Finite Group Databasebeta

← #13176448

Subgroups — #13176448

Order 1728.

1664 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723931C211yesmaximal, characteristic, complemented
2C3×#64104C211yesmaximal, characteristic, complemented
3C32×#893C211yesmaximal, characteristic, complemented
4#11723529C211yesmaximal, characteristic, complemented
5#11723523C212yesmaximal, complemented
6#11723522C211yesmaximal, characteristic, complemented
7#11723519C212yesmaximal, complemented
8#11723515C212yesmaximal, complemented
9#11723513C212yesmaximal, complemented
10Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11#11723437C211yesmaximal, characteristic, complemented
12#10596033—31nomaximal
13#10594496—34nomaximal
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21C2×#6290C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23C33×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2212yes—
26#88857C2212yes—
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3752C2212yes—
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4×Dic3C2211yescharacteristic
31#88807C2211yescharacteristic
32#88807C2211yescharacteristic
33#88803C2211yescharacteristic
34#88803C2211yescharacteristic
35#88803C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64147—31no—
40#64139—32no—
41#64138—32no—
42#64104D311yescharacteristic, complemented
43#64099—31no—
44#64099—34no—
45#64099—64no—
46#64093—32no—
47#64092—32no—
48C4×C4⋊(C3⋊D3)—31no—
49#64081—31no—
50C3×#895—34no—
51C3×#893D314yescomplemented
52#63914—34no—
53#63908—34no—
54#63908—38no—
55#63905—38no—
56Dic3×D12—34no—
57#63891—38no—
58#63885—38no—
59#63855—34no—
60C22×C3×C3⋊D3C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4C2312yes—
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C3×C4⋊(C3⋊D3)—22no—
70C3×C4⋊(C3⋊D3)—42no—
71C3×C4×C3⋊D3—22no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×D12—22no—
75C32×C4×D3—42no—
76#6290—42no—
77#6290—42no—
78#6290—42no—
79Dic3×C3⋊D3—42no—
80#5619—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)C2×D311yescharacteristic
88C2×C4⋊(C3⋊D3)C2×D312yes—
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×C3×D12—34no—
92C2×C3×D12—38no—
93C2×C3×D12C2×D314yes—
94C2×C3×C4×D3—34no—
95C2×C3×C4×D3C2×D318yes—
96C2×#625—38no—
97C2×#625—38no—
98C2×#625—38no—
99C2×D3×Dic3—38no—
100C32×C4⋊C4—24no—
101C32×C4⋊C4C2×D314yes—
102C32×C4⋊C4C2×D311yescharacteristic
103C32×C22⋊C4—32no—
104C32×C42—31no—
105#3757—32no—
106#3752—31no—
107#3752—31no—
108#3752—32no—
109#3752C2×D312yes—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—32no—
112C4⋊(C3⋊Dic3)—38no—
113C4⋊(C3⋊Dic3)—34no—
114C4⋊(C3⋊Dic3)—68no—
115C4⋊(C3⋊Dic3)—64no—
116#3750—32no—
117#3750—38no—
118#3750—31no—
119#3750—31no—
120#3750—68no—
121C4×C3⋊Dic3—31no—
122C4×C3⋊Dic3—34no—
123C4×C3⋊Dic3—64no—
124C3×(C2×D3)⋊C4—38no—
125C3×(C2×D3)⋊C4C2×D318yes—
126C3×C4×Dic3C2×D314yes—
127Dic3⋊Dic3—34no—
128Dic3⋊Dic3—34no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—34no—
131(C2×D3)⋊Dic3—34no—
132(C2×D3)⋊Dic3—38no—
133C22×C33C2411yescharacteristic
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—21no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143C33×C4—22no—
144C33×C4#4112yes—
145C3⋊(C3⋊Dic3)—22no—
146C3⋊(C3⋊Dic3)—21no—
147C3⋊(C3⋊Dic3)—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—21no—
150C32×Dic3—21no—
151C3×#121—91no—
152C3×C4×D4—34no—
153#940—94no—
154#940—91no—
155#939—92no—
156#938—94no—
157#938—92no—
158D4×Dic3—91no—
159#930—98no—
160#930—91no—
161#929—98no—
162#926—91no—
163#895—34no—
164#893C3⋊D311yescharacteristic, complemented
165#890—94no—
166#890—91no—
167#890—184no—
168#886—92no—
169#884—98no—
170#883—98no—
171#883—92no—
172C4×D12—94no—
173#872—94no—
174C23×C32—32no—
175C22×C3⋊D3C22×D311yescharacteristic
176C22×C3⋊D3C22×D312yes—
177C22×C3×D3—34no—
178C22×C3×D3—38no—
179C22×C3×D3C22×D318yes—
180C32×D4—62no—
181C2×C32×C4—24no—
182C2×C32×C4—28no—
183C2×C32×C4—31no—
184C2×C32×C4—31no—
185C2×C32×C4—122no—
186C2×C32×C4C22×D318yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190#638—122no—
191#638—122no—
192#638—122no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—34no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C4⋊(C3⋊D3)—22no—
204C4⋊(C3⋊D3)—42no—
205C4×C3⋊D3—22no—
206C4×C3⋊D3—122no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3C22×D314yes—
209C2×C3×Dic3C22×D314yes—
210C3×D12—28no—
211C3×D12—68no—
212C3×D12—128no—
213C3×C4×D3—48no—
214C3×C4×D3—68no—
215#625—128no—
216#625—128no—
217#625—128no—
218D3×Dic3—128no—
219#540—271noSylow
220C2×C33C2×#4111yescharacteristic
221C2×C33H(5,𝔽2)11yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—41no—
224C3×C3⋊D3—42no—
225C32×D3—42no—
226C2×C3×D4—34no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C22×C3×C4—38no—
230C22×C3×C4—91no—
231C2×#67—98no—
232C2×#67—98no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67—92no—
236C2×#67—92no—
237C22×Dic3—92no—
238C2×D12—34no—
239C2×D12—38no—
240C2×D12C2×C3⋊D311yescharacteristic
241C2×C4×D3—34no—
242C2×C4×D3—98no—
243C2×C4×D3C2×C3⋊D312yes—
244C3×C4⋊C4—24no—
245C3×C4⋊C4D3214yes—
246C3×C4⋊C4C2×C3⋊D311yescharacteristic
247C3×C22⋊C4—38no—
248C3×C22⋊C4—92no—
249C3×C42—34no—
250#217—98no—
251#217—92no—
252#217—91no—
253#217—91no—
254(C2×D3)⋊C4—38no—
255(C2×D3)⋊C4—94no—
256(C2×D3)⋊C4—94no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4C2×C3⋊D312yes—
260C4⋊Dic3—94no—
261C4⋊Dic3—98no—
262C4⋊Dic3—92no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—184no—
267C4⋊Dic3—188no—
268Dic3⋊C4—94no—
269Dic3⋊C4—94no—
270Dic3⋊C4—98no—
271Dic3⋊C4—92no—
272Dic3⋊C4—188no—
273C4×Dic3—94no—
274C4×Dic3—91no—
275C4×Dic3—184no—
276C4×Dic3C2×C3⋊D311yescharacteristic
277C22×C32—24no—
278C22×C32—62no—
279C22×C32—62no—
280C22×C32—62no—
281C22×C32C23×D314yes—
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—21no—
284C2×C3⋊D3—22no—
285C2×C3⋊D3—21no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—21no—
289C2×C3×D3—28no—
290C2×C3×D3—28no—
291C2×C3×D3—28no—
292C2×C3×D3—64no—
293C2×C3×D3—68no—
294C2×C3×D3—68no—
295C2×C3×D3—64no—
296C2×C3×D3—68no—
297C2×C3×D3—64no—
298C32×C4—22no—
299C32×C4—28no—
300C32×C4—28no—
301C32×C4—48no—
302C32×C4—61no—
303C32×C4—61no—
304C32×C4#23518yes—
305C32×C4#23712yes—
306C3⋊Dic3—21no—
307C3⋊Dic3—61no—
308C3⋊Dic3—62no—
309C3⋊Dic3—61no—
310C3⋊Dic3—64no—
311C3⋊Dic3—68no—
312C3⋊Dic3—64no—
313C3⋊Dic3—124no—
314C3⋊Dic3—128no—
315C3⋊Dic3—124no—
316C3×Dic3—24no—
317C3×Dic3—24no—
318C3×Dic3—64no—
319#127—271no—
320#125—271no—
321#123—272no—
322#123—272no—
323#121—91no—
324#121—272no—
325#121—272no—
326#121—271no—
327C4×D4—31no—
328C4×D4—271no—
329#117—271no—
330C33#54011yescharacteristic, complemented, Sylow
331C23×C3—38no—
332C23×C3—91no—
333C23×C3—92no—
334C22×D3—38no—
335C22×D3—34no—
336C22×D3C22×C3⋊D312yes—
337C3×D4—68no—
338C3×D4—182no—
339C3×D4—362no—
340C2×C3×C4—24no—
341C2×C3×C4—28no—
342C2×C3×C4—34no—
343C2×C3×C4—34no—
344C2×C3×C4—91no—
345C2×C3×C4—128no—
346C2×C3×C4—182no—
347C2×C3×C4C2×D3218yes—
348C2×C3×C4C2×D3214yes—
349C2×C3×C4C22×C3⋊D312yes—
350C2×C3×C4C22×C3⋊D311yescharacteristic
351#67—362no—
352#67—362no—
353#67—362no—
354#67—368no—
355#67—368no—
356#67—368no—
357C2×Dic3—34no—
358C2×Dic3—94no—
359C2×Dic3—94no—
360C2×Dic3—98no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—92no—
364C2×Dic3—184no—
365C2×Dic3—184no—
366C2×Dic3—188no—
367C2×Dic3—362no—
368C2×Dic3C22×C3⋊D311yescharacteristic
369C2×Dic3C22×C3⋊D311yescharacteristic
370D12—22no—
371D12—68no—
372D12—128no—
373C4×D3—42no—
374C4×D3—68no—
375C4×D3—368no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—122no—
380C2×C32C2×#23514yes—
381C2×C32C2×#23711yescharacteristic
382C2×C32#100414yes—
383C2×C32#100411yescharacteristic
384C2×C32#100914yes—
385C2×C32#100911yescharacteristic
386C3⋊D3—41no—
387C3⋊D3—42no—
388C3×D3—48no—
389C3×D3—124no—
390C3×D3—128no—
391C2×D4—31no—
392C2×D4—91no—
393C2×D4—92no—
394C2×D4—272no—
395C2×D4—272no—
396C2×D4—272no—
397C22×C4—32no—
398C22×C4—91no—
399C22×C4—272no—
400C4⋊C4—271no—
401C4⋊C4—271no—
402C4⋊C4—271no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C4⋊C4D3×C3⋊D311yescharacteristic
406C22⋊C4—32no—
407C22⋊C4—92no—
408C22⋊C4—271no—
409C22⋊C4—271no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C42—31no—
413C42—271no—
414C22×C3—24no—
415C22×C3—68no—
416C22×C3—68no—
417C22×C3—68no—
418C22×C3—181no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—181no—
422C22×C3—182no—
423C22×C3—181no—
424C22×C3C22×D3214yes—
425C22×C3C23×C3⋊D311yescharacteristic
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—64no—
430C2×D3—68no—
431C2×D3—64no—
432C2×D3—68no—
433C2×D3—68no—
434C2×D3—64no—
435C3×C4—28no—
436C3×C4—22no—
437C3×C4—28no—
438C3×C4—48no—
439C3×C4—64no—
440C3×C4—64no—
441C3×C4—181no—
442C3×C4#379518yes—
443C3×C4#382812yes—
444Dic3—21no—
445Dic3—21no—
446Dic3—64no—
447Dic3—184no—
448Dic3—188no—
449Dic3—184no—
450Dic3—181no—
451Dic3—182no—
452Dic3—181no—
453Dic3—364no—
454Dic3—368no—
455Dic3—364no—
456C32—24no—
457C32#556214yescomplemented
458C32#561911yescharacteristic, complemented
459C23—32no—
460C23—92no—
461C23—91no—
462D4—62no—
463D4—182no—
464D4—362no—
465D4—1082no—
466D4—1082no—
467D4—1082no—
468C2×C4—31no—
469C2×C4—31no—
470C2×C4—91no—
471C2×C4—122no—
472C2×C4—182no—
473C2×C4—271no—
474C2×C4—271no—
475C2×C4—272no—
476C2×C4—1082no—
477C2×C4C2×D3×C3⋊D312yes—
478C2×C4C2×D3×C3⋊D311yescharacteristic
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—128no—
483C2×C3—361no—
484C2×C3—362no—
485C2×C3C2×#379514yes—
486C2×C3#6431314yes—
487C2×C3#6432014yes—
488C2×C3C2×#382811yescharacteristic
489C2×C3#6436111yescharacteristic
490C2×C3#6436611yescharacteristic
491D3—42no—
492D3—124no—
493D3—128no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—181no—
498C22—182no—
499C22—181no—
500C22—182no—
501C22—182no—
502C22—181no—
503C22C22×D3×C3⋊D311yescharacteristic
504C4—22no—
505C4—61no—
506C4—61no—
507C4—181no—
508C4—541no—
509C4—542no—
510C4—541no—
511C4#8902112yes—
512C3—24no—
513C3#1059449614yescomplemented
514C3#1059603311yescharacteristic, complemented
515C2—122no—
516C2—361no—
517C2—362no—
518C2C2×#8902111yescharacteristic
519C2#1172458711yescharacteristic
520C2#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. #11723931C2semidirect
2. C3×#64104C2semidirect
3. C32×#893C2semidirect
4. #11723529C2semidirect
5. #11723523C2semidirect
6. #11723522C2semidirect
7. #11723519C2semidirect
8. #11723515C2semidirect
9. #11723513C2semidirect
10. Dic3×C4⋊(C3⋊D3)C2semidirect
11. #11723437C2semidirect
42. #64104D3semidirect
51. C3×#893D3semidirect
164. #893C3⋊D3semidirect
330. C33#540semidirect
457. C32#5562semidirect
458. C32#5619semidirect
513. C3#10594496semidirect
514. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
#117239312
C3×#641042
C32×#8932
#117235292
#117235232
#117235222
#117235192
#117235152
#117235132
Dic3×C4⋊(C3⋊D3)2
#117234372
#105960333
#105944963