Finite Group Databasebeta

← #13177245

Subgroups — #13177245

Order 1728.

1744 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#11723967C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4#11723631C212yesmaximal, complemented
5#11723615C212yesmaximal, complemented
6#11723529C211yesmaximal, characteristic, complemented
7#11723523C212yesmaximal, complemented
8#11723501C211yesmaximal, characteristic, complemented
9#11723499C211yesmaximal, characteristic, complemented
10#11723480C212yesmaximal, complemented
11#11723450C211yesmaximal, characteristic, complemented
12#10596090—31nomaximal
13#10595041—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#638C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×D12C2211yescharacteristic
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C2×D3×C3⋊Dic3C2212yes—
23#88864C2211yescharacteristic
24#88864C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#3752C2212yes—
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C32×(C2×D3)⋊C4C2212yes—
32C32×C4⋊Dic3C2211yescharacteristic
33#88807C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88803C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#121—31no—
38#64149—31no—
39#64148—32no—
40#64147—32no—
41D4×C3⋊Dic3—31no—
42#64139D311yescharacteristic, complemented
43#64135—31no—
44#64135—34no—
45#64135—64no—
46#64102—31no—
47#64099—31no—
48#64095—32no—
49#64092—32no—
50C3×#930—34no—
51C3×#930D314yescomplemented
52#63970—38no—
53#63970—38no—
54#63914—34no—
55#63898—34no—
56#63898—34no—
57#63891—38no—
58#63891—38no—
59#63872—34no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C22×C32×D3C2312yes—
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C3×#638—42no—
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—22no—
73C32×#67—42no—
74C2×C32×Dic3C2312yes—
75C32×D12—22no—
76#6290—42no—
77#6290—42no—
78Dic3×C3⋊D3—42no—
79D3×C3⋊Dic3—42no—
80#5619—94no—
81#5619—91no—
82C2×C32×D4—32no—
83C2×C32×D4—31no—
84C22×C32×C4—24no—
85C22×C32×C4C2×D314yes—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88C2×#638—32no—
89C2×#638C2×D312yes—
90C22×C3⋊Dic3—32no—
91C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
92C2×C4×C3⋊D3—32no—
93C2×C3×#67—38no—
94C2×C3×#67C2×D318yes—
95C2×C3×D12—34no—
96C2×C3×D12C2×D314yes—
97C2×#625—38no—
98C2×#625—38no—
99C2×D3×Dic3—38no—
100C2×D3×Dic3—38no—
101C32×C4⋊C4—31no—
102C32×C22⋊C4—32no—
103#3757—31no—
104#3757—31no—
105#3757—34no—
106#3757—34no—
107#3757—32no—
108#3757—64no—
109#3757—64no—
110#3752—32no—
111#3752C2×D312yes—
112C4⋊(C3⋊Dic3)—31no—
113C4⋊(C3⋊Dic3)—31no—
114C4⋊(C3⋊Dic3)—34no—
115C4⋊(C3⋊Dic3)—34no—
116C4⋊(C3⋊Dic3)—32no—
117C4⋊(C3⋊Dic3)—64no—
118C4⋊(C3⋊Dic3)—64no—
119C4⋊(C3⋊Dic3)C2×D311yescharacteristic
120#3750—32no—
121C4×C3⋊Dic3—31no—
122C4×C3⋊Dic3—31no—
123C4×C3⋊Dic3—34no—
124C4×C3⋊Dic3—34no—
125C4×C3⋊Dic3—64no—
126C4×C3⋊Dic3—64no—
127C3×(C2×D3)⋊C4—38no—
128C3×(C2×D3)⋊C4C2×D318yes—
129C3×C4⋊Dic3—34no—
130C3×C4⋊Dic3C2×D314yes—
131Dic3⋊Dic3—38no—
132Dic3⋊Dic3—38no—
133(C2×D3)⋊Dic3—38no—
134(C2×D3)⋊Dic3—38no—
135C22×C33—21no—
136C22×C33—21no—
137C22×C33—21no—
138C22×C33C2411yescharacteristic
139C2×C3×C3⋊D3—22no—
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3—22no—
142C2×C32×D3—22no—
143C2×C32×D3—22no—
144C2×C32×D3—22no—
145C33×C4—21no—
146C33×C4#4112yes—
147C3⋊(C3⋊Dic3)—21no—
148C3⋊(C3⋊Dic3)—21no—
149C3⋊(C3⋊Dic3)—21no—
150C3⋊(C3⋊Dic3)—21no—
151C3×C3⋊Dic3—22no—
152C32×Dic3—22no—
153C3×#121—34no—
154C3×#121—91no—
155#940—94no—
156#940—91no—
157#939—98no—
158#939—92no—
159#938—98no—
160#938—92no—
161D4×Dic3—94no—
162D4×Dic3—91no—
163#930—34no—
164#930C3⋊D311yescharacteristic, complemented
165#926—94no—
166#926—91no—
167#926—184no—
168#893—94no—
169#893—91no—
170#890—94no—
171#890—91no—
172#886—98no—
173#886—92no—
174#883—98no—
175#883—92no—
176C23×C32—24no—
177C23×C32—32no—
178C23×C32C22×D314yes—
179C23×C32C22×D311yescharacteristic
180C22×C3⋊D3C22×D312yes—
181C22×C3×D3—38no—
182C22×C3×D3C22×D318yes—
183C32×D4—62no—
184C32×D4—122no—
185C2×C32×C4—216no—
186C2×C32×C4—28no—
187C2×C32×C4—22no—
188C2×C32×C4—24no—
189C2×C32×C4—24no—
190C2×C32×C4—32no—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—42no—
196#638—122no—
197#638—122no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—31no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—31no—
202C2×C3⋊Dic3—34no—
203C2×C3⋊Dic3—34no—
204C2×C3⋊Dic3—34no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—122no—
211C2×C3⋊Dic3C22×D312yes—
212C4⋊(C3⋊D3)—22no—
213C4×C3⋊D3—122no—
214C3×#67—48no—
215C3×#67—128no—
216C2×C3×Dic3—38no—
217C2×C3×Dic3C22×D318yes—
218C3×D12—28no—
219C3×D12—68no—
220#625—128no—
221#625—128no—
222D3×Dic3—128no—
223D3×Dic3—128no—
224#540—271noSylow
225C2×C33—41no—
226C2×C33C2×#4111yescharacteristic
227C2×C33H(5,𝔽2)11yescharacteristic
228C2×C33H(5,𝔽2)11yescharacteristic
229C3×C3⋊D3—42no—
230C32×D3—42no—
231C2×C3×D4—38no—
232C2×C3×D4—34no—
233C2×C3×D4—92no—
234C2×C3×D4—91no—
235C22×C3×C4—24no—
236C22×C3×C4D3214yes—
237C22×C3×C4C2×C3⋊D311yescharacteristic
238C2×#67—38no—
239C2×#67—98no—
240C2×#67—98no—
241C2×#67—92no—
242C2×#67—92no—
243C2×#67C2×C3⋊D312yes—
244C22×Dic3—98no—
245C22×Dic3—92no—
246C2×D12—34no—
247C2×D12C2×C3⋊D311yescharacteristic
248C2×C4×D3—98no—
249C2×C4×D3—92no—
250C3×C4⋊C4—34no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—38no—
253C3×C22⋊C4—92no—
254#217—98no—
255#217—94no—
256#217—94no—
257#217—91no—
258#217—91no—
259#217—92no—
260#217—184no—
261#217—184no—
262(C2×D3)⋊C4—38no—
263(C2×D3)⋊C4—98no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4C2×C3⋊D312yes—
266C4⋊Dic3—34no—
267C4⋊Dic3—98no—
268C4⋊Dic3—94no—
269C4⋊Dic3—94no—
270C4⋊Dic3—91no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—184no—
274C4⋊Dic3—184no—
275C4⋊Dic3C2×C3⋊D311yescharacteristic
276Dic3⋊C4—98no—
277Dic3⋊C4—92no—
278C4×Dic3—94no—
279C4×Dic3—94no—
280C4×Dic3—91no—
281C4×Dic3—91no—
282C4×Dic3—184no—
283C4×Dic3—184no—
284C22×C32—28no—
285C22×C32—24no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—24no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—44no—
293C22×C32—44no—
294C22×C32—62no—
295C22×C32—62no—
296C22×C32—62no—
297C22×C32C23×D314yes—
298C22×C32C23×D311yescharacteristic
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3×D3—28no—
303C2×C3×D3—28no—
304C2×C3×D3—28no—
305C2×C3×D3—68no—
306C2×C3×D3—68no—
307C2×C3×D3—68no—
308C32×C4—28no—
309C32×C4—24no—
310C32×C4—21no—
311C32×C4—28no—
312C32×C4—62no—
313C32×C4#23718yes—
314C32×C4#23712yes—
315C3⋊Dic3—22no—
316C3⋊Dic3—61no—
317C3⋊Dic3—61no—
318C3⋊Dic3—61no—
319C3⋊Dic3—61no—
320C3⋊Dic3—64no—
321C3⋊Dic3—64no—
322C3⋊Dic3—64no—
323C3⋊Dic3—64no—
324C3⋊Dic3—124no—
325C3⋊Dic3—124no—
326C3⋊Dic3—124no—
327C3⋊Dic3—124no—
328C3×Dic3—28no—
329C3×Dic3—68no—
330#127—271no—
331#125—271no—
332#123—272no—
333#123—272no—
334#121—31no—
335#121—91no—
336#121—272no—
337#121—272no—
338C4×D4—271no—
339C4×D4—271no—
340#117—271no—
341C33#54011yescharacteristic, complemented, Sylow
342C23×C3—24no—
343C23×C3—38no—
344C23×C3—92no—
345C23×C3C2×D3214yes—
346C23×C3C22×C3⋊D311yescharacteristic
347C22×D3—38no—
348C22×D3C22×C3⋊D312yes—
349C3×D4—68no—
350C3×D4—128no—
351C3×D4—182no—
352C3×D4—362no—
353C2×C3×C4—216no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—38no—
359C2×C3×C4—92no—
360C2×C3×C4C2×D3214yes—
361C2×C3×C4C2×D3214yes—
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×C3×C4C22×C3⋊D311yescharacteristic
364#67—42no—
365#67—128no—
366#67—362no—
367#67—362no—
368#67—368no—
369#67—368no—
370C2×Dic3—38no—
371C2×Dic3—94no—
372C2×Dic3—94no—
373C2×Dic3—94no—
374C2×Dic3—94no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—184no—
380C2×Dic3—184no—
381C2×Dic3—184no—
382C2×Dic3—184no—
383C2×Dic3—362no—
384C2×Dic3—368no—
385C2×Dic3C22×C3⋊D312yes—
386D12—22no—
387D12—68no—
388C4×D3—368no—
389C4×D3—362no—
390C2×C32—24no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—48no—
394C2×C32—44no—
395C2×C32—41no—
396C2×C32—122no—
397C2×C32C2×#23714yes—
398C2×C32C2×#23711yescharacteristic
399C2×C32#100414yes—
400C2×C32#100411yescharacteristic
401C2×C32#100914yes—
402C2×C32#100911yescharacteristic
403C3⋊D3—42no—
404C3×D3—48no—
405C3×D3—128no—
406C2×D4—32no—
407C2×D4—31no—
408C2×D4—92no—
409C2×D4—91no—
410C2×D4—272no—
411C2×D4—272no—
412C22×C4—272no—
413C22×C4—272no—
414C22×C4D3×C3⋊D311yescharacteristic
415C4⋊C4—31no—
416C4⋊C4—91no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—271no—
420C4⋊C4—271no—
421C22⋊C4—32no—
422C22⋊C4—92no—
423C22⋊C4—272no—
424C22⋊C4—272no—
425C22⋊C4—271no—
426C22⋊C4—271no—
427C42—271no—
428C42—271no—
429C22×C3—28no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—24no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—68no—
440C22×C3—68no—
441C22×C3—68no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3C22×D3214yes—
446C22×C3C23×C3⋊D311yescharacteristic
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—22no—
450C2×D3—68no—
451C2×D3—68no—
452C2×D3—68no—
453C3×C4—28no—
454C3×C4—24no—
455C3×C4—21no—
456C3×C4—28no—
457C3×C4—68no—
458C3×C4—182no—
459C3×C4#380518yes—
460C3×C4#383012yes—
461Dic3—22no—
462Dic3—68no—
463Dic3—184no—
464Dic3—184no—
465Dic3—184no—
466Dic3—184no—
467Dic3—181no—
468Dic3—181no—
469Dic3—181no—
470Dic3—181no—
471Dic3—364no—
472Dic3—364no—
473Dic3—364no—
474Dic3—364no—
475C32—24no—
476C32#561914yescomplemented
477C32#561911yescharacteristic, complemented
478C23—32no—
479C23—92no—
480C23C2×D3×C3⋊D311yescharacteristic
481D4—62no—
482D4—122no—
483D4—182no—
484D4—362no—
485D4—1082no—
486D4—1082no—
487C2×C4—22no—
488C2×C4—32no—
489C2×C4—92no—
490C2×C4—271no—
491C2×C4—271no—
492C2×C4—271no—
493C2×C4—271no—
494C2×C4—1082no—
495C2×C4—1082no—
496C2×C4C2×D3×C3⋊D311yescharacteristic
497C2×C4C2×D3×C3⋊D311yescharacteristic
498C2×C3—24no—
499C2×C3—24no—
500C2×C3—24no—
501C2×C3—48no—
502C2×C3—44no—
503C2×C3—41no—
504C2×C3—128no—
505C2×C3—362no—
506C2×C3#6430714yes—
507C2×C3#6430714yes—
508C2×C3C2×#380514yes—
509C2×C3C2×#383011yescharacteristic
510C2×C3#6436111yescharacteristic
511C2×C3#6436611yescharacteristic
512D3—42no—
513D3—128no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—182no—
521C22—182no—
522C22—182no—
523C22C22×D3×C3⋊D311yescharacteristic
524C4—21no—
525C4—62no—
526C4—182no—
527C4—541no—
528C4—541no—
529C4—541no—
530C4—541no—
531C4#8902912yes—
532C3—24no—
533C3#1059504114yescomplemented
534C3#1059609011yescharacteristic, complemented
535C2—41no—
536C2—122no—
537C2—362no—
538C2#1172457911yescharacteristic
539C2#1172458011yescharacteristic
540C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723967C2semidirect
2. C3×#64139C2semidirect
3. C32×#930C2semidirect
4. #11723631C2semidirect
5. #11723615C2semidirect
6. #11723529C2semidirect
7. #11723523C2semidirect
8. #11723501C2semidirect
9. #11723499C2semidirect
10. #11723480C2semidirect
11. #11723450C2semidirect
42. #64139D3semidirect
51. C3×#930D3semidirect
164. #930C3⋊D3semidirect
341. C33#540semidirect
476. C32#5619semidirect
477. C32#5619semidirect
533. C3#10595041semidirect
534. C3#10596090semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×#641392
C32×#9302
#117236312
#117236152
#117235292
#117235232
#117235012
#117234992
#117234802
#117234502
#105960903
#105950413