Finite Group Databasebeta

← #13161804

Subgroups — #13161804

Order 1728.

1002 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C3×#64135C212yesmaximal, complemented
3#11722083C214yesmaximal, complemented
4#10591232—31nomaximal
5#10591231—32nomaximal
6C22×C33×C4C2211yescharacteristic
7C2×C3×#638C2212yes—
8C3×#3828—22no—
9C3×#3828—22no—
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C412yescomplemented
12C2×C3×#634C411yescharacteristic, complemented
13C3×#3757C2214yes—
14C3×C4⋊(C3⋊Dic3)—22no—
15C3×C4×C3⋊Dic3—22no—
16C2×#3828D311yescharacteristic
17C2×C3×#235—32no—
18C2×C3×#235—61no—
19#64135D312yescomplemented
20C3×#926—32no—
21C3×#926—32no—
22C3×#926—62no—
23#63581—34no—
24#63580—38no—
25C23×C33C2311yescharacteristic
26C22×C3×C3⋊D3C2×C412yes—
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4D412yes—
30C2×C33×C4D412yes—
31C3×#638—24no—
32C2×C3×C3⋊Dic3—24no—
33C2×C3×C3⋊Dic3C2×C412yes—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3—22no—
37C3×#634—22no—
38#5240—91no—
39#5006—92no—
40C22×C32×C4—22no—
41C22×C32×C4—21no—
42C22×C32×C4—41no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C2×#638C2×D312yes—
46#3828—22no—
47#3828—22no—
48C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
49C2×C4×C3⋊D3Dic312yescomplemented
50C2×#634Dic311yescharacteristic, complemented
51C2×C3×#67—34no—
52C2×C3×#67—62no—
53C3×#235—64no—
54C3×#235—64no—
55C3×#235—62no—
56C3×#235—62no—
57C3×#235—62no—
58C3×#235—62no—
59C2×C3×D12—32no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—34no—
62C2×C3×C4×D3—62no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—61no—
65#3757C2×D314yes—
66C4⋊(C3⋊Dic3)—22no—
67C4×C3⋊Dic3—22no—
68C3×#217—34no—
69C3×#217—34no—
70C3×#217—64no—
71C3×C4⋊Dic3—64no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3—62no—
74C3×C4×Dic3—64no—
75C3×C4×Dic3—62no—
76C3×C4×Dic3—62no—
77C22×C33—41no—
78C22×C33C22×C411yescharacteristic
79C22×C33C2×D412yes—
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3—42no—
82C33×C4C22⋊C412yes—
83C33×C4C22⋊C411yescharacteristic
84C33×C4C22⋊C411yescharacteristic
85C3×C3⋊Dic3—22no—
86C3×C3⋊Dic3—44no—
87C2×C3×#41—91no—
88C2×#235—32no—
89C2×#235—61no—
90C3×#117—92no—
91#926—32no—
92#926—32no—
93#926—62no—
94#834—94no—
95#806—98no—
96C23×C32—22no—
97C23×C32—21no—
98C23×C32—41no—
99C23×C32C22×D312yes—
100C23×C32C22×D311yescharacteristic
101C22×C3⋊D3C2×Dic312yes—
102C22×C3×D3—34no—
103C22×C3×D3—62no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—42no—
117C2×C32×C4—42no—
118C2×C32×C4C4×D312yes—
119C2×C32×C4C4×D312yes—
120C2×C32×C4D1212yes—
121C2×C32×C4D1212yes—
122C2×C32×C4C2×Dic311yescharacteristic
123C2×C32×C4C2×Dic311yescharacteristic
124C2×C32×C4#6712yes—
125C2×C32×C4#6712yes—
126C2×C32×C4#6712yes—
127C2×C32×C4#6712yes—
128#638—24no—
129C2×C3⋊Dic3—24no—
130C2×C3⋊Dic3C2×Dic312yes—
131C4⋊(C3⋊D3)—22no—
132C4×C3⋊D3—22no—
133C4×C3⋊D3—22no—
134#634—22no—
135C3×#67—68no—
136C3×#67—64no—
137C3×#67—64no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—68no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C3×D12—64no—
144C3×D12—62no—
145C3×D12—62no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148C3×C4×D3—64no—
149C3×C4×D3—64no—
150C3×Dic6—64no—
151C3×Dic6—62no—
152C3×Dic6—62no—
153#410—271noSylow
154C2×C33—22no—
155C2×C33—21no—
156C2×C33C2×C22⋊C411yescharacteristic
157C3×C3⋊D3—42no—
158C3×#41—182no—
159C3×#41—182no—
160C2×C3×Q8—91no—
161C2×C3×D4—92no—
162C2×C3×D4—91no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—41no—
166C22×C3×C4—92no—
167C22×C3×C4D3212yes—
168C22×C3×C4D3211yescharacteristic
169C2×#67—34no—
170C2×#67—62no—
171#235—64no—
172#235—64no—
173#235—62no—
174#235—62no—
175#235—62no—
176#235—62no—
177C2×D12—32no—
178C2×D12—61no—
179C2×C4×D3—34no—
180C2×C4×D3—62no—
181C2×Dic6—32no—
182C2×Dic6—61no—
183C3×C4⋊C4—182no—
184C3×C22⋊C4—94no—
185C3×C42—182no—
186#217—34no—
187#217—34no—
188#217—64no—
189C4⋊Dic3—64no—
190C4⋊Dic3—62no—
191C4⋊Dic3—62no—
192C4×Dic3—64no—
193C4×Dic3—62no—
194C4×Dic3—62no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—42no—
204C22×C32—44no—
205C22×C32—42no—
206C22×C32—44no—
207C22×C32C2×C4×D312yes—
208C22×C32C2×D1212yes—
209C22×C32C22×Dic311yescharacteristic
210C22×C32C2×#6712yes—
211C22×C32C2×#6712yes—
212C2×C3⋊D3—22no—
213C2×C3⋊D3—42no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—124no—
217C2×C3×D3—124no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—42no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4(C2×D3)⋊C412yes—
229C32×C4(C2×D3)⋊C412yes—
230C32×C4(C2×D3)⋊C412yes—
231C32×C4(C2×D3)⋊C412yes—
232C32×C4#21712yes—
233C32×C4#21711yescharacteristic
234C32×C4#21711yescharacteristic
235C3⋊Dic3—22no—
236C3⋊Dic3—44no—
237C3×Dic3—64no—
238C3×Dic3—64no—
239C3×Dic3—128no—
240C3×Dic3—124no—
241C3×Dic3—124no—
242C2×#41—91no—
243#117—92no—
244#99—274no—
245C33#41011yescharacteristic, complemented, Sylow
246C23×C3—22no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—92no—
250C23×C3C2×D3212yes—
251C23×C3C2×D3211yescharacteristic
252C22×D3—34no—
253C22×D3—62no—
254C3×Q8—182no—
255C3×D4—184no—
256C3×D4—182no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—92no—
272C2×C3×C4—182no—
273C2×C3×C4—182no—
274C2×C3×C4—184no—
275C2×C3×C4D3×Dic312yes—
276C2×C3×C4D3×Dic312yes—
277C2×C3×C4#62411yescharacteristic
278C2×C3×C4#62411yescharacteristic
279C2×C3×C4#62512yes—
280C2×C3×C4#62512yes—
281C2×C3×C4#62612yes—
282C2×C3×C4#62612yes—
283C2×C3×C4#62612yes—
284C2×C3×C4#62612yes—
285#67—68no—
286#67—64no—
287#67—64no—
288C2×Dic3—34no—
289C2×Dic3—62no—
290C2×Dic3—68no—
291C2×Dic3—64no—
292C2×Dic3—64no—
293D12—64no—
294D12—62no—
295D12—62no—
296C4×D3—64no—
297C4×D3—64no—
298C4×D3—64no—
299C4×D3—64no—
300Dic6—64no—
301Dic6—62no—
302Dic6—62no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—24no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—41no—
313C2×C32—42no—
314C2×C32—42no—
315C2×C32—42no—
316C2×C32—44no—
317C2×C32—42no—
318C2×C32C2×(C2×D3)⋊C412yes—
319C2×C32C2×#21711yescharacteristic
320C3⋊D3—42no—
321C3×D3—124no—
322C3×D3—124no—
323#41—182no—
324#41—182no—
325C2×Q8—91no—
326C2×D4—92no—
327C2×D4—91no—
328C22×C4—92no—
329C22×C4#111311yescharacteristic
330C4⋊C4—182no—
331C22⋊C4—94no—
332C42—182no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—41no—
339C22×C3—42no—
340C22×C3—44no—
341C22×C3—42no—
342C22×C3—44no—
343C22×C3—41no—
344C22×C3—42no—
345C22×C3—182no—
346C22×C3—362no—
347C22×C3C2×D3×Dic312yes—
348C22×C3C2×#62411yescharacteristic
349C22×C3C2×#62512yes—
350C22×C3C2×#62612yes—
351C22×C3C2×#62612yes—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—124no—
355C2×D3—124no—
356C3×C4—22no—
357C3×C4—21no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—22no—
363C3×C4—41no—
364C3×C4—42no—
365C3×C4—41no—
366C3×C4—182no—
367C3×C4—364no—
368C3×C4(C2×D3)⋊Dic312yes—
369C3×C4(C2×D3)⋊Dic312yes—
370C3×C4(C2×D3)⋊Dic312yes—
371C3×C4(C2×D3)⋊Dic312yes—
372C3×C4#372211yescharacteristic
373C3×C4#372212yes—
374C3×C4#372211yescharacteristic
375Dic3—64no—
376Dic3—64no—
377Dic3—128no—
378Dic3—124no—
379Dic3—124no—
380C32—21no—
381C32—22no—
382C32—41no—
383C32#500612yescomplemented
384C32#524011yescharacteristic, complemented
385C23—92no—
386C23C2×#111311yescharacteristic
387Q8—182no—
388D4—184no—
389D4—182no—
390C2×C4—92no—
391C2×C4—182no—
392C2×C4—182no—
393C2×C4—184no—
394C2×C4#629411yescharacteristic
395C2×C4#629411yescharacteristic
396C2×C4#629512yes—
397C2×C4#629512yes—
398C2×C3—24no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3—21no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—44no—
411C2×C3—42no—
412C2×C3—41no—
413C2×C3—362no—
414C2×C3C2×(C2×D3)⋊Dic312yes—
415C2×C3C2×#372211yescharacteristic
416D3—124no—
417D3—124no—
418C22—41no—
419C22—182no—
420C22—362no—
421C22C2×#629411yescharacteristic
422C22C2×#629512yes—
423C4—182no—
424C4—364no—
425C4#8881411yescharacteristic
426C4#8881412yes—
427C4#8881411yescharacteristic
428C3—21no—
429C3—22no—
430C3—41no—
431C3#1059123112yescomplemented
432C3#1059123211yescharacteristic, complemented
433C2—22no—
434C2—21no—
435C2—362no—
436C2C2×#8881411yescharacteristic
subgroupG/H
centerC4#88814
commutatorC22×C33C22×C4
FrattiniC23C2×#1113
FittingC22×C33×C4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC23×C33C23
hypercenterC22×C4#1113
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. #11722083C2semidirect
10. C2×C3×C4⋊(C3⋊D3)C4semidirect
11. C2×C3×C4×C3⋊D3C4semidirect
12. C2×C3×#634C4semidirect
19. #64135D3semidirect
48. C2×C4⋊(C3⋊D3)Dic3semidirect
49. C2×C4×C3⋊D3Dic3semidirect
50. C2×#634Dic3semidirect
245. C33#410semidirect
383. C32#5006semidirect
384. C32#5240semidirect
431. C3#10591231semidirect
432. C3#10591232semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C3×#641352
#117220832
#105912323
#105912313