Finite Group Databasebeta

← #13177641

Subgroups — #13177641

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#11723979C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723632C212yesmaximal, complemented
5#11723624C212yesmaximal, complemented
6#11723621C212yesmaximal, complemented
7#11723605C212yesmaximal, complemented
8#11723532C211yesmaximal, characteristic, complemented
9Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
10#11723452C211yesmaximal, characteristic, complemented
11#11723447C211yesmaximal, characteristic, complemented
12#10596033—31nomaximal
13#10595218—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C22×C32×Dic3C2212yes—
21C2×#6292C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23#88864C2212yes—
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88806C2211yescharacteristic
30#88806C2212yes—
31#88806C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2211yescharacteristic
34#88804C2211yescharacteristic
35#88803C2212yes—
36#88803C2212yes—
37C32×C4×D4—31no—
38#64149D311yescharacteristic, complemented
39#64147—31no—
40#64147—34no—
41#64147—64no—
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×#940—34no—
51C3×D4×Dic3D314yescomplemented
52#63978—38no—
53#63971—38no—
54#63969—38no—
55#63961—38no—
56#63916—34no—
57Dic3×D12—34no—
58#63874—34no—
59#63869—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65#6312—42no—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4×C3⋊(C3⋊D3)—22no—
69C3×#638—42no—
70C3×#638—42no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
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—
79Dic3×C3⋊D3—42no—
80#5719—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—38no—
88C2×#638—68no—
89C2×#638C2×D312yes—
90C2×#638C2×D312yes—
91C2×C4⋊(C3⋊D3)—32no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—34no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—64no—
97C2×C3×#67—38no—
98C2×C3×#67—38no—
99C22×C3×Dic3C2×D318yes—
100C2×C3×D12—34no—
101C2×#626—38no—
102C2×D3×Dic3—38no—
103C32×C4⋊C4—31no—
104C32×C22⋊C4—32no—
105C32×C42—31no—
106#3757—32no—
107#3757—38no—
108#3757—68no—
109#3752—32no—
110#3752—32no—
111#3752—31no—
112#3752—31no—
113C4⋊(C3⋊Dic3)—31no—
114C4⋊(C3⋊Dic3)—34no—
115C4⋊(C3⋊Dic3)—32no—
116C4⋊(C3⋊Dic3)—64no—
117#3750—31no—
118#3750—31no—
119#3750—32no—
120C4×C3⋊Dic3C2×D311yescharacteristic
121C3×#217C2×D318yes—
122C3×C4⋊Dic3C2×D314yes—
123C3×C4×Dic3—34no—
124C3×C4×Dic3C2×D314yes—
125#3724—38no—
126#3724—34no—
127#3724—34no—
128Dic3⋊Dic3—38no—
129#3722—34no—
130#3722—34no—
131(C2×D3)⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33—22no—
136C22×C33C2411yescharacteristic
137C2×C3⋊(C3⋊D3)—21no—
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3⋊(C3⋊D3)—21no—
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—22no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C32×Dic3—21no—
149C32×Dic3—22no—
150C32×Dic3—21no—
151C3×#127—91no—
152C3×C4×D4—34no—
153#940—34no—
154#939—92no—
155#939—92no—
156#938—94no—
157#938—91no—
158#938—91no—
159#938—184no—
160#935—92no—
161#935—92no—
162D4×Dic3—91no—
163D4×Dic3C3⋊D311yescharacteristic, complemented
164#930—98no—
165#929—98no—
166#895—94no—
167#890—94no—
168#884—98no—
169#883—98no—
170C4×D12—94no—
171#872—94no—
172#872—91no—
173#870—91no—
174C23×C32—28no—
175C23×C32C22×D318yes—
176C23×C32C22×D312yes—
177C22×C3⋊D3—31no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—64no—
180C22×C3⋊D3C22×D312yes—
181C22×C3×D3—38no—
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—42no—
194#638—122no—
195#638—128no—
196#638—1216no—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3—64no—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4⋊(C3⋊D3)—22no—
206C4⋊(C3⋊D3)—122no—
207C4×C3⋊D3—62no—
208C4×C3⋊D3—68no—
209C4×C3⋊D3—128no—
210C4×C3⋊D3—122no—
211C3×#67—128no—
212C3×#67—128no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—48no—
216C2×C3×Dic3C22×D314yes—
217C2×C3×Dic3C22×D318yes—
218C2×C3×Dic3C22×D314yes—
219C3×D12—68no—
220#626—128no—
221D3×Dic3—128no—
222#540—271noSylow
223C2×C33—42no—
224C2×C33C2×#4111yescharacteristic
225C2×C33H(5,𝔽2)11yescharacteristic
226C2×C33H(5,𝔽2)11yescharacteristic
227C3⋊(C3⋊D3)—41no—
228C3×C3⋊D3—42no—
229C2×C3×D4—24no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—91no—
233C2×C3×D4D3214yes—
234C2×C3×D4C2×C3⋊D311yescharacteristic
235C22×C3×C4—38no—
236C2×#67—38no—
237C2×#67—38no—
238C2×#67—98no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—188no—
242C22×Dic3—92no—
243C22×Dic3C2×C3⋊D312yes—
244C2×D12—34no—
245C2×D12—98no—
246C2×C4×D3—94no—
247C2×C4×D3—91no—
248C2×C4×D3—98no—
249C2×C4×D3—184no—
250C3×C4⋊C4—34no—
251C3×C22⋊C4—38no—
252C3×C42—34no—
253C3×C42—91no—
254#217—98no—
255#217—92no—
256#217—92no—
257#217—92no—
258#217—188no—
259#217C2×C3⋊D312yes—
260(C2×D3)⋊C4—94no—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—98no—
263(C2×D3)⋊C4—98no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266C4⋊Dic3—98no—
267C4⋊Dic3—94no—
268C4⋊Dic3—91no—
269C4⋊Dic3—184no—
270C4⋊Dic3C2×C3⋊D311yescharacteristic
271Dic3⋊C4—94no—
272Dic3⋊C4—94no—
273Dic3⋊C4—98no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278C4×Dic3—34no—
279C4×Dic3C2×C3⋊D311yescharacteristic
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—24no—
284C22×C32—28no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—48no—
288C22×C32—48no—
289C22×C32—48no—
290C22×C32C23×D314yes—
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—64no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—124no—
303C2×C3⋊D3—124no—
304C2×C3×D3—68no—
305C2×C3×D3—68no—
306C2×C3×D3—68no—
307C32×C4—28no—
308C32×C4—61no—
309C32×C4—62no—
310C32×C4—61no—
311C32×C4#23518yes—
312C32×C4#23712yes—
313C3⋊Dic3—21no—
314C3⋊Dic3—21no—
315C3⋊Dic3—61no—
316C3⋊Dic3—62no—
317C3⋊Dic3—64no—
318C3⋊Dic3—68no—
319C3⋊Dic3—124no—
320C3⋊Dic3—128no—
321C3×Dic3—28no—
322C3×Dic3—24no—
323C3×Dic3—24no—
324C3×Dic3—64no—
325C3×Dic3—64no—
326#127—91no—
327#125—271no—
328#123—272no—
329#123—272no—
330#121—272no—
331#121—271no—
332#121—272no—
333#121—271no—
334C4×D4—31no—
335C4×D4—271no—
336#117—271no—
337C33#54011yescharacteristic, complemented, Sylow
338C23×C3—28no—
339C23×C3—92no—
340C23×C3C2×D3218yes—
341C23×C3C22×C3⋊D312yes—
342C22×D3—38no—
343C22×D3—94no—
344C22×D3—91no—
345C22×D3—184no—
346C3×D4—28no—
347C3×D4—22no—
348C3×D4—48no—
349C3×D4—182no—
350C3×D4—362no—
351C3×D4—362no—
352C2×C3×C4—24no—
353C2×C3×C4—38no—
354C2×C3×C4—34no—
355C2×C3×C4—34no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—128no—
359C2×C3×C4C2×D3214yes—
360C2×C3×C4C22×C3⋊D311yescharacteristic
361#67—128no—
362#67—128no—
363#67—362no—
364#67—368no—
365#67—362no—
366#67—3616no—
367C2×Dic3—34no—
368C2×Dic3—34no—
369C2×Dic3—42no—
370C2×Dic3—94no—
371C2×Dic3—98no—
372C2×Dic3—92no—
373C2×Dic3—91no—
374C2×Dic3—188no—
375C2×Dic3—184no—
376C2×Dic3—362no—
377C2×Dic3C22×C3⋊D311yescharacteristic
378C2×Dic3C22×C3⋊D311yescharacteristic
379C2×Dic3C22×C3⋊D312yes—
380D12—68no—
381D12—368no—
382C4×D3—188no—
383C4×D3—182no—
384C4×D3—368no—
385C4×D3—368no—
386C2×C32—24no—
387C2×C32—24no—
388C2×C32—24no—
389C2×C32—42no—
390C2×C32—48no—
391C2×C32—88no—
392C2×C32C2×#23514yes—
393C2×C32C2×#23711yescharacteristic
394C2×C32#100414yes—
395C2×C32#100411yescharacteristic
396C2×C32#100411yescharacteristic
397C2×C32#100914yes—
398C3⋊D3—42no—
399C3⋊D3—128no—
400C3⋊D3—121no—
401C3⋊D3—124no—
402C3×D3—128no—
403C2×D4—92no—
404C2×D4—92no—
405C2×D4—91no—
406C2×D4—272no—
407C2×D4—272no—
408C2×D4D3×C3⋊D311yescharacteristic
409C22×C4—32no—
410C22×C4—272no—
411C22×C4—271no—
412C4⋊C4—31no—
413C4⋊C4—271no—
414C4⋊C4—271no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C4⋊C4—271no—
418C22⋊C4—32no—
419C22⋊C4—271no—
420C22⋊C4—271no—
421C22⋊C4—272no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C42—31no—
425C42—91no—
426C22×C3—28no—
427C22×C3—28no—
428C22×C3—28no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—48no—
434C22×C3—48no—
435C22×C3—48no—
436C22×C3—182no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C22×D3214yes—
440C22×C3C23×C3⋊D311yescharacteristic
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—28no—
454C3×C4—68no—
455C3×C4—64no—
456C3×C4—64no—
457C3×C4—181no—
458C3×C4—181no—
459C3×C4#380418yes—
460C3×C4#382812yes—
461Dic3—21no—
462Dic3—22no—
463Dic3—21no—
464Dic3—64no—
465Dic3—64no—
466Dic3—188no—
467Dic3—184no—
468Dic3—181no—
469Dic3—182no—
470Dic3—364no—
471Dic3—368no—
472C32—24no—
473C32#556214yescomplemented
474C32#571911yescharacteristic, complemented
475C23—92no—
476C23—271no—
477C23C2×D3×C3⋊D312yes—
478D4—22no—
479D4—182no—
480D4—362no—
481D4—362no—
482D4—1082no—
483D4—1082no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—32no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—122no—
490C2×C4—271no—
491C2×C4—272no—
492C2×C4—542no—
493C2×C4—1082no—
494C2×C4C2×D3×C3⋊D311yescharacteristic
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—24no—
498C2×C3—48no—
499C2×C3—42no—
500C2×C3—88no—
501C2×C3—362no—
502C2×C3#6431314yes—
503C2×C3#6431414yes—
504C2×C3C2×#380414yes—
505C2×C3C2×#382811yescharacteristic
506C2×C3#6436111yescharacteristic
507C2×C3#6436611yescharacteristic
508D3—128no—
509D3—368no—
510D3—364no—
511D3—361no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—182no—
516C22—182no—
517C22—182no—
518C22—541no—
519C22—541no—
520C22—541no—
521C22C22×D3×C3⋊D311yescharacteristic
522C4—61no—
523C4—62no—
524C4—61no—
525C4—181no—
526C4—181no—
527C4—542no—
528C4—541no—
529C4#8903112yes—
530C3—24no—
531C3#1059521814yescomplemented
532C3#1059603311yescharacteristic, complemented
533C2—42no—
534C2—362no—
535C2—1081no—
536C2#1172458711yescharacteristic
537C2#1172458911yescharacteristic
538C2C2×#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. #11723979C2semidirect
2. C3×#64149C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723632C2semidirect
5. #11723624C2semidirect
6. #11723621C2semidirect
7. #11723605C2semidirect
8. #11723532C2semidirect
9. Dic3×C4⋊(C3⋊D3)C2semidirect
10. #11723452C2semidirect
11. #11723447C2semidirect
38. #64149D3semidirect
51. C3×D4×Dic3D3semidirect
163. D4×Dic3C3⋊D3semidirect
337. C33#540semidirect
473. C32#5562semidirect
474. C32#5719semidirect
531. C3#10595218semidirect
532. C3#10596033semidirect

Maximal subgroups

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

subgroupindex
#117239792
C3×#641492
C32×D4×Dic32
#117236322
#117236242
#117236212
#117236052
#117235322
Dic3×C4⋊(C3⋊D3)2
#117234522
#117234472
#105960333
#105952183