Finite Group Databasebeta

← #13177626

Subgroups — #13177626

Order 1728.

1944 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723981C211yesmaximal, characteristic, complemented
2C3×#64146C211yesmaximal, characteristic, complemented
3C32×#935C211yesmaximal, characteristic, complemented
4#11723636C212yesmaximal, complemented
5#11723632C212yesmaximal, complemented
6#11723628C212yesmaximal, complemented
7#11723623C212yesmaximal, complemented
8#11723486C212yesmaximal, complemented
9#11723433C212yesmaximal, complemented
10#10596137—31nomaximal
11#10595261—34nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2212yes—
15C2×#6312C2212yescomplemented
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
17C2×C3×#638C2212yes—
18C22×C32×Dic3C2211yescharacteristic, complemented
19C2×#6292C2212yes—
20C2×#6292C2212yes—
21C2×#6289C2212yescomplemented
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C3×#3757C2211yescharacteristic
24C3×#3752C2212yes—
25C32×#217C2212yes—
26C32×#217C2211yescharacteristic, complemented
27C32×Dic3⋊C4C2212yescomplemented
28#88807C2212yescomplemented
29Dic3⋊(C3⋊Dic3)C2212yescomplemented
30#88804C2212yescomplemented
31#88803C2212yes—
32#88803C2212yes—
33C32×#123—31no—
34#64149—31no—
35#64149—34no—
36#64149—64no—
37#64146D311yescharacteristic, complemented
38#64139—32no—
39#64107—32no—
40#64104—32no—
41#64093—32no—
42#64092—32no—
43#64091—32no—
44C3×#937—34no—
45C3×#935D314yescomplemented
46#63978—38no—
47#63977—38no—
48#63975—38no—
49#63958—38no—
50#63910—38no—
51#63862—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)C2312yes—
55C22×C3×C3⋊D3—41no—
56C22×C3×C3⋊D3—42no—
57C22×C3×C3⋊D3C2312yes—
58C33×D4—41no—
59C2×C33×C4C2311yescharacteristic
60#6312—42no—
61#6312—42no—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C4⋊(C3⋊(C3⋊D3))—41no—
64C3×#638—42no—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C32×Dic3—41no—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69#6292—42no—
70#6292—42no—
71#6289—42no—
72#5666—94no—
73#5595—91no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3×D3—34no—
76C2×C32×D4—24no—
77C2×C32×D4C2×D314yes—
78C2×C32×D4C2×D311yescharacteristic
79C22×C32×C4—31no—
80C2×#638—32no—
81C2×#638—32no—
82C2×#638—38no—
83C2×#638—38no—
84C2×#638—68no—
85C2×#638—68no—
86C2×#638C2×D312yes—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)—32no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4⋊(C3⋊D3)—34no—
91C2×C4⋊(C3⋊D3)—64no—
92C2×C4×C3⋊D3—32no—
93C2×C3×#67—38no—
94C22×C3×Dic3C2×D314yescomplemented
95C2×#626—38no—
96C2×#626—38no—
97C2×#624—38no—
98C32×C4⋊C4—32no—
99C32×C22⋊C4—32no—
100C32×C22⋊C4—31no—
101#3757C2×D311yescharacteristic
102#3752—32no—
103#3752—32no—
104#3752—32no—
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)—32no—
107#3750—32no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—64no—
111C3×#217—34no—
112C3×#217C2×D318yes—
113C3×#217C2×D314yescomplemented
114C3×(C2×D3)⋊C4—38no—
115C3×Dic3⋊C4C2×D318yescomplemented
116Dic3⋊Dic3—38no—
117Dic3⋊Dic3—38no—
118#3722—38no—
119(C2×D3)⋊Dic3—38no—
120(C2×D3)⋊Dic3—38no—
121C22×C33—21no—
122C22×C33—22no—
123C22×C33—22no—
124C22×C33—21no—
125C22×C33C2411yescharacteristic
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—44no—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C32×Dic3—22no—
137C32×Dic3—22no—
138C3×#123—34no—
139C3×#120—91no—
140#953—92no—
141#940—94no—
142#940—91no—
143#940—184no—
144#939—92no—
145#938—92no—
146#937—34no—
147#935C3⋊D311yescharacteristic, complemented
148#930—98no—
149#898—98no—
150#895—98no—
151#884—98no—
152#884—92no—
153#883—98no—
154#883—92no—
155#882—98no—
156#879—92no—
157C23×C32—24no—
158C23×C32—24no—
159C23×C32C22×D314yes—
160C23×C32C22×D314yes—
161C23×C32C22×D311yescharacteristic
162C23×C32C22×D311yescharacteristic
163C22×C3⋊D3—32no—
164C22×C3⋊D3—38no—
165C22×C3⋊D3—41no—
166C22×C3⋊D3—42no—
167C22×C3⋊D3—68no—
168C22×C3⋊D3C22×D312yes—
169C22×C3×D3—38no—
170C22×C3×D3—124no—
171C22×C3×D3—128no—
172C32×D4—41no—
173C32×D4—44no—
174C32×D4—48no—
175C2×C32×C4—24no—
176C2×C32×C4—32no—
177C2×C32×C4—32no—
178C2×C32×C4—121no—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D311yescharacteristic
181#638—42no—
182#638—122no—
183#638—122no—
184#638—128no—
185#638—128no—
186#638—1216no—
187#638—1216no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3C22×D312yes—
192C4⋊(C3⋊D3)—121no—
193C4⋊(C3⋊D3)—124no—
194C4⋊(C3⋊D3)—128no—
195C4⋊(C3⋊D3)—122no—
196C4⋊(C3⋊D3)—122no—
197C4×C3⋊D3—122no—
198C3×#67—128no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—44no—
201C2×C3×Dic3C22×D318yes—
202C2×C3×Dic3C22×D318yes—
203#626—128no—
204#626—128no—
205#624—128no—
206#560—271noSylow
207C2×C33—41no—
208C2×C33—41no—
209C2×C33H(5,𝔽2)11yescharacteristic
210C2×C33H(5,𝔽2)12yes—
211C3⋊(C3⋊D3)—42no—
212C3×C3⋊D3—42no—
213C24×C3—91no—
214C23×D3—34no—
215C2×C3×D4—24no—
216C2×C3×D4—92no—
217C2×C3×D4D3214yes—
218C2×C3×D4C2×C3⋊D311yescharacteristic
219C22×C3×C4—34no—
220C2×#67—38no—
221C2×#67—98no—
222C2×#67—98no—
223C2×#67—92no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—188no—
228C2×#67—188no—
229C22×Dic3C2×C3⋊D311yescharacteristic, complemented
230C2×D12—98no—
231C2×D12—98no—
232C2×D12—94no—
233C2×D12—91no—
234C2×D12—184no—
235C2×C4×D3—98no—
236C2×C4×D3—92no—
237C3×C4⋊C4—38no—
238C3×C22⋊C4—38no—
239C3×C22⋊C4—34no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242#217—34no—
243#217—92no—
244#217—92no—
245#217C2×C3⋊D312yes—
246#217C2×C3⋊D311yescharacteristic, complemented
247(C2×D3)⋊C4—38no—
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—98no—
251(C2×D3)⋊C4—92no—
252C4⋊Dic3—98no—
253C4⋊Dic3—92no—
254Dic3⋊C4—98no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×C3⋊D312yescomplemented
257C4×Dic3—94no—
258C4×Dic3—91no—
259C4×Dic3—184no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—28no—
268C22×C32—28no—
269C22×C32—48no—
270C22×C32—48no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—44no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C2×C3⋊D3—128no—
288C2×C3×D3—68no—
289C2×C3×D3—68no—
290C2×C3×D3—68no—
291C2×C3×D3—1216no—
292C32×C4—28no—
293C32×C4—24no—
294C32×C4—21no—
295C32×C4—62no—
296C32×C4—62no—
297C3⋊Dic3—22no—
298C3⋊Dic3—616no—
299C3⋊Dic3—62no—
300C3⋊Dic3—68no—
301C3×Dic3—28no—
302C3×Dic3—28no—
303C3×Dic3—68no—
304#127—271no—
305#126—272no—
306#123—31no—
307#123—272no—
308#121—272no—
309#121—272no—
310#121—272no—
311#120—91no—
312#120—272no—
313C33#56011yescharacteristic, complemented, Sylow
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—92no—
317C23×C3—361no—
318C23×C3—362no—
319C23×C3C2×D3214yes—
320C23×C3C2×D3214yes—
321C23×C3C22×C3⋊D311yescharacteristic
322C23×C3C22×C3⋊D311yescharacteristic
323C22×D3—38no—
324C22×D3—98no—
325C22×D3—92no—
326C22×D3—128no—
327C22×D3—124no—
328C22×D3—188no—
329C3×D4—44no—
330C3×D4—41no—
331C3×D4—48no—
332C3×D4—362no—
333C2×C3×C4—24no—
334C2×C3×C4—38no—
335C2×C3×C4—38no—
336C2×C3×C4—92no—
337C2×C3×C4—124no—
338C2×C3×C4C2×D3214yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—128no—
341#67—362no—
342#67—362no—
343#67—368no—
344#67—368no—
345#67—362no—
346#67—362no—
347#67—3616no—
348#67—3616no—
349C2×Dic3—38no—
350C2×Dic3—41no—
351C2×Dic3—98no—
352C2×Dic3—92no—
353C2×Dic3—188no—
354C2×Dic3C22×C3⋊D312yes—
355C2×Dic3C22×C3⋊D312yes—
356D12—364no—
357D12—361no—
358D12—368no—
359D12—368no—
360D12—368no—
361C4×D3—362no—
362C4×D3—368no—
363C2×C32—28no—
364C2×C32—24no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—44no—
368C2×C32—44no—
369C2×C32—84no—
370C2×C32—84no—
371C2×C32#100414yes—
372C2×C32#100412yes—
373C2×C32#100411yescharacteristic
374C2×C32#100918yes—
375C3⋊D3—42no—
376C3⋊D3—122no—
377C3⋊D3—128no—
378C3⋊D3—248no—
379C3×D3—128no—
380C24—91no—
381C2×D4—92no—
382C2×D4—272no—
383C2×D4—272no—
384C2×D4—272no—
385C2×D4—272no—
386C2×D4—271no—
387C2×D4D3×C3⋊D311yescharacteristic
388C22×C4—31no—
389C22×C4—272no—
390C4⋊C4—32no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C22⋊C4—32no—
394C22⋊C4—31no—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C42—271no—
401C22×C3—28no—
402C22×C3—28no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—24no—
410C22×C3—48no—
411C22×C3—48no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—364no—
418C22×C3C22×D3214yes—
419C22×C3C23×C3⋊D311yescharacteristic
420C2×D3—68no—
421C2×D3—68no—
422C2×D3—68no—
423C2×D3—1216no—
424C2×D3—188no—
425C2×D3—188no—
426C2×D3—188no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—368no—
431C2×D3—368no—
432C2×D3—368no—
433C3×C4—28no—
434C3×C4—24no—
435C3×C4—21no—
436C3×C4—68no—
437C3×C4—68no—
438C3×C4—182no—
439Dic3—22no—
440Dic3—22no—
441Dic3—68no—
442Dic3—1816no—
443Dic3—188no—
444Dic3—182no—
445C32—24no—
446C32#559511yescharacteristic, complemented
447C32#566614yescomplemented
448C23—92no—
449C23—272no—
450C23—362no—
451C23—361no—
452C23C2×D3×C3⋊D311yescharacteristic
453C23C2×D3×C3⋊D311yescharacteristic
454D4—41no—
455D4—362no—
456D4—1081no—
457D4—1082no—
458D4—1082no—
459D4—1082no—
460D4—1082no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—92no—
464C2×C4—121no—
465C2×C4—272no—
466C2×C4—1082no—
467C2×C4C2×D3×C3⋊D311yescharacteristic
468C2×C3—28no—
469C2×C3—24no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—44no—
473C2×C3—44no—
474C2×C3—84no—
475C2×C3—84no—
476C2×C3—362no—
477C2×C3#6431418yes—
478C2×C3#6433014yes—
479C2×C3#6436111yescharacteristic
480C2×C3#6436612yes—
481D3—128no—
482D3—362no—
483D3—368no—
484D3—728no—
485C22—22no—
486C22—22no—
487C22—21no—
488C22—21no—
489C22—182no—
490C22—182no—
491C22—182no—
492C22—364no—
493C22—542no—
494C22—542no—
495C22—542no—
496C22C22×D3×C3⋊D311yescharacteristic
497C4—21no—
498C4—62no—
499C4—62no—
500C4—182no—
501C4—542no—
502C3—24no—
503C3#1059526114yescomplemented
504C3#1059613711yescharacteristic, complemented
505C2—41no—
506C2—41no—
507C2—362no—
508C2—1082no—
509C2#1172458912yes—
510C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#560
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723981C2semidirect
2. C3×#64146C2semidirect
3. C32×#935C2semidirect
4. #11723636C2semidirect
5. #11723632C2semidirect
6. #11723628C2semidirect
7. #11723623C2semidirect
8. #11723486C2semidirect
9. #11723433C2semidirect
15. C2×#6312C22semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C22×C32×Dic3C22semidirect
21. C2×#6289C22semidirect
22. C4×C3⋊(C3⋊Dic3)C22semidirect
26. C32×#217C22semidirect
27. C32×Dic3⋊C4C22semidirect
28. #88807C22semidirect
29. Dic3⋊(C3⋊Dic3)C22semidirect
30. #88804C22semidirect
37. #64146D3semidirect
45. C3×#935D3semidirect
94. C22×C3×Dic3C2×D3semidirect
113. C3×#217C2×D3semidirect
115. C3×Dic3⋊C4C2×D3semidirect
147. #935C3⋊D3semidirect
229. C22×Dic3C2×C3⋊D3semidirect
246. #217C2×C3⋊D3semidirect
256. Dic3⋊C4C2×C3⋊D3semidirect
313. C33#560semidirect
446. C32#5595semidirect
447. C32#5666semidirect
503. C3#10595261semidirect
504. C3#10596137semidirect

Maximal subgroups

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

subgroupindex
#117239812
C3×#641462
C32×#9352
#117236362
#117236322
#117236282
#117236232
#117234862
#117234332
#105961373
#105952613