Finite Group Databasebeta

← #13178052

Subgroups — #13178052

Order 1728.

2524 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic, complemented
2C2×#89031C211yesmaximal, characteristic, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C3×#64162C211yesmaximal, characteristic, complemented
5#11723633C212yesmaximal, complemented
6#11723631C212yesmaximal, complemented
7#11723630C211yesmaximal, characteristic, complemented
8#11723629C211yesmaximal, characteristic, complemented
9#11723600C212yesmaximal, complemented
10#11723599C212yesmaximal, complemented
11#11723594C211yesmaximal, characteristic, complemented
12#10596276—31nomaximal
13#10595513—34nomaximal
14C24×C33C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic
16C2×#6312C2211yescharacteristic
17C2×#6312C2212yescomplemented
18C2×C3×#638C2211yescharacteristic
19C2×C3×#638C2212yescomplemented
20C2×C32×#67—24no—
21C2×C32×#67C2212yes—
22C2×C32×#67C2212yes—
23C22×C32×Dic3C2211yescharacteristic
24C2×#6293C2211yescharacteristic, complemented
25C2×#6292C2211yescharacteristic, complemented
26C2×#6291C2212yescomplemented
27C2×#6290C2212yescomplemented
28C2×#6289C2211yescharacteristic, complemented
29#89031—24no—
30C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
31#88864C2211yescharacteristic
32#88864C2212yescomplemented
33C3×#3757C2211yescharacteristic
34C3×#3757C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C22×C32×D4—31no—
40C2×#3828—31no—
41C22×C3×#67D314yescomplemented
42C2×#3804—34no—
43#64162—31no—
44#64162—31no—
45#64162—31no—
46#64162—34no—
47#64162—64no—
48#64162D311yescharacteristic, complemented
49#64149—32no—
50#64147—32no—
51#64147—32no—
52#64145—32no—
53#64135—31no—
54C3×#953—34no—
55#63979—38no—
56#63977—34no—
57#63976—34no—
58#63970—38no—
59#63959—38no—
60#63958—38no—
61#63955—34no—
62C23×C33—22no—
63C23×C33—42no—
64C23×C33C2312yes—
65C23×C33C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3—22no—
69C22×C32×D3—42no—
70C22×C32×D3C2312yes—
71#6312—22no—
72#6312—42no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C3×#638—22no—
76C3×#638—42no—
77C2×C3×C3⋊Dic3C2312yes—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C32×#67—24no—
80C32×#67—24no—
81C2×C32×Dic3D414yescomplemented
82C2×C32×Dic3C2311yescharacteristic
83C2×C32×Dic3C2311yescharacteristic
84#6293—22no—
85#6292—22no—
86#6291—42no—
87#6290—42no—
88#6289—22no—
89Dic3×C3⋊D3—22no—
90#5805—94no—
91#5592—91no—
92C24×C32—24no—
93C24×C32—31no—
94C24×C32C2×D314yes—
95C24×C32C2×D311yescharacteristic
96C23×C3×D3C2×D314yes—
97C2×C32×D4—32no—
98C2×C32×D4—32no—
99C2×C32×D4—64no—
100C22×C32×C4—31no—
101C2×#638—31no—
102C2×#638—34no—
103C2×#638—32no—
104C2×#638—38no—
105C2×#638—32no—
106C2×#638—32no—
107C2×#638—64no—
108C2×#638—68no—
109C2×#638C2×D311yescharacteristic
110C2×#638C2×D312yescomplemented
111#3828—64no—
112C2×C4⋊(C3⋊D3)—31no—
113C2×C4×C3⋊D3—31no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×C3×#67—216no—
117C2×C3×#67—34no—
118C2×C3×#67—38no—
119C2×C3×#67C2×D318yes—
120C2×C3×#67C2×D318yes—
121C22×C3×Dic3C2×D314yes—
122C2×#627—34no—
123C2×#626—34no—
124C2×#626—38no—
125C2×#625—38no—
126C2×#624—34no—
127#3804—616no—
128C2×D3×Dic3—34no—
129#3757—31no—
130#3757—32no—
131#3757—34no—
132#3757—38no—
133#3757—32no—
134#3757—32no—
135#3757—64no—
136#3757—68no—
137#3757C2×D311yescharacteristic
138#3757C2×D312yescomplemented
139C4⋊(C3⋊Dic3)—32no—
140C4×C3⋊Dic3—32no—
141C3×#217—34no—
142C3×#217—38no—
143Dic3⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146Dic32—38no—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33—22no—
151C22×C33—44no—
152C22×C33C2×D412yes—
153C22×C33C2411yescharacteristic
154C2×C3⋊(C3⋊D3)—21no—
155C2×C3⋊(C3⋊D3)—22no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—22no—
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3—22no—
161C2×C32×D3—44no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)—22no—
164C3×C3⋊Dic3—21no—
165C3×C3⋊Dic3—22no—
166C32×Dic3C2×D412yes—
167C32×Dic3C2×D412yes—
168C22×C3×D4—34no—
169C22×#67C3⋊D311yescharacteristic, complemented
170C2×#237—91no—
171C2×#235—94no—
172C3×#120—91no—
173#953—34no—
174#953—94no—
175#953—94no—
176#953—94no—
177#953—91no—
178#953—184no—
179#940—98no—
180#940—92no—
181#939—92no—
182#938—98no—
183#938—98no—
184#937—91no—
185#936—98no—
186#926—94no—
187#885—92no—
188#884—92no—
189#882—91no—
190#877—91no—
191C23×C32—22no—
192C23×C32—28no—
193C23×C32—24no—
194C23×C32—28no—
195C23×C32—32no—
196C23×C32—48no—
197C23×C32—42no—
198C23×C32—48no—
199C23×C32—416no—
200C23×C32—62no—
201C23×C32—122no—
202C23×C32C22×D318yes—
203C23×C32C22×D314yes—
204C23×C32C22×D312yes—
205C23×C32C22×D311yescharacteristic
206C22×C3⋊D3—31no—
207C22×C3⋊D3—34no—
208C22×C3⋊D3—64no—
209C22×C3⋊D3C22×D311yescharacteristic
210C22×C3×D3—28no—
211C22×C3×D3—34no—
212C22×C3×D3—48no—
213C22×C3×D3C22×D318yes—
214C32×D4—64no—
215C32×D4—64no—
216C2×C32×C4—34no—
217C2×C32×C4—31no—
218C2×C32×C4—31no—
219#638—22no—
220#638—42no—
221#638—62no—
222#638—68no—
223#638—128no—
224#638—122no—
225#638—122no—
226#638—122no—
227#638—128no—
228#638—1216no—
229C2×C3⋊Dic3—32no—
230C2×C3⋊Dic3—31no—
231C2×C3⋊Dic3—38no—
232C2×C3⋊Dic3—34no—
233C2×C3⋊Dic3—68no—
234C2×C3⋊Dic3—64no—
235C2×C3⋊Dic3C22×D312yes—
236C2×C3⋊Dic3C22×D311yescharacteristic
237C4⋊(C3⋊D3)—62no—
238C4×C3⋊D3—62no—
239C4×C3⋊D3—62no—
240#634—62no—
241C3×#67—216no—
242C3×#67—216no—
243C3×#67—68no—
244C3×#67—128no—
245C2×C3×Dic3—38no—
246C2×C3×Dic3—34no—
247C2×C3×Dic3#67116yescomplemented
248C2×C3×Dic3C22×D314yes—
249C2×C3×Dic3C22×D314yes—
250#627—68no—
251#626—68no—
252#626—128no—
253#625—128no—
254#624—68no—
255D3×Dic3—68no—
256#535—271noSylow
257C2×C33—22no—
258C2×C33—42no—
259C2×C33C22×D411yescharacteristic
260C2×C33H(5,𝔽2)12yes—
261C3⋊(C3⋊D3)—41no—
262C3×C3⋊D3—41no—
263C32×D3—42no—
264C24×C3—24no—
265C24×C3—34no—
266C24×C3D3214yes—
267C24×C3C2×C3⋊D311yescharacteristic
268C23×D3C2×C3⋊D311yescharacteristic
269C2×C3×D4—38no—
270C2×C3×D4—38no—
271C2×C3×D4—616no—
272C2×C3×D4—91no—
273C2×C3×D4—92no—
274C22×C3×C4—34no—
275C2×#67—24no—
276C2×#67—34no—
277C2×#67—38no—
278C2×#67—98no—
279C2×#67—94no—
280C2×#67—91no—
281C2×#67—98no—
282C2×#67—98no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—91no—
286C2×#67—184no—
287C2×#67—188no—
288C2×#67C2×C3⋊D312yes—
289C2×#67C2×C3⋊D312yes—
290C22×Dic3—91no—
291C22×Dic3C2×C3⋊D311yescharacteristic
292#237—184no—
293#235—1816no—
294C2×D12—94no—
295C2×D12—92no—
296C2×C4×D3—94no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×Dic6—94no—
300C2×Dic6—91no—
301C3×C22⋊C4—91no—
302C3×C22⋊C4—92no—
303#217—34no—
304#217—38no—
305#217—98no—
306#217—98no—
307#217—94no—
308#217—98no—
309#217—91no—
310#217—92no—
311#217—184no—
312#217—188no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315C4⋊Dic3—98no—
316Dic3⋊C4—92no—
317C4×Dic3—98no—
318C4×Dic3—92no—
319C22×C32—22no—
320C22×C32—216no—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—28no—
324C22×C32—22no—
325C22×C32—24no—
326C22×C32—28no—
327C22×C32—28no—
328C22×C32—28no—
329C22×C32—28no—
330C22×C32—48no—
331C22×C32—48no—
332C22×C32—48no—
333C22×C32—44no—
334C22×C32—416no—
335C22×C32—432no—
336C22×C32—62no—
337C22×C32—62no—
338C22×C32—62no—
339C22×C32—124no—
340C22×C32D3×D412yes—
341C22×C32C2×#6718yes—
342C22×C32C23×D314yes—
343C22×C32C23×D311yescharacteristic
344C2×C3⋊D3—21no—
345C2×C3⋊D3—22no—
346C2×C3⋊D3—61no—
347C2×C3⋊D3—62no—
348C2×C3⋊D3—68no—
349C2×C3⋊D3—64no—
350C2×C3⋊D3—124no—
351C2×C3⋊D3—128no—
352C2×C3×D3—28no—
353C2×C3×D3—28no—
354C2×C3×D3—28no—
355C2×C3×D3—416no—
356C2×C3×D3—68no—
357C2×C3×D3—64no—
358C32×C4—32no—
359C32×C4—32no—
360C3⋊Dic3—21no—
361C3⋊Dic3—22no—
362C3⋊Dic3—61no—
363C3⋊Dic3—62no—
364C3⋊Dic3—68no—
365C3⋊Dic3—64no—
366C3⋊Dic3—124no—
367C3⋊Dic3—128no—
368C3×Dic3—68no—
369C3×Dic3—64no—
370C3×Dic3C2×#6718yes—
371C3×Dic3C2×#6718yes—
372C2×#41—271no—
373C22×D4—31no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—91no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—271no—
383C33#53511yescharacteristic, complemented, Sylow
384C23×C3—28no—
385C23×C3—22no—
386C23×C3—28no—
387C23×C3—24no—
388C23×C3—38no—
389C23×C3—48no—
390C23×C3—48no—
391C23×C3—42no—
392C23×C3—416no—
393C23×C3—68no—
394C23×C3—91no—
395C23×C3—128no—
396C23×C3C2×D3218yes—
397C23×C3C2×D3214yes—
398C23×C3C22×C3⋊D312yes—
399C23×C3C22×C3⋊D311yescharacteristic
400C22×D3—22no—
401C22×D3—34no—
402C22×D3—42no—
403C22×D3—94no—
404C22×D3—91no—
405C22×D3—184no—
406C22×D3C22×C3⋊D312yes—
407C3×D4—616no—
408C3×D4—616no—
409C3×D4—182no—
410C3×D4—362no—
411C2×C3×C4—34no—
412C2×C3×C4—34no—
413C2×C3×C4—316no—
414C2×C3×C4—92no—
415C2×C3×C4—91no—
416#67—24no—
417#67—24no—
418#67—68no—
419#67—128no—
420#67—188no—
421#67—182no—
422#67—182no—
423#67—368no—
424#67—368no—
425#67—362no—
426#67—368no—
427#67—368no—
428#67—362no—
429#67—3616no—
430C2×Dic3—38no—
431C2×Dic3—34no—
432C2×Dic3—98no—
433C2×Dic3—94no—
434C2×Dic3—92no—
435C2×Dic3—91no—
436C2×Dic3—188no—
437C2×Dic3—184no—
438C2×Dic3—182no—
439C2×Dic3#63814yescomplemented
440C2×Dic3C22×C3⋊D311yescharacteristic
441C2×Dic3C22×C3⋊D311yescharacteristic
442D12—188no—
443D12—362no—
444C4×D3—188no—
445C4×D3—188no—
446C4×D3—182no—
447Dic6—188no—
448Dic6—182no—
449C2×C32—22no—
450C2×C32—24no—
451C2×C32—28no—
452C2×C32—28no—
453C2×C32—48no—
454C2×C32—416no—
455C2×C32—48no—
456C2×C32—42no—
457C2×C32—122no—
458C2×C32C2×D3×D411yescharacteristic
459C2×C32#100418yes—
460C2×C32#100412yes—
461C2×C32C22×#6714yes—
462C3⋊D3—41no—
463C3⋊D3—121no—
464C3⋊D3—124no—
465C3⋊D3—244no—
466C3×D3—48no—
467C3×D3—124no—
468C24—31no—
469C24D3×C3⋊D311yescharacteristic
470#41—544no—
471C2×Q8—271no—
472C2×D4—32no—
473C2×D4—32no—
474C2×D4—64no—
475C2×D4—91no—
476C2×D4—92no—
477C2×D4—271no—
478C2×D4—272no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—272no—
482C22×C4—31no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—272no—
486C22⋊C4—91no—
487C22⋊C4—92no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—271no—
491C22⋊C4—272no—
492C42—272no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—216no—
496C22×C3—28no—
497C22×C3—28no—
498C22×C3—28no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—24no—
503C22×C3—28no—
504C22×C3—48no—
505C22×C3—48no—
506C22×C3—48no—
507C22×C3—44no—
508C22×C3—416no—
509C22×C3—432no—
510C22×C3—68no—
511C22×C3—68no—
512C22×C3—68no—
513C22×C3—1216no—
514C22×C3—182no—
515C22×C3—181no—
516C22×C3D3×#6718yes—
517C22×C3C2×#63812yes—
518C22×C3C22×D3214yes—
519C22×C3C23×C3⋊D311yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—44no—
524C2×D3—64no—
525C2×D3—68no—
526C2×D3—184no—
527C2×D3—188no—
528C2×D3—182no—
529C2×D3—181no—
530C2×D3—364no—
531C2×D3—368no—
532C3×C4—38no—
533C3×C4—38no—
534C3×C4—182no—
535C3×C4—181no—
536Dic3—64no—
537Dic3—68no—
538Dic3—184no—
539Dic3—188no—
540Dic3—182no—
541Dic3—181no—
542Dic3—364no—
543Dic3—368no—
544Dic3C2×#63812yes—
545Dic3C2×#63812yes—
546C32—24no—
547C32#559211yescharacteristic, complemented
548C32#580514yescomplemented
549C23—22no—
550C23—32no—
551C23—42no—
552C23—62no—
553C23—91no—
554C23—122no—
555C23—271no—
556C23C2×D3×C3⋊D312yes—
557C23C2×D3×C3⋊D311yescharacteristic
558Q8—542no—
559D4—64no—
560D4—64no—
561D4—182no—
562D4—362no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—31no—
570C2×C4—34no—
571C2×C4—92no—
572C2×C4—91no—
573C2×C4—272no—
574C2×C4—271no—
575C2×C4—542no—
576C2×C4—542no—
577C2×C3—22no—
578C2×C3—28no—
579C2×C3—24no—
580C2×C3—28no—
581C2×C3—48no—
582C2×C3—416no—
583C2×C3—42no—
584C2×C3—48no—
585C2×C3—128no—
586C2×C3—361no—
587C2×C3C2×D3×#6714yes—
588C2×C3#6433018yes—
589C2×C3#6436112yes—
590C2×C3C22×#63811yescharacteristic
591D3—42no—
592D3—124no—
593D3—361no—
594D3—364no—
595D3—724no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—62no—
602C22—62no—
603C22—62no—
604C22—124no—
605C22—181no—
606C22—182no—
607C22—541no—
608C22—542no—
609C22D3×#63812yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—32no—
612C4—32no—
613C4—181no—
614C4—182no—
615C4—541no—
616C4—542no—
617C3—24no—
618C3#1059551314yescomplemented
619C3#1059627611yescharacteristic, complemented
620C2—22no—
621C2—42no—
622C2—122no—
623C2—361no—
624C2—1081no—
625C2C2×D3×#63811yescharacteristic
626C2#1172461112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C32×#67C2semidirect
2. C2×#89031C2semidirect
3. #11723994C2semidirect
4. C3×#64162C2semidirect
5. #11723633C2semidirect
6. #11723631C2semidirect
7. #11723630C2semidirect
8. #11723629C2semidirect
9. #11723600C2semidirect
10. #11723599C2semidirect
11. #11723594C2semidirect
17. C2×#6312C22semidirect
19. C2×C3×#638C22semidirect
24. C2×#6293C22semidirect
25. C2×#6292C22semidirect
26. C2×#6291C22semidirect
27. C2×#6290C22semidirect
28. C2×#6289C22semidirect
30. C2×Dic3×C3⋊D3C22semidirect
32. #88864C22semidirect
34. C3×#3757C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
41. C22×C3×#67D3semidirect
48. #64162D3semidirect
81. C2×C32×Dic3D4semidirect
110. C2×#638C2×D3semidirect
138. #3757C2×D3semidirect
169. C22×#67C3⋊D3semidirect
247. C2×C3×Dic3#67semidirect
383. C33#535semidirect
439. C2×Dic3#638semidirect
547. C32#5592semidirect
548. C32#5805semidirect
618. C3#10595513semidirect
619. C3#10596276semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#672
C2×#890312
#117239942
C3×#641622
#117236332
#117236312
#117236302
#117236292
#117236002
#117235992
#117235942
#105962763
#105955133