Finite Group Databasebeta

← #13177229

Subgroups — #13177229

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6312C211yesmaximal, characteristic, complemented
2C3×#64135C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4#11723614C212yesmaximal, complemented
5#11723599C212yesmaximal, complemented
6#11723531C211yesmaximal, characteristic, complemented
7#11723503C211yesmaximal, characteristic, complemented
8#11723499C211yesmaximal, characteristic, complemented
9#11723461C212yesmaximal, complemented
10#11723450C211yesmaximal, characteristic, complemented
11#11723428C212yesmaximal, complemented
12#10596085—31nomaximal
13#10594967—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×D12C2211yescharacteristic
19C2×#6291C2212yes—
20C2×#6289C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22#88864C2211yescharacteristic
23#88859C2211yescharacteristic
24#88857C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×#3757C2212yes—
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4×C3⋊Dic3C2212yes—
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#121—31no—
38#64147—32no—
39#64147—31no—
40#64144—32no—
41D4×C3⋊Dic3—31no—
42C4×#638—31no—
43C4×#638—34no—
44C4×#638—64no—
45#64135D311yescharacteristic, complemented
46C4⋊C4×C3⋊D3—31no—
47#64099—31no—
48#64090—32no—
49#64088—32no—
50C3×#930D314yescomplemented
51C3×#926—34no—
52#63969—38no—
53#63958—38no—
54#63916—34no—
55#63900—34no—
56#63898—34no—
57#63880—38no—
58#63872—34no—
59#63859—38no—
60C23×C33C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66#6312—22no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4×C3⋊(C3⋊D3)—22no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C32×#67—42no—
74C2×C32×Dic3C2312yes—
75C32×D12—22no—
76#6291—24no—
77#6289—42no—
78D3×C3⋊Dic3—42no—
79#5614—94no—
80#5530—91no—
81C2×C32×D4—32no—
82C2×C32×D4—31no—
83C22×C32×C4—24no—
84C22×C32×C4C2×D314yes—
85C22×C32×C4C2×D311yescharacteristic
86C2×#638—31no—
87C2×#638—34no—
88C2×#638—32no—
89C2×#638—64no—
90C22×C3⋊Dic3—32no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—34no—
93C2×C4×C3⋊D3—32no—
94C2×C4×C3⋊D3—64no—
95C2×C3×#67C2×D318yes—
96C2×C3×D12C2×D314yes—
97C2×#626—38no—
98C2×#624—38no—
99C2×D3×Dic3—38no—
100C32×C4⋊C4—31no—
101C32×C22⋊C4—32no—
102#3757—31no—
103#3757—34no—
104#3757—32no—
105#3757—32no—
106#3757—64no—
107#3757C2×D312yes—
108#3752—31no—
109#3752—34no—
110#3752—64no—
111C4⋊(C3⋊Dic3)—32no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114#3750—31no—
115#3750—34no—
116#3750—32no—
117#3750—32no—
118#3750—64no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—34no—
121C4×C3⋊Dic3—64no—
122C4×C3⋊Dic3C2×D312yes—
123C3×#217—38no—
124C3×(C2×D3)⋊C4C2×D318yes—
125C3×C4⋊Dic3—34no—
126C3×C4⋊Dic3—34no—
127C3×C4⋊Dic3C2×D314yes—
128C3×C4×Dic3—38no—
129#3724—38no—
130Dic3⋊Dic3—38no—
131Dic3⋊Dic3—38no—
132(C2×D3)⋊Dic3—38no—
133(C2×D3)⋊Dic3—38no—
134C22×C33—21no—
135C22×C33—21no—
136C22×C33—21no—
137C22×C33C2411yescharacteristic
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3⋊(C3⋊D3)—21no—
140C2×C3⋊(C3⋊D3)#4112yes—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143C2×C32×D3—22no—
144C33×C4—21no—
145C33×C4#4112yes—
146C3⋊(C3⋊Dic3)—21no—
147C3⋊(C3⋊Dic3)—21no—
148C3⋊(C3⋊Dic3)—21no—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3—22no—
151C32×Dic3—22no—
152C3×#121—34no—
153C3×#117—91no—
154#938—94no—
155#938—98no—
156#935—98no—
157D4×Dic3—94no—
158#930C3⋊D311yescharacteristic, complemented
159C4×#67—94no—
160C4×#67—91no—
161C4×#67—184no—
162#926—34no—
163#895—91no—
164#893—91no—
165D3×C4⋊C4—94no—
166D3×C4⋊C4—91no—
167#890—94no—
168#890—91no—
169#884—92no—
170#883—92no—
171#67⋊C4—98no—
172#879—98no—
173#876—92no—
174C4×D12—92no—
175C23×C32—24no—
176C23×C32—32no—
177C23×C32C22×D314yes—
178C23×C32C22×D311yescharacteristic
179C22×C3⋊D3—31no—
180C22×C3⋊D3—34no—
181C22×C3⋊D3—64no—
182C22×C3×D3C22×D318yes—
183C32×D4—62no—
184C32×D4—122no—
185C2×C32×C4—22no—
186C2×C32×C4—24no—
187C2×C32×C4—24no—
188C2×C32×C4—28no—
189C2×C32×C4—32no—
190C2×C32×C4—48no—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—64no—
196#638—62no—
197#638—68no—
198#638—616no—
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—64no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—122no—
209C2×C3⋊Dic3C22×D312yes—
210C2×C3⋊Dic3C22×D312yes—
211C4×C3⋊D3—62no—
212C4×C3⋊D3—68no—
213C4×C3⋊D3—128no—
214C4×C3⋊D3—122no—
215C3×#67—48no—
216C2×C3×Dic3—38no—
217C2×C3×Dic3—38no—
218C2×C3×Dic3C22×D318yes—
219C3×D12—28no—
220#626—616no—
221#624—128no—
222D3×Dic3—128no—
223#553—271noSylow
224C2×C33—41no—
225C2×C33C2×#4111yescharacteristic
226C2×C33C2×#4111yescharacteristic
227C2×C33H(5,𝔽2)11yescharacteristic
228C3⋊(C3⋊D3)—22no—
229C32×D3—42no—
230C2×C3×D4—38no—
231C2×C3×D4—34no—
232C22×C3×C4—24no—
233C22×C3×C4D3214yes—
234C22×C3×C4C2×C3⋊D311yescharacteristic
235C2×#67—98no—
236C2×#67—94no—
237C2×#67—91no—
238C2×#67—184no—
239C2×#67C2×C3⋊D312yes—
240C22×Dic3—98no—
241C2×D12—92no—
242C2×D12C2×C3⋊D311yescharacteristic
243C2×C4×D3—94no—
244C2×C4×D3—91no—
245C2×C4×D3—98no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—184no—
249C3×C4⋊C4—34no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—38no—
253C3×C22⋊C4—92no—
254C3×C42—92no—
255#217—38no—
256#217—98no—
257#217—98no—
258#217—94no—
259#217—91no—
260#217—184no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—184no—
266(C2×D3)⋊C4C2×C3⋊D312yes—
267C4⋊Dic3—34no—
268C4⋊Dic3—34no—
269C4⋊Dic3—98no—
270C4⋊Dic3—92no—
271C4⋊Dic3C2×C3⋊D311yescharacteristic
272Dic3⋊C4—98no—
273Dic3⋊C4—98no—
274Dic3⋊C4—94no—
275Dic3⋊C4—91no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—184no—
279C4×Dic3—38no—
280C4×Dic3—94no—
281C4×Dic3—91no—
282C4×Dic3—184no—
283C22×C32—28no—
284C22×C32—21no—
285C22×C32—24no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—24no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—62no—
294C22×C32—62no—
295C22×C32—62no—
296C22×C32C23×D314yes—
297C22×C32C23×D311yescharacteristic
298C2×C3⋊D3—32no—
299C2×C3⋊D3—38no—
300C2×C3⋊D3—68no—
301C2×C3⋊D3—61no—
302C2×C3⋊D3—61no—
303C2×C3⋊D3—64no—
304C2×C3⋊D3—64no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C2×C3×D3—28no—
308C2×C3×D3—28no—
309C2×C3×D3—28no—
310C32×C4—21no—
311C32×C4—28no—
312C32×C4—24no—
313C32×C4—44no—
314C32×C4—62no—
315C32×C4#23512yes—
316C32×C4#23718yes—
317C3⋊Dic3—22no—
318C3⋊Dic3—22no—
319C3⋊Dic3—68no—
320C3⋊Dic3—61no—
321C3⋊Dic3—64no—
322C3⋊Dic3—61no—
323C3⋊Dic3—61no—
324C3⋊Dic3—64no—
325C3⋊Dic3—64no—
326C3⋊Dic3—124no—
327C3⋊Dic3—124no—
328C3×Dic3—28no—
329C3×Dic3—68no—
330C3×Dic3—68no—
331#126—272no—
332#125—271no—
333#123—272no—
334#121—31no—
335#121—272no—
336#121—271no—
337C4×D4—272no—
338C4×D4—271no—
339C4×D4—271no—
340#117—91no—
341C2×C4⋊C4—271no—
342C33#55311yescharacteristic, complemented, Sylow
343C23×C3—24no—
344C23×C3—38no—
345C23×C3C2×D3214yes—
346C23×C3C22×C3⋊D311yescharacteristic
347C22×D3—94no—
348C22×D3—91no—
349C22×D3—184no—
350C22×D3C22×C3⋊D312yes—
351C3×D4—68no—
352C3×D4—128no—
353C2×C3×C4—28no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—38no—
358C2×C3×C4—48no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4C2×D3214yes—
362C2×C3×C4C2×D3214yes—
363C2×C3×C4C22×C3⋊D311yescharacteristic
364C2×C3×C4C22×C3⋊D311yescharacteristic
365#67—42no—
366#67—1816no—
367#67—188no—
368#67—182no—
369#67—1816no—
370C2×Dic3—38no—
371C2×Dic3—38no—
372C2×Dic3—94no—
373C2×Dic3—94no—
374C2×Dic3—94no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—184no—
379C2×Dic3—184no—
380C2×Dic3—184no—
381C2×Dic3—368no—
382C2×Dic3C22×C3⋊D312yes—
383D12—22no—
384D12—184no—
385C4×D3—188no—
386C4×D3—182no—
387C4×D3—368no—
388C4×D3—362no—
389C4×D3—368no—
390C4×D3—362no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—24no—
394C2×C32—48no—
395C2×C32—41no—
396C2×C32—44no—
397C2×C32—122no—
398C2×C32C2×#23511yescharacteristic
399C2×C32C2×#23714yes—
400C2×C32#100414yes—
401C2×C32D3×#4114yes—
402C2×C32D3×#4111yescharacteristic
403C2×C32#100911yescharacteristic
404C3⋊D3—62no—
405C3⋊D3—68no—
406C3⋊D3—128no—
407C3×D3—48no—
408C2×D4—32no—
409C2×D4—31no—
410C2×D4—272no—
411C2×D4—271no—
412C22×C4—272no—
413C22×C4—272no—
414C22×C4—271no—
415C22×C4D3×C3⋊D311yescharacteristic
416C4⋊C4—31no—
417C4⋊C4—91no—
418C4⋊C4—91no—
419C4⋊C4—272no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—271no—
423C22⋊C4—32no—
424C22⋊C4—92no—
425C22⋊C4—272no—
426C22⋊C4—272no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C42—92no—
430C42—271no—
431C22×C3—28no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—24no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—68no—
442C22×C3—68no—
443C22×C3—68no—
444C22×C3C22×D3214yes—
445C22×C3C23×C3⋊D311yescharacteristic
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—98no—
450C2×D3—92no—
451C2×D3—188no—
452C2×D3—184no—
453C2×D3—184no—
454C2×D3—181no—
455C2×D3—181no—
456C2×D3—364no—
457C2×D3—364no—
458C3×C4—24no—
459C3×C4—21no—
460C3×C4—28no—
461C3×C4—44no—
462C3×C4—68no—
463C3×C4—182no—
464C3×C4—182no—
465C3×C4#380418yes—
466C3×C4#383012yes—
467Dic3—22no—
468Dic3—68no—
469Dic3—68no—
470Dic3—188no—
471Dic3—184no—
472Dic3—184no—
473Dic3—184no—
474Dic3—181no—
475Dic3—181no—
476Dic3—181no—
477Dic3—364no—
478Dic3—364no—
479C32—24no—
480C32#553011yescharacteristic, complemented
481C32#561414yescomplemented
482C23—32no—
483C23—271no—
484C23C2×D3×C3⋊D311yescharacteristic
485D4—62no—
486D4—122no—
487D4—544no—
488D4—542no—
489C2×C4—22no—
490C2×C4—32no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—271no—
494C2×C4—271no—
495C2×C4—271no—
496C2×C4—542no—
497C2×C4—1082no—
498C2×C4—1082no—
499C2×C4C2×D3×C3⋊D311yescharacteristic
500C2×C4C2×D3×C3⋊D311yescharacteristic
501C2×C3—24no—
502C2×C3—24no—
503C2×C3—24no—
504C2×C3—48no—
505C2×C3—44no—
506C2×C3—41no—
507C2×C3—128no—
508C2×C3#6430614yes—
509C2×C3#6430714yes—
510C2×C3C2×#380414yes—
511C2×C3C2×#383011yescharacteristic
512C2×C3#6436111yescharacteristic
513C2×C3#41×C3⋊D311yescharacteristic
514D3—42no—
515D3—188no—
516D3—182no—
517D3—368no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—272no—
525C22—541no—
526C22—541no—
527C22C22×D3×C3⋊D311yescharacteristic
528C4—21no—
529C4—62no—
530C4—182no—
531C4—182no—
532C4—541no—
533C4—541no—
534C4—541no—
535C4#8902812yes—
536C3—24no—
537C3#1059496714yescomplemented
538C3#1059608511yescharacteristic, complemented
539C2—41no—
540C2—122no—
541C2—542no—
542C2#1172457811yescharacteristic
543C2#1172457911yescharacteristic
544C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#6312C2semidirect
2. C3×#64135C2semidirect
3. C32×#930C2semidirect
4. #11723614C2semidirect
5. #11723599C2semidirect
6. #11723531C2semidirect
7. #11723503C2semidirect
8. #11723499C2semidirect
9. #11723461C2semidirect
10. #11723450C2semidirect
11. #11723428C2semidirect
45. #64135D3semidirect
50. C3×#930D3semidirect
158. #930C3⋊D3semidirect
342. C33#553semidirect
480. C32#5530semidirect
481. C32#5614semidirect
537. C3#10594967semidirect
538. C3#10596085semidirect

Maximal subgroups

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

subgroupindex
C4×#63122
C3×#641352
C32×#9302
#117236142
#117235992
#117235312
#117235032
#117234992
#117234612
#117234502
#117234282
#105960853
#105949673