Finite Group Databasebeta

← #13178047

Subgroups — #13178047

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×C3×#638C211yesmaximal, characteristic, complemented
2C2×#89028C211yesmaximal, characteristic, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C32×#953C211yesmaximal, characteristic, complemented
5#11723636C211yesmaximal, characteristic, complemented
6#11723634C211yesmaximal, characteristic, complemented
7#11723633C212yesmaximal, complemented
8#11723623C212yesmaximal, complemented
9#11723615C212yesmaximal, complemented
10#11723595C211yesmaximal, characteristic, complemented
11#11723591C212yesmaximal, complemented
12#10596063—31nomaximal
13#10595513—34nomaximal
14C24×C33C2211yescharacteristic
15C23×C3×C3⋊D3C2211yescharacteristic
16C2×#6312C2211yescharacteristic
17C2×#6312C2212yescomplemented
18C2×C3×#638—24no—
19C2×C3×#638C2212yes—
20C2×C3×#638C2212yes—
21C22×C3×C3⋊Dic3C2211yescharacteristic
22C2×C32×#67C2211yescharacteristic
23C2×C32×#67C2212yescomplemented
24C2×#6293C2211yescharacteristic, complemented
25C2×#6292C2212yescomplemented
26C2×#6291C2211yescharacteristic, complemented
27C2×#6290C2212yescomplemented
28C2×#6289C2211yescharacteristic, complemented
29#89028—24no—
30C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
31#88864C2211yescharacteristic
32#88864C2212yescomplemented
33C32×#217C2211yescharacteristic
34C32×#217C2212yescomplemented
35#88807C2212yescomplemented
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C22×#638D311yescharacteristic, complemented
40C2×#3830—31no—
41C22×C3×#67—34no—
42C2×#3804—34no—
43C32×#120—31no—
44#64162—31no—
45#64162—34no—
46#64162—64no—
47#64149—32no—
48#64148—32no—
49#64146—31no—
50#64094—32no—
51#64093—32no—
52#64091—31no—
53#64086—31no—
54C3×#953D314yescomplemented
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⋊D3—22no—
68C22×C3×C3⋊D3—42no—
69C22×C3×C3⋊D3C2312yes—
70C22×C32×D3C2311yescharacteristic
71#6312—22no—
72#6312—42no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C3×#638—24no—
76C3×#638—24no—
77C2×C3×C3⋊Dic3D414yescomplemented
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C32×#67—22no—
81C32×#67—42no—
82C2×C32×Dic3C2312yes—
83C2×C32×Dic3C2311yescharacteristic
84#6293—22no—
85#6292—42no—
86#6291—22no—
87#6290—42no—
88#6289—22no—
89D3×C3⋊Dic3—22no—
90#5805—91no—
91#5592—94no—
92C24×C32—24no—
93C24×C32C2×D314yes—
94C24×C32C2×D311yescharacteristic
95C23×C3⋊D3C2×D311yescharacteristic
96C23×C3×D3—34no—
97C2×C32×D4—31no—
98C2×C32×D4—32no—
99C2×#638—24no—
100C2×#638—31no—
101C2×#638—34no—
102C2×#638—32no—
103C2×#638—38no—
104C2×#638—32no—
105C2×#638—31no—
106C2×#638—64no—
107C2×#638—68no—
108C2×#638C2×D312yes—
109C2×#638C2×D312yes—
110C22×C3⋊Dic3—31no—
111C22×C3⋊Dic3C2×D311yescharacteristic
112#3830—64no—
113C2×C4⋊(C3⋊D3)—32no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×C3×#67—38no—
117C2×C3×#67—38no—
118C2×C3×#67—616no—
119C2×C3×#67C2×D314yes—
120C2×C3×#67C2×D318yescomplemented
121C22×C3×Dic3—34no—
122C2×#627—34no—
123C2×#626—34no—
124C2×#626—38no—
125C2×#625—38no—
126C2×#624—34no—
127#3804—616no—
128C2×D3×Dic3—34no—
129C32×C22⋊C4—31no—
130C32×C22⋊C4—32no—
131#3757—31no—
132#3757—32no—
133#3757—34no—
134#3757—38no—
135#3757—64no—
136#3757—68no—
137#3752—32no—
138#3752—32no—
139#3750—32no—
140C4×C3⋊Dic3—32no—
141C3×#217C2×D314yes—
142C3×#217C2×D318yescomplemented
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—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—44no—
160C2×C32×D3—21no—
161C2×C32×D3—22no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)—22no—
164C3×C3⋊Dic3C2×D412yes—
165C3×C3⋊Dic3C2×D412yes—
166C32×Dic3—21no—
167C32×Dic3—22no—
168C22×C3×D4—91no—
169C22×#67—34no—
170C2×#237—94no—
171C2×#235—91no—
172C3×#120—34no—
173#953—94no—
174#953—91no—
175#953—91no—
176#953—91no—
177#953—184no—
178#953C3⋊D311yescharacteristic, complemented
179#940—98no—
180#940—92no—
181#939—98no—
182#938—92no—
183#938—92no—
184#937—94no—
185#936—92no—
186#926—91no—
187#885—98no—
188#884—98no—
189#882—94no—
190#877—94no—
191C23×C32—22no—
192C23×C32—28no—
193C23×C32—24no—
194C23×C32—28no—
195C23×C32—31no—
196C23×C32—48no—
197C23×C32—42no—
198C23×C32—48no—
199C23×C32—416no—
200C23×C32C22×D318yes—
201C23×C32C22×D314yes—
202C23×C32C22×D312yes—
203C23×C32C22×D311yescharacteristic
204C22×C3⋊D3—22no—
205C22×C3⋊D3—31no—
206C22×C3⋊D3—34no—
207C22×C3⋊D3—42no—
208C22×C3⋊D3—64no—
209C22×C3⋊D3C22×D312yes—
210C22×C3×D3—38no—
211C22×C3×D3—68no—
212C22×C3×D3—128no—
213C22×C3×D3C22×D314yes—
214C32×D4—62no—
215C32×D4—122no—
216C2×C32×C4—32no—
217C2×C32×C4—31no—
218#638—24no—
219#638—24no—
220#638—62no—
221#638—62no—
222#638—68no—
223#638—128no—
224#638—122no—
225#638—122no—
226#638—128no—
227#638—1216no—
228C2×C3⋊Dic3—32no—
229C2×C3⋊Dic3—31no—
230C2×C3⋊Dic3—38no—
231C2×C3⋊Dic3—34no—
232C2×C3⋊Dic3—68no—
233C2×C3⋊Dic3—64no—
234C2×C3⋊Dic3—62no—
235C2×C3⋊Dic3#6714yescomplemented
236C2×C3⋊Dic3C22×D311yescharacteristic
237C2×C3⋊Dic3C22×D311yescharacteristic
238C4⋊(C3⋊D3)—122no—
239C4×C3⋊D3—62no—
240#634—62no—
241C3×#67—28no—
242C3×#67—48no—
243C3×#67—616no—
244C3×#67—616no—
245C2×C3×Dic3—316no—
246C2×C3×Dic3—34no—
247C2×C3×Dic3—34no—
248C2×C3×Dic3C22×D318yes—
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—42no—
263C32×D3—41no—
264C24×C3—24no—
265C24×C3—91no—
266C24×C3D3214yes—
267C24×C3C2×C3⋊D311yescharacteristic
268C23×D3—34no—
269C2×C3×D4—34no—
270C2×C3×D4—38no—
271C2×C3×D4—92no—
272C2×C3×D4—92no—
273C2×C3×D4—184no—
274C22×C3×C4—91no—
275C2×#67—38no—
276C2×#67—38no—
277C2×#67—616no—
278C2×#67—94no—
279C2×#67—94no—
280C2×#67—91no—
281C2×#67—98no—
282C2×#67—98no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—92no—
286C2×#67—184no—
287C2×#67—188no—
288C2×#67C2×C3⋊D311yescharacteristic
289C2×#67C2×C3⋊D312yescomplemented
290C22×Dic3—34no—
291C22×Dic3—94no—
292#237—1816no—
293#235—184no—
294C2×D12—91no—
295C2×D12—98no—
296C2×C4×D3—94no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×Dic6—94no—
300C2×Dic6—91no—
301C3×C22⋊C4—34no—
302C3×C22⋊C4—38no—
303#217—94no—
304#217—98no—
305#217—91no—
306#217—92no—
307#217—92no—
308#217—92no—
309#217—184no—
310#217—188no—
311#217C2×C3⋊D311yescharacteristic
312#217C2×C3⋊D312yescomplemented
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—98no—
315C4⋊Dic3—92no—
316Dic3⋊C4—98no—
317C4×Dic3—98no—
318C4×Dic3—92no—
319C22×C32—22no—
320C22×C32—216no—
321C22×C32—28no—
322C22×C32—22no—
323C22×C32—22no—
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—61no—
338C22×C32D3×D418yes—
339C22×C32C2×#6712yes—
340C22×C32C23×D314yes—
341C22×C32C23×D311yescharacteristic
342C2×C3⋊D3—22no—
343C2×C3⋊D3—22no—
344C2×C3⋊D3—22no—
345C2×C3⋊D3—44no—
346C2×C3⋊D3—62no—
347C2×C3⋊D3—61no—
348C2×C3⋊D3—64no—
349C2×C3⋊D3—68no—
350C2×C3⋊D3—124no—
351C2×C3⋊D3—128no—
352C2×C3×D3—24no—
353C2×C3×D3—28no—
354C2×C3×D3—68no—
355C2×C3×D3—68no—
356C2×C3×D3—68no—
357C2×C3×D3—1216no—
358C32×C4—62no—
359C32×C4—61no—
360C3⋊Dic3—62no—
361C3⋊Dic3—61no—
362C3⋊Dic3—64no—
363C3⋊Dic3—68no—
364C3⋊Dic3—124no—
365C3⋊Dic3—128no—
366C3⋊Dic3C2×#6712yes—
367C3⋊Dic3C2×#6712yes—
368C3×Dic3—24no—
369C3×Dic3—28no—
370C3×Dic3—38no—
371C3×Dic3—38no—
372C2×#41—271no—
373C22×D4—91no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—31no—
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—34no—
389C23×C3—48no—
390C23×C3—48no—
391C23×C3—42no—
392C23×C3—416no—
393C23×C3—92no—
394C23×C3—182no—
395C23×C3—362no—
396C23×C3C2×D3218yes—
397C23×C3C2×D3214yes—
398C23×C3C22×C3⋊D312yes—
399C23×C3C22×C3⋊D311yescharacteristic
400C22×D3—38no—
401C22×D3—68no—
402C22×D3—94no—
403C22×D3—91no—
404C22×D3—128no—
405C22×D3—184no—
406C22×D3C22×C3⋊D311yescharacteristic
407C3×D4—68no—
408C3×D4—128no—
409C3×D4—184no—
410C3×D4—184no—
411C2×C3×C4—38no—
412C2×C3×C4—34no—
413C2×C3×C4—91no—
414C2×C3×C4—91no—
415C2×C3×C4—94no—
416#67—22no—
417#67—42no—
418#67—616no—
419#67—616no—
420#67—188no—
421#67—188no—
422#67—182no—
423#67—368no—
424#67—362no—
425#67—362no—
426#67—368no—
427#67—368no—
428#67—362no—
429#67—3616no—
430C2×Dic3—34no—
431C2×Dic3—34no—
432C2×Dic3—316no—
433C2×Dic3—98no—
434C2×Dic3—94no—
435C2×Dic3—92no—
436C2×Dic3—91no—
437C2×Dic3—188no—
438C2×Dic3—184no—
439C2×Dic3—188no—
440C2×Dic3C22×C3⋊D312yes—
441C2×Dic3C22×C3⋊D311yescharacteristic
442D12—182no—
443D12—368no—
444C4×D3—182no—
445C4×D3—188no—
446C4×D3—182no—
447Dic6—182no—
448Dic6—188no—
449C2×C32—22no—
450C2×C32—24no—
451C2×C32—28no—
452C2×C32—28no—
453C2×C32—48no—
454C2×C32—416no—
455C2×C32—42no—
456C2×C32—48no—
457C2×C32—121no—
458C2×C32C2×D3×D414yes—
459C2×C32#100418yes—
460C2×C32#100412yes—
461C2×C32C22×#6711yescharacteristic
462C3⋊D3—42no—
463C3⋊D3—121no—
464C3⋊D3—124no—
465C3⋊D3—244no—
466C3×D3—44no—
467C3×D3—128no—
468C24—91no—
469C24D3×C3⋊D311yescharacteristic
470#41—544no—
471C2×Q8—271no—
472C2×D4—31no—
473C2×D4—32no—
474C2×D4—92no—
475C2×D4—92no—
476C2×D4—184no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—272no—
482C22×C4—91no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—272no—
486C22⋊C4—31no—
487C22⋊C4—32no—
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—416no—
508C22×C3—44no—
509C22×C3—432no—
510C22×C3—64no—
511C22×C3—68no—
512C22×C3—182no—
513C22×C3—182no—
514C22×C3—182no—
515C22×C3—364no—
516C22×C3D3×#6718yes—
517C22×C3D4×C3⋊D312yes—
518C22×C3C22×D3214yes—
519C22×C3C23×C3⋊D311yescharacteristic
520C2×D3—21no—
521C2×D3—22no—
522C2×D3—68no—
523C2×D3—68no—
524C2×D3—68no—
525C2×D3—1216no—
526C2×D3—184no—
527C2×D3—188no—
528C2×D3—181no—
529C2×D3—182no—
530C2×D3—364no—
531C2×D3—368no—
532C3×C4—64no—
533C3×C4—68no—
534C3×C4—92no—
535C3×C4—92no—
536Dic3—21no—
537Dic3—22no—
538Dic3—38no—
539Dic3—38no—
540Dic3—184no—
541Dic3—188no—
542Dic3—181no—
543Dic3—182no—
544Dic3—364no—
545Dic3—368no—
546C32—24no—
547C32#559214yescomplemented
548C32#580511yescharacteristic, complemented
549C23—22no—
550C23—31no—
551C23—42no—
552C23—92no—
553C23—182no—
554C23—271no—
555C23—362no—
556C23C2×D3×C3⋊D312yes—
557C23C2×D3×C3⋊D311yescharacteristic
558Q8—542no—
559D4—62no—
560D4—122no—
561D4—184no—
562D4—184no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—32no—
569C2×C4—31no—
570C2×C4—91no—
571C2×C4—91no—
572C2×C4—94no—
573C2×C4—272no—
574C2×C4—271no—
575C2×C4—542no—
576C2×C4—542no—
577C2×C3—28no—
578C2×C3—22no—
579C2×C3—24no—
580C2×C3—28no—
581C2×C3—48no—
582C2×C3—416no—
583C2×C3—48no—
584C2×C3—42no—
585C2×C3—124no—
586C2×C3—362no—
587C2×C3C2×D3×#6714yes—
588C2×C3#6433018yes—
589C2×C3C2×D4×C3⋊D311yescharacteristic
590C2×C3#6436112yes—
591D3—41no—
592D3—128no—
593D3—364no—
594D3—361no—
595D3—724no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—61no—
602C22—62no—
603C22—182no—
604C22—182no—
605C22—182no—
606C22—364no—
607C22—541no—
608C22—542no—
609C22C3⋊D3×#6712yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—61no—
612C4—62no—
613C4—92no—
614C4—92no—
615C4—541no—
616C4—542no—
617C3—24no—
618C3#1059551314yescomplemented
619C3#1059606311yescharacteristic, complemented
620C2—22no—
621C2—42no—
622C2—121no—
623C2—362no—
624C2—1081no—
625C2C2×C3⋊D3×#6711yescharacteristic
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×C3×#638C2semidirect
2. C2×#89028C2semidirect
3. #11723994C2semidirect
4. C32×#953C2semidirect
5. #11723636C2semidirect
6. #11723634C2semidirect
7. #11723633C2semidirect
8. #11723623C2semidirect
9. #11723615C2semidirect
10. #11723595C2semidirect
11. #11723591C2semidirect
17. C2×#6312C22semidirect
23. C2×C32×#67C22semidirect
24. C2×#6293C22semidirect
25. C2×#6292C22semidirect
26. C2×#6291C22semidirect
27. C2×#6290C22semidirect
28. C2×#6289C22semidirect
30. C2×D3×C3⋊Dic3C22semidirect
32. #88864C22semidirect
34. C32×#217C22semidirect
35. #88807C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
39. C22×#638D3semidirect
54. C3×#953D3semidirect
77. C2×C3×C3⋊Dic3D4semidirect
120. C2×C3×#67C2×D3semidirect
142. C3×#217C2×D3semidirect
178. #953C3⋊D3semidirect
235. C2×C3⋊Dic3#67semidirect
289. C2×#67C2×C3⋊D3semidirect
312. #217C2×C3⋊D3semidirect
383. C33#535semidirect
547. C32#5592semidirect
548. C32#5805semidirect
618. C3#10595513semidirect
619. C3#10596063semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×#890282
#117239942
C32×#9532
#117236362
#117236342
#117236332
#117236232
#117236152
#117235952
#117235912
#105960633
#105955133