Finite Group Databasebeta

← C3⋊D3×#835 #13161124

Subgroups — C3⋊D3×#835 #13161124

Order 1728.

1140 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722205C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8C3⋊D3×#103—31nomaximal
9D3×#835—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3C414yescomplemented
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C2×Dic3×C3⋊D3—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23C3⋊D3×C3⋊C8—22no—
24C2×Q8×C3⋊D3D311yescharacteristic
25C2×C3×D3×Q8—34no—
26C2×C8×C3⋊D3—31no—
27C4⋊C4×C3⋊D3—31no—
28D3×C4⋊Dic3—34no—
29C2×D3×C3⋊C8—34no—
30C32×#103—31no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63643—31no—
35#63637—31no—
36C3×#835D314yescomplemented
37#63566—34no—
38#63565—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yescomplemented
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—24no—
48C3×C4×C3⋊D3C2×C412yes—
49C3×C4×C3⋊D3D412yes—
50C3×#634—21no—
51C3×#634—24no—
52C3×#634C2×C412yescomplemented
53C2×C32×Dic3—21no—
54Dic3×C3⋊D3—42no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57C2×#835—91no—
58D3×#103—94no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62Q8×C3⋊D3—22no—
63Q8×C3⋊D3Dic314yescomplemented
64C2×C4×C3⋊D3—21no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—21no—
68C2×#634C2×D311yescharacteristic
69C3×D3×Q8—316no—
70C3×D3×Q8—68no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—64no—
73C2×C3×Dic6—34no—
74C2×C3×Dic6—64no—
75C2×D3×Dic3—64no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—64no—
81#3750—61no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85C8×C3⋊D3—62no—
86C3×C4⋊Dic3C2×D314yes—
87C2×C3×C3⋊C8C2×D314yes—
88Dic3⋊Dic3—64no—
89D3×C3⋊C8—68no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3C22⋊C412yes—
92C2×C3×C3⋊D3SD1612yes—
93C2×C3×C3⋊D3Q1612yes—
94C33×C4—22no—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3⋊(C3⋊Dic3)—41no—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3C22⋊C412yes—
100C32×Dic3—41no—
101C22×C3×Q8—91no—
102C2×D3×Q8—34no—
103C2×C4⋊Dic3—91no—
104C22×C3⋊C8—91no—
105C2×D3×C8—94no—
106D3×C4⋊C4—94no—
107C3×#103—34no—
108#835—94no—
109#835—91no—
110#835—91no—
111#835—91no—
112#835—184no—
113#835C3⋊D311yescharacteristic, direct product factor
114#816—94no—
115#810—94no—
116C22×C3⋊D3#6711yescharacteristic
117C22×C3×D3—34no—
118C32×Q8—21no—
119C32×Q8—28no—
120C32×Q8—24no—
121C32×Q8—44no—
122C32×Q8C4×D318yescomplemented
123C32×Q8C2×Dic312yescomplemented
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C2×C3⋊Dic3#6711yescharacteristic
136C4×C3⋊D3—24no—
137C4×C3⋊D3—122no—
138C4×C3⋊D3C2×Dic312yes—
139C4×C3⋊D3#6712yes—
140#634—21no—
141#634—24no—
142#634C2×Dic312yescomplemented
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C3×C4×D3—38no—
147C3×C4×D3—38no—
148C3×C4×D3—616no—
149C3×Dic6—38no—
150C3×Dic6—616no—
151C3×Dic6—64no—
152D3×Dic3—128no—
153C32×C8—61no—
154C3⋊(C3⋊C8)—61no—
155C3⋊(C3⋊C8)—64no—
156C3⋊(C3⋊C8)—124no—
157C3×C3⋊C8—24no—
158C2×#103—271noSylow
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33C2×SD1611yescharacteristic
161C2×C33C2×Q1611yescharacteristic
162C3×C3⋊D3#10314yescomplemented
163C2×C3×Q8—24no—
164C2×C3×Q8—94no—
165C2×C3×Q8—91no—
166C2×C3×Q8—181no—
167C2×C3×Q8—182no—
168C2×C3×Q8D3214yes—
169C2×C3×Q8C2×C3⋊D311yescharacteristic
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172C22×Dic3—181no—
173D3×Q8—316no—
174D3×Q8—68no—
175C2×C4×D3—34no—
176C2×C4×D3—64no—
177C2×C4×D3—184no—
178C2×Dic6—34no—
179C2×Dic6—64no—
180C2×C3×C8—34no—
181C3×C4⋊C4—34no—
182C4⋊Dic3—94no—
183C4⋊Dic3—91no—
184C4⋊Dic3—184no—
185C4⋊Dic3—181no—
186C4⋊Dic3C2×C3⋊D311yescharacteristic
187Dic3⋊C4—184no—
188C2×C3⋊C8—94no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—184no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193D3×C8—188no—
194C22×C32—24no—
195C22×C32D3×D414yes—
196C22×C32C2×#6711yescharacteristic
197C2×C3⋊D3#21512yes—
198C2×C3⋊D3#21612yes—
199C2×C3⋊D3#21712yes—
200C2×C3×D3—38no—
201C2×C3×D3—38no—
202C2×C3×D3—38no—
203C32×C4—22no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—28no—
207C32×C4—48no—
208C32×C4—121no—
209C32×C4C2×C4×D314yes—
210C32×C4D3×D414yes—
211C32×C4C22×Dic311yescharacteristic
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—22no—
214C3⋊Dic3—121no—
215C3⋊Dic3—124no—
216C3⋊Dic3—244no—
217C3⋊Dic3#21712yes—
218C3×Dic3—38no—
219C3×Dic3—44no—
220C3×Dic3—68no—
221C22×Q8—91no—
222C22×C8—271no—
223C2×C4⋊C4—271no—
224#103—31no—
225#103—271no—
226#103—271no—
227#103—271no—
228C33C2×#10311yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C22×D3—34no—
231C3×Q8—24no—
232C3×Q8—21no—
233C3×Q8—28no—
234C3×Q8—44no—
235C3×Q8—92no—
236C3×Q8—184no—
237C3×Q8—181no—
238C3×Q8D3×Dic318yescomplemented
239C3×Q8C4×C3⋊D312yescomplemented
240C2×C3×C4—24no—
241C2×C3×C4—21no—
242C2×C3×C4—24no—
243C2×C3×C4—44no—
244C2×C3×C4—64no—
245C2×C3×C4—91no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4—184no—
249C2×C3×C4—181no—
250C2×C3×C4C2×D3214yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252C2×Dic3—21no—
253C2×Dic3—34no—
254C2×Dic3—64no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—364no—
258C2×Dic3—362no—
259C4×D3—38no—
260C4×D3—38no—
261C4×D3—616no—
262C4×D3—368no—
263Dic6—38no—
264Dic6—616no—
265Dic6—64no—
266C3×C8—64no—
267C3⋊C8—21no—
268C3⋊C8—184no—
269C3⋊C8—181no—
270C3⋊C8—364no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32D3×C22⋊C414yes—
275C2×C32D3×SD1614yes—
276C2×C32D3×Q1614yes—
277C2×C32C2×#21511yescharacteristic
278C2×C32C2×#21611yescharacteristic
279C2×C32C2×#21711yescharacteristic
280C3⋊D3#83514yesdirect product factor
281C3×D3—316no—
282C2×Q8—94no—
283C2×Q8—91no—
284C2×Q8—181no—
285C2×Q8—182no—
286C2×Q8D3×C3⋊D311yescharacteristic
287C22×C4—91no—
288C22×C4—181no—
289C22×C4—541no—
290C2×C8—31no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—271no—
295C4⋊C4—541no—
296C22×C3—24no—
297C22×C3—92no—
298C22×C3—92no—
299C22×C3—92no—
300C22×C3D3×#6714yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—38no—
303C2×D3—38no—
304C2×D3—38no—
305C3×C4—22no—
306C3×C4—28no—
307C3×C4—24no—
308C3×C4—24no—
309C3×C4—48no—
310C3×C4—92no—
311C3×C4—124no—
312C3×C4—182no—
313C3×C4C2×D3×Dic314yes—
314C3×C4D3×#6714yes—
315C3×C4C2×C4×C3⋊D311yescharacteristic
316C3×C4D4×C3⋊D311yescharacteristic
317Dic3—38no—
318Dic3—41no—
319Dic3—68no—
320Dic3—361no—
321Dic3—364no—
322Dic3—724no—
323C32—24no—
324C32D3×#10314yescomplemented
325C32C2×#83511yescharacteristic, complemented
326C23—91no—
327Q8—21no—
328Q8—92no—
329Q8—184no—
330Q8—181no—
331Q8Dic3×C3⋊D312yescomplemented
332C2×C4—21no—
333C2×C4—61no—
334C2×C4—91no—
335C2×C4—92no—
336C2×C4—92no—
337C2×C4—184no—
338C2×C4—181no—
339C2×C4—541no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—61no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—94no—
348C2×C3D3×#21514yes—
349C2×C3D3×#21614yes—
350C2×C3D3×#21714yes—
351C2×C3C22⋊C4×C3⋊D311yescharacteristic
352C2×C3SD16×C3⋊D311yescharacteristic
353C2×C3Q16×C3⋊D311yescharacteristic
354D3—316no—
355C22—92no—
356C22—92no—
357C22—92no—
358C22C3⋊D3×#6711yescharacteristic
359C4—22no—
360C4—92no—
361C4—121no—
362C4—182no—
363C4—1081no—
364C4C2×Dic3×C3⋊D311yescharacteristic
365C4C3⋊D3×#6711yescharacteristic
366C3—24no—
367C3D3×#83514yescomplemented
368C3C3⋊D3×#10311yescharacteristic, complemented
369C2—94no—
370C2C3⋊D3×#21511yescharacteristic
371C2C3⋊D3×#21611yescharacteristic
372C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11722205C2semidirect
5. C32×#835C2semidirect
6. #11722040C2semidirect
7. #11722039C2semidirect
12. C3×Q8×C3⋊D3C4semidirect
36. C3×#835D3semidirect
41. C33×Q8C2×C4semidirect
52. C3×#634C2×C4semidirect
63. Q8×C3⋊D3Dic3semidirect
113. #835C3⋊D3direct (N × H)
122. C32×Q8C4×D3semidirect
123. C32×Q8C2×Dic3semidirect
142. #634C2×Dic3semidirect
162. C3×C3⋊D3#103semidirect
228. C33C2×#103semidirect
238. C3×Q8D3×Dic3semidirect
239. C3×Q8C4×C3⋊D3semidirect
280. C3⋊D3#835direct (N × H)
324. C32D3×#103semidirect
325. C32C2×#835semidirect
331. Q8Dic3×C3⋊D3semidirect
367. C3D3×#835semidirect
368. C3C3⋊D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C3⋊D3×C4⋊Dic32
C2×C3⋊D3×C3⋊C82
#117222052
C32×#8352
#117220402
#117220392
C3⋊D3×#1033
D3×#8353