Finite Group Databasebeta

← #10588663⋊C3 #13142652

Subgroups — #10588663⋊C3 #13142652

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
2C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
3C2×(3−1+2⋊C4)⋊C4C212yesmaximal
43−1+2⋊#95C212yesmaximal
53−1+2⋊#95C211yesmaximal, characteristic
6C3×#4983—31nomaximal
7#10588663C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2212yes—
9C22×C4×3−1+2C2211yescharacteristic
10C22×3−1+2⋊C4C2211yescharacteristic
11C22×3−1+2⋊C4C2211yescharacteristic
12C22×3−1+2⋊C4C2212yes—
13C4⋊C4×3−1+2—22no—
14(3−1+2⋊C4)⋊C4—22no—
15(3−1+2⋊C4)⋊C4—24no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C3×Dic3⋊C4—32no—
18C2×C3×Dic3⋊C4—31no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—32no—
21C2×C9×C4⋊C4—21no—
22C2×C9×C4⋊C4C2×C311yescharacteristic
23C2×Dic9⋊C4C2×C311yescharacteristic
24C2×Dic9⋊C4C2×C312yes—
25C9⋊#95C2×C312yes—
26C9⋊#95C2×C311yescharacteristic
27C23×3−1+2C2311yescharacteristic
28C2×C4×3−1+2—22no—
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2D412yes—
33C2×3−1+2⋊C4—21no—
34C2×3−1+2⋊C4—22no—
35C2×3−1+2⋊C4—21no—
36C2×3−1+2⋊C4—22no—
37C2×3−1+2⋊C4—21no—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4D412yes—
40C2×3−1+2⋊C4Q814yes—
41C3×#398—91no—
42#4983—31no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3×Dic3—31no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—31no—
48C32×C4⋊C4—22no—
49C3×Dic3⋊C4—64no—
50C3×Dic3⋊C4—62no—
51C22×C4×C9—21no—
52C22×C4×C9—22no—
53C22×C4×C9C22×C312yes—
54C22×C4×C9C22×C311yescharacteristic
55C22×Dic9C22×C311yescharacteristic
56C22×Dic9C22×C311yescharacteristic
57C22×Dic9C22×C312yes—
58C9×C4⋊C4—24no—
59C9×C4⋊C4—22no—
60Dic9⋊C4—22no—
61Dic9⋊C4—24no—
62C22×3−1+2C2×D411yescharacteristic
63C22×3−1+2C2×D411yescharacteristic
64C22×3−1+2C2×Q812yes—
65C22×3−1+2#4111yescharacteristic
66C22×3−1+2#4112yes—
67C4×3−1+2—22no—
68C4×3−1+2—42no—
693−1+2⋊C4—22no—
703−1+2⋊C4—24no—
713−1+2⋊C4—41no—
72C2×C3×C4⋊C4—31no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C4⋊C4—92no—
75C2×C3×C4⋊C4C3×D311yescharacteristic
76C2×Dic3⋊C4—32no—
77C2×Dic3⋊C4—31no—
78C3×#95—91no—
79C3×#95—92no—
80C3⋊#95—31no—
81C3⋊#95—32no—
82C23×C32C22×D311yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4#6712yes—
88C2×C3×Dic3—32no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—61no—
96C23×C9—21no—
97C23×C9C23×C311yescharacteristic
98C2×C4×C9—24no—
99C2×C4×C9—22no—
100C2×C4×C9—24no—
101C2×C4×C9—22no—
102C2×C4×C9—22no—
103C2×C4×C9—22no—
104C2×C4×C9—22no—
105C2×C4×C9—42no—
106C2×C4×C9—42no—
107C2×C4×C9C3×D412yes—
108C2×Dic9—22no—
109C2×Dic9—21no—
110C2×Dic9—22no—
111C2×Dic9—21no—
112C2×Dic9—21no—
113C2×Dic9—22no—
114C2×Dic9C3×D412yes—
115C2×Dic9C3×Q814yes—
116#398—91noSylow
117C2×3−1+2#12111yescharacteristic
118C2×3−1+2C22⋊Q812yes—
119C2×3−1+2#12311yescharacteristic
120C2×3−1+2#12512yes—
121C2×3−1+2C4⋊Q811yescharacteristic
122C22×C3×C4—31no—
123C22×C3×C4—32no—
124C22×C3×C4—92no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4C2×C3×D312yes—
128C22×C3×C4C2×C3×D311yescharacteristic
129C22×Dic3—32no—
130C22×Dic3—31no—
131C22×Dic3—31no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—182no—
135C3×C4⋊C4—184no—
136Dic3⋊C4—64no—
137Dic3⋊C4—62no—
138C22×C32#23512yes—
139C22×C32D3×D411yescharacteristic
140C22×C32#23711yescharacteristic
141C22×C32D3×Q812yes—
142C22×C32C2×#6711yescharacteristic
143C32×C4—22no—
144C32×C4—42no—
145C3×Dic3—64no—
146C3×Dic3—62no—
147C3×Dic3—121no—
148C22×C9—22no—
149C22×C9—21no—
150C22×C9—21no—
151C22×C9—22no—
152C22×C9—21no—
153C22×C9C2×C3×D411yescharacteristic
154C22×C9C2×C3×D411yescharacteristic
155C22×C9C2×C3×Q812yes—
156C22×C9C3×#4112yes—
157C22×C9C3×#4111yescharacteristic
158C4×C9—24no—
159C4×C9—22no—
160C4×C9—44no—
161C4×C9—42no—
162Dic9—22no—
163Dic9—24no—
164Dic9—41no—
165C2×C4⋊C4—92no—
166C2×C4⋊C4—91no—
167C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
168#95—91no—
169#95—92no—
1703−1+2#39811yescharacteristic, complemented, Sylow
171C23×C3—31no—
172C23×C3C22×C3×D311yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—32no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—62no—
182C2×C3×C4—92no—
183C2×C3×C4—94no—
184C2×C3×C4—182no—
185C2×C3×C4—181no—
186C2×C3×C4—182no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—182no—
190C2×C3×C4C3×#6712yes—
191C2×Dic3—32no—
192C2×Dic3—34no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197C2×Dic3—62no—
198C2×Dic3—61no—
199C2×C32#88912yes—
200C2×C32#89612yes—
201C2×C32#92911yescharacteristic
202C2×C32#93811yescharacteristic
203C2×C32#94411yescharacteristic
204C2×C9—22no—
205C2×C9—22no—
206C2×C9—21no—
207C2×C9—21no—
208C2×C9—21no—
209C2×C9C3×#12111yescharacteristic
210C2×C9C3×C22⋊Q812yes—
211C2×C9C3×#12311yescharacteristic
212C2×C9C3×#12512yes—
213C2×C9C3×C4⋊Q811yescharacteristic
214C22×C4—92no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4C2×AGL(1,ℤ/9)12yes—
218C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—184no—
221C4⋊C4—182no—
222C22×C3—32no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—32no—
226C22×C3—31no—
227C22×C3C3×#23512yes—
228C22×C3C3×D3×D411yescharacteristic
229C22×C3C3×#23711yescharacteristic
230C22×C3C3×D3×Q812yes—
231C22×C3C2×C3×#6711yescharacteristic
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—62no—
235C3×C4—122no—
236C3×C4—184no—
237C3×C4—182no—
238C3×C4—361no—
239Dic3—64no—
240Dic3—62no—
241Dic3—121no—
242C32#498311yescharacteristic
243C9—21no—
244C9C3×#39811yescharacteristic, complemented
245C23C22×AGL(1,ℤ/9)11yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—92no—
251C2×C4—94no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4—182no—
257C2×C4—181no—
258C2×C4#611⋊C312yes—
259C2×C3—32no—
260C2×C3—32no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3C3×#88912yes—
265C2×C3C3×#89612yes—
266C2×C3C3×#92911yescharacteristic
267C2×C3C3×#93811yescharacteristic
268C2×C3C3×#94411yescharacteristic
269C22#3697⋊C312yes—
270C22D4×AGL(1,ℤ/9)11yescharacteristic
271C22#3699⋊C311yescharacteristic
272C22Q8×AGL(1,ℤ/9)12yes—
273C22C2×#611⋊C311yescharacteristic
274C4—22no—
275C4—42no—
276C4—184no—
277C4—182no—
278C4—361no—
279C3—31no—
280C3C3×#498311yescharacteristic
281C2#63451⋊C312yes—
282C2#63458⋊C312yes—
283C2#63491⋊C311yescharacteristic
284C2#63500⋊C311yescharacteristic
285C2#63506⋊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×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588663C3semidirect
170. 3−1+2#398semidirect
244. C9C3×#398semidirect

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×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49833
#105886633