Finite Group Databasebeta

← #13176466

Subgroups — #13176466

Order 1728.

1816 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723936C211yesmaximal, characteristic, complemented
2C3×#64099C211yesmaximal, characteristic, complemented
3C32×#893C211yesmaximal, characteristic, complemented
4#11723531C211yesmaximal, characteristic, complemented
5#11723525C212yesmaximal, complemented
6#11723524C211yesmaximal, characteristic, complemented
7#11723516C212yesmaximal, complemented
8#11723484C211yesmaximal, characteristic, complemented
9#11723452C211yesmaximal, characteristic, complemented
10#11723422C212yesmaximal, complemented
11#11723414C212yesmaximal, complemented
12#10596033—31nomaximal
13#10594594—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6291C2212yes—
20C2×#6291C2212yes—
21C2×#6291C2212yes—
22C2×#6289C2212yes—
23C33×C4⋊C4C2211yescharacteristic
24#88859C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2212yes—
27C3×#3750C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4×Dic3C2211yescharacteristic
31#88806C2211yescharacteristic
32#88806C2211yescharacteristic
33#88804C2211yescharacteristic
34#88804C2212yes—
35#88804C2211yescharacteristic
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64147—31no—
40#64139—32no—
41#64138—32no—
42#64104—31no—
43#64104—34no—
44#64104—64no—
45#64099D311yescharacteristic, complemented
46#64093—32no—
47#64092—32no—
48C4×C4⋊(C3⋊D3)—31no—
49#64081—31no—
50C3×#893D314yescomplemented
51C3×#890—34no—
52#63916—34no—
53#63911—34no—
54#63910—38no—
55#63909—38no—
56#63894—34no—
57#63874—34no—
58#63854—38no—
59#63850—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4C2312yes—
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C4⋊(C3⋊(C3⋊D3))—22no—
67C4⋊(C3⋊(C3⋊D3))—42no—
68C4×C3⋊(C3⋊D3)—22no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×D12—22no—
75C32×C4×D3—42no—
76#6291—42no—
77#6291—42no—
78#6291—42no—
79#6289—42no—
80#5697—91no—
81#5562—94no—
82C2×C32×D4—31no—
83C22×C32×C4—32no—
84C2×#638—32no—
85C2×#638—32no—
86C2×#638—32no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—32no—
90C2×C4⋊(C3⋊D3)—38no—
91C2×C4⋊(C3⋊D3)—64no—
92C2×C4⋊(C3⋊D3)—68no—
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—34no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—64no—
97C2×C3×D12C2×D314yes—
98C2×C3×C4×D3C2×D318yes—
99C2×#626—38no—
100C2×#626—38no—
101C2×#626—38no—
102C2×#624—38no—
103C32×C4⋊C4—24no—
104C32×C4⋊C4C2×D314yes—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C22⋊C4—32no—
107C32×C42—31no—
108#3757—32no—
109#3752—32no—
110#3752—38no—
111#3752—31no—
112#3752—31no—
113#3752—32no—
114#3752—68no—
115C4⋊(C3⋊Dic3)C2×D311yescharacteristic
116C4⋊(C3⋊Dic3)C2×D312yes—
117#3750—31no—
118#3750—31no—
119#3750C2×D312yes—
120C4×C3⋊Dic3C2×D311yescharacteristic
121C3×(C2×D3)⋊C4C2×D318yes—
122C3×C4⋊Dic3—38no—
123C3×C4⋊Dic3—34no—
124C3×Dic3⋊C4—38no—
125C3×C4×Dic3—34no—
126C3×C4×Dic3C2×D314yes—
127#3724—34no—
128#3724—34no—
129#3722—38no—
130#3722—34no—
131#3722—34no—
132(C2×D3)⋊Dic3—38no—
133C22×C33C2411yescharacteristic
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3⋊(C3⋊D3)—22no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143C33×C4—22no—
144C33×C4#4112yes—
145C3⋊(C3⋊Dic3)—21no—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—21no—
150C32×Dic3—21no—
151C3×#125—91no—
152C3×C4×D4—34no—
153#940—94no—
154#938—94no—
155#930—98no—
156#929—98no—
157#895—91no—
158#895—92no—
159#895—94no—
160#895—91no—
161#895—92no—
162#895—184no—
163#894—92no—
164#894—92no—
165#893—91no—
166#893C3⋊D311yescharacteristic, complemented
167#890—34no—
168#884—98no—
169#883—98no—
170#874—91no—
171C4×D12—94no—
172#872—94no—
173#872—91no—
174C23×C32—32no—
175C22×C3⋊D3—31no—
176C22×C3⋊D3—32no—
177C22×C3⋊D3—34no—
178C22×C3⋊D3—38no—
179C22×C3⋊D3—64no—
180C22×C3⋊D3—68no—
181C22×C3×D3C22×D318yes—
182C32×D4—62no—
183C2×C32×C4—24no—
184C2×C32×C4—28no—
185C2×C32×C4—31no—
186C2×C32×C4—31no—
187C2×C32×C4—122no—
188C2×C32×C4C22×D318yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192#638—122no—
193#638—122no—
194#638—122no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C4⋊(C3⋊D3)—62no—
202C4⋊(C3⋊D3)—68no—
203C4⋊(C3⋊D3)—128no—
204C4⋊(C3⋊D3)—122no—
205C4⋊(C3⋊D3)—128no—
206C4⋊(C3⋊D3)—1216no—
207C4×C3⋊D3—62no—
208C4×C3⋊D3—68no—
209C4×C3⋊D3—128no—
210C4×C3⋊D3—122no—
211C2×C3×Dic3—38no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3C22×D314yes—
215C2×C3×Dic3C22×D314yes—
216C3×D12—28no—
217C3×C4×D3—48no—
218#626—128no—
219#626—128no—
220#626—128no—
221#624—128no—
222#540—271noSylow
223C2×C33C2×#4111yescharacteristic
224C2×C33H(5,𝔽2)11yescharacteristic
225C2×C33H(5,𝔽2)11yescharacteristic
226C3⋊(C3⋊D3)—41no—
227C3⋊(C3⋊D3)—42no—
228C32×D3—42no—
229C2×C3×D4—34no—
230C22×C3×C4—38no—
231C2×#67—98no—
232C2×#67—98no—
233C2×#67—98no—
234C2×D12—94no—
235C2×D12—91no—
236C2×D12—92no—
237C2×D12—92no—
238C2×D12—92no—
239C2×D12—98no—
240C2×D12—92no—
241C2×D12—184no—
242C2×D12—188no—
243C2×D12C2×C3⋊D311yescharacteristic
244C2×C4×D3—94no—
245C2×C4×D3—91no—
246C2×C4×D3—98no—
247C2×C4×D3—92no—
248C2×C4×D3—184no—
249C2×C4×D3C2×C3⋊D312yes—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4D3214yes—
255C3×C4⋊C4C2×C3⋊D311yescharacteristic
256C3×C22⋊C4—38no—
257C3×C42—34no—
258C3×C42—91no—
259#217—98no—
260(C2×D3)⋊C4—94no—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—98no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—98no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—91no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—188no—
270(C2×D3)⋊C4C2×C3⋊D312yes—
271C4⋊Dic3—34no—
272C4⋊Dic3—38no—
273Dic3⋊C4—38no—
274Dic3⋊C4—94no—
275Dic3⋊C4—94no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278C4×Dic3—34no—
279C4×Dic3C2×C3⋊D311yescharacteristic
280C22×C32—24no—
281C22×C32—62no—
282C22×C32—62no—
283C22×C32—62no—
284C22×C32C23×D314yes—
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—68no—
287C2×C3⋊D3—64no—
288C2×C3⋊D3—61no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—61no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—61no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3—68no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—124no—
300C2×C3⋊D3—124no—
301C2×C3⋊D3—128no—
302C2×C3⋊D3—128no—
303C2×C3⋊D3—128no—
304C2×C3×D3—28no—
305C2×C3×D3—28no—
306C2×C3×D3—28no—
307C32×C4—22no—
308C32×C4—28no—
309C32×C4—28no—
310C32×C4—48no—
311C32×C4—61no—
312C32×C4—61no—
313C32×C4#23518yes—
314C32×C4#23912yes—
315C3⋊Dic3—22no—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3⋊Dic3—61no—
319C3⋊Dic3—64no—
320C3⋊Dic3—124no—
321C3×Dic3—24no—
322C3×Dic3—24no—
323C3×Dic3—64no—
324C3×Dic3—68no—
325C3×Dic3—64no—
326#127—271no—
327#125—91no—
328#123—272no—
329#123—272no—
330#121—272no—
331#121—272no—
332#121—271no—
333#121—271no—
334C4×D4—31no—
335C4×D4—271no—
336#117—271no—
337C33#54011yescharacteristic, complemented, Sylow
338C23×C3—38no—
339C22×D3—94no—
340C22×D3—98no—
341C22×D3—91no—
342C22×D3—92no—
343C22×D3—184no—
344C22×D3—188no—
345C22×D3C22×C3⋊D312yes—
346C3×D4—68no—
347C2×C3×C4—24no—
348C2×C3×C4—28no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—128no—
355C2×C3×C4C2×D3218yes—
356C2×C3×C4C2×D3214yes—
357C2×C3×C4C22×C3⋊D312yes—
358C2×C3×C4C22×C3⋊D311yescharacteristic
359#67—368no—
360#67—368no—
361#67—368no—
362C2×Dic3—34no—
363C2×Dic3—34no—
364C2×Dic3—38no—
365C2×Dic3—94no—
366C2×Dic3—91no—
367C2×Dic3—184no—
368C2×Dic3C22×C3⋊D311yescharacteristic
369C2×Dic3C22×C3⋊D311yescharacteristic
370D12—22no—
371D12—188no—
372D12—182no—
373D12—368no—
374D12—368no—
375D12—362no—
376D12—3616no—
377D12—362no—
378D12—362no—
379D12—362no—
380C4×D3—42no—
381C4×D3—188no—
382C4×D3—182no—
383C4×D3—368no—
384C4×D3—362no—
385C4×D3—368no—
386C2×C32—24no—
387C2×C32—24no—
388C2×C32—24no—
389C2×C32—122no—
390C2×C32C2×#23514yes—
391C2×C32#100414yes—
392C2×C32C2×#23911yescharacteristic
393C2×C32#100914yes—
394C2×C32#100911yescharacteristic
395C2×C32#100911yescharacteristic
396C3⋊D3—128no—
397C3⋊D3—121no—
398C3⋊D3—122no—
399C3⋊D3—124no—
400C3⋊D3—128no—
401C3⋊D3—248no—
402C3×D3—48no—
403C2×D4—31no—
404C2×D4—272no—
405C2×D4—272no—
406C2×D4—272no—
407C2×D4—271no—
408C2×D4—272no—
409C22×C4—32no—
410C22×C4—272no—
411C22×C4—271no—
412C4⋊C4—91no—
413C4⋊C4—92no—
414C4⋊C4—92no—
415C4⋊C4—271no—
416C4⋊C4—271no—
417C4⋊C4D3×C3⋊D311yescharacteristic
418C22⋊C4—32no—
419C22⋊C4—271no—
420C22⋊C4—271no—
421C22⋊C4—272no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C42—31no—
425C42—91no—
426C22×C3—24no—
427C22×C3—68no—
428C22×C3—68no—
429C22×C3—68no—
430C22×C3C22×D3214yes—
431C22×C3C23×C3⋊D311yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—188no—
436C2×D3—184no—
437C2×D3—188no—
438C2×D3—184no—
439C2×D3—188no—
440C2×D3—188no—
441C2×D3—184no—
442C2×D3—181no—
443C2×D3—182no—
444C2×D3—181no—
445C2×D3—182no—
446C2×D3—182no—
447C2×D3—181no—
448C2×D3—364no—
449C2×D3—364no—
450C2×D3—368no—
451C2×D3—368no—
452C2×D3—368no—
453C3×C4—28no—
454C3×C4—22no—
455C3×C4—28no—
456C3×C4—48no—
457C3×C4—64no—
458C3×C4—64no—
459C3×C4—181no—
460C3×C4—182no—
461C3×C4—181no—
462C3×C4#379918yes—
463C3×C4#382812yes—
464Dic3—21no—
465Dic3—21no—
466Dic3—64no—
467Dic3—68no—
468Dic3—64no—
469Dic3—184no—
470Dic3—181no—
471Dic3—364no—
472C32—24no—
473C32#556214yescomplemented
474C32#569711yescharacteristic, complemented
475C23—32no—
476C23—271no—
477C23—272no—
478D4—62no—
479D4—542no—
480D4—1082no—
481D4—1082no—
482D4—1082no—
483D4—1082no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—92no—
489C2×C4—122no—
490C2×C4—271no—
491C2×C4—542no—
492C2×C4—1082no—
493C2×C4C2×D3×C3⋊D312yes—
494C2×C4C2×D3×C3⋊D311yescharacteristic
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—24no—
498C2×C3—128no—
499C2×C3C2×#379914yes—
500C2×C3#6431414yes—
501C2×C3#6432014yes—
502C2×C3C2×#382811yescharacteristic
503C2×C3#6436111yescharacteristic
504C2×C3#6436611yescharacteristic
505D3—42no—
506D3—368no—
507D3—364no—
508D3—368no—
509D3—361no—
510D3—362no—
511D3—728no—
512C22—62no—
513C22—62no—
514C22—62no—
515C22—541no—
516C22—542no—
517C22—541no—
518C22—542no—
519C22—542no—
520C22—541no—
521C22C22×D3×C3⋊D311yescharacteristic
522C4—22no—
523C4—61no—
524C4—61no—
525C4—181no—
526C4—182no—
527C4—181no—
528C4—541no—
529C4#8902212yes—
530C3—24no—
531C3#1059459414yescomplemented
532C3#1059603311yescharacteristic, complemented
533C2—122no—
534C2—1081no—
535C2—1082no—
536C2C2×#8902211yescharacteristic
537C2#1172458811yescharacteristic
538C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723936C2semidirect
2. C3×#64099C2semidirect
3. C32×#893C2semidirect
4. #11723531C2semidirect
5. #11723525C2semidirect
6. #11723524C2semidirect
7. #11723516C2semidirect
8. #11723484C2semidirect
9. #11723452C2semidirect
10. #11723422C2semidirect
11. #11723414C2semidirect
45. #64099D3semidirect
50. C3×#893D3semidirect
166. #893C3⋊D3semidirect
337. C33#540semidirect
473. C32#5562semidirect
474. C32#5697semidirect
531. C3#10594594semidirect
532. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
#117239362
C3×#640992
C32×#8932
#117235312
#117235252
#117235242
#117235162
#117234842
#117234522
#117234222
#117234142
#105960333
#105945943