Finite Group Databasebeta

← #13175355

Subgroups — #13175355

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#11723913C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723532C211yesmaximal, characteristic, complemented
5#11723531C211yesmaximal, characteristic, complemented
6#11723529C211yesmaximal, characteristic, complemented
7#11723520C212yesmaximal, complemented
8#11723518C212yesmaximal, complemented
9#11723514C212yesmaximal, complemented
10#11723513C212yesmaximal, complemented
11#11723450C211yesmaximal, characteristic, complemented
12#10596033—31nomaximal
13#10593984—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6292C2212yes—
20C2×#6291C2212yes—
21C2×#6290C2212yes—
22C2×#6290C2212yes—
23C33×C42C2211yescharacteristic
24#88859C2211yescharacteristic
25#88859C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
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×C4×D4—31no—
38#64149—31no—
39#64147—31no—
40#64139—32no—
41#64138—32no—
42#64104—31no—
43#64099—31no—
44#64093—32no—
45#64092—32no—
46C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
47#64081—31no—
48#64081—34no—
49#64081—64no—
50C3×C4×D12—34no—
51C3×C4×D12D314yescomplemented
52#63916—34no—
53#63916—34no—
54#63914—34no—
55#63906—38no—
56#63906—38no—
57#63905—38no—
58#63905—38no—
59#63872—34no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C22×C32×D3C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)—22no—
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4⋊(C3⋊D3)—22no—
72C3×C4×C3⋊D3—42no—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×C4×D3—42no—
76#6292—42no—
77#6291—42no—
78#6290—42no—
79#6290—42no—
80#5562—94no—
81#5562—91no—
82C2×C32×D4—31no—
83C22×C32×C4—32no—
84C2×#638—32no—
85C2×#638—32no—
86C2×#638—32no—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—64no—
92C2×C4×C3⋊D3C2×D312yes—
93C2×C3×D12—34no—
94C2×C3×D12C2×D314yes—
95C2×C3×C4×D3—38no—
96C2×C3×C4×D3C2×D318yes—
97C2×#626—38no—
98C2×#626—38no—
99C2×#625—38no—
100C2×#625—38no—
101C32×C4⋊C4—31no—
102C32×C22⋊C4—32no—
103C32×C42—24no—
104C32×C42C2×D314yes—
105C32×C42C2×D311yescharacteristic
106#3757—32no—
107#3752—32no—
108#3752—31no—
109#3752—31no—
110#3752—34no—
111#3752—34no—
112#3752—64no—
113#3752—64no—
114#3752C2×D312yes—
115C4⋊(C3⋊Dic3)—32no—
116C4⋊(C3⋊Dic3)C2×D311yescharacteristic
117#3750—31no—
118#3750—31no—
119#3750—34no—
120#3750—34no—
121#3750—32no—
122#3750—64no—
123#3750—64no—
124C4×C3⋊Dic3—31no—
125C4×C3⋊Dic3—34no—
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×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C2×C3⋊(C3⋊D3)—21no—
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—21no—
147C33×C4#4112yes—
148C3⋊(C3⋊Dic3)—21no—
149C3⋊(C3⋊Dic3)—21no—
150C3⋊(C3⋊Dic3)—21no—
151C3×C3⋊Dic3—22no—
152C32×Dic3—22no—
153C3×C4×D4—34no—
154C3×C4×D4—91no—
155#940—94no—
156#940—91no—
157#938—94no—
158#938—91no—
159#930—98no—
160#930—92no—
161#929—98no—
162#929—92no—
163#895—94no—
164#895—91no—
165#890—94no—
166#890—91no—
167#884—98no—
168#884—92no—
169#883—98no—
170#883—92no—
171C4×D12—34no—
172C4×D12C3⋊D311yescharacteristic, complemented
173#872—94no—
174#872—91no—
175#872—184no—
176C23×C32—32no—
177C22×C3⋊D3—31no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—64no—
180C22×C3⋊D3C22×D312yes—
181C22×C3×D3—38no—
182C22×C3×D3C22×D318yes—
183C32×D4—62no—
184C2×C32×C4—24no—
185C2×C32×C4—24no—
186C2×C32×C4—24no—
187C2×C32×C4—32no—
188C2×C32×C4—122no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—122no—
196#638—122no—
197#638—122no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—31no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—34no—
203C2×C3⋊Dic3—34no—
204C2×C3⋊Dic3—64no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3C22×D312yes—
208C4⋊(C3⋊D3)—22no—
209C4⋊(C3⋊D3)—122no—
210C4×C3⋊D3—42no—
211C4×C3⋊D3—616no—
212C4×C3⋊D3—68no—
213C4×C3⋊D3—62no—
214C2×C3×Dic3—38no—
215C2×C3×Dic3C22×D318yes—
216C3×D12—28no—
217C3×D12—68no—
218C3×C4×D3—48no—
219C3×C4×D3—128no—
220#626—128no—
221#626—128no—
222#625—128no—
223#625—128no—
224#540—271noSylow
225C2×C33C2×#4111yescharacteristic
226C2×C33H(5,𝔽2)11yescharacteristic
227C2×C33H(5,𝔽2)11yescharacteristic
228C3⋊(C3⋊D3)—41no—
229C3×C3⋊D3—42no—
230C32×D3—42no—
231C2×C3×D4—34no—
232C2×C3×D4—91no—
233C22×C3×C4—38no—
234C22×C3×C4—92no—
235C2×#67—98no—
236C2×#67—98no—
237C2×#67—98no—
238C2×#67—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×D12—34no—
242C2×D12—98no—
243C2×D12—92no—
244C2×D12C2×C3⋊D311yescharacteristic
245C2×C4×D3—38no—
246C2×C4×D3—94no—
247C2×C4×D3—91no—
248C2×C4×D3—184no—
249C2×C4×D3C2×C3⋊D312yes—
250C3×C4⋊C4—34no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—38no—
253C3×C22⋊C4—92no—
254C3×C42—24no—
255C3×C42D3214yes—
256C3×C42C2×C3⋊D311yescharacteristic
257#217—98no—
258#217—92no—
259(C2×D3)⋊C4—38no—
260(C2×D3)⋊C4—98no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—184no—
267(C2×D3)⋊C4—184no—
268(C2×D3)⋊C4C2×C3⋊D312yes—
269C4⋊Dic3—34no—
270C4⋊Dic3—98no—
271C4⋊Dic3—92no—
272C4⋊Dic3C2×C3⋊D311yescharacteristic
273Dic3⋊C4—98no—
274Dic3⋊C4—94no—
275Dic3⋊C4—94no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278Dic3⋊C4—92no—
279Dic3⋊C4—184no—
280Dic3⋊C4—184no—
281C4×Dic3—94no—
282C4×Dic3—91no—
283C4×Dic3—184no—
284C22×C32—24no—
285C22×C32—62no—
286C22×C32—62no—
287C22×C32—62no—
288C22×C32C23×D314yes—
289C22×C32C23×D311yescharacteristic
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—68no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—61no—
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—64no—
300C2×C3⋊D3—124no—
301C2×C3⋊D3—124no—
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—21no—
309C32×C4—21no—
310C32×C4—28no—
311C32×C4—24no—
312C32×C4—24no—
313C32×C4—44no—
314C32×C4—44no—
315C32×C4—62no—
316C32×C4#23518yes—
317C32×C4#23512yes—
318C3⋊Dic3—22no—
319C3⋊Dic3—68no—
320C3⋊Dic3—64no—
321C3⋊Dic3—61no—
322C3⋊Dic3—61no—
323C3⋊Dic3—61no—
324C3⋊Dic3—64no—
325C3⋊Dic3—64no—
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—272no—
335#121—272no—
336#121—271no—
337#121—271no—
338C4×D4—31no—
339C4×D4—91no—
340#117—271no—
341C33#54011yescharacteristic, complemented, Sylow
342C23×C3—38no—
343C23×C3—92no—
344C22×D3—38no—
345C22×D3—94no—
346C22×D3—91no—
347C22×D3—184no—
348C22×D3C22×C3⋊D312yes—
349C3×D4—68no—
350C3×D4—182no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—38no—
355C2×C3×C4—92no—
356C2×C3×C4—128no—
357C2×C3×C4—362no—
358C2×C3×C4C2×D3214yes—
359C2×C3×C4C2×D3214yes—
360C2×C3×C4C2×D3214yes—
361C2×C3×C4C22×C3⋊D311yescharacteristic
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×C3×C4C22×C3⋊D311yescharacteristic
364#67—362no—
365#67—368no—
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—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—184no—
378C2×Dic3—184no—
379C2×Dic3—184no—
380C2×Dic3C22×C3⋊D312yes—
381D12—22no—
382D12—68no—
383D12—368no—
384D12—362no—
385C4×D3—42no—
386C4×D3—128no—
387C4×D3—1816no—
388C4×D3—188no—
389C4×D3—182no—
390C2×C32—24no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—122no—
394C2×C32C2×#23514yes—
395C2×C32C2×#23511yescharacteristic
396C2×C32#100414yes—
397C2×C32#100411yescharacteristic
398C2×C32#100914yes—
399C2×C32#100911yescharacteristic
400C3⋊D3—42no—
401C3⋊D3—128no—
402C3⋊D3—124no—
403C3⋊D3—121no—
404C3×D3—48no—
405C3×D3—128no—
406C2×D4—31no—
407C2×D4—91no—
408C2×D4—272no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—272no—
412C22×C4—32no—
413C22×C4—92no—
414C22×C4—271no—
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—
428C42D3×C3⋊D311yescharacteristic
429C22×C3—24no—
430C22×C3—68no—
431C22×C3—68no—
432C22×C3—68no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3C22×D3214yes—
437C22×C3C23×C3⋊D311yescharacteristic
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—68no—
442C2×D3—68no—
443C2×D3—68no—
444C2×D3—188no—
445C2×D3—184no—
446C2×D3—184no—
447C2×D3—184no—
448C2×D3—181no—
449C2×D3—181no—
450C2×D3—181no—
451C2×D3—364no—
452C2×D3—364no—
453C3×C4—24no—
454C3×C4—24no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—28no—
458C3×C4—44no—
459C3×C4—44no—
460C3×C4—68no—
461C3×C4—182no—
462C3×C4#379818yes—
463C3×C4#382812yes—
464Dic3—22no—
465Dic3—68no—
466Dic3—188no—
467Dic3—184no—
468Dic3—184no—
469Dic3—184no—
470Dic3—181no—
471Dic3—181no—
472Dic3—181no—
473Dic3—364no—
474Dic3—364no—
475C32—24no—
476C32#556214yescomplemented
477C32#556211yescharacteristic, complemented
478C23—32no—
479C23—92no—
480C23—271no—
481D4—62no—
482D4—182no—
483D4—1082no—
484D4—1082no—
485D4—1082no—
486D4—1082no—
487C2×C4—32no—
488C2×C4—92no—
489C2×C4—122no—
490C2×C4—271no—
491C2×C4—271no—
492C2×C4—271no—
493C2×C4—362no—
494C2×C4—542no—
495C2×C4C2×D3×C3⋊D311yescharacteristic
496C2×C4C2×D3×C3⋊D311yescharacteristic
497C2×C4C2×D3×C3⋊D311yescharacteristic
498C2×C3—24no—
499C2×C3—24no—
500C2×C3—24no—
501C2×C3—128no—
502C2×C3—362no—
503C2×C3C2×#379814yes—
504C2×C3#6430714yes—
505C2×C3#6430714yes—
506C2×C3C2×#382811yescharacteristic
507C2×C3#6436111yescharacteristic
508C2×C3#6436611yescharacteristic
509D3—42no—
510D3—128no—
511D3—368no—
512D3—364no—
513D3—361no—
514C22—62no—
515C22—62no—
516C22—62no—
517C22—182no—
518C22—182no—
519C22—182no—
520C22—541no—
521C22—541no—
522C22—541no—
523C22C22×D3×C3⋊D311yescharacteristic
524C4—21no—
525C4—21no—
526C4—62no—
527C4—182no—
528C4—541no—
529C4—541no—
530C4—541no—
531C4#8902012yes—
532C3—24no—
533C3#1059398414yescomplemented
534C3#1059603311yescharacteristic, complemented
535C2—122no—
536C2—362no—
537C2—1081no—
538C2C2×#8902011yescharacteristic
539C2#1172457911yescharacteristic
540C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723913C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×D12C2semidirect
4. #11723532C2semidirect
5. #11723531C2semidirect
6. #11723529C2semidirect
7. #11723520C2semidirect
8. #11723518C2semidirect
9. #11723514C2semidirect
10. #11723513C2semidirect
11. #11723450C2semidirect
46. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C4×D12D3semidirect
172. C4×D12C3⋊D3semidirect
341. C33#540semidirect
476. C32#5562semidirect
477. C32#5562semidirect
533. C3#10593984semidirect
534. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
#117239132
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
#117235322
#117235312
#117235292
#117235202
#117235182
#117235142
#117235132
#117234502
#105960333
#105939843