Finite Group Databasebeta

← #13178791

Subgroups — #13178791

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
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×#89046C211yesmaximal, characteristic, complemented
3C3×#64162C212yesmaximal, complemented
4#11723726C214yesmaximal, complemented
5#11723721C212yesmaximal, complemented
6#11723720C212yesmaximal, complemented
7#11723717C212yesmaximal, complemented
8#11723715C211yesmaximal, characteristic, complemented
9#10595501—31nomaximal
10#10595498—32nomaximal
11C24×C33C2211yescharacteristic
12C2×C3×#638C2212yes—
13C2×C3×#638C2214yes—
14C22×C3×C3⋊Dic3—24no—
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C2×#6296C2211yescharacteristic, complemented
20C2×#6295C2211yescharacteristic, complemented
21#89046—24no—
22C2×#6294C2212yescomplemented
23C3×#3757C2212yes—
24C3×#3757C2214yes—
25#88815C2212yescomplemented
26#88815C2214yescomplemented
27#88814C2214yescomplemented
28#88813C2212yescomplemented
29C23×C3⋊Dic3D311yescharacteristic, complemented
30C23×C3×Dic3—32no—
31C23×C3×Dic3—61no—
32C2×#3805—31no—
33C2×#3804—32no—
34#64162D312yescomplemented
35C3×#953—32no—
36C3×#953—32no—
37C3×#953—62no—
38#63982—32no—
39#63980—32no—
40#63978—34no—
41#67⋊Dic3—34no—
42Dic3×#67—34no—
43#63966—32no—
44#63964—34no—
45#63962—32no—
46#63961—32no—
47#63957—31no—
48#63955—32no—
49#63952—34no—
50C23×C33—22no—
51C23×C33—24no—
52C23×C33C2311yescharacteristic
53C23×C33C2312yes—
54C22×C3×C3⋊D3C2312yes—
55C3×#638—24no—
56C3×#638—28no—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3—24no—
59C2×C3×C3⋊Dic3—24no—
60C2×C3×C3⋊Dic3D414yescomplemented
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2314yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2311yescharacteristic
66#6296—22no—
67#6295—22no—
68#6294—24no—
69#5791—91no—
70#5586—92no—
71C24×C32—21no—
72C24×C32—22no—
73C24×C32—41no—
74C24×C32C2×D311yescharacteristic
75C24×C32C2×D312yes—
76C2×#638C2×D312yes—
77C2×#638C2×D314yes—
78C22×C3⋊Dic3—24no—
79C22×C3⋊Dic3C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D312yes—
81C22×C3⋊Dic3C2×D312yes—
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×C3×#67—32no—
84C2×C3×#67—32no—
85C2×C3×#67—34no—
86C2×C3×#67—34no—
87C2×C3×#67—64no—
88C2×C3×#67—62no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—34no—
93C22×C3×Dic3—68no—
94C22×C3×Dic3—61no—
95C22×C3×Dic3—62no—
96C22×C3×Dic3—62no—
97C22×C3×Dic3—61no—
98C22×C3×Dic3—68no—
99C2×#627—32no—
100C2×#627—31no—
101C2×#626—32no—
102C2×#625—31no—
103C2×#624—32no—
104#3805—64no—
105#3804—68no—
106C2×D3×Dic3—32no—
107C2×D3×Dic3—32no—
108#3757C2×D312yes—
109#3757C2×D314yes—
110C3×#217—34no—
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—34no—
114C3×#217—62no—
115C3×#217—64no—
116#3724—34no—
117#3724—32no—
118Dic3⋊Dic3—34no—
119Dic3⋊Dic3—34no—
120Dic3⋊Dic3—34no—
121#3722—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124Dic32—34no—
125Dic32—32no—
126C22×C33—28no—
127C22×C33—22no—
128C22×C33—22no—
129C22×C33—22no—
130C22×C33C2×D412yes—
131C22×C33#4114yescomplemented
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—24no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—24no—
138C3×C3⋊Dic3C2×D412yes—
139C3×C3⋊Dic3C2×D412yes—
140C23×C3×C4—91no—
141C23×Dic3—32no—
142C23×Dic3—61no—
143C2×#237—92no—
144C2×#235—91no—
145C3×#120—92no—
146#953—32no—
147#953—32no—
148#953—62no—
149#939—94no—
150#935—92no—
151D4×Dic3—94no—
152#930—92no—
153#929—92no—
154C4×#67—94no—
155#926—91no—
156#924—92no—
157#886—92no—
158#67⋊C4—94no—
159#878—94no—
160#877—92no—
161C23×C32—24no—
162C23×C32—22no—
163C23×C32—28no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—28no—
167C23×C32—22no—
168C23×C32—21no—
169C23×C32—24no—
170C23×C32—22no—
171C23×C32—41no—
172C23×C32—42no—
173C23×C32—48no—
174C23×C32—44no—
175C23×C32—44no—
176C23×C32—48no—
177C23×C32C22×D312yes—
178C23×C32C22×D312yes—
179C23×C32C22×D314yes—
180C23×C32C22×D311yescharacteristic
181C22×C3⋊D3C22×D312yes—
182C22×C3×D3—32no—
183C22×C3×D3—32no—
184C22×C3×D3—62no—
185#638—24no—
186#638—28no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—24no—
189C2×C3⋊Dic3—24no—
190C2×C3⋊Dic3#6714yescomplemented
191C2×C3⋊Dic3C22×D314yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C3×#67—68no—
197C3×#67—64no—
198C3×#67—68no—
199C3×#67—64no—
200C3×#67—128no—
201C3×#67—124no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—64no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—64no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—68no—
217C2×C3×Dic3—68no—
218C2×C3×Dic3—68no—
219C2×C3×Dic3—68no—
220C2×C3×Dic3—64no—
221C2×C3×Dic3—64no—
222#627—64no—
223#627—62no—
224#626—64no—
225#625—62no—
226#624—64no—
227D3×Dic3—64no—
228D3×Dic3—64no—
229#525—271noSylow
230C2×C33—22no—
231C2×C33—24no—
232C2×C33C22×D411yescharacteristic
233C2×C33C2×#4112yes—
234C3×C3⋊D3—42no—
235C24×C3—21no—
236C24×C3—22no—
237C24×C3—41no—
238C24×C3D3211yescharacteristic
239C24×C3D3212yes—
240C2×C3×D4—94no—
241C2×C3×D4—92no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C22×C3×C4—92no—
245C22×C3×C4—91no—
246C22×C3×C4—184no—
247C2×#67—34no—
248C2×#67—32no—
249C2×#67—34no—
250C2×#67—32no—
251C2×#67—64no—
252C2×#67—62no—
253C2×#67—92no—
254C22×Dic3—32no—
255C22×Dic3—34no—
256C22×Dic3—34no—
257C22×Dic3—32no—
258C22×Dic3—61no—
259C22×Dic3—62no—
260C22×Dic3—62no—
261C22×Dic3—61no—
262C22×Dic3—68no—
263C22×Dic3—68no—
264C22×Dic3—92no—
265#237—188no—
266#235—184no—
267C2×D12—91no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×Dic6—92no—
271C2×Dic6—91no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—94no—
274#217—32no—
275#217—34no—
276#217—34no—
277#217—32no—
278#217—62no—
279#217—64no—
280#217—94no—
281(C2×D3)⋊C4—94no—
282(C2×D3)⋊C4—94no—
283C4⋊Dic3—94no—
284C4⋊Dic3—92no—
285Dic3⋊C4—94no—
286Dic3⋊C4—94no—
287Dic3⋊C4—94no—
288C4×Dic3—94no—
289C4×Dic3—92no—
290C22×C32—28no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—24no—
295C22×C32—216no—
296C22×C32—216no—
297C22×C32—24no—
298C22×C32—24no—
299C22×C32—24no—
300C22×C32—24no—
301C22×C32—24no—
302C22×C32—21no—
303C22×C32—22no—
304C22×C32—28no—
305C22×C32—24no—
306C22×C32—24no—
307C22×C32—22no—
308C22×C32—42no—
309C22×C32—44no—
310C22×C32—41no—
311C22×C32—44no—
312C22×C32—44no—
313C22×C32—44no—
314C22×C32—44no—
315C22×C32—44no—
316C22×C32—44no—
317C22×C32—416no—
318C22×C32—416no—
319C22×C32#23514yescomplemented
320C22×C32D3×D414yes—
321C22×C32#23718yescomplemented
322C22×C32C2×#6712yes—
323C22×C32C23×D311yescharacteristic
324C22×C32C23×D312yes—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—24no—
327C2×C3×D3—64no—
328C2×C3×D3—62no—
329C2×C3×D3—64no—
330C2×C3×D3—62no—
331C2×C3×D3—124no—
332C2×C3×D3—122no—
333C3⋊Dic3—22no—
334C3⋊Dic3—22no—
335C3⋊Dic3—24no—
336C3⋊Dic3C2×#6712yes—
337C3⋊Dic3C2×#6712yes—
338C3×Dic3—34no—
339C3×Dic3—34no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—64no—
343C3×Dic3—64no—
344C3×Dic3—62no—
345C3×Dic3—64no—
346C3×Dic3—64no—
347C3×Dic3—62no—
348C3×Dic3—124no—
349C3×Dic3—122no—
350C2×#41—271no—
351C23×C4—91no—
352#123—272no—
353C22⋊Q8—272no—
354#121—272no—
355#120—92no—
356C4×D4—274no—
357#117—271no—
358C33#52511yescharacteristic, complemented, Sylow
359C23×C3—24no—
360C23×C3—28no—
361C23×C3—28no—
362C23×C3—24no—
363C23×C3—24no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—48no—
370C23×C3—44no—
371C23×C3—48no—
372C23×C3—44no—
373C23×C3—42no—
374C23×C3—41no—
375C23×C3—92no—
376C23×C3C2×D3214yes—
377C23×C3C2×D3212yes—
378C23×C3C2×D3212yes—
379C23×C3C2×D3211yescharacteristic
380C22×D3—32no—
381C22×D3—32no—
382C22×D3—62no—
383C3×D4—188no—
384C3×D4—184no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—94no—
388C2×C3×C4—94no—
389C2×C3×C4—92no—
390C2×C3×C4—92no—
391C2×C3×C4—184no—
392C2×C3×C4—184no—
393C2×C3×C4—182no—
394#67—68no—
395#67—68no—
396#67—64no—
397#67—64no—
398#67—128no—
399#67—124no—
400#67—184no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—38no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—32no—
407C2×Dic3—34no—
408C2×Dic3—32no—
409C2×Dic3—64no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—62no—
413C2×Dic3—64no—
414C2×Dic3—62no—
415C2×Dic3—68no—
416C2×Dic3—68no—
417C2×Dic3—68no—
418C2×Dic3—68no—
419C2×Dic3—64no—
420C2×Dic3—64no—
421C2×Dic3—184no—
422D12—182no—
423C4×D3—184no—
424C4×D3—184no—
425Dic6—184no—
426Dic6—182no—
427C2×C32—24no—
428C2×C32—22no—
429C2×C32—28no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—28no—
433C2×C32—24no—
434C2×C32—22no—
435C2×C32—22no—
436C2×C32—21no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32—48no—
440C2×C32—44no—
441C2×C32—44no—
442C2×C32—48no—
443C2×C32C2×#23512yes—
444C2×C32C2×D3×D412yes—
445C2×C32C2×#23714yes—
446C2×C32C22×#6711yescharacteristic
447C3⋊D3—42no—
448C3×D3—122no—
449C3×D3—122no—
450C3×D3—242no—
451C24#111311yescharacteristic
452#41—544no—
453C2×Q8—271no—
454C2×D4—94no—
455C2×D4—92no—
456C2×D4—271no—
457C22×C4—91no—
458C22×C4—92no—
459C22×C4—92no—
460C22×C4—91no—
461C22×C4—184no—
462C22×C4—272no—
463C4⋊C4—272no—
464C4⋊C4—274no—
465C22⋊C4—92no—
466C22⋊C4—94no—
467C22⋊C4—274no—
468C42—272no—
469C22×C3—24no—
470C22×C3—28no—
471C22×C3—216no—
472C22×C3—216no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—24no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—22no—
483C22×C3—24no—
484C22×C3—28no—
485C22×C3—22no—
486C22×C3—24no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—416no—
491C22×C3—416no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—41no—
496C22×C3—44no—
497C22×C3—42no—
498C22×C3—184no—
499C22×C3—182no—
500C22×C3#380418yescomplemented
501C22×C3#380514yescomplemented
502C22×C3D3×#6714yes—
503C22×C3#381112yes—
504C22×C3C22×D3212yes—
505C22×C3C22×D3211yescharacteristic
506C2×D3—64no—
507C2×D3—64no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—124no—
511C2×D3—122no—
512C3×C4—92no—
513C3×C4—92no—
514C3×C4—184no—
515C3×C4—182no—
516C3×C4—182no—
517Dic3—34no—
518Dic3—34no—
519Dic3—62no—
520Dic3—62no—
521Dic3—64no—
522Dic3—64no—
523Dic3—64no—
524Dic3—62no—
525Dic3—64no—
526Dic3—62no—
527Dic3—124no—
528Dic3—122no—
529C32—22no—
530C32—21no—
531C32—41no—
532C32#558612yescomplemented
533C32#579111yescharacteristic, complemented
534C23—24no—
535C23—22no—
536C23—92no—
537C23C2×#111312yes—
538C23C2×#111311yescharacteristic
539Q8—542no—
540D4—188no—
541D4—184no—
542D4—542no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—94no—
546C2×C4—94no—
547C2×C4—92no—
548C2×C4—92no—
549C2×C4—184no—
550C2×C4—184no—
551C2×C4—182no—
552C2×C4—544no—
553C2×C3—28no—
554C2×C3—24no—
555C2×C3—28no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—22no—
559C2×C3—24no—
560C2×C3—22no—
561C2×C3—21no—
562C2×C3—22no—
563C2×C3—48no—
564C2×C3—44no—
565C2×C3—48no—
566C2×C3—44no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—362no—
570C2×C3C2×#380414yes—
571C2×C3C2×#380512yes—
572C2×C3C2×D3×#6712yes—
573C2×C3C2×#381111yescharacteristic
574D3—122no—
575D3—122no—
576D3—242no—
577C22—28no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—184no—
582C22—182no—
583C22#8904614yescomplemented
584C22#8904912yes—
585C22C22×#111311yescharacteristic
586C4—92no—
587C4—92no—
588C4—184no—
589C4—182no—
590C4—182no—
591C3—22no—
592C3—21no—
593C3—41no—
594C3#1059549812yescomplemented
595C3#1059550111yescharacteristic, complemented
596C2—24no—
597C2—22no—
598C2—362no—
599C2C2×#8904612yes—
600C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24#1113
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
2. C2×#89046C2semidirect
3. C3×#64162C2semidirect
4. #11723726C2semidirect
5. #11723721C2semidirect
6. #11723720C2semidirect
7. #11723717C2semidirect
8. #11723715C2semidirect
19. C2×#6296C22semidirect
20. C2×#6295C22semidirect
22. C2×#6294C22semidirect
25. #88815C22semidirect
26. #88815C22semidirect
27. #88814C22semidirect
28. #88813C22semidirect
29. C23×C3⋊Dic3D3semidirect
34. #64162D3semidirect
60. C2×C3×C3⋊Dic3D4semidirect
131. C22×C33#41semidirect
190. C2×C3⋊Dic3#67semidirect
319. C22×C32#235semidirect
321. C22×C32#237semidirect
358. C33#525semidirect
500. C22×C3#3804semidirect
501. C22×C3#3805semidirect
532. C32#5586semidirect
533. C32#5791semidirect
583. C22#89046semidirect
594. C3#10595498semidirect
595. C3#10595501semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×#890462
C3×#641622
#117237262
#117237212
#117237202
#117237172
#117237152
#105955013
#105954983