Finite Group Databasebeta

← C2×#11722071 #13161800

Subgroups — C2×#11722071 #13161800

Order 1728.

898 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C212yesmaximal
3#11722071C214yesmaximal, direct product factor
4C2×#63555—31nomaximal
5C2×#63554—32nomaximal
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C412yes—
8C3×C4×C3⋊Dic3C412yes—
9C3×C4×C3⋊Dic3C2212yes—
10C2×C3×C3⋊(C3⋊C8)C418yescomplemented
11C2×C3×C3⋊(C3⋊C8)C2214yes—
12C2×C4×C3⋊Dic3D311yescharacteristic
13C22×C3⋊(C3⋊C8)D312yes—
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C22×C3×C3⋊C8—32no—
17C22×C3×C3⋊C8—32no—
18C22×C3×C3⋊C8—62no—
19#63555—34no—
20#63554—38no—
21C23×C33C2×C411yescharacteristic
22C2×C33×C4C2×C411yescharacteristic
23C2×C33×C4C2×C414yes—
24C2×C33×C4C2311yescharacteristic
25C2×C3×C3⋊Dic3—24no—
26C2×C3×C3⋊Dic3C2×C414yes—
27C3×C3⋊(C3⋊C8)C2×C418yescomplemented
28C2×#815—92no—
29C2×#803—91no—
30C22×C32×C4—22no—
31C22×C32×C4—21no—
32C22×C32×C4—41no—
33C22×C32×C4C2×D311yescharacteristic
34C22×C32×C4C2×D312yes—
35C22×C3⋊Dic3Dic312yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—62no—
38C4×C3⋊Dic3Dic312yes—
39C4×C3⋊Dic3C2×D312yes—
40C2×C3⋊(C3⋊C8)Dic318yescomplemented
41C2×C3⋊(C3⋊C8)C2×D314yes—
42C3×C4×Dic3—34no—
43C3×C4×Dic3—34no—
44C3×C4×Dic3—62no—
45C3×C4×Dic3—62no—
46C2×C3×C3⋊C8—38no—
47C2×C3×C3⋊C8—38no—
48C2×C3×C3⋊C8—34no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—68no—
51C2×C3×C3⋊C8—64no—
52C22×C33C4212yes—
53C22×C33M1614yes—
54C22×C33C22×C411yescharacteristic
55C33×C4C4212yes—
56C33×C4C22×C412yes—
57C3×C3⋊Dic3—24no—
58C22×C3×C8—92no—
59C2×C3×C42—91no—
60C2×C4×Dic3—32no—
61C2×C4×Dic3—61no—
62C22×C3⋊C8—32no—
63C22×C3⋊C8—32no—
64C22×C3⋊C8—62no—
65#815—98no—
66#803—94no—
67C23×C32—22no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32C4×D312yes—
71C23×C32C2×Dic311yescharacteristic
72C2×C32×C4—22no—
73C2×C32×C4—24no—
74C2×C32×C4—24no—
75C2×C32×C4—24no—
76C2×C32×C4—21no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4—44no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4C4×D312yes—
83C2×C32×C4C4×D314yes—
84C2×C32×C4C4×D314yes—
85C2×C32×C4C2×Dic311yescharacteristic
86C2×C32×C4C2×Dic314yes—
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C22×D312yes—
89C2×C3⋊Dic3—24no—
90C2×C3⋊Dic3C2×Dic314yes—
91C2×C3×Dic3—38no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—68no—
94C2×C3×Dic3—68no—
95C3⋊(C3⋊C8)C2×Dic318yescomplemented
96C3×C3⋊C8—38no—
97C3×C3⋊C8—38no—
98C3×C3⋊C8—68no—
99C2×#97—271noSylow
100C2×C33#9714yescomplemented
101C2×C33C2×C4211yescharacteristic
102C2×C33C2×M1612yes—
103C22×C3×C4—22no—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
106C22×C3×C4—92no—
107C22×C3×C4D3211yescharacteristic
108C22×C3×C4D3212yes—
109C22×Dic3—34no—
110C22×Dic3—62no—
111C2×C3×C8—98no—
112C2×C3×C8—94no—
113C3×C42—92no—
114C3×C42—92no—
115C4×Dic3—34no—
116C4×Dic3—34no—
117C4×Dic3—62no—
118C4×Dic3—62no—
119C2×C3⋊C8—38no—
120C2×C3⋊C8—38no—
121C2×C3⋊C8—34no—
122C2×C3⋊C8—34no—
123C2×C3⋊C8—68no—
124C2×C3⋊C8—64no—
125C22×C32—28no—
126C22×C32—24no—
127C22×C32—24no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—21no—
131C22×C32—42no—
132C22×C32—44no—
133C22×C32—41no—
134C22×C32ΓL(1,𝔽25)18yes—
135C22×C32#20814yes—
136C22×C32C4×Dic312yes—
137C22×C32C4×Dic314yes—
138C22×C32C2×C4×D312yes—
139C22×C32C22×Dic311yescharacteristic
140C32×C4—24no—
141C32×C4—22no—
142C32×C4—22no—
143C32×C4—22no—
144C32×C4—22no—
145C32×C4—42no—
146C32×C4—42no—
147C32×C4C4×Dic312yes—
148C32×C4C4×Dic314yes—
149C32×C4C2×C4×D312yes—
150C32×C4C2×C4×D312yes—
151C32×C4C22×Dic312yes—
152C3⋊Dic3—24no—
153C3×Dic3—68no—
154C3×Dic3—68no—
155C22×C8—92no—
156C2×C42—91no—
157#97—274no—
158C33C2×#9711yescharacteristic, complemented, Sylow
159C23×C3—22no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3D3×Dic312yes—
163C23×C3#62411yescharacteristic
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—24no—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—44no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—94no—
175C2×C3×C4—184no—
176C2×C3×C4D3×Dic312yes—
177C2×C3×C4D3×Dic314yes—
178C2×C3×C4D3×Dic314yes—
179C2×C3×C4#62411yescharacteristic
180C2×C3×C4#62414yes—
181C2×C3×C4C2×D3211yescharacteristic
182C2×C3×C4C2×D3212yes—
183C2×Dic3—38no—
184C2×Dic3—64no—
185C2×Dic3—68no—
186C2×Dic3—68no—
187C3×C8—98no—
188C3⋊C8—38no—
189C3⋊C8—38no—
190C3⋊C8—68no—
191C2×C32—28no—
192C2×C32—24no—
193C2×C32—24no—
194C2×C32—22no—
195C2×C32—22no—
196C2×C32—21no—
197C2×C32—44no—
198C2×C32—41no—
199C2×C32—42no—
200C2×C32#80314yescomplemented
201C2×C32#81518yescomplemented
202C2×C32C2×ΓL(1,𝔽25)14yes—
203C2×C32C2×#20812yes—
204C2×C32C2×C4×Dic311yescharacteristic
205C2×C32C2×C4×Dic312yes—
206C22×C4—92no—
207C22×C4#111311yescharacteristic
208C2×C8—98no—
209C2×C8—94no—
210C42—92no—
211C42—92no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—28no—
216C22×C3—22no—
217C22×C3—24no—
218C22×C3—41no—
219C22×C3—44no—
220C22×C3—42no—
221C22×C3#371118yes—
222C22×C3#371214yes—
223C22×C3Dic3214yes—
224C22×C3Dic3212yes—
225C22×C3C2×D3×Dic312yes—
226C22×C3C2×#62411yescharacteristic
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—22no—
230C3×C4—24no—
231C3×C4—22no—
232C3×C4—42no—
233C3×C4—42no—
234C3×C4—184no—
235C3×C4Dic3214yes—
236C3×C4Dic3212yes—
237C3×C4C2×D3×Dic312yes—
238C3×C4C2×D3×Dic312yes—
239C3×C4C2×#62412yes—
240Dic3—68no—
241Dic3—68no—
242C32—22no—
243C32—21no—
244C32—41no—
245C32C2×#80311yescharacteristic, complemented
246C32C2×#81512yescomplemented
247C23#629411yescharacteristic
248C2×C4—94no—
249C2×C4—184no—
250C2×C4#629414yes—
251C2×C4#629411yescharacteristic
252C2×C4C2×#111311yescharacteristic
253C8—98no—
254C2×C3—24no—
255C2×C3—28no—
256C2×C3—22no—
257C2×C3—24no—
258C2×C3—21no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—42no—
262C2×C3—41no—
263C2×C3#6355418yescomplemented
264C2×C3#6355514yescomplemented
265C2×C3C2×#371114yes—
266C2×C3C2×#371212yes—
267C2×C3C2×Dic3212yes—
268C2×C3C2×Dic3211yescharacteristic
269C22#8880914yes—
270C22#8881312yes—
271C22C2×#629411yescharacteristic
272C4—184no—
273C4#8881312yes—
274C4C2×#629412yes—
275C3—21no—
276C3—22no—
277C3—41no—
278C3C2×#6355412yescomplemented
279C3C2×#6355511yescharacteristic, complemented
280C2#1172207114yesdirect product factor
281C2C2×#8880912yes—
282C2C2×#8881311yescharacteristic
subgroupG/H
center, hypercenterC22×C4#1113
commutatorC2×C33C2×C42
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socle, prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypocenterC33C2×#97
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722071C2direct (N × H)
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
27. C3×C3⋊(C3⋊C8)C2×C4semidirect
40. C2×C3⋊(C3⋊C8)Dic3semidirect
95. C3⋊(C3⋊C8)C2×Dic3semidirect
100. C2×C33#97semidirect
158. C33C2×#97semidirect
200. C2×C32#803semidirect
201. C2×C32#815semidirect
245. C32C2×#803semidirect
246. C32C2×#815semidirect
263. C2×C3#63554semidirect
264. C2×C3#63555semidirect
278. C3C2×#63554semidirect
279. C3C2×#63555semidirect
280. C2#11722071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C22×C3×C3⋊(C3⋊C8)2
#117220712
C2×#635553
C2×#635543