Finite Group Databasebeta

← #13177617

Subgroups — #13177617

Order 1728.

1864 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723977C211yesmaximal, characteristic, complemented
2C3×#64146C211yesmaximal, characteristic, complemented
3C32×#937C211yesmaximal, characteristic, complemented
4#11723634C212yesmaximal, complemented
5#11723629C212yesmaximal, complemented
6#11723600C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723530C212yesmaximal, complemented
9#11723528C212yesmaximal, complemented
10#10596065—31nomaximal
11#10595256—34nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic, complemented
13C23×C32×D3C2211yescharacteristic, complemented
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C2×C3×#638C2212yes—
17C2×C32×#67C2212yes—
18C2×#6293C2212yescomplemented
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21#88864C2212yescomplemented
22#88864C2212yescomplemented
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
24C3×#3757C2211yescharacteristic, complemented
25C3×#3752C2212yescomplemented
26C32×#217C2211yescharacteristic, complemented
27C32×(C2×D3)⋊C4C2212yescomplemented
28#88803C2212yescomplemented
29#88803C2212yescomplemented
30(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
31(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#120—31no—
34#64162—32no—
35#64146—32no—
36#64146D311yescharacteristic, complemented
37#64145—31no—
38#64145—32no—
39#64145—34no—
40#64145—64no—
41#64094—32no—
42#64094—32no—
43#64094—32no—
44C3×#937—34no—
45C3×#937D314yescomplemented
46#63976—38no—
47#63976—38no—
48#63959—38no—
49#63959—38no—
50#63915—38no—
51#63913—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3×C3⋊D3—41no—
55C22×C3×C3⋊D3—42no—
56C22×C3×C3⋊D3C2312yes—
57C22×C32×D3—41no—
58C22×C32×D3—42no—
59C22×C32×D3C2312yes—
60C33×D4—41no—
61C2×C33×C4C2311yescharacteristic
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64#6308—41no—
65C3×#638—42no—
66C2×C3×C3⋊Dic3C2312yes—
67C32×#67—42no—
68C2×C32×Dic3C2312yes—
69#6293—42no—
70#6290—42no—
71#6290—42no—
72#5594—94no—
73#5594—91no—
74C24×C32—31no—
75C23×C3⋊D3C2×D311yescharacteristic, complemented
76C23×C3×D3—34no—
77C23×C3×D3C2×D314yescomplemented
78C2×C32×D4—24no—
79C2×C32×D4—32no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C2×D311yescharacteristic
82C2×#638—32no—
83C2×#638—32no—
84C2×#638C2×D312yes—
85C2×#634—31no—
86C2×#634—34no—
87C2×#634—32no—
88C2×#634—64no—
89C2×C3×#67—38no—
90C2×C3×#67C2×D318yes—
91C2×#627—38no—
92C2×#625—38no—
93C2×#625—38no—
94C32×C22⋊C4—31no—
95C32×C22⋊C4—32no—
96#3757—32no—
97#3757—32no—
98#3757—38no—
99#3757—38no—
100#3757—32no—
101#3757—32no—
102#3757—68no—
103#3757—68no—
104#3757C2×D311yescharacteristic, complemented
105#3752—32no—
106#3752—32no—
107#3752C2×D312yescomplemented
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3—64no—
112C3×#217—34no—
113C3×#217C2×D314yescomplemented
114C3×(C2×D3)⋊C4—38no—
115C3×(C2×D3)⋊C4C2×D318yescomplemented
116(C2×D3)⋊Dic3—38no—
117(C2×D3)⋊Dic3—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—22no—
122C22×C33—22no—
123C22×C33—21no—
124C22×C33—21no—
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—44no—
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C2×C32×D3—22no—
133C2×C32×D3—44no—
134C33×C4—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3⋊(C3⋊Dic3)—22no—
137C3×C3⋊Dic3—22no—
138C32×Dic3—22no—
139C3×#120—34no—
140C3×#120—91no—
141#953—98no—
142#953—92no—
143#937—34no—
144#937—98no—
145#937—92no—
146#937C3⋊D311yescharacteristic, complemented
147#936—98no—
148#936—94no—
149#936—91no—
150#936—92no—
151#936—184no—
152#885—98no—
153#885—98no—
154#885—98no—
155#885—92no—
156#885—92no—
157#885—92no—
158C23×C32—24no—
159C23×C32—24no—
160C23×C32—32no—
161C23×C32—122no—
162C23×C32—121no—
163C23×C32C22×D314yes—
164C23×C32C22×D314yes—
165C23×C32C22×D311yescharacteristic
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—42no—
168C22×C3⋊D3—41no—
169C22×C3⋊D3C22×D312yes—
170C22×C3×D3—38no—
171C22×C3×D3—44no—
172C22×C3×D3—48no—
173C22×C3×D3—124no—
174C22×C3×D3—128no—
175C22×C3×D3C22×D318yes—
176C32×D4—41no—
177C32×D4—44no—
178C32×D4—48no—
179C32×D4—122no—
180C2×C32×C4—24no—
181C2×C32×C4—32no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184#638—42no—
185#638—122no—
186#638—122no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D312yes—
194#634—122no—
195#634—121no—
196#634—124no—
197#634—128no—
198C3×#67—48no—
199C3×#67—128no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3C22×D318yes—
202#627—128no—
203#625—128no—
204#625—128no—
205H(3,𝔽4)—271noSylow
206C2×C33—41no—
207C2×C33—41no—
208C2×C33H(5,𝔽2)12yes—
209C2×C33H(5,𝔽2)11yescharacteristic
210C3×C3⋊D3—42no—
211C32×D3—42no—
212C24×C3—34no—
213C24×C3—91no—
214C23×D3—34no—
215C23×D3C2×C3⋊D311yescharacteristic, complemented
216C2×C3×D4—24no—
217C2×C3×D4—38no—
218C2×C3×D4—92no—
219C2×C3×D4D3214yes—
220C2×C3×D4C2×C3⋊D311yescharacteristic
221C2×#67—38no—
222C2×#67—98no—
223C2×#67—98no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67C2×C3⋊D312yes—
227C2×Dic6—98no—
228C2×Dic6—94no—
229C2×Dic6—91no—
230C2×Dic6—92no—
231C2×Dic6—184no—
232C3×C22⋊C4—34no—
233C3×C22⋊C4—38no—
234C3×C22⋊C4—91no—
235C3×C22⋊C4—92no—
236#217—34no—
237#217—98no—
238#217—98no—
239#217—98no—
240#217—98no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—188no—
246#217—188no—
247#217C2×C3⋊D311yescharacteristic, complemented
248(C2×D3)⋊C4—38no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—98no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4C2×C3⋊D312yescomplemented
254C4×Dic3—98no—
255C4×Dic3—94no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—184no—
259C22×C32—22no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—48no—
271C22×C32—48no—
272C22×C32—62no—
273C22×C32—62no—
274C22×C32—62no—
275C22×C32—124no—
276C22×C32C23×D314yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—44no—
282C2×C3×D3—28no—
283C2×C3×D3—28no—
284C2×C3×D3—28no—
285C2×C3×D3—416no—
286C2×C3×D3—68no—
287C2×C3×D3—68no—
288C2×C3×D3—68no—
289C2×C3×D3—1216no—
290C32×C4—28no—
291C32×C4—24no—
292C32×C4—21no—
293C32×C4—62no—
294C3⋊Dic3—22no—
295C3⋊Dic3—62no—
296C3⋊Dic3—62no—
297C3⋊Dic3—68no—
298C3⋊Dic3—68no—
299C3⋊Dic3—128no—
300C3⋊Dic3—128no—
301C3×Dic3—28no—
302C3×Dic3—68no—
303#124—272no—
304#124—272no—
305#124—272no—
306#124—272no—
307#124—271no—
308#120—31no—
309#120—91no—
310#120—272no—
311#120—272no—
312C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—38no—
316C23×C3—92no—
317C23×C3—128no—
318C23×C3—124no—
319C23×C3—362no—
320C23×C3—361no—
321C23×C3C2×D3214yes—
322C23×C3C2×D3214yes—
323C23×C3C22×C3⋊D311yescharacteristic
324C23×C3C22×C3⋊D311yescharacteristic
325C22×D3—38no—
326C22×D3—42no—
327C22×D3—41no—
328C22×D3—128no—
329C22×D3—124no—
330C22×D3C22×C3⋊D312yes—
331C3×D4—44no—
332C3×D4—41no—
333C3×D4—48no—
334C3×D4—128no—
335C3×D4—362no—
336C2×C3×C4—24no—
337C2×C3×C4—38no—
338C2×C3×C4—92no—
339C2×C3×C4C2×D3214yes—
340C2×C3×C4C22×C3⋊D311yescharacteristic
341#67—42no—
342#67—128no—
343#67—362no—
344#67—362no—
345#67—368no—
346#67—368no—
347C2×Dic3—38no—
348C2×Dic3—98no—
349C2×Dic3—98no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—188no—
353C2×Dic3—188no—
354C2×Dic3C22×C3⋊D312yes—
355Dic6—362no—
356Dic6—368no—
357Dic6—364no—
358Dic6—361no—
359Dic6—368no—
360C2×C32—28no—
361C2×C32—24no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—44no—
365C2×C32—44no—
366C2×C32—84no—
367C2×C32—84no—
368C2×C32—122no—
369C2×C32#100418yes—
370C2×C32#100414yes—
371C2×C32#100412yes—
372C2×C32#100411yescharacteristic
373C3⋊D3—42no—
374C3×D3—48no—
375C3×D3—128no—
376C24—31no—
377C24—91no—
378C2×Q8—272no—
379C2×Q8—271no—
380C2×D4—32no—
381C2×D4—92no—
382C2×D4—272no—
383C2×D4—272no—
384C2×D4D3×C3⋊D311yescharacteristic
385C22⋊C4—31no—
386C22⋊C4—32no—
387C22⋊C4—91no—
388C22⋊C4—92no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C42—272no—
396C42—271no—
397C22×C3—28no—
398C22×C3—28no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—22no—
405C22×C3—24no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—48no—
409C22×C3—48no—
410C22×C3—68no—
411C22×C3—68no—
412C22×C3—68no—
413C22×C3—1216no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—364no—
418C22×C3C22×D3214yes—
419C22×C3C23×C3⋊D311yescharacteristic
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—44no—
424C2×D3—68no—
425C2×D3—68no—
426C2×D3—68no—
427C2×D3—1216no—
428C3×C4—28no—
429C3×C4—24no—
430C3×C4—21no—
431C3×C4—68no—
432C3×C4—182no—
433Dic3—22no—
434Dic3—68no—
435Dic3—188no—
436Dic3—188no—
437Dic3—182no—
438Dic3—182no—
439Dic3—368no—
440Dic3—368no—
441C32—24no—
442C32#559414yescomplemented
443C32#559411yescharacteristic, complemented
444C23—32no—
445C23—92no—
446C23—122no—
447C23—121no—
448C23—362no—
449C23—361no—
450C23C2×D3×C3⋊D311yescharacteristic
451C23C2×D3×C3⋊D311yescharacteristic
452Q8—1081no—
453Q8—1082no—
454D4—41no—
455D4—122no—
456D4—362no—
457D4—1082no—
458D4—1082no—
459C2×C4—32no—
460C2×C4—92no—
461C2×C4—272no—
462C2×C4—272no—
463C2×C4C2×D3×C3⋊D311yescharacteristic
464C2×C3—28no—
465C2×C3—24no—
466C2×C3—44no—
467C2×C3—44no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—84no—
471C2×C3—84no—
472C2×C3—128no—
473C2×C3—362no—
474C2×C3#6431318yes—
475C2×C3#6433014yes—
476C2×C3#6436112yes—
477C2×C3#6436111yescharacteristic
478D3—42no—
479D3—128no—
480C22—22no—
481C22—22no—
482C22—21no—
483C22—21no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22—124no—
488C22—182no—
489C22—182no—
490C22—182no—
491C22—364no—
492C22C22×D3×C3⋊D311yescharacteristic
493C4—21no—
494C4—62no—
495C4—182no—
496C4—542no—
497C4—542no—
498C3—24no—
499C3#1059525614yescomplemented
500C3#1059606511yescharacteristic, complemented
501C2—41no—
502C2—41no—
503C2—122no—
504C2—362no—
505C2#1172458712yes—
506C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723977C2semidirect
2. C3×#64146C2semidirect
3. C32×#937C2semidirect
4. #11723634C2semidirect
5. #11723629C2semidirect
6. #11723600C2semidirect
7. #11723591C2semidirect
8. #11723530C2semidirect
9. #11723528C2semidirect
12. C23×C3×C3⋊D3C22semidirect
13. C23×C32×D3C22semidirect
15. C2×#6308C22semidirect
18. C2×#6293C22semidirect
21. #88864C22semidirect
22. #88864C22semidirect
23. C4×C3⋊(C3⋊Dic3)C22semidirect
24. C3×#3757C22semidirect
25. C3×#3752C22semidirect
26. C32×#217C22semidirect
27. C32×(C2×D3)⋊C4C22semidirect
28. #88803C22semidirect
29. #88803C22semidirect
30. (C2×D3)⋊(C3⋊Dic3)C22semidirect
31. (C2×D3)⋊(C3⋊Dic3)C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
36. #64146D3semidirect
45. C3×#937D3semidirect
75. C23×C3⋊D3C2×D3semidirect
77. C23×C3×D3C2×D3semidirect
104. #3757C2×D3semidirect
107. #3752C2×D3semidirect
113. C3×#217C2×D3semidirect
115. C3×(C2×D3)⋊C4C2×D3semidirect
146. #937C3⋊D3semidirect
215. C23×D3C2×C3⋊D3semidirect
247. #217C2×C3⋊D3semidirect
253. (C2×D3)⋊C4C2×C3⋊D3semidirect
312. C33H(3,𝔽4)semidirect
442. C32#5594semidirect
443. C32#5594semidirect
499. C3#10595256semidirect
500. C3#10596065semidirect

Maximal subgroups

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

subgroupindex
#117239772
C3×#641462
C32×#9372
#117236342
#117236292
#117236002
#117235912
#117235302
#117235282
#105960653
#105952563