Finite Group Databasebeta

← #10588666⋊C3 #13142656

Subgroups — #10588666⋊C3 #13142656

Order 1728.

448 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
33−1+2⋊#95C211yesmaximal, characteristic
43−1+2⋊#95C212yesmaximal
53−1+2⋊#95C212yesmaximal
6C3×#4986—31nomaximal
7#10588666C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4C2212yes—
12C22×3−1+2⋊C4C2212yes—
13C4⋊C4×3−1+2—22no—
14C4⋊(3−1+2⋊C4)—22no—
15C2×C32×C4⋊C4D311yescharacteristic
16C2×C3×C4⋊Dic3—31no—
17C3×C3⋊#95—32no—
18C3×C3⋊#95—32no—
19C3×C3⋊#95—31no—
20C2×C9×C4⋊C4—21no—
21C2×C9×C4⋊C4C2×C311yescharacteristic
22C2×C4⋊Dic9C2×C311yescharacteristic
23C9⋊#95C2×C311yescharacteristic
24C9⋊#95C2×C312yes—
25C9⋊#95C2×C312yes—
26C23×3−1+2C2311yescharacteristic
27C2×C4×3−1+2—21no—
28C2×C4×3−1+2—21no—
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2—21no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2D412yes—
33C2×C4×3−1+2Q812yes—
34C2×3−1+2⋊C4—22no—
35C2×3−1+2⋊C4—24no—
36C2×3−1+2⋊C4—22no—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4—22no—
39C3×#400—91no—
40#4986—31no—
41C22×C32×C4C2×D311yescharacteristic
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C32×C4⋊C4—22no—
47C3×C4⋊Dic3—62no—
48C22×C4×C9—21no—
49C22×C4×C9—21no—
50C22×C4×C9—21no—
51C22×C4×C9C22×C311yescharacteristic
52C22×C4×C9C22×C311yescharacteristic
53C22×C4×C9C22×C311yescharacteristic
54C22×Dic9C22×C312yes—
55C22×Dic9C22×C312yes—
56C9×C4⋊C4—24no—
57C9×C4⋊C4—22no—
58C4⋊Dic9—22no—
59C22×3−1+2C2×D411yescharacteristic
60C22×3−1+2C2×Q811yescharacteristic
61C22×3−1+2#4111yescharacteristic
62C22×3−1+2#4112yes—
63C22×3−1+2#4112yes—
64C4×3−1+2—22no—
65C4×3−1+2—22no—
66C4×3−1+2—41no—
673−1+2⋊C4—42no—
683−1+2⋊C4—42no—
69C2×C3×C4⋊C4—31no—
70C2×C3×C4⋊C4—91no—
71C2×C3×C4⋊C4C3×D311yescharacteristic
72C2×C4⋊Dic3—31no—
73C3×#95—92no—
74C3×#95—91no—
75C3×#95—92no—
76C3⋊#95—31no—
77C3⋊#95—32no—
78C3⋊#95—32no—
79C23×C32C22×D311yescharacteristic
80C2×C32×C4—21no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—21no—
85C2×C32×C4Dic612yes—
86C2×C32×C4#6712yes—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—62no—
92C23×C9—21no—
93C23×C9C23×C311yescharacteristic
94C2×C4×C9—22no—
95C2×C4×C9—22no—
96C2×C4×C9—21no—
97C2×C4×C9—21no—
98C2×C4×C9—22no—
99C2×C4×C9—22no—
100C2×C4×C9—21no—
101C2×C4×C9—22no—
102C2×C4×C9—22no—
103C2×C4×C9—22no—
104C2×C4×C9—42no—
105C2×C4×C9—42no—
106C2×C4×C9C3×D412yes—
107C2×C4×C9C3×Q812yes—
108C2×Dic9—22no—
109C2×Dic9—22no—
110C2×Dic9—22no—
111C2×Dic9—24no—
112C2×Dic9—22no—
113#400—91noSylow
114C2×3−1+2C22⋊Q811yescharacteristic
115C2×3−1+2#12311yescharacteristic
116C2×3−1+2#12411yescharacteristic
117C2×3−1+2#12512yes—
118C2×3−1+2#12612yes—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—92no—
123C22×C3×C4—92no—
124C22×C3×C4C2×C3×D311yescharacteristic
125C22×C3×C4C2×C3×D311yescharacteristic
126C22×C3×C4C2×C3×D311yescharacteristic
127C22×Dic3—32no—
128C22×Dic3—32no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—182no—
132C4⋊Dic3—62no—
133C22×C32C2×Dic611yescharacteristic
134C22×C32#23511yescharacteristic
135C22×C32#23712yes—
136C22×C32#23912yes—
137C22×C32C2×#6711yescharacteristic
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—41no—
141C3×Dic3—122no—
142C3×Dic3—122no—
143C22×C9—21no—
144C22×C9—21no—
145C22×C9—22no—
146C22×C9—22no—
147C22×C9—21no—
148C22×C9C2×C3×D411yescharacteristic
149C22×C9C2×C3×Q811yescharacteristic
150C22×C9C3×#4111yescharacteristic
151C22×C9C3×#4112yes—
152C22×C9C3×#4112yes—
153C4×C9—24no—
154C4×C9—22no—
155C4×C9—22no—
156C4×C9—42no—
157C4×C9—42no—
158C4×C9—41no—
159Dic9—42no—
160Dic9—42no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
163#95—91no—
164#95—92no—
165#95—92no—
1663−1+2#40011yescharacteristic, complemented, Sylow
167C23×C3—31no—
168C23×C3C22×C3×D311yescharacteristic
169C2×C3×C4—21no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—62no—
181C2×C3×C4—182no—
182C2×C3×C4—182no—
183C2×C3×C4—182no—
184C2×C3×C4—182no—
185C2×C3×C4—184no—
186C2×C3×C4C3×Dic612yes—
187C2×C3×C4C3×#6712yes—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—64no—
192C2×Dic3—62no—
193C2×C32#89012yes—
194C2×C32#89812yes—
195C2×C32#92411yescharacteristic
196C2×C32#93511yescharacteristic
197C2×C32#94711yescharacteristic
198C2×C9—22no—
199C2×C9—22no—
200C2×C9—21no—
201C2×C9—21no—
202C2×C9—21no—
203C2×C9C3×C22⋊Q811yescharacteristic
204C2×C9C3×#12311yescharacteristic
205C2×C9C3×#12411yescharacteristic
206C2×C9C3×#12512yes—
207C2×C9C3×#12612yes—
208C22×C4—92no—
209C22×C4—92no—
210C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
211C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
212C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—182no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—32no—
218C22×C3—32no—
219C22×C3—31no—
220C22×C3C2×C3×Dic611yescharacteristic
221C22×C3C3×#23511yescharacteristic
222C22×C3C3×#23712yes—
223C22×C3C3×#23912yes—
224C22×C3C2×C3×#6711yescharacteristic
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—41no—
228C3×C4—62no—
229C3×C4—62no—
230C3×C4—121no—
231C3×C4—362no—
232C3×C4—362no—
233Dic3—122no—
234Dic3—122no—
235C32#498611yescharacteristic
236C9—21no—
237C9C3×#40011yescharacteristic, complemented
238C23C22×AGL(1,ℤ/9)11yescharacteristic
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—182no—
245C2×C4—182no—
246C2×C4—182no—
247C2×C4—184no—
248C2×C4—182no—
249C2×C43−1+2⋊Q812yes—
250C2×C4#611⋊C312yes—
251C2×C3—32no—
252C2×C3—32no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3C3×#89012yes—
257C2×C3C3×#89812yes—
258C2×C3C3×#92411yescharacteristic
259C2×C3C3×#93511yescharacteristic
260C2×C3C3×#94711yescharacteristic
261C22C2×3−1+2⋊Q811yescharacteristic
262C22#3697⋊C311yescharacteristic
263C22#3699⋊C312yes—
264C22#8872412yes—
265C22C2×#611⋊C311yescharacteristic
266C4—22no—
267C4—22no—
268C4—41no—
269C4—362no—
270C4—362no—
271C3—31no—
272C3C3×#498611yescharacteristic
273C2#63452⋊C312yes—
274C2#63460⋊C312yes—
275C2#63486⋊C311yescharacteristic
276C2#63497⋊C311yescharacteristic
277C2#63509⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#400
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588666C3semidirect
166. 3−1+2#400semidirect
237. C9C3×#400semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×C4⋊(3−1+2⋊C4)2
3−1+2⋊#952
C3×#49863
#105886663