Finite Group Databasebeta

← #13167990

Subgroups — #13167990

Order 1728.

1646 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3802C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic, complemented
3#11723499C212yesmaximal, complemented
4#11723437C211yesmaximal, characteristic, complemented
5C3×Dic3×D12C212yesmaximal, complemented
6#11722924C214yesmaximal, complemented
7D3×(C2×D3)⋊Dic3C214yesmaximal, complemented
8#10595215—32nomaximal
9#10593996—31nomaximal
10C22×C3×D32C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×D3×C3⋊Dic3C2214yescomplemented
14C2×C3×#625C2212yes—
15C2×C3×D3×Dic3C2214yescomplemented
16C3×#3802C418yescomplemented
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C3×C4⋊(C3⋊Dic3)C2212yescomplemented
19C32×C4×Dic3C2211yescharacteristic, complemented
20#88814C2212yes—
21#88813C2212yescomplemented
22#88807C2212yescomplemented
23#88803C2212yes—
24(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
25C3×(C2×D3)⋊Dic3C2214yescomplemented
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic
28D4×C3⋊Dic3—32no—
29#64081—31no—
30C3×D4×Dic3—32no—
31C3×C4×D12—32no—
32#67⋊Dic3—34no—
33Dic3×#67—34no—
34D3×#217—34no—
35C2×(C2×D3)⋊Dic3—34no—
36D3×(C2×D3)⋊C4—34no—
37#63898—32no—
38#63898—32no—
39#63898—62no—
40Dic3×D12D312yescomplemented
41#63879—34no—
42#63855—32no—
43#63855—61no—
44#63849—31no—
45#63841—32no—
46C22×C3×C3⋊D3C2311yescharacteristic
47C22×C32×D3C2314yes—
48C2×C3×D32—44no—
49C2×C3×D32C2×C418yescomplemented
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2312yes—
52C2×C3×C3⋊Dic3C2314yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×C4×C3⋊D3C2×C414yescomplemented
55C2×C32×Dic3C2312yes—
56C32×D12C2×C418yescomplemented
57D3×C3⋊Dic3—44no—
58C3×#625C2×C418yescomplemented
59C3×D3×Dic3—44no—
60#5798—91no—
61#5545—92no—
62C23×C3×D3—34no—
63C22×D32C2×D312yes—
64C2×C32×D4—32no—
65C22×C3⋊Dic3—34no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×#67—34no—
68C22×C3×Dic3—34no—
69C3×D3×D4—316no—
70C2×C3×D12—22no—
71C2×C3×D12C2×D312yes—
72C2×C3×D12C2×D312yes—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—61no—
76C2×#625C2×D312yes—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79C2×D3×Dic3—64no—
80C2×D3×Dic3C2×D314yescomplemented
81#3802Dic318yescomplemented
82C32×C42—31no—
83#3757—34no—
84#3752—32no—
85C4⋊(C3⋊Dic3)C2×D312yescomplemented
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—32no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—62no—
91C4×C3⋊Dic3—121no—
92C3×#217—34no—
93C3×(C2×D3)⋊C4—34no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—62no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D312yescomplemented
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—62no—
102#3722—32no—
103(C2×D3)⋊Dic3—34no—
104(C2×D3)⋊Dic3—34no—
105(C2×D3)⋊Dic3—34no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—64no—
110(C2×D3)⋊Dic3C2×D314yescomplemented
111Dic32—34no—
112Dic32—32no—
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3C22×C412yes—
116C2×C32×D3—24no—
117C2×C32×D3—24no—
118C2×C32×D3C22×C418yes—
119C3×D32—48no—
120C33×C4C22×C412yes—
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3—24no—
123C3×C3⋊Dic3C22×C412yes—
124C32×Dic3—22no—
125C22×C3×D4—91no—
126C2×D3×D4—32no—
127C3×C4×D4—92no—
128C2×#217—94no—
129D4×Dic3—32no—
130D4×Dic3—92no—
131D4×Dic3—92no—
132D4×Dic3—94no—
133D4×Dic3—182no—
134C4×#67—94no—
135C2×(C2×D3)⋊C4—94no—
136#926—91no—
137#926—91no—
138#893—92no—
139#892—92no—
140#67⋊C4—94no—
141D3×C22⋊C4—94no—
142C4×D12—32no—
143#872—92no—
144#872—181no—
145C23×C32—34no—
146C22×C3⋊D3C22×D311yescharacteristic
147C22×C3×D3—24no—
148C22×C3×D3—32no—
149C22×C3×D3—316no—
150C22×C3×D3—61no—
151C22×C3×D3—124no—
152C22×C3×D3—124no—
153C22×C3×D3C22×D314yes—
154C22×C3×D3C22×D314yes—
155C2×D32—44no—
156C2×D32C2×Dic318yescomplemented
157C32×D4—38no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—32no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—62no—
170C2×C3⋊Dic3—122no—
171C2×C3⋊Dic3—124no—
172C2×C3⋊Dic3C22×D314yes—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C4×C3⋊D3C2×Dic314yescomplemented
175C3×#67—316no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—64no—
182C2×C3×Dic3—124no—
183C2×C3×Dic3C22×D312yes—
184C2×C3×Dic3C22×D312yes—
185C3×D12—28no—
186C3×D12C4×D318yescomplemented
187C3×D12C2×Dic318yescomplemented
188C3×C4×D3—38no—
189C3×C4×D3—64no—
190C3×C4×D3—124no—
191#625C2×Dic318yescomplemented
192D3×Dic3—44no—
193D3×Dic3—124no—
194D3×Dic3—124no—
195D3×Dic3—128no—
196#518—271noSylow
197C2×C33C23×C411yescharacteristic
198C2×C33H(5,𝔽2)12yes—
199C3×C3⋊D3—22no—
200C32×D3—28no—
201C24×C3—92no—
202C23×D3—34no—
203C2×C3×D4—32no—
204C2×C3×D4—32no—
205C2×C3×D4—62no—
206C2×C3×D4—98no—
207C2×C3×D4—92no—
208C22×C3×C4—94no—
209C22×C3×C4—91no—
210C2×#67—34no—
211C22×Dic3—34no—
212C22×Dic3—94no—
213C22×Dic3—94no—
214C22×Dic3—184no—
215D3×D4—316no—
216C2×D12D3212yes—
217C2×C4×D3—32no—
218C2×C4×D3—34no—
219C2×C4×D3—61no—
220C2×C4×D3—94no—
221C3×C4⋊C4—92no—
222C3×C22⋊C4—94no—
223C3×C42—32no—
224C3×C42—61no—
225#217—34no—
226#217—92no—
227#217—94no—
228#217—94no—
229#217—92no—
230#217—184no—
231(C2×D3)⋊C4—34no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—94no—
236(C2×D3)⋊C4—182no—
237C4⋊Dic3—32no—
238C4⋊Dic3—32no—
239C4⋊Dic3—62no—
240C4⋊Dic3—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—182no—
244C4×Dic3—94no—
245C4×Dic3—92no—
246C4×Dic3—92no—
247C4×Dic3—91no—
248C4×Dic3—181no—
249C4×Dic3—182no—
250C4×Dic3—361no—
251C4×Dic3D3211yescharacteristic, complemented
252C22×C32—21no—
253C22×C32—22no—
254C22×C32—38no—
255C22×C32—41no—
256C22×C32—64no—
257C22×C32—64no—
258C22×C32C23×D312yes—
259C22×C32C23×D311yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3C22×Dic312yes—
262C2×C3×D3—24no—
263C2×C3×D3—24no—
264C2×C3×D3—24no—
265C2×C3×D3—24no—
266C2×C3×D3—28no—
267C2×C3×D3—34no—
268C2×C3×D3—44no—
269C2×C3×D3—44no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—1216no—
275C2×C3×D3C2×C4×D318yes—
276C2×C3×D3C22×Dic318yes—
277D32—48no—
278C32×C4—22no—
279C32×C4—24no—
280C32×C4—42no—
281C32×C4—62no—
282C32×C4C2×C4×D314yes—
283C32×C4C22×Dic312yes—
284C3⋊Dic3—24no—
285C3⋊Dic3—62no—
286C3⋊Dic3—64no—
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3⋊Dic3—122no—
290C3⋊Dic3—122no—
291C3⋊Dic3—124no—
292C3⋊Dic3C22×Dic312yes—
293C3×Dic3—22no—
294C3×Dic3—22no—
295C3×Dic3—34no—
296C3×Dic3—42no—
297C3×Dic3—62no—
298C3×Dic3—64no—
299C3×Dic3—64no—
300C3×Dic3—124no—
301C22×D4—91no—
302C4×D4—92no—
303C4×D4—274no—
304C4×D4—272no—
305#117—271no—
306#117—271no—
307C2×C22⋊C4—274no—
308C33#51811yescharacteristic, complemented, Sylow
309C23×C3—34no—
310C23×C3—34no—
311C23×C3—64no—
312C23×C3—98no—
313C23×C3—91no—
314C23×C3—364no—
315C22×D3—32no—
316C22×D3—316no—
317C22×D3—61no—
318C22×D3—124no—
319C22×D3—124no—
320C22×D3C2×D3214yes—
321C3×D4—38no—
322C3×D4—38no—
323C3×D4—68no—
324C3×D4—98no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—41no—
330C2×C3×C4—62no—
331C2×C3×C4—91no—
332C2×C3×C4—94no—
333C2×C3×C4—94no—
334C2×C3×C4—364no—
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3212yes—
337#67—316no—
338C2×Dic3—32no—
339C2×Dic3—34no—
340C2×Dic3—34no—
341C2×Dic3—61no—
342C2×Dic3—64no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—124no—
347C2×Dic3—182no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3—362no—
351C2×Dic3—364no—
352C2×Dic3—364no—
353C2×Dic3—368no—
354C2×Dic3C2×D3212yes—
355D12D3×Dic318yescomplemented
356C4×D3—38no—
357C4×D3—64no—
358C4×D3—124no—
359C4×D3—364no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—41no—
366C2×C32—42no—
367C2×C32—68no—
368C2×C32C22×C4×D312yes—
369C2×C32#100412yes—
370C2×C32#100412yes—
371C2×C32#100912yes—
372C2×C32C23×Dic311yescharacteristic
373C3⋊D3—22no—
374C3×D3—28no—
375C3×D3—28no—
376C3×D3—48no—
377C3×D3—64no—
378C3×D3—122no—
379C24—92no—
380C2×D4—32no—
381C2×D4—98no—
382C2×D4—92no—
383C22×C4—94no—
384C22×C4—91no—
385C22×C4—274no—
386C4⋊C4—92no—
387C4⋊C4—272no—
388C22⋊C4—94no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—274no—
392C42—31no—
393C42—272no—
394C42—271no—
395C22×C3—22no—
396C22×C3—21no—
397C22×C3—38no—
398C22×C3—38no—
399C22×C3—41no—
400C22×C3—68no—
401C22×C3—64no—
402C22×C3—64no—
403C22×C3—64no—
404C22×C3—64no—
405C22×C3—92no—
406C22×C3—124no—
407C22×C3—124no—
408C22×C3—182no—
409C22×C3—368no—
410C22×C3C22×D3211yescharacteristic
411C22×C3C22×D3212yes—
412C2×D3—24no—
413C2×D3—24no—
414C2×D3—34no—
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—122no—
419C2×D3—1216no—
420C2×D3C2×D3×Dic318yes—
421C3×C4—22no—
422C3×C4—24no—
423C3×C4—42no—
424C3×C4—62no—
425C3×C4—62no—
426C3×C4—92no—
427C3×C4—122no—
428C3×C4—184no—
429C3×C4C4×D3212yes—
430C3×C4C2×D3×Dic314yes—
431Dic3—22no—
432Dic3—34no—
433Dic3—62no—
434Dic3—64no—
435Dic3—64no—
436Dic3—124no—
437Dic3—184no—
438Dic3—182no—
439Dic3—182no—
440Dic3—182no—
441Dic3—362no—
442Dic3—362no—
443Dic3—364no—
444C32—22no—
445C32—21no—
446C32—41no—
447C32#554512yescomplemented
448C32#579811yescharacteristic, complemented
449C23—34no—
450C23—98no—
451C23—91no—
452C23—364no—
453D4—38no—
454D4—98no—
455C2×C4—32no—
456C2×C4—91no—
457C2×C4—94no—
458C2×C4—94no—
459C2×C4—272no—
460C2×C4—364no—
461C2×C4—1084no—
462C2×C4D3311yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—42no—
469C2×C3—41no—
470C2×C3—68no—
471C2×C3—68no—
472C2×C3—128no—
473C2×C3—182no—
474C2×C3C2×C4×D3211yescharacteristic
475C2×C3#6430712yes—
476C2×C3#6431312yes—
477C2×C3#6431412yes—
478C2×C3C22×D3×Dic312yes—
479D3—28no—
480D3—64no—
481D3—122no—
482C22—38no—
483C22—64no—
484C22—64no—
485C22—92no—
486C22—182no—
487C22—368no—
488C22C2×D3311yescharacteristic
489C4—62no—
490C4—92no—
491C4—184no—
492C4—542no—
493C4D32×Dic312yes—
494C3—21no—
495C3—22no—
496C3—41no—
497C3#1059399611yescharacteristic, complemented
498C3#1059521512yescomplemented
499C2—68no—
500C2—182no—
501C2#1172438112yes—
502C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C3×#3802C2
hypercenterC2×C4D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723667C2semidirect
3. #11723499C2semidirect
4. #11723437C2semidirect
5. C3×Dic3×D12C2semidirect
6. #11722924C2semidirect
7. D3×(C2×D3)⋊Dic3C2semidirect
13. C2×D3×C3⋊Dic3C22semidirect
15. C2×C3×D3×Dic3C22semidirect
16. C3×#3802C4semidirect
17. C4×C3⋊(C3⋊Dic3)C22semidirect
18. C3×C4⋊(C3⋊Dic3)C22semidirect
19. C32×C4×Dic3C22semidirect
21. #88813C22semidirect
22. #88807C22semidirect
24. (C2×D3)⋊(C3⋊Dic3)C22semidirect
25. C3×(C2×D3)⋊Dic3C22semidirect
40. Dic3×D12D3semidirect
49. C2×C3×D32C2×C4semidirect
54. C3×C4×C3⋊D3C2×C4semidirect
56. C32×D12C2×C4semidirect
58. C3×#625C2×C4semidirect
80. C2×D3×Dic3C2×D3semidirect
81. #3802Dic3semidirect
85. C4⋊(C3⋊Dic3)C2×D3semidirect
98. C3×C4×Dic3C2×D3semidirect
110. (C2×D3)⋊Dic3C2×D3semidirect
156. C2×D32C2×Dic3semidirect
174. C4×C3⋊D3C2×Dic3semidirect
186. C3×D12C4×D3semidirect
187. C3×D12C2×Dic3semidirect
191. #625C2×Dic3semidirect
251. C4×Dic3D32semidirect
308. C33#518semidirect
355. D12D3×Dic3semidirect
447. C32#5545semidirect
448. C32#5798semidirect
497. C3#10593996semidirect
498. C3#10595215semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
#117236672
#117234992
#117234372
C3×Dic3×D122
#117229242
D3×(C2×D3)⋊Dic32
#105952153
#105939963