Finite Group Databasebeta

← #13168311

Subgroups — #13168311

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×C4×#1113C211yesmaximal, characteristic
2C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723437C212yesmaximal, complemented
4C3×C4×#624C211yesmaximal, characteristic, complemented
5C3×#63874C212yesmaximal, complemented
6#11722922C214yesmaximal, complemented
7#11722910C212yesmaximal, complemented
8#11722905C212yesmaximal
9#10593992—31nomaximal
10D3×#872—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×#6294C2211yescharacteristic
15C2×#6294C2212yes—
16C4×#1113C418yescomplemented
17C2×Dic3×C3⋊D3C2212yes—
18C2×C3×#624C2212yes—
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
20C32×C4×Dic3C2212yescomplemented
21C32×C4×Dic3C2211yescharacteristic, complemented
22#88807C2214yescomplemented
23#88803C2214yes—
24Dic3×C3⋊Dic3C2212yescomplemented
25C3×#3724C2214yescomplemented
26C3×#3722C2214yes—
27C3×Dic32C2212yescomplemented
28C2×C4×D32—32no—
29C2×C4×D32—31no—
30#64081—32no—
31C42×C3⋊D3—31no—
32C3×#872—32no—
33C3×#872—32no—
34C3×C42×D3—32no—
35#63921—32no—
36D3×(C2×D3)⋊C4—34no—
37#63883—32no—
38C4×#624D311yescharacteristic, complemented
39#63878—34no—
40#63877—34no—
41D3×Dic3⋊C4—34no—
42C4×D3×Dic3—32no—
43C4×D3×Dic3—61no—
44#63874D312yescomplemented
45#63855—32no—
46#63855—32no—
47#63855—62no—
48#63836—34no—
49C22×C3×C3⋊D3C2312yes—
50C22×C3×C3⋊D3C2311yescharacteristic
51C2×#1113—24no—
52C2×#1113C2×C414yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C2×C3×C3⋊Dic3C2312yes—
57C3×C4×C3⋊D3C2×C414yes—
58C3×C4×C3⋊D3C2×C418yescomplemented
59C2×C32×Dic3C2314yes—
60C2×C32×Dic3C2312yes—
61#6294C2×C414yes—
62#6294C2×C418yescomplemented
63Dic3×C3⋊D3—24no—
64C3×#624—24no—
65D3×#117—92no—
66C2×#872—91no—
67C22×D32—32no—
68C22×D32—31no—
69C2×C4×C3⋊D3—32no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×C4×C3⋊D3C2×D312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—62no—
77C2×C3×C4×D3—61no—
78C2×#624—32no—
79C2×#624—31no—
80C2×#624C2×D312yes—
81C2×D3×Dic3—32no—
82C2×D3×Dic3—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3—62no—
86C4×D32—38no—
87C4×D32—316no—
88C32×C42—31no—
89C32×C42—32no—
90#3752—34no—
91#3750—34no—
92C4×C3⋊Dic3—31no—
93C4×C3⋊Dic3—32no—
94C4×C3⋊Dic3—32no—
95C4×C3⋊Dic3—62no—
96C4×C3⋊Dic3—61no—
97C4×C3⋊Dic3—121no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—34no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C3×C4×Dic3—22no—
103C3×C4×Dic3—21no—
104C3×C4×Dic3—34no—
105C3×C4×Dic3C2×D312yescomplemented
106C3×C4×Dic3C2×D312yescomplemented
107C3×C4×Dic3C2×D312yescomplemented
108#3724C2×D314yescomplemented
109Dic3⋊Dic3—34no—
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—64no—
112#3722C2×D314yes—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—34no—
115(C2×D3)⋊Dic3—64no—
116Dic32—34no—
117Dic32—62no—
118Dic32C2×D312yescomplemented
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—24no—
121C2×C3×C3⋊D3C22×C414yes—
122C2×C3×C3⋊D3C22×C412yes—
123C2×C3×C3⋊D3#4114yes—
124#1113—28no—
125C33×C4C22×C412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3C22×C414yes—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—22no—
130C32×Dic3—24no—
131C22×C4×D3—91no—
132C22×C4×D3—92no—
133C3×#117—92no—
134C2×C3×C42—91no—
135C2×(C2×D3)⋊C4—92no—
136#926—92no—
137C2×Dic3⋊C4—92no—
138C2×C4×Dic3—91no—
139#892—94no—
140D3×C4⋊C4—94no—
141D3×C22⋊C4—94no—
142#877—94no—
143#872—32no—
144#872—32no—
145#872—94no—
146#872—92no—
147#872—92no—
148#872—182no—
149C42×D3—32no—
150C42×D3—92no—
151C42×D3—181no—
152C22×C3⋊D3C22×D312yes—
153C22×C3⋊D3C22×D311yescharacteristic
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—61no—
158C22×C3×D3—62no—
159C2×D32—38no—
160C2×D32—34no—
161C2×D32—68no—
162C2×D32—64no—
163C2×C32×C4—22no—
164C2×C32×C4—21no—
165C2×C32×C4—34no—
166C2×C32×C4—32no—
167C2×C32×C4—41no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D312yes—
170C2×C3⋊Dic3—34no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—122no—
175C2×C3⋊Dic3C22×D311yescharacteristic
176C2×C3⋊Dic3C22×D312yes—
177C4×C3⋊D3—64no—
178C4×C3⋊D3C4×D314yes—
179C4×C3⋊D3C4×D318yescomplemented
180C2×C3×Dic3—22no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3C22×D314yes—
188C2×C3×Dic3C22×D314yes—
189C2×C3×Dic3C22×D314yes—
190C3×C4×D3—38no—
191C3×C4×D3—38no—
192C3×C4×D3—38no—
193C3×C4×D3—64no—
194C3×C4×D3—68no—
195C3×C4×D3—68no—
196#624—24no—
197#624—34no—
198#624—38no—
199D3×Dic3—38no—
200D3×Dic3—38no—
201D3×Dic3—38no—
202D3×Dic3—68no—
203D3×Dic3—68no—
204C2×#117—271noSylow
205C2×C33C23×C411yescharacteristic
206C2×C33C2×#4112yes—
207C3×C3⋊D3—24no—
208C3×C3⋊D3#11714yescomplemented
209C23×D3—92no—
210C23×D3—91no—
211C22×C3×C4—92no—
212C22×C3×C4—91no—
213C22×C3×C4—92no—
214C22×Dic3—92no—
215C22×Dic3—91no—
216C22×Dic3—92no—
217C2×C4×D3—32no—
218C2×C4×D3—32no—
219C2×C4×D3—32no—
220C2×C4×D3—34no—
221C2×C4×D3—61no—
222C2×C4×D3—62no—
223C2×C4×D3—916no—
224C2×C4×D3—98no—
225C2×C4×D3—92no—
226C2×C4×D3—92no—
227C2×C4×D3—92no—
228C2×C4×D3—94no—
229C2×C4×D3—182no—
230C3×C4⋊C4—94no—
231C3×C22⋊C4—94no—
232C3×C42—32no—
233C3×C42—32no—
234C3×C42—32no—
235C3×C42—61no—
236C3×C42—62no—
237C3×C42—92no—
238#217—94no—
239(C2×D3)⋊C4—34no—
240(C2×D3)⋊C4—34no—
241(C2×D3)⋊C4—94no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—184no—
244C4⋊Dic3—94no—
245Dic3⋊C4—34no—
246Dic3⋊C4—34no—
247Dic3⋊C4—94no—
248Dic3⋊C4—94no—
249Dic3⋊C4—184no—
250C4×Dic3—34no—
251C4×Dic3—94no—
252C4×Dic3—92no—
253C4×Dic3—91no—
254C4×Dic3—92no—
255C4×Dic3—182no—
256C4×Dic3—181no—
257C4×Dic3—182no—
258C4×Dic3—361no—
259C4×Dic3D3212yescomplemented
260C4×Dic3D3211yescharacteristic, complemented
261C22×C32—22no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32C23×D311yescharacteristic
265C22×C32C23×D312yes—
266C2×C3⋊D3—24no—
267C2×C3⋊D3C2×C4×D314yes—
268C2×C3⋊D3C2×C4×D312yes—
269C2×C3⋊D3#23514yes—
270C2×C3×D3—34no—
271C2×C3×D3—38no—
272C2×C3×D3—34no—
273C2×C3×D3—34no—
274C2×C3×D3—62no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—124no—
280D32—616no—
281D32—68no—
282C32×C4—22no—
283C32×C4—24no—
284C32×C4—42no—
285C32×C4—62no—
286C32×C4—64no—
287C32×C4C2×C4×D312yes—
288C32×C4C2×C4×D314yes—
289C3⋊Dic3—62no—
290C3⋊Dic3—64no—
291C3⋊Dic3—64no—
292C3⋊Dic3—124no—
293C3⋊Dic3—124no—
294C3⋊Dic3C2×C4×D312yes—
295C3⋊Dic3C2×C4×D314yes—
296C3×Dic3—24no—
297C3×Dic3—24no—
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—34no—
301C3×Dic3—34no—
302C3×Dic3—34no—
303C3×Dic3—44no—
304C3×Dic3—64no—
305C3×Dic3—62no—
306C23×C4—271no—
307#117—92no—
308#117—272no—
309#117—274no—
310C2×C4⋊C4—272no—
311C2×C22⋊C4—272no—
312C2×C42—91no—
313C2×C42—271no—
314C33C2×#11711yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C23×C3—92no—
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—32no—
320C22×D3—61no—
321C22×D3—62no—
322C22×D3—98no—
323C22×D3—94no—
324C22×D3—188no—
325C22×D3—184no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—41no—
332C2×C3×C4—62no—
333C2×C3×C4—64no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4—94no—
337C2×C3×C4—98no—
338C2×C3×C4—184no—
339C2×C3×C4C2×D3212yes—
340C2×C3×C4C2×D3211yescharacteristic
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—61no—
345C2×Dic3—62no—
346C2×Dic3—94no—
347C2×Dic3—98no—
348C2×Dic3—92no—
349C2×Dic3—94no—
350C2×Dic3—182no—
351C2×Dic3—184no—
352C2×Dic3—184no—
353C2×Dic3—362no—
354C2×Dic3C2×D3214yes—
355C2×Dic3C2×D3212yes—
356C4×D3—38no—
357C4×D3—38no—
358C4×D3—38no—
359C4×D3—64no—
360C4×D3—68no—
361C4×D3—68no—
362C4×D3—98no—
363C4×D3—98no—
364C4×D3—98no—
365C4×D3—188no—
366C4×D3—188no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—24no—
370C2×C32—22no—
371C2×C32—41no—
372C2×C32—42no—
373C2×C32C22×C4×D311yescharacteristic
374C2×C32C22×C4×D312yes—
375C2×C32C2×#23512yes—
376C2×C32D3×#4114yes—
377C3⋊D3—24no—
378C3⋊D3#87214yescomplemented
379C3×D3—38no—
380C3×D3—64no—
381C3×D3—64no—
382C3×D3—64no—
383C3×D3—124no—
384C24—271no—
385C22×C4—92no—
386C22×C4—91no—
387C22×C4—92no—
388C22×C4—272no—
389C22×C4—278no—
390C22×C4—271no—
391C22×C4—272no—
392C4⋊C4—94no—
393C4⋊C4—274no—
394C22⋊C4—94no—
395C22⋊C4—274no—
396C42—32no—
397C42—31no—
398C42—92no—
399C42—272no—
400C42—271no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—92no—
405C22×C3—94no—
406C22×C3—94no—
407C22×C3—184no—
408C22×C3C22×D3212yes—
409C22×C3C22×D3211yescharacteristic
410C2×D3—34no—
411C2×D3—38no—
412C2×D3—34no—
413C2×D3—34no—
414C2×D3—62no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—124no—
420C2×D3—1816no—
421C2×D3—188no—
422C3×C4—22no—
423C3×C4—24no—
424C3×C4—42no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—94no—
430C3×C4—92no—
431C3×C4—124no—
432C3×C4C4×D3214yes—
433C3×C4C4×D3212yes—
434Dic3—24no—
435Dic3—22no—
436Dic3—34no—
437Dic3—34no—
438Dic3—34no—
439Dic3—62no—
440Dic3—64no—
441Dic3—184no—
442Dic3—184no—
443Dic3—182no—
444Dic3—364no—
445Dic3—364no—
446C32—22no—
447C32—21no—
448C32—41no—
449C32C2×#87211yescharacteristic, complemented
450C32D3×#11712yescomplemented
451C23—91no—
452C23—92no—
453C23—274no—
454C23—544no—
455C2×C4—34no—
456C2×C4—32no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—94no—
460C2×C4—98no—
461C2×C4—184no—
462C2×C4—274no—
463C2×C4—278no—
464C2×C4—272no—
465C2×C4—544no—
466C2×C4D3311yescharacteristic
467C2×C3—22no—
468C2×C3—24no—
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—42no—
472C2×C3—41no—
473C2×C3—94no—
474C2×C3—184no—
475C2×C3C2×C4×D3212yes—
476C2×C3C2×C4×D3211yescharacteristic
477C2×C3D3×#23514yes—
478C2×C3#6430612yes—
479D3—38no—
480D3—64no—
481D3—64no—
482D3—64no—
483D3—124no—
484C22—94no—
485C22—92no—
486C22—94no—
487C22—184no—
488C22—548no—
489C22C2×D3311yescharacteristic
490C4—64no—
491C4—62no—
492C4—92no—
493C4—94no—
494C4—542no—
495C4#8894912yes—
496C3—21no—
497C3—22no—
498C3—41no—
499C3D3×#87212yescomplemented
500C3#1059399211yescharacteristic, complemented
501C2—94no—
502C2—184no—
503C2#1172438712yes—
504C2C2×#8894911yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×#1113C22
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×Dic3×C3⋊D3C2semidirect
3. #11723437C2semidirect
4. C3×C4×#624C2semidirect
5. C3×#63874C2semidirect
6. #11722922C2semidirect
7. #11722910C2semidirect
16. C4×#1113C4semidirect
19. C4×C3⋊(C3⋊Dic3)C22semidirect
20. C32×C4×Dic3C22semidirect
21. C32×C4×Dic3C22semidirect
22. #88807C22semidirect
24. Dic3×C3⋊Dic3C22semidirect
25. C3×#3724C22semidirect
27. C3×Dic32C22semidirect
38. C4×#624D3semidirect
44. #63874D3semidirect
58. C3×C4×C3⋊D3C2×C4semidirect
62. #6294C2×C4semidirect
105. C3×C4×Dic3C2×D3semidirect
106. C3×C4×Dic3C2×D3semidirect
107. C3×C4×Dic3C2×D3semidirect
108. #3724C2×D3semidirect
118. Dic32C2×D3semidirect
179. C4×C3⋊D3C4×D3semidirect
208. C3×C3⋊D3#117semidirect
259. C4×Dic3D32semidirect
260. C4×Dic3D32semidirect
314. C33C2×#117semidirect
378. C3⋊D3#872semidirect
449. C32C2×#872semidirect
450. C32D3×#117semidirect
499. C3D3×#872semidirect
500. C3#10593992semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C4×Dic3×C3⋊D32
#117234372
C3×C4×#6242
C3×#638742
#117229222
#117229102
#117229052
#105939923
D3×#8723