Finite Group Databasebeta

← #13172194

Subgroups — #13172194

Order 1728.

1950 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic, complemented
3#11723715C211yesmaximal, characteristic, complemented
4#11723638C212yesmaximal, complemented
5C3×#63961C212yesmaximal, complemented
6#11722977C212yesmaximal, complemented
7#11722964C212yesmaximal, complemented
8#11722935C214yesmaximal, complemented
9#10595159—32nomaximal
10#10594920—31nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C32×#67C2212yes—
14C22×C32×Dic3C2211yescharacteristic
15C2×#6291C2214yes—
16C2×#6289—24no—
17C2×#6289C2212yes—
18C2×#6289C2212yes—
19C2×C3×#627C2211yescharacteristic, complemented
20C2×C3×#625C2211yescharacteristic, complemented
21C3×#3805—24no—
22C2×C3×D3×Dic3C2212yescomplemented
23C3×#3757C2212yes—
24#88815C2212yescomplemented
25#88813C2212yescomplemented
26#88804C2214yes—
27C3×#3724C2214yescomplemented
28C3×(C2×D3)⋊Dic3C2214yescomplemented
29C22×C4×C3⋊D3—31no—
30C2×C3×#237—32no—
31C22×#624—32no—
32C22×#624—61no—
33C2×#3805D311yescharacteristic, complemented
34#64146—32no—
35C3×#935—32no—
36C3×#929—32no—
37#63981—32no—
38#63981—32no—
39#63981—62no—
40#63961D312yescomplemented
41#63958—34no—
42#63957—31no—
43#63955—32no—
44#63952—34no—
45#63912—32no—
46#63904—34no—
47#63888—34no—
48#63881—34no—
49#63865—32no—
50C23×C33C2311yescharacteristic
51C22×C3⋊(C3⋊D3)—22no—
52C22×C3⋊(C3⋊D3)—24no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C32×D3C2312yes—
55C2×C3×C3⋊Dic3D414yescomplemented
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C32×#67—24no—
60C2×C32×Dic3—22no—
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63#6291—28no—
64#6289—24no—
65#6289—24no—
66C3×#627—22no—
67C3×#625—22no—
68C3×D3×Dic3—24no—
69#5828—91no—
70#5656—92no—
71C23×C3⋊D3—32no—
72C23×C3⋊D3—31no—
73C23×C3⋊D3—61no—
74C23×C3⋊D3—62no—
75C23×C3⋊D3—121no—
76C2×C32×D4—32no—
77C22×C32×C4—31no—
78C2×#638—34no—
79C22×C3⋊Dic3C2×D311yescharacteristic
80C2×C4×C3⋊D3—32no—
81C2×C4×C3⋊D3—32no—
82C2×C4×C3⋊D3—64no—
83C2×C3×#67—22no—
84C2×C3×#67—32no—
85C2×C3×#67C2×D312yes—
86C2×C3×#67C2×D312yes—
87C22×C3×Dic3—21no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—61no—
91C22×C3×Dic3C2×D312yes—
92C3×#237—68no—
93C2×C3×C4×D3—32no—
94C2×C3×Dic6—32no—
95C2×#627C2×D311yescharacteristic, complemented
96C2×#626—34no—
97C2×#626—34no—
98C2×#626—64no—
99C2×#625C2×D311yescharacteristic, complemented
100C2×#624—34no—
101C2×#624—34no—
102C2×#624—68no—
103C2×#624—62no—
104C2×#624—62no—
105C2×#624—68no—
106#3805—24no—
107C2×D3×Dic3C2×D312yescomplemented
108#3757C2×D312yes—
109#3752—34no—
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—34no—
113C3×#217—62no—
114C3×(C2×D3)⋊C4—34no—
115C3×Dic3⋊C4—34no—
116C3×Dic3⋊C4—34no—
117#3724—32no—
118#3724C2×D314yescomplemented
119Dic3⋊Dic3—34no—
120#3722—34no—
121#3722—34no—
122#3722—64no—
123(C2×D3)⋊Dic3C2×D314yescomplemented
124Dic32—32no—
125Dic32—34no—
126C22×C33—22no—
127C22×C33C2×D412yes—
128C22×C33C2411yescharacteristic
129C2×C3⋊(C3⋊D3)—28no—
130C2×C3⋊(C3⋊D3)—22no—
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)#4114yescomplemented
133C2×C32×D3—22no—
134C2×C32×D3—24no—
135C3×C3⋊Dic3—24no—
136C3×C3⋊Dic3C2×D412yes—
137C3×C3⋊Dic3C2×D412yes—
138C32×Dic3—22no—
139C32×Dic3—22no—
140C2×C3×#41—91no—
141C2×#237—32no—
142C22×C4×D3—92no—
143C22×C4×D3—91no—
144C22×C4×D3—181no—
145C3×#123—92no—
146#946—92no—
147#938—92no—
148#937—92no—
149#937—92no—
150#937—182no—
151#935—32no—
152#929—32no—
153C4×#67—94no—
154#926—91no—
155#896—94no—
156#893—94no—
157#884—94no—
158#882—92no—
159#67⋊C4—94no—
160#877—92no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—32no—
164C23×C32—41no—
165C23×C32C22×D312yes—
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—32no—
168C22×C3⋊D3—34no—
169C22×C3⋊D3—62no—
170C22×C3⋊D3—64no—
171C22×C3⋊D3—64no—
172C22×C3⋊D3—62no—
173C22×C3⋊D3—68no—
174C22×C3⋊D3—68no—
175C22×C3⋊D3—64no—
176C22×C3⋊D3—64no—
177C22×C3⋊D3—122no—
178C22×C3⋊D3—128no—
179C22×C3⋊D3—124no—
180C22×C3⋊D3—124no—
181C22×C3⋊D3—128no—
182C22×C3×D3—22no—
183C22×C3×D3C22×D312yes—
184C22×C3×D3C22×D312yes—
185C32×D4—64no—
186C2×C32×C4—32no—
187C2×C32×C4—32no—
188C2×C32×C4—62no—
189#638—68no—
190C2×C3⋊Dic3#6714yescomplemented
191C2×C3⋊Dic3C22×D314yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4×C3⋊D3—64no—
195C4×C3⋊D3—64no—
196C3×#67—24no—
197C3×#67—24no—
198C3×#67—44no—
199C3×#67—64no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—24no—
202C2×C3×Dic3—22no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3—38no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—64no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3C22×D314yes—
215C2×C3×Dic3C22×D312yes—
216C2×C3×Dic3C22×D312yes—
217C3×C4×D3—64no—
218C3×Dic6—64no—
219#627—22no—
220#626—68no—
221#626—68no—
222#626—128no—
223#625—22no—
224#624—68no—
225#624—68no—
226#624—68no—
227#624—68no—
228D3×Dic3—24no—
229#525—271noSylow
230C2×C33—22no—
231C2×C33C22×D411yescharacteristic
232C2×C33C2×#4112yes—
233C3⋊(C3⋊D3)—24no—
234C32×D3—42no—
235C23×D3—92no—
236C23×D3—91no—
237C23×D3—181no—
238C23×D3—182no—
239C23×D3—361no—
240C3×#41—184no—
241C2×C3×Q8—91no—
242C2×C3×D4—32no—
243C2×C3×D4—32no—
244C2×C3×D4—62no—
245C2×C3×D4—91no—
246C22×C3×C4—32no—
247C22×C3×C4—61no—
248C22×C3×C4—92no—
249C22×C3×C4—91no—
250C2×#67—32no—
251C2×#67—94no—
252C2×#67—94no—
253C2×#67—184no—
254C2×#67D3212yes—
255C22×Dic3—32no—
256C22×Dic3—32no—
257C22×Dic3—61no—
258C22×Dic3D3211yescharacteristic
259#237—68no—
260C2×D12—94no—
261C2×C4×D3—32no—
262C2×C4×D3—94no—
263C2×C4×D3—94no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—182no—
267C2×C4×D3—182no—
268C2×C4×D3—184no—
269C2×C4×D3—188no—
270C2×C4×D3—188no—
271C2×Dic6—32no—
272C3×C4⋊C4—94no—
273C3×C22⋊C4—94no—
274C3×C22⋊C4—92no—
275#217—32no—
276#217—32no—
277#217—34no—
278#217—62no—
279(C2×D3)⋊C4—34no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—94no—
282(C2×D3)⋊C4—94no—
283(C2×D3)⋊C4—184no—
284C4⋊Dic3—92no—
285Dic3⋊C4—34no—
286Dic3⋊C4—34no—
287Dic3⋊C4—94no—
288C4×Dic3—94no—
289C4×Dic3—92no—
290C22×C32—22no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—21no—
294C22×C32—22no—
295C22×C32—24no—
296C22×C32—22no—
297C22×C32—42no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—62no—
302C22×C32—64no—
303C22×C32D3×D414yes—
304C22×C32C2×#6712yes—
305C22×C32C23×D312yes—
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—34no—
308C2×C3⋊D3—38no—
309C2×C3⋊D3—64no—
310C2×C3⋊D3—68no—
311C2×C3⋊D3—68no—
312C2×C3⋊D3—62no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—64no—
315C2×C3⋊D3—616no—
316C2×C3⋊D3—616no—
317C2×C3⋊D3—64no—
318C2×C3⋊D3—64no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—124no—
321C2×C3⋊D3—124no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3—124no—
324C2×C3⋊D3—124no—
325C2×C3⋊D3—1216no—
326C2×C3⋊D3—1216no—
327C2×C3×D3—22no—
328C2×C3×D3—24no—
329C2×C3×D3—24no—
330C2×C3×D3—22no—
331C2×C3×D3—44no—
332C2×C3×D3—42no—
333C32×C4—62no—
334C32×C4—62no—
335C3⋊Dic3—24no—
336C3⋊Dic3C2×#6712yes—
337C3⋊Dic3C2×#6712yes—
338C3×Dic3—24no—
339C3×Dic3—24no—
340C3×Dic3—22no—
341C3×Dic3—22no—
342C3×Dic3—34no—
343C3×Dic3—34no—
344C3×Dic3—42no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—64no—
348C3×Dic3—64no—
349C3×Dic3—124no—
350C2×#41—91no—
351C23×C4—271no—
352#123—92no—
353C22⋊Q8—272no—
354#121—272no—
355#120—272no—
356C4×D4—274no—
357#117—271no—
358C33#52511yescharacteristic, complemented, Sylow
359C23×C3—21no—
360C23×C3—22no—
361C23×C3—32no—
362C23×C3—32no—
363C23×C3—41no—
364C23×C3—62no—
365C23×C3C2×D3212yes—
366C23×C3C2×D3211yescharacteristic
367C22×D3—94no—
368C22×D3—92no—
369C22×D3—182no—
370C22×D3—184no—
371C22×D3—184no—
372C22×D3—188no—
373C22×D3—184no—
374C22×D3—188no—
375C22×D3—184no—
376C22×D3—182no—
377C22×D3—362no—
378C22×D3—368no—
379C22×D3—364no—
380C22×D3—368no—
381C22×D3—364no—
382C22×D3C2×D3212yes—
383C3×Q8—182no—
384C3×D4—64no—
385C3×D4—64no—
386C3×D4—124no—
387C3×D4—182no—
388C2×C3×C4—32no—
389C2×C3×C4—34no—
390C2×C3×C4—32no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—64no—
394C2×C3×C4—64no—
395C2×C3×C4—91no—
396C2×C3×C4—91no—
397C2×C3×C4—94no—
398C2×C3×C4—94no—
399C2×C3×C4—184no—
400#67—24no—
401#67—64no—
402#67—188no—
403#67—188no—
404#67—368no—
405C2×Dic3—22no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—38no—
409C2×Dic3—34no—
410C2×Dic3—34no—
411C2×Dic3—64no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—64no—
415C2×Dic3—64no—
416C2×Dic3C2×D3212yes—
417C2×Dic3C2×D3212yes—
418D12—188no—
419C4×D3—64no—
420C4×D3—184no—
421C4×D3—184no—
422C4×D3—188no—
423C4×D3—188no—
424C4×D3—188no—
425C4×D3—188no—
426Dic6—64no—
427C2×C32—22no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—22no—
431C2×C32—24no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—44no—
437C2×C32—44no—
438C2×C32—122no—
439C2×C32C2×D3×D412yes—
440C2×C32D3×#4114yes—
441C2×C32D3×#4112yes—
442C2×C32C22×#6711yescharacteristic
443C3⋊D3—64no—
444C3⋊D3—68no—
445C3⋊D3—68no—
446C3⋊D3—128no—
447C3⋊D3—128no—
448C3×D3—42no—
449C3×D3—42no—
450C3×D3—82no—
451C24—271no—
452#41—184no—
453C2×Q8—91no—
454C2×D4—32no—
455C2×D4—91no—
456C2×D4—274no—
457C22×C4—31no—
458C22×C4—92no—
459C22×C4—91no—
460C22×C4—272no—
461C22×C4—272no—
462C22×C4—544no—
463C4⋊C4—94no—
464C4⋊C4—272no—
465C22⋊C4—94no—
466C22⋊C4—92no—
467C22⋊C4—274no—
468C42—272no—
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—21no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—64no—
481C22×C3—64no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—124no—
485C22×C3—122no—
486C22×C3D3×#6714yes—
487C22×C3#381112yes—
488C22×C3C22×D3211yescharacteristic
489C22×C3C22×D3212yes—
490C2×D3—24no—
491C2×D3—22no—
492C2×D3—94no—
493C2×D3—98no—
494C2×D3—184no—
495C2×D3—188no—
496C2×D3—184no—
497C2×D3—188no—
498C2×D3—1816no—
499C2×D3—1816no—
500C2×D3—184no—
501C2×D3—182no—
502C2×D3—184no—
503C2×D3—184no—
504C2×D3—182no—
505C2×D3—364no—
506C2×D3—364no—
507C2×D3—364no—
508C2×D3—3616no—
509C2×D3—3616no—
510C2×D3—364no—
511C2×D3—364no—
512C3×C4—64no—
513C3×C4—62no—
514C3×C4—64no—
515C3×C4—62no—
516C3×C4—92no—
517C3×C4—92no—
518C3×C4—122no—
519C3×C4—184no—
520Dic3—22no—
521Dic3—22no—
522Dic3—34no—
523Dic3—34no—
524Dic3—62no—
525Dic3—62no—
526Dic3—64no—
527Dic3—64no—
528Dic3—124no—
529C32—22no—
530C32—21no—
531C32—41no—
532C32#565612yescomplemented
533C32#582811yescharacteristic, complemented
534C23—32no—
535C23—272no—
536C23—544no—
537C23—542no—
538C23D3311yescharacteristic
539Q8—182no—
540D4—64no—
541D4—182no—
542D4—548no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—62no—
546C2×C4—91no—
547C2×C4—91no—
548C2×C4—94no—
549C2×C4—94no—
550C2×C4—184no—
551C2×C4—544no—
552C2×C4—544no—
553C2×C3—24no—
554C2×C3—24no—
555C2×C3—22no—
556C2×C3—24no—
557C2×C3—22no—
558C2×C3—21no—
559C2×C3—22no—
560C2×C3—44no—
561C2×C3—44no—
562C2×C3—42no—
563C2×C3—41no—
564C2×C3—122no—
565C2×C3—122no—
566C2×C3—242no—
567C2×C3#6430612yes—
568C2×C3#6431214yes—
569C2×C3C2×D3×#6712yes—
570C2×C3C2×#381111yescharacteristic
571D3—42no—
572D3—188no—
573D3—188no—
574D3—184no—
575D3—368no—
576D3—368no—
577C22—22no—
578C22—64no—
579C22—62no—
580C22—274no—
581C22—548no—
582C22—542no—
583C22—542no—
584C22#8895212yes—
585C22C2×D3311yescharacteristic
586C4—62no—
587C4—62no—
588C4—92no—
589C4—92no—
590C4—184no—
591C3—21no—
592C3—22no—
593C3—41no—
594C3#1059492011yescharacteristic, complemented
595C3#1059515912yescomplemented
596C2—22no—
597C2—122no—
598C2—544no—
599C2#1172441112yes—
600C2C2×#8895211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC23×C33C23
hypercenterC23D33
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6289C2semidirect
2. C2×C3×#3805C2semidirect
3. #11723715C2semidirect
4. #11723638C2semidirect
5. C3×#63961C2semidirect
6. #11722977C2semidirect
7. #11722964C2semidirect
8. #11722935C2semidirect
19. C2×C3×#627C22semidirect
20. C2×C3×#625C22semidirect
22. C2×C3×D3×Dic3C22semidirect
24. #88815C22semidirect
25. #88813C22semidirect
27. C3×#3724C22semidirect
28. C3×(C2×D3)⋊Dic3C22semidirect
33. C2×#3805D3semidirect
40. #63961D3semidirect
55. C2×C3×C3⋊Dic3D4semidirect
95. C2×#627C2×D3semidirect
99. C2×#625C2×D3semidirect
107. C2×D3×Dic3C2×D3semidirect
118. #3724C2×D3semidirect
123. (C2×D3)⋊Dic3C2×D3semidirect
132. C2×C3⋊(C3⋊D3)#41semidirect
190. C2×C3⋊Dic3#67semidirect
358. C33#525semidirect
532. C32#5656semidirect
533. C32#5828semidirect
594. C3#10594920semidirect
595. C3#10595159semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×C3×#38052
#117237152
#117236382
C3×#639612
#117229772
#117229642
#117229352
#105951593
#105949203