Finite Group Databasebeta

← C3⋊D3×#816 #13160048

Subgroups — C3⋊D3×#816 #13160048

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⋊D3×Dic6C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
4#11722186C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8C3⋊D3×#103—31nomaximal
9D3×#816—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C2×Dic3×C3⋊D3—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×C8×C3⋊D3—22no—
22C32×C4⋊Dic3C2211yescharacteristic
23#88807—21no—
24C2×Q8×C3⋊D3—31no—
25C2×D3×Dic6—34no—
26C2×C8×C3⋊D3D311yescharacteristic
27C4⋊C4×C3⋊D3—31no—
28C2×C3×D3×C8—34no—
29D3×C4⋊Dic3—34no—
30C32×#103—31no—
31#63662—31no—
32#63643—31no—
33#63643—34no—
34#63643—64no—
35#63637—31no—
36C3×#816D314yescomplemented
37#63567—34no—
38#63566—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308C2×C412yescomplemented
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3C2×C412yes—
47C3×C4×C3⋊D3D412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yescomplemented
52#6293—24no—
53Dic3×C3⋊D3—24no—
54Dic3×C3⋊D3—42no—
55C33×C8—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C2×#816—91no—
58D3×#103—94no—
59C2×C32×Q8—31no—
60Q8×C3⋊D3—34no—
61Q8×C3⋊D3—62no—
62C2×C4×C3⋊D3—61no—
63C2×C4×C3⋊D3—61no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×#634—31no—
66C2×#634—34no—
67C2×#634—61no—
68C2×#634—64no—
69C2×C3×C4×D3—34no—
70C2×C3×Dic6C2×D314yes—
71C2×#627—64no—
72C2×D3×Dic3—64no—
73C2×D3×Dic3—64no—
74D3×Dic6—316no—
75D3×Dic6—68no—
76C2×C32×C8—24no—
77C2×C32×C8C2×D314yes—
78C2×C32×C8C2×D311yescharacteristic
79C32×C4⋊C4—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—34no—
82C4⋊(C3⋊Dic3)—64no—
83#3750—61no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C8×C3⋊D3—22no—
86C3×C4⋊Dic3C2×D314yes—
87C2×C3×C3⋊C8—34no—
88C3×D3×C8—68no—
89Dic3⋊Dic3—64no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3C22⋊C412yes—
92C2×C3×C3⋊D3SD1612yes—
93C2×C3×C3⋊D3Q1612yes—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—22no—
97C3⋊(C3⋊Dic3)—41no—
98C3×C3⋊Dic3C22⋊C412yes—
99C32×Dic3—22no—
100C32×Dic3—41no—
101C2×D3×Q8—94no—
102C22×Dic6—91no—
103C22×C3×C8—91no—
104C2×C4⋊Dic3—91no—
105C2×D3×C8—34no—
106D3×C4⋊C4—94no—
107C3×#103—34no—
108#835—94no—
109#816—91no—
110#816—94no—
111#816—91no—
112#816—91no—
113#816—184no—
114#816C3⋊D311yescharacteristic, direct product factor
115#810—94no—
116C22×C3⋊D3D1211yescharacteristic
117C22×C3×D3—34no—
118C32×Q8—32no—
119C32×Q8—61no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3—124no—
131C2×C3⋊Dic3D1211yescharacteristic
132C4×C3⋊D3—64no—
133C4×C3⋊D3—122no—
134C4×C3⋊D3C4×D312yes—
135C4×C3⋊D3#6712yes—
136#634—32no—
137#634—38no—
138#634—68no—
139#634—61no—
140#634—64no—
141#634—64no—
142#634—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—34no—
146C3×C4×D3—38no—
147C3×C4×D3—38no—
148C3×Dic6—24no—
149C3×Dic6C4×D318yescomplemented
150#627—616no—
151D3×Dic3—616no—
152D3×Dic3—128no—
153C32×C8—21no—
154C32×C8—24no—
155C32×C8—44no—
156C3⋊(C3⋊C8)—21no—
157C3×C3⋊C8—64no—
158C2×#103—271noSylow
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33C2×SD1611yescharacteristic
161C2×C33C2×Q1611yescharacteristic
162C3×C3⋊D3#10314yescomplemented
163C2×C3×Q8—34no—
164C22×C3×C4—91no—
165C22×Dic3—181no—
166C22×Dic3—181no—
167D3×Q8—916no—
168D3×Q8—188no—
169C2×C4×D3—34no—
170C2×C4×D3—184no—
171C2×C4×D3—184no—
172C2×Dic6—94no—
173C2×Dic6—91no—
174C2×Dic6—94no—
175C2×Dic6—184no—
176C2×Dic6—182no—
177C2×Dic6—181no—
178C2×Dic6—184no—
179C2×Dic6C2×C3⋊D311yescharacteristic
180C2×C3×C8—24no—
181C2×C3×C8—91no—
182C2×C3×C8—182no—
183C2×C3×C8D3214yes—
184C2×C3×C8C2×C3⋊D311yescharacteristic
185C3×C4⋊C4—34no—
186C4⋊Dic3—94no—
187C4⋊Dic3—91no—
188C4⋊Dic3—184no—
189C4⋊Dic3—181no—
190C4⋊Dic3C2×C3⋊D311yescharacteristic
191Dic3⋊C4—184no—
192C2×C3⋊C8—34no—
193D3×C8—68no—
194C22×C32—24no—
195C22×C32C2×D1211yescharacteristic
196C22×C32D3×D414yes—
197C2×C3⋊D3#20412yes—
198C2×C3⋊D3Dic1212yes—
199C2×C3⋊D3(C2×D3)⋊C412yes—
200C2×C3×D3—38no—
201C2×C3×D3—38no—
202C2×C3×D3—38no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—62no—
206C32×C4—121no—
207C32×C4C2×C4×D314yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4D3×D414yes—
210C32×C4C2×#6711yescharacteristic
211C3⋊Dic3—62no—
212C3⋊Dic3—68no—
213C3⋊Dic3—128no—
214C3⋊Dic3—121no—
215C3⋊Dic3—124no—
216C3⋊Dic3—244no—
217C3⋊Dic3(C2×D3)⋊C412yes—
218C3×Dic3—28no—
219C3×Dic3—38no—
220C3×Dic3—44no—
221C22×Q8—271no—
222C22×C8—91no—
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—38no—
232C3×Q8—64no—
233C2×C3×C4—24no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4—91no—
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—34no—
244C2×Dic3—184no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247C2×Dic3—184no—
248C2×Dic3—181no—
249C2×Dic3—364no—
250C2×Dic3—364no—
251C2×Dic3—362no—
252C4×D3—38no—
253C4×D3—38no—
254C4×D3—1816no—
255C4×D3—368no—
256Dic6—21no—
257Dic6—98no—
258Dic6—92no—
259Dic6—188no—
260Dic6—184no—
261Dic6—1816no—
262Dic6—184no—
263Dic6—181no—
264Dic6—364no—
265Dic6C4×C3⋊D312yescomplemented
266C3×C8—24no—
267C3×C8—21no—
268C3×C8—44no—
269C3×C8—181no—
270C3⋊C8—64no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32D3×C22⋊C414yes—
275C2×C32C2×#20411yescharacteristic
276C2×C32C2×Dic1211yescharacteristic
277C2×C32D3×SD1614yes—
278C2×C32D3×Q1614yes—
279C2×C32C2×(C2×D3)⋊C411yescharacteristic
280C3⋊D3#81614yesdirect product factor
281C3×D3—316no—
282C2×Q8—31no—
283C2×Q8—274no—
284C2×Q8—271no—
285C2×Q8—541no—
286C2×Q8—542no—
287C22×C4—91no—
288C22×C4—541no—
289C22×C4—541no—
290C2×C8—91no—
291C2×C8—182no—
292C2×C8D3×C3⋊D311yescharacteristic
293C4⋊C4—31no—
294C4⋊C4—271no—
295C4⋊C4—541no—
296C22×C3—24no—
297C22×C3—92no—
298C22×C3—92no—
299C22×C3—92no—
300C22×C3D3×D1214yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—38no—
303C2×D3—38no—
304C2×D3—38no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—68no—
308C3×C4—92no—
309C3×C4—124no—
310C3×C4C4×D3214yes—
311C3×C4D3×#6714yes—
312C3×C4C2×C4×C3⋊D311yescharacteristic
313C3×C4D4×C3⋊D311yescharacteristic
314Dic3—22no—
315Dic3—38no—
316Dic3—41no—
317Dic3—182no—
318Dic3—188no—
319Dic3—368no—
320Dic3—364no—
321Dic3—361no—
322Dic3—724no—
323C32—24no—
324C32D3×#10314yescomplemented
325C32C2×#81611yescharacteristic, complemented
326C23—91no—
327Q8—32no—
328Q8—61no—
329Q8—272no—
330Q8—544no—
331Q8—541no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—91no—
335C2×C4—92no—
336C2×C4—92no—
337C2×C4—544no—
338C2×C4—541no—
339C2×C4—541no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—21no—
343C8—181no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—94no—
348C2×C3D3×#20414yes—
349C2×C3D3×Dic1214yes—
350C2×C3D3×(C2×D3)⋊C414yes—
351C2×C3C22⋊C4×C3⋊D311yescharacteristic
352C2×C3SD16×C3⋊D311yescharacteristic
353C2×C3Q16×C3⋊D311yescharacteristic
354D3—316no—
355C22—92no—
356C22—92no—
357C22—92no—
358C22C3⋊D3×D1211yescharacteristic
359C4—62no—
360C4—92no—
361C4—121no—
362C4—542no—
363C4—1081no—
364C4C4×D3×C3⋊D311yescharacteristic
365C4C3⋊D3×#6711yescharacteristic
366C3—24no—
367C3D3×#81614yescomplemented
368C3C3⋊D3×#10311yescharacteristic, complemented
369C2—94no—
370C2C3⋊D3×#20411yescharacteristic
371C2C3⋊D3×Dic1211yescharacteristic
372C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11722186C2semidirect
5. C32×#816C2semidirect
6. #11722041C2semidirect
7. #11722038C2semidirect
17. C3⋊D3×Dic6C4semidirect
36. C3×#816D3semidirect
44. #6308C2×C4semidirect
51. C32×Dic6C2×C4semidirect
114. #816C3⋊D3direct (N × H)
149. C3×Dic6C4×D3semidirect
162. C3×C3⋊D3#103semidirect
228. C33C2×#103semidirect
265. Dic6C4×C3⋊D3semidirect
280. C3⋊D3#816direct (N × H)
324. C32D3×#103semidirect
325. C32C2×#816semidirect
367. C3D3×#816semidirect
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⋊D3×Dic62
C2×C3×C8×C3⋊D32
C3⋊D3×C4⋊Dic32
#117221862
C32×#8162
#117220412
#117220382
C3⋊D3×#1033
D3×#8163