Finite Group Databasebeta

← C2×#11720959 #13150117

Subgroups — C2×#11720959 #13150117

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊Dic9C212yesmaximal
5C2×Dic3⋊Dic9C212yesmaximal, complemented
6#11720959C218yesmaximal, direct product factor
7C2×#63855—31nomaximal
8C2×#63488—31nomaximal
9C23×D3×C9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3⋊Dic9C2212yes—
12C22×C9×Dic3C2211yescharacteristic
13C22×C3×Dic9C2212yes—
14C2×C4×D3×C9C418yescomplemented
15C2×C4×D3×C9C2214yes—
16C4×C3⋊Dic9C2214yescomplemented
17C3×C4×Dic9C2214yescomplemented
18(C2×D3)⋊Dic9C2218yes—
19Dic3⋊Dic9C2218yescomplemented
20C22×C3×C4×D3D311yescharacteristic
21C2×C4×C3⋊Dic3—31no—
22C2×C3×C4×Dic3—31no—
23C2×Dic3⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—32no—
25#63855—38no—
26C23×C4×C9—31no—
27C2×#3676—32no—
28#63488—38no—
29C2×C4⋊Dic9—32no—
30C2×C4×Dic9—31no—
31C2×C4×Dic9—61no—
32C2×C4×Dic9D311yescharacteristic, complemented
33C23×C3×C9C2311yescharacteristic
34C22×D3×C9—24no—
35C22×D3×C9C2×C414yes—
36C22×D3×C9C2312yes—
37C2×C3×C4×C9C2×C414yes—
38C2×C3×C4×C9C2312yes—
39C2×C3⋊Dic9—24no—
40C2×C3⋊Dic9C2314yes—
41C2×C9×Dic3C2×C414yes—
42C2×C9×Dic3C2312yes—
43C2×C3×Dic9—24no—
44C2×C3×Dic9C2314yes—
45C4×D3×C9C2×C4116yescomplemented
46C2×#926—91no—
47C2×#872—91no—
48C23×C3×D3C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3C2×D311yescharacteristic
53C2×C3×C4×D3Dic318yes—
54C2×C3×C4×D3C2×D314yes—
55C24×C9—31no—
56C4×C3⋊Dic3—34no—
57C3×C4×Dic3—34no—
58Dic3⋊Dic3—38no—
59(C2×D3)⋊Dic3—38no—
60C22×C4×C9—21no—
61C22×C4×C9—31no—
62C22×C4×C9—38no—
63C22×C4×C9—34no—
64C22×C4×C9C2×D311yescharacteristic
65C22×Dic9—32no—
66C22×Dic9—62no—
67C22×Dic9C2×D312yes—
68#3676—38no—
69C4⋊Dic9—38no—
70C4×Dic9—34no—
71C4×Dic9—64no—
72C4×Dic9C2×D314yescomplemented
73C22×C3×C9C22×C412yes—
74C22×C3×C9#4114yes—
75C22×C3×C9C2411yescharacteristic
76C2×D3×C9—28no—
77C2×D3×C9—24no—
78C2×D3×C9C22×C414yes—
79C3×C4×C9C22×C414yes—
80C3⋊Dic9—24no—
81C9×Dic3C22×C414yes—
82C3×Dic9—24no—
83C23×C3×C4—31no—
84C22×C4×D3D911yescharacteristic
85C2×C3×C42—91no—
86C2×#217—92no—
87C2×(C2×D3)⋊C4—92no—
88#926—98no—
89C2×C4⋊Dic3—92no—
90C2×Dic3⋊C4—92no—
91C2×C4×Dic3—31no—
92C2×C4×Dic3—91no—
93C2×C4×Dic3—91no—
94C2×C4×Dic3—181no—
95#872—98no—
96C23×C32C22×D311yescharacteristic
97C22×C3×D3—24no—
98C22×C3×D3C2×Dic314yes—
99C22×C3×D3C22×D312yes—
100C2×C32×C4C2×Dic314yes—
101C2×C32×C4C22×D312yes—
102C2×C3⋊Dic3—34no—
103C2×C3⋊Dic3—64no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3C2×Dic314yes—
107C2×C3×Dic3C22×D312yes—
108C3×C4×D3C2×Dic3116yes—
109C23×C9—21no—
110C23×C9—34no—
111C23×C9—32no—
112C23×C9—64no—
113C23×C9C22×D311yescharacteristic
114C2×C4×C9—24no—
115C2×C4×C9—22no—
116C2×C4×C9—34no—
117C2×C4×C9—32no—
118C2×C4×C9—316no—
119C2×C4×C9C4×D314yes—
120C2×C4×C9C22×D312yes—
121C2×Dic9—24no—
122C2×Dic9—34no—
123C2×Dic9—64no—
124C2×Dic9—68no—
125C2×Dic9—64no—
126C2×Dic9C22×D314yes—
127C2×#117—271noSylow
128C2×C3×C9#11714yescomplemented
129C2×C3×C9C23×C411yescharacteristic
130C2×C3×C9C2×#4112yes—
131D3×C9—24no—
132C24×C3—31no—
133C23×D3C2×D911yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—31no—
136C22×C3×C4—38no—
137C22×C3×C4—34no—
138C22×C3×C4—92no—
139C22×C3×C4C2×D911yescharacteristic
140C22×C3×C4D3211yescharacteristic
141C22×Dic3—32no—
142C22×Dic3—92no—
143C22×Dic3—92no—
144C22×Dic3—182no—
145C22×Dic3C2×D911yescharacteristic
146C2×C4×D3Dic918yescomplemented
147C2×C4×D3C2×D914yes—
148C3×C42—94no—
149#217—98no—
150(C2×D3)⋊C4—98no—
151C4⋊Dic3—98no—
152Dic3⋊C4—98no—
153C4×Dic3—34no—
154C4×Dic3—94no—
155C4×Dic3—94no—
156C4×Dic3—184no—
157C22×C32#23514yes—
158C22×C32C22×Dic312yes—
159C22×C32C23×D311yescharacteristic
160C2×C3×D3—28no—
161C2×C3×D3—24no—
162C2×C3×D3C22×Dic314yes—
163C32×C4C22×Dic314yes—
164C3⋊Dic3—64no—
165C3×Dic3—64no—
166C3×Dic3C22×Dic314yes—
167C22×C9—21no—
168C22×C9—24no—
169C22×C9—22no—
170C22×C9—34no—
171C22×C9—68no—
172C22×C9—64no—
173C22×C9C2×C4×D312yes—
174C22×C9#23514yes—
175C22×C9C23×D311yescharacteristic
176C4×C9—24no—
177C4×C9—34no—
178C4×C9C2×C4×D314yes—
179Dic9—24no—
180Dic9—68no—
181Dic9—64no—
182C23×C4—31no—
183#117—278no—
184C2×C4⋊C4—272no—
185C2×C22⋊C4—272no—
186C2×C42—91no—
187C2×C42—271no—
188C3×C9C2×#11711yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3—32no—
191C23×C3—34no—
192C23×C3—64no—
193C23×C3C22×D911yescharacteristic
194C23×C3C2×D3211yescharacteristic
195C22×D3—24no—
196C22×D3C2×Dic914yes—
197C22×D3C22×D912yes—
198C2×C3×C4—24no—
199C2×C3×C4—22no—
200C2×C3×C4—32no—
201C2×C3×C4—34no—
202C2×C3×C4—316no—
203C2×C3×C4—94no—
204C2×C3×C4—184no—
205C2×C3×C4C2×Dic914yes—
206C2×C3×C4C22×D912yes—
207C2×C3×C4D3×Dic314yes—
208C2×C3×C4C2×D3212yes—
209C2×Dic3—34no—
210C2×Dic3—64no—
211C2×Dic3—94no—
212C2×Dic3—94no—
213C2×Dic3—184no—
214C2×Dic3—188no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3C2×Dic914yes—
218C2×Dic3C22×D912yes—
219C4×D3C2×Dic9116yescomplemented
220C2×C32#92614yes—
221C2×C32C2×#23512yes—
222C2×C32C23×Dic311yescharacteristic
223C3×D3—24no—
224C2×C9—22no—
225C2×C9—21no—
226C2×C9—24no—
227C2×C9—64no—
228C2×C9#87214yescomplemented
229C2×C9C22×C4×D311yescharacteristic
230C2×C9C2×#23512yes—
231C24—31no—
232C22×C4—31no—
233C22×C4—38no—
234C22×C4—34no—
235C22×C4—92no—
236C22×C4—272no—
237C22×C4D3×D911yescharacteristic
238C4⋊C4—278no—
239C22⋊C4—278no—
240C42—94no—
241C42—274no—
242C22×C3—24no—
243C22×C3—21no—
244C22×C3—22no—
245C22×C3—34no—
246C22×C3—64no—
247C22×C3—68no—
248C22×C3#369714yes—
249C22×C3C22×Dic912yes—
250C22×C3C23×D911yescharacteristic
251C22×C3#379814yes—
252C22×C3C2×D3×Dic312yes—
253C22×C3C22×D3211yescharacteristic
254C2×D3—28no—
255C2×D3—24no—
256C2×D3C22×Dic914yes—
257C3×C4—24no—
258C3×C4—34no—
259C3×C4—184no—
260C3×C4C22×Dic914yes—
261C3×C4C2×D3×Dic314yes—
262Dic3—64no—
263Dic3—188no—
264Dic3—184no—
265Dic3—184no—
266Dic3C22×Dic914yes—
267C32C2×#92611yescharacteristic
268C9—21no—
269C9C2×#87211yescharacteristic, complemented
270C23—32no—
271C23—34no—
272C23—64no—
273C23C2×D3×D911yescharacteristic
274C2×C4—32no—
275C2×C4—34no—
276C2×C4—316no—
277C2×C4—94no—
278C2×C4—184no—
279C2×C4—274no—
280C2×C4—544no—
281C2×C4D3×Dic914yes—
282C2×C4C2×D3×D912yes—
283C2×C3—21no—
284C2×C3—22no—
285C2×C3—24no—
286C2×C3—64no—
287C2×C3#6348814yescomplemented
288C2×C3C2×#369712yes—
289C2×C3C23×Dic911yescharacteristic
290C2×C3#6385514yes—
291C2×C3C2×#379812yes—
292C2×C3C22×D3×Dic311yescharacteristic
293D3—24no—
294C22—34no—
295C22—64no—
296C22—68no—
297C22#8864014yes—
298C22C2×D3×Dic912yes—
299C22C22×D3×D911yescharacteristic
300C4—34no—
301C4—184no—
302C4—544no—
303C4C2×D3×Dic914yes—
304C3—21no—
305C3C2×#6348811yescharacteristic, complemented
306C3C2×#6385511yescharacteristic
307C2—64no—
308C2#1172095914yesdirect product factor
309C2C2×#8864012yes—
310C2C22×D3×Dic911yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D3C2×D3
cosocleC22×C3×C9C24
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊Dic9C2semidirect
3. C2×C3×C4×Dic9C2semidirect
5. C2×Dic3⋊Dic9C2semidirect
6. #11720959C2direct (N × H)
14. C2×C4×D3×C9C4semidirect
16. C4×C3⋊Dic9C22semidirect
17. C3×C4×Dic9C22semidirect
19. Dic3⋊Dic9C22semidirect
32. C2×C4×Dic9D3semidirect
45. C4×D3×C9C2×C4semidirect
72. C4×Dic9C2×D3semidirect
128. C2×C3×C9#117semidirect
146. C2×C4×D3Dic9semidirect
188. C3×C9C2×#117semidirect
219. C4×D3C2×Dic9semidirect
228. C2×C9#872semidirect
269. C9C2×#872semidirect
287. C2×C3#63488semidirect
305. C3C2×#63488semidirect
308. C2#11720959direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×C4×C3⋊Dic92
C2×C3×C4×Dic92
C2×(C2×D3)⋊Dic92
C2×Dic3⋊Dic92
#117209592
C2×#638553
C2×#634883