Finite Group Databasebeta

← #13177500

Subgroups — #13177500

Order 1728.

1872 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723981C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723632C212yesmaximal, complemented
5#11723628C212yesmaximal, complemented
6#11723623C212yesmaximal, complemented
7#11723612C212yesmaximal, complemented
8#11723526C211yesmaximal, characteristic, complemented
9#11723484C211yesmaximal, characteristic, complemented
10#11723450C211yesmaximal, characteristic, complemented
11#11723437C211yesmaximal, characteristic, complemented
12#10596033—31nomaximal
13#10595125—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2212yes—
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C22×C32×Dic3C2212yes—
21C2×#6292C2212yes—
22C2×#6289C2212yes—
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3757C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2211yescharacteristic
30#88807C2212yes—
31#88807C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88803C2211yescharacteristic
36#88803C2211yescharacteristic
37C32×C4×D4—31no—
38#64149—31no—
39#64149—34no—
40#64149—64no—
41#64147D311yescharacteristic, complemented
42#64139—32no—
43#64138—32no—
44#64104—31no—
45#64099—31no—
46#64093—32no—
47#64092—32no—
48C4×C4⋊(C3⋊D3)—31no—
49#64081—31no—
50C3×#938—34no—
51C3×D4×Dic3D314yescomplemented
52#63978—38no—
53#63975—38no—
54#63966—38no—
55#63958—38no—
56#63909—34no—
57#63894—34no—
58#63872—34no—
59#63855—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65#6312—42no—
66#6312—42no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4⋊(C3⋊(C3⋊D3))—22no—
70C3×#638—42no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3—42no—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78#6292—42no—
79#6289—42no—
80#5619—91no—
81#5562—94no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C2×D311yescharacteristic
85C22×C32×C4—32no—
86C2×#638—32no—
87C2×#638—32no—
88C2×#638—38no—
89C2×#638—38no—
90C2×#638—68no—
91C2×#638—68no—
92C2×#638C2×D312yes—
93C2×C4⋊(C3⋊D3)—31no—
94C2×C4⋊(C3⋊D3)—34no—
95C2×C4⋊(C3⋊D3)—32no—
96C2×C4⋊(C3⋊D3)—64no—
97C2×C4×C3⋊D3—32no—
98C2×C4×C3⋊D3C2×D311yescharacteristic
99C2×C3×#67—38no—
100C22×C3×Dic3C2×D318yes—
101C2×C3×C4×D3—34no—
102C2×#626—38no—
103C2×#624—38no—
104C32×C4⋊C4—31no—
105C32×C22⋊C4—32no—
106C32×C42—31no—
107#3757C2×D312yes—
108#3752—31no—
109#3752—31no—
110#3752—32no—
111#3752—32no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114#3750—31no—
115#3750—31no—
116#3750—32no—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—34no—
119C4×C3⋊Dic3—64no—
120C3×#217—38no—
121C3×#217C2×D318yes—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124C3×C4×Dic3C2×D314yes—
125Dic3⋊Dic3—34no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—38no—
128Dic3⋊Dic3—38no—
129#3722—38no—
130#3722—38no—
131(C2×D3)⋊Dic3—34no—
132(C2×D3)⋊Dic3—34no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33—22no—
136C22×C33C2411yescharacteristic
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)—22no—
139C2×C3⋊(C3⋊D3)—22no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3—21no—
142C2×C3×C3⋊D3—21no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—21no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—21no—
149C32×Dic3—22no—
150C32×Dic3—21no—
151C3×#121—91no—
152C3×C4×D4—34no—
153#940—94no—
154#940—91no—
155#940—184no—
156#939—92no—
157#938—34no—
158#938—92no—
159D4×Dic3C3⋊D311yescharacteristic, complemented
160#930—98no—
161#930—91no—
162#929—98no—
163#926—91no—
164#895—94no—
165#893—91no—
166#890—94no—
167#890—91no—
168#886—92no—
169#884—98no—
170#883—98no—
171#883—92no—
172C4×D12—94no—
173#872—94no—
174C23×C32—28no—
175C23×C32C22×D318yes—
176C23×C32C22×D312yes—
177C22×C3⋊D3—32no—
178C22×C3⋊D3—38no—
179C22×C3⋊D3—68no—
180C22×C3⋊D3C22×D311yescharacteristic
181C22×C3×D3—34no—
182C32×D4—22no—
183C32×D4—28no—
184C32×D4—48no—
185C2×C32×C4—24no—
186C2×C32×C4—32no—
187C2×C32×C4—31no—
188C2×C32×C4—31no—
189C2×C32×C4—122no—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192#638—42no—
193#638—122no—
194#638—122no—
195#638—128no—
196#638—128no—
197#638—1216no—
198#638—1216no—
199C2×C3⋊Dic3—31no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—34no—
203C2×C3⋊Dic3—64no—
204C2×C3⋊Dic3—64no—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D312yes—
207C4⋊(C3⋊D3)—62no—
208C4⋊(C3⋊D3)—68no—
209C4⋊(C3⋊D3)—128no—
210C4⋊(C3⋊D3)—122no—
211C4×C3⋊D3—22no—
212C4×C3⋊D3—122no—
213C3×#67—128no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—38no—
216C2×C3×Dic3—48no—
217C2×C3×Dic3C22×D314yes—
218C2×C3×Dic3C22×D314yes—
219C2×C3×Dic3C22×D318yes—
220C3×C4×D3—68no—
221#626—128no—
222#624—128no—
223#540—271noSylow
224C2×C33—42no—
225C2×C33C2×#4111yescharacteristic
226C2×C33H(5,𝔽2)11yescharacteristic
227C2×C33H(5,𝔽2)11yescharacteristic
228C3⋊(C3⋊D3)—42no—
229C3×C3⋊D3—41no—
230C2×C3×D4—24no—
231C2×C3×D4—92no—
232C2×C3×D4D3214yes—
233C2×C3×D4C2×C3⋊D311yescharacteristic
234C22×C3×C4—38no—
235C22×C3×C4—91no—
236C2×#67—38no—
237C2×#67—98no—
238C2×#67—98no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—92no—
242C2×#67—188no—
243C2×#67—188no—
244C22×Dic3C2×C3⋊D312yes—
245C2×D12—94no—
246C2×D12—91no—
247C2×D12—98no—
248C2×D12—184no—
249C2×C4×D3—34no—
250C2×C4×D3—98no—
251C2×C4×D3—92no—
252C3×C4⋊C4—34no—
253C3×C4⋊C4—91no—
254C3×C22⋊C4—38no—
255C3×C22⋊C4—92no—
256C3×C42—34no—
257#217—38no—
258#217—91no—
259#217—91no—
260#217C2×C3⋊D312yes—
261(C2×D3)⋊C4—98no—
262(C2×D3)⋊C4—98no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—34no—
268C4⋊Dic3—98no—
269C4⋊Dic3—91no—
270C4⋊Dic3—91no—
271C4⋊Dic3—92no—
272C4⋊Dic3C2×C3⋊D311yescharacteristic
273Dic3⋊C4—94no—
274Dic3⋊C4—94no—
275Dic3⋊C4—98no—
276Dic3⋊C4—92no—
277C4×Dic3—94no—
278C4×Dic3—91no—
279C4×Dic3—184no—
280C4×Dic3C2×C3⋊D311yescharacteristic
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—24no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—28no—
288C22×C32—48no—
289C22×C32—48no—
290C22×C32—48no—
291C22×C32C23×D314yes—
292C22×C32C23×D311yescharacteristic
293C2×C3⋊D3—21no—
294C2×C3⋊D3—21no—
295C2×C3⋊D3—21no—
296C2×C3⋊D3—62no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—68no—
300C2×C3⋊D3—68no—
301C2×C3⋊D3—68no—
302C2×C3⋊D3—128no—
303C2×C3⋊D3—128no—
304C2×C3⋊D3—128no—
305C2×C3×D3—64no—
306C2×C3×D3—64no—
307C2×C3×D3—64no—
308C32×C4—28no—
309C32×C4—61no—
310C32×C4—62no—
311C32×C4—61no—
312C32×C4#23518yes—
313C32×C4#23712yes—
314C3⋊Dic3—21no—
315C3⋊Dic3—22no—
316C3⋊Dic3—68no—
317C3⋊Dic3—68no—
318C3⋊Dic3—64no—
319C3⋊Dic3—61no—
320C3⋊Dic3—64no—
321C3⋊Dic3—61no—
322C3×Dic3—28no—
323C3×Dic3—24no—
324C3×Dic3—24no—
325C3×Dic3—64no—
326C3×Dic3—68no—
327#127—271no—
328#125—271no—
329#123—272no—
330#123—272no—
331#121—91no—
332#121—272no—
333#121—271no—
334#121—272no—
335C4×D4—31no—
336C4×D4—271no—
337#117—271no—
338C33#54011yescharacteristic, complemented, Sylow
339C23×C3—28no—
340C23×C3—91no—
341C23×C3C2×D3218yes—
342C23×C3C22×C3⋊D312yes—
343C22×D3—34no—
344C22×D3—98no—
345C22×D3—92no—
346C22×D3—188no—
347C3×D4—28no—
348C3×D4—22no—
349C3×D4—48no—
350C3×D4—362no—
351C2×C3×C4—24no—
352C2×C3×C4—38no—
353C2×C3×C4—34no—
354C2×C3×C4—34no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—128no—
358C2×C3×C4—182no—
359C2×C3×C4C2×D3214yes—
360C2×C3×C4C22×C3⋊D311yescharacteristic
361#67—128no—
362#67—362no—
363#67—368no—
364#67—368no—
365#67—362no—
366#67—362no—
367#67—3616no—
368#67—3616no—
369C2×Dic3—34no—
370C2×Dic3—38no—
371C2×Dic3—42no—
372C2×Dic3—94no—
373C2×Dic3—94no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—184no—
377C2×Dic3—184no—
378C2×Dic3C22×C3⋊D311yescharacteristic
379C2×Dic3C22×C3⋊D311yescharacteristic
380C2×Dic3C22×C3⋊D312yes—
381D12—188no—
382D12—182no—
383D12—368no—
384D12—368no—
385C4×D3—68no—
386C4×D3—362no—
387C4×D3—368no—
388C2×C32—24no—
389C2×C32—24no—
390C2×C32—24no—
391C2×C32—42no—
392C2×C32—48no—
393C2×C32—88no—
394C2×C32C2×#23514yes—
395C2×C32C2×#23711yescharacteristic
396C2×C32#100414yes—
397C2×C32#100411yescharacteristic
398C2×C32#100914yes—
399C2×C32#100911yescharacteristic
400C3⋊D3—41no—
401C3⋊D3—122no—
402C3⋊D3—128no—
403C3⋊D3—248no—
404C3×D3—124no—
405C2×D4—92no—
406C2×D4—272no—
407C2×D4—272no—
408C2×D4—272no—
409C2×D4—271no—
410C2×D4D3×C3⋊D311yescharacteristic
411C22×C4—32no—
412C22×C4—91no—
413C22×C4—272no—
414C4⋊C4—31no—
415C4⋊C4—91no—
416C4⋊C4—271no—
417C4⋊C4—271no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C22⋊C4—32no—
421C22⋊C4—92no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C22⋊C4—271no—
425C22⋊C4—271no—
426C42—31no—
427C42—271no—
428C22×C3—28no—
429C22×C3—28no—
430C22×C3—28no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—48no—
436C22×C3—48no—
437C22×C3—48no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3C22×D3214yes—
442C22×C3C23×C3⋊D311yescharacteristic
443C2×D3—64no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—188no—
447C2×D3—188no—
448C2×D3—188no—
449C2×D3—182no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—368no—
453C2×D3—368no—
454C2×D3—368no—
455C3×C4—28no—
456C3×C4—64no—
457C3×C4—68no—
458C3×C4—64no—
459C3×C4—181no—
460C3×C4—182no—
461C3×C4#380418yes—
462C3×C4#382812yes—
463Dic3—21no—
464Dic3—22no—
465Dic3—21no—
466Dic3—68no—
467Dic3—64no—
468Dic3—188no—
469Dic3—188no—
470Dic3—184no—
471Dic3—184no—
472Dic3—181no—
473Dic3—181no—
474C32—24no—
475C32#556214yescomplemented
476C32#561911yescharacteristic, complemented
477C23—91no—
478C23—272no—
479C23C2×D3×C3⋊D312yes—
480D4—22no—
481D4—362no—
482D4—542no—
483D4—1082no—
484D4—1082no—
485D4—1082no—
486C2×C4—31no—
487C2×C4—31no—
488C2×C4—32no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—122no—
492C2×C4—182no—
493C2×C4—271no—
494C2×C4—271no—
495C2×C4—1082no—
496C2×C4C2×D3×C3⋊D311yescharacteristic
497C2×C3—24no—
498C2×C3—24no—
499C2×C3—24no—
500C2×C3—48no—
501C2×C3—42no—
502C2×C3—88no—
503C2×C3—361no—
504C2×C3#6431414yes—
505C2×C3#6431414yes—
506C2×C3C2×#380414yes—
507C2×C3C2×#382811yescharacteristic
508C2×C3#6436111yescharacteristic
509C2×C3#6436611yescharacteristic
510D3—124no—
511D3—368no—
512D3—362no—
513D3—728no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—181no—
518C22—181no—
519C22—181no—
520C22—542no—
521C22—542no—
522C22—542no—
523C22C22×D3×C3⋊D311yescharacteristic
524C4—61no—
525C4—62no—
526C4—61no—
527C4—182no—
528C4—181no—
529C4—541no—
530C4—541no—
531C4#8903112yes—
532C3—24no—
533C3#1059512514yescomplemented
534C3#1059603311yescharacteristic, complemented
535C2—42no—
536C2—361no—
537C2—1082no—
538C2#1172458811yescharacteristic
539C2#1172458911yescharacteristic
540C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723981C2semidirect
2. C3×#64147C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723632C2semidirect
5. #11723628C2semidirect
6. #11723623C2semidirect
7. #11723612C2semidirect
8. #11723526C2semidirect
9. #11723484C2semidirect
10. #11723450C2semidirect
11. #11723437C2semidirect
41. #64147D3semidirect
51. C3×D4×Dic3D3semidirect
159. D4×Dic3C3⋊D3semidirect
338. C33#540semidirect
475. C32#5562semidirect
476. C32#5619semidirect
533. C3#10595125semidirect
534. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
#117239812
C3×#641472
C32×D4×Dic32
#117236322
#117236282
#117236232
#117236122
#117235262
#117234842
#117234502
#117234372
#105960333
#105951253