidirect
31. #88855C22semidirect
32. C3×#3756C22semidirect
33. C3×#3755C22semidirect
34. C3×#3754C22semidirect
35. C3×#3753C22semidirect
36. C2×C3×C3⋊(C3⋊C8)C22semidirect
37. #88792C22semidirect
38. #88790C22semidirect
39. #88790C22semidirect
40. #88789C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1009D3semidirect
48. #64159D3semidirect
102. C32×#41C2×D3semidirect
103. C32×#41C2×D3semidirect
112. #3828C2×D3semidirect
117. C3×#239C2×D3semidirect
119. C3×D3×D4C2×D3semidirect
124. C2×C3×D12C2×D3semidirect
130. #3756C2×D3semidirect
134. #3755C2×D3semidirect
136. #3754C2×D3semidirect
143. #3753C2×D3semidirect
149. C2×C3⋊(C3⋊C8)C2×D3semidirect
179. #1009C3⋊D3semidirect
281. C3×#41D32semidirect
282. C3×#41C2×C3⋊D3semidirect
289. #239C2×C3⋊D3semidirect
295. D3×D4C2×C3⋊D3semidirect
307. C2×D12C2×C3⋊D3semidirect
387. C33#576semidirect
481. #41D3×C3⋊D3semidirect
542. C32#5769semidirect
543. C32#5835semidirect
608. C3#10595456semidirect
609. C3#10596306semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
#117245812
#117239902
C3×#641592
#117235822
#117235682
#117235572
D3×#37532
C2×#887902
#117233642
#117233492
#105963063
#105954563