Finite Group Databasebeta

← #13167992

Subgroups — #13167992

Order 1728.

1610 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D32C211yesmaximal, characteristic
2C4×#6294C211yesmaximal, characteristic, complemented
3C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
4#11723423C212yesmaximal, complemented
5C3×#63855C212yesmaximal, complemented
6#11722914C214yesmaximal, complemented
7#11722908C212yesmaximal, complemented
8#11722902C212yesmaximal
9D3×#926—32nomaximal
10#10593992—31nomaximal
11C22×C3×D32C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×#6294C2212yes—
15C2×Dic3×C3⋊D3C2212yes—
16C2×C3×#624C2211yescharacteristic
17C2×C3×D3×Dic3C2212yes—
18C3×C4×D32C418yescomplemented
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
20C3×C4×C3⋊Dic3C2212yescomplemented
21C32×C4×Dic3C2211yescharacteristic, complemented
22#88813C2212yescomplemented
23Dic3⋊(C3⋊Dic3)C2214yescomplemented
24(C2×D3)⋊(C3⋊Dic3)C2214yes—
25Dic3×C3⋊Dic3C2212yescomplemented
26C3×Dic3⋊Dic3C2214yescomplemented
27C3×(C2×D3)⋊Dic3C2214yes—
28C22×C3×C4×D3—32no—
29C2×C4×D32D311yescharacteristic
30#64135—32no—
31C42×C3⋊D3—31no—
32C3×#926—32no—
33C3×#872—32no—
34D3×#217—34no—
35#63955—34no—
36#63921—32no—
37D3×C4⋊Dic3—34no—
38#63883—32no—
39C4×#624—31no—
40#63878—34no—
41C4×D3×Dic3—32no—
42C4×D3×Dic3—32no—
43C4×D3×Dic3—61no—
44#63855—32no—
45#63855—32no—
46#63855—62no—
47#63855D312yescomplemented
48#63841—34no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C32×D3C2312yes—
51C2×C3×D32—24no—
52C2×C3×D32C2×C414yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3C2×C414yes—
58C2×C32×Dic3C2312yes—
59C2×C32×Dic3C2312yes—
60C32×C4×D3C2×C418yescomplemented
61#6294—24no—
62Dic3×C3⋊D3—24no—
63C3×#624C2×C414yes—
64C3×D3×Dic3C2×C418yescomplemented
65C2×#926—91no—
66D3×#117—92no—
67C23×C3×D3—32no—
68C22×D32C2×D311yescharacteristic
69C22×C32×C4—32no—
70C2×C4×C3⋊D3—32no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C22×C3×Dic3—32no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—316no—
78C2×C3×C4×D3—61no—
79C2×C3×C4×D3C2×D312yes—
80C2×C3×C4×D3C2×D312yes—
81C2×#624—32no—
82C2×#624C2×D311yescharacteristic
83C2×D3×Dic3—34no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3—62no—
86C2×D3×Dic3C2×D312yes—
87C4×D32Dic318yescomplemented
88C32×C42—31no—
89#3757—34no—
90C4⋊(C3⋊Dic3)—34no—
91C4×C3⋊Dic3—31no—
92C4×C3⋊Dic3—32no—
93C4×C3⋊Dic3—32no—
94C4×C3⋊Dic3—62no—
95C4×C3⋊Dic3—61no—
96C4×C3⋊Dic3—121no—
97C4×C3⋊Dic3C2×D312yescomplemented
98C3×#217—34no—
99C3×(C2×D3)⋊C4—34no—
100C3×C4⋊Dic3—34no—
101C3×Dic3⋊C4—34no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3—32no—
104C3×C4×Dic3—32no—
105C3×C4×Dic3—62no—
106C3×C4×Dic3C2×D312yescomplemented
107Dic3⋊Dic3—34no—
108Dic3⋊Dic3—34no—
109Dic3⋊Dic3—64no—
110Dic3⋊Dic3C2×D314yescomplemented
111(C2×D3)⋊Dic3—34no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—64no—
114(C2×D3)⋊Dic3C2×D314yes—
115Dic32—34no—
116Dic32—32no—
117Dic32—34no—
118Dic32—62no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3C22×C412yes—
121C2×C3×C3⋊D3#4114yes—
122C2×C32×D3—24no—
123C2×C32×D3C22×C414yes—
124C3×D32—28no—
125C33×C4C22×C412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3—24no—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—22no—
130C32×Dic3C22×C414yes—
131C23×C3×C4—91no—
132C22×C4×D3—32no—
133C3×#117—92no—
134C2×#217—92no—
135#926—32no—
136#926—92no—
137#926—92no—
138#926—94no—
139#926—182no—
140C2×C4⋊Dic3—92no—
141C2×C4×Dic3—91no—
142C2×C4×Dic3—91no—
143#892—94no—
144D3×C4⋊C4—94no—
145D3×C22⋊C4—94no—
146#877—94no—
147#872—32no—
148#872—92no—
149C42×D3—92no—
150C42×D3—92no—
151C42×D3—181no—
152C23×C32—32no—
153C22×C3⋊D3C22×D311yescharacteristic
154C22×C3×D3—22no—
155C22×C3×D3—32no—
156C22×C3×D3—38no—
157C22×C3×D3—61no—
158C22×C3×D3—68no—
159C22×C3×D3C22×D312yes—
160C22×C3×D3C22×D312yes—
161C2×D32—24no—
162C2×D32C2×Dic314yes—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—32no—
166C2×C32×C4—38no—
167C2×C32×C4—32no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—34no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—62no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—122no—
176C2×C3⋊Dic3C22×D314yes—
177C2×C3⋊Dic3C22×D311yescharacteristic
178C4×C3⋊D3—64no—
179C4×C3⋊D3C2×Dic314yes—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3C22×D314yes—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D312yes—
191C3×C4×D3—28no—
192C3×C4×D3—38no—
193C3×C4×D3—38no—
194C3×C4×D3—38no—
195C3×C4×D3—64no—
196C3×C4×D3C4×D318yescomplemented
197C3×C4×D3C2×Dic318yescomplemented
198#624—64no—
199#624C2×Dic314yes—
200D3×Dic3—68no—
201D3×Dic3—68no—
202D3×Dic3—68no—
203D3×Dic3C2×Dic318yescomplemented
204C2×#117—271noSylow
205C2×C33C23×C411yescharacteristic
206C2×C33C2×#4112yes—
207C3×C3⋊D3#11714yescomplemented
208C32×D3—24no—
209C24×C3—91no—
210C23×D3—32no—
211C22×C3×C4—32no—
212C22×C3×C4—32no—
213C22×C3×C4—62no—
214C22×C3×C4—91no—
215C22×C3×C4—98no—
216C22×C3×C4—92no—
217C22×C3×C4—91no—
218C22×Dic3—32no—
219C22×Dic3—92no—
220C22×Dic3—92no—
221C2×C4×D3—32no—
222C2×C4×D3—32no—
223C2×C4×D3—32no—
224C2×C4×D3—316no—
225C2×C4×D3—61no—
226C2×C4×D3—94no—
227C2×C4×D3—94no—
228C2×C4×D3—182no—
229C2×C4×D3D3212yes—
230C3×C4⋊C4—94no—
231C3×C22⋊C4—94no—
232C3×C42—32no—
233C3×C42—61no—
234C3×C42—92no—
235#217—34no—
236#217—94no—
237#217—94no—
238#217—184no—
239(C2×D3)⋊C4—34no—
240(C2×D3)⋊C4—94no—
241C4⋊Dic3—34no—
242C4⋊Dic3—94no—
243C4⋊Dic3—94no—
244C4⋊Dic3—184no—
245Dic3⋊C4—34no—
246Dic3⋊C4—94no—
247C4×Dic3—32no—
248C4×Dic3—32no—
249C4×Dic3—62no—
250C4×Dic3—94no—
251C4×Dic3—92no—
252C4×Dic3—94no—
253C4×Dic3—92no—
254C4×Dic3—91no—
255C4×Dic3—92no—
256C4×Dic3—182no—
257C4×Dic3—181no—
258C4×Dic3—182no—
259C4×Dic3—361no—
260C4×Dic3D3211yescharacteristic, complemented
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—34no—
264C22×C32—41no—
265C22×C32—64no—
266C22×C32C23×D312yes—
267C22×C32C23×D311yescharacteristic
268C2×C3⋊D3#23514yes—
269C2×C3⋊D3C22×Dic312yes—
270C2×C3×D3—24no—
271C2×C3×D3—24no—
272C2×C3×D3—24no—
273C2×C3×D3—34no—
274C2×C3×D3—38no—
275C2×C3×D3—44no—
276C2×C3×D3—62no—
277C2×C3×D3—64no—
278C2×C3×D3—616no—
279C2×C3×D3C2×C4×D314yes—
280C2×C3×D3C22×Dic314yes—
281D32—28no—
282C32×C4—22no—
283C32×C4—24no—
284C32×C4—34no—
285C32×C4—42no—
286C32×C4—62no—
287C32×C4C2×C4×D314yes—
288C32×C4C22×Dic312yes—
289C3⋊Dic3—24no—
290C3⋊Dic3—62no—
291C3⋊Dic3—64no—
292C3⋊Dic3—64no—
293C3⋊Dic3—124no—
294C3⋊Dic3—124no—
295C3⋊Dic3C22×Dic312yes—
296C3×Dic3—24no—
297C3×Dic3—24no—
298C3×Dic3—24no—
299C3×Dic3—34no—
300C3×Dic3—62no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303C3×Dic3—124no—
304C3×Dic3C2×C4×D314yes—
305C3×Dic3C22×Dic314yes—
306C23×C4—91no—
307#117—92no—
308#117—272no—
309#117—274no—
310C2×C4⋊C4—272no—
311C2×C22⋊C4—272no—
312C2×C42—271no—
313C2×C42—271no—
314C33C2×#11711yescharacteristic, complemented, Sylow
315C23×C3—32no—
316C23×C3—32no—
317C23×C3—62no—
318C23×C3—94no—
319C23×C3—91no—
320C23×C3—184no—
321C22×D3—32no—
322C22×D3—38no—
323C22×D3—61no—
324C22×D3—68no—
325C22×D3C2×D3212yes—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—38no—
331C2×C3×C4—38no—
332C2×C3×C4—34no—
333C2×C3×C4—41no—
334C2×C3×C4—68no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—94no—
338C2×C3×C4—98no—
339C2×C3×C4—91no—
340C2×C3×C4—94no—
341C2×C3×C4—94no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3212yes—
344C2×Dic3—32no—
345C2×Dic3—38no—
346C2×Dic3—34no—
347C2×Dic3—34no—
348C2×Dic3—61no—
349C2×Dic3—64no—
350C2×Dic3—94no—
351C2×Dic3—92no—
352C2×Dic3—182no—
353C2×Dic3—184no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356C2×Dic3—362no—
357C2×Dic3C2×D3212yes—
358C2×Dic3C2×D3212yes—
359C4×D3—38no—
360C4×D3—38no—
361C4×D3—38no—
362C4×D3—64no—
363C4×D3—188no—
364C4×D3—188no—
365C4×D3—188no—
366C4×D3D3×Dic318yescomplemented
367C2×C32—21no—
368C2×C32—24no—
369C2×C32—22no—
370C2×C32—22no—
371C2×C32—41no—
372C2×C32—42no—
373C2×C32—64no—
374C2×C32C22×C4×D312yes—
375C2×C32C2×#23512yes—
376C2×C32D3×#4114yes—
377C2×C32C23×Dic311yescharacteristic
378C3⋊D3#92614yescomplemented
379C3×D3—24no—
380C3×D3—24no—
381C3×D3—38no—
382C3×D3—44no—
383C3×D3—64no—
384C24—91no—
385C22×C4—32no—
386C22×C4—91no—
387C22×C4—98no—
388C22×C4—92no—
389C22×C4—91no—
390C22×C4—272no—
391C22×C4—272no—
392C4⋊C4—94no—
393C4⋊C4—274no—
394C22⋊C4—94no—
395C22⋊C4—274no—
396C42—31no—
397C42—92no—
398C42—272no—
399C42—272no—
400C42—271no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—34no—
404C22×C3—34no—
405C22×C3—41no—
406C22×C3—64no—
407C22×C3—64no—
408C22×C3—64no—
409C22×C3—92no—
410C22×C3—94no—
411C22×C3—124no—
412C22×C3—188no—
413C22×C3C22×D3211yescharacteristic
414C22×C3C22×D3212yes—
415C2×D3—24no—
416C2×D3—38no—
417C2×D3—34no—
418C2×D3—62no—
419C2×D3—64no—
420C2×D3—616no—
421C2×D3C2×D3×Dic314yes—
422C3×C4—22no—
423C3×C4—24no—
424C3×C4—34no—
425C3×C4—34no—
426C3×C4—42no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—64no—
430C3×C4—92no—
431C3×C4—184no—
432C3×C4C4×D3212yes—
433C3×C4C2×D3×Dic314yes—
434Dic3—22no—
435Dic3—34no—
436Dic3—62no—
437Dic3—64no—
438Dic3—64no—
439Dic3—124no—
440Dic3—184no—
441Dic3—184no—
442Dic3—182no—
443Dic3—364no—
444Dic3—364no—
445Dic3C2×D3×Dic314yes—
446C32—22no—
447C32—21no—
448C32—41no—
449C32D3×#11712yescomplemented
450C32C2×#92611yescharacteristic, complemented
451C23—32no—
452C23—94no—
453C23—91no—
454C23—184no—
455C2×C4—32no—
456C2×C4—38no—
457C2×C4—32no—
458C2×C4—94no—
459C2×C4—98no—
460C2×C4—91no—
461C2×C4—94no—
462C2×C4—94no—
463C2×C4—272no—
464C2×C4—544no—
465C2×C4—544no—
466C2×C4D3311yescharacteristic
467C2×C3—24no—
468C2×C3—22no—
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—42no—
472C2×C3—41no—
473C2×C3—64no—
474C2×C3—64no—
475C2×C3—94no—
476C2×C3—124no—
477C2×C3C2×C4×D3211yescharacteristic
478C2×C3D3×#23514yes—
479C2×C3#6430612yes—
480C2×C3C22×D3×Dic312yes—
481D3—24no—
482D3—38no—
483D3—64no—
484C22—34no—
485C22—64no—
486C22—94no—
487C22—92no—
488C22—188no—
489C22C2×D3311yescharacteristic
490C4—34no—
491C4—62no—
492C4—92no—
493C4—184no—
494C4—542no—
495C4D32×Dic312yes—
496C3—21no—
497C3—22no—
498C3—41no—
499C3#1059399211yescharacteristic, complemented
500C3D3×#92612yescomplemented
501C2—64no—
502C2—94no—
503C2#1172438712yes—
504C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×C3×D32C22
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6294C2semidirect
3. C4×Dic3×C3⋊D3C2semidirect
4. #11723423C2semidirect
5. C3×#63855C2semidirect
6. #11722914C2semidirect
7. #11722908C2semidirect
18. C3×C4×D32C4semidirect
19. C4×C3⋊(C3⋊Dic3)C22semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C32×C4×Dic3C22semidirect
22. #88813C22semidirect
23. Dic3⋊(C3⋊Dic3)C22semidirect
25. Dic3×C3⋊Dic3C22semidirect
26. C3×Dic3⋊Dic3C22semidirect
47. #63855D3semidirect
60. C32×C4×D3C2×C4semidirect
64. C3×D3×Dic3C2×C4semidirect
87. C4×D32Dic3semidirect
97. C4×C3⋊Dic3C2×D3semidirect
106. C3×C4×Dic3C2×D3semidirect
110. Dic3⋊Dic3C2×D3semidirect
196. C3×C4×D3C4×D3semidirect
197. C3×C4×D3C2×Dic3semidirect
203. D3×Dic3C2×Dic3semidirect
207. C3×C3⋊D3#117semidirect
260. C4×Dic3D32semidirect
314. C33C2×#117semidirect
366. C4×D3D3×Dic3semidirect
378. C3⋊D3#926semidirect
449. C32D3×#117semidirect
450. C32C2×#926semidirect
499. C3#10593992semidirect
500. C3D3×#926semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
C4×#62942
C4×Dic3×C3⋊D32
#117234232
C3×#638552
#117229142
#117229082
#117229022
D3×#9263
#105939923