Finite Group Databasebeta

← #10588661⋊C3 #13142657

Subgroups — #10588661⋊C3 #13142657

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 291 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
3C2×(3−1+2⋊C4)⋊C4C212yesmaximal
43−1+2⋊#95C212yesmaximal
53−1+2⋊#95C211yesmaximal, characteristic
6C3×#4981—31nomaximal
7#10588661C311yesmaximal, 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—
15(3−1+2⋊C4)⋊C4—24no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—32no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—32no—
21C2×C9×C4⋊C4—21no—
22C2×C9×C4⋊C4C2×C311yescharacteristic
23C2×C4⋊Dic9C2×C311yescharacteristic
24C2×Dic9⋊C4C2×C312yes—
25C9⋊#95C2×C312yes—
26C9⋊#95C2×C311yescharacteristic
27C23×3−1+2C2311yescharacteristic
28C2×C4×3−1+2—21no—
29C2×C4×3−1+2—21no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—21no—
32C2×C4×3−1+2—22no—
33C2×C4×3−1+2D412yes—
34C2×C4×3−1+2D412yes—
35C2×3−1+2⋊C4—22no—
36C2×3−1+2⋊C4—22no—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4—22no—
40C2×3−1+2⋊C4Q814yes—
41C3×#398—91no—
42#4981—31no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C32×C4⋊C4—22no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—64no—
51C22×C4×C9—21no—
52C22×C4×C9—21no—
53C22×C4×C9—21no—
54C22×C4×C9C22×C311yescharacteristic
55C22×C4×C9C22×C311yescharacteristic
56C22×C4×C9C22×C311yescharacteristic
57C22×Dic9C22×C312yes—
58C22×Dic9C22×C312yes—
59C9×C4⋊C4—24no—
60C9×C4⋊C4—22no—
61C4⋊Dic9—22no—
62Dic9⋊C4—24no—
63C22×3−1+2C2×D411yescharacteristic
64C22×3−1+2C2×D411yescharacteristic
65C22×3−1+2C2×Q812yes—
66C22×3−1+2#4111yescharacteristic
67C22×3−1+2#4112yes—
68C4×3−1+2—22no—
69C4×3−1+2—22no—
70C4×3−1+2—41no—
713−1+2⋊C4—24no—
723−1+2⋊C4—42no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—92no—
75C2×C3×C4⋊C4—91no—
76C2×C3×C4⋊C4C3×D311yescharacteristic
77C2×C4⋊Dic3—31no—
78C2×Dic3⋊C4—32no—
79C3×#95—91no—
80C3×#95—92no—
81C3⋊#95—31no—
82C3⋊#95—32no—
83C23×C32C22×D311yescharacteristic
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4D1212yes—
90C2×C32×C4#6712yes—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C23×C9—21no—
98C23×C9C23×C311yescharacteristic
99C2×C4×C9—22no—
100C2×C4×C9—22no—
101C2×C4×C9—21no—
102C2×C4×C9—21no—
103C2×C4×C9—22no—
104C2×C4×C9—22no—
105C2×C4×C9—21no—
106C2×C4×C9—22no—
107C2×C4×C9—22no—
108C2×C4×C9—22no—
109C2×C4×C9—42no—
110C2×C4×C9—42no—
111C2×C4×C9C3×D412yes—
112C2×C4×C9C3×D412yes—
113C2×Dic9—22no—
114C2×Dic9—22no—
115C2×Dic9—22no—
116C2×Dic9—22no—
117C2×Dic9—22no—
118C2×Dic9C3×Q814yes—
119#398—91noSylow
120C2×3−1+2#12111yescharacteristic
121C2×3−1+2C22⋊Q812yes—
122C2×3−1+2#12311yescharacteristic
123C2×3−1+2#12512yes—
124C2×3−1+2C4⋊Q811yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—92no—
129C22×C3×C4—92no—
130C22×C3×C4C2×C3×D311yescharacteristic
131C22×C3×C4C2×C3×D311yescharacteristic
132C22×C3×C4C2×C3×D311yescharacteristic
133C22×Dic3—32no—
134C22×Dic3—32no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—62no—
137C3×C4⋊C4—184no—
138C3×C4⋊C4—182no—
139C4⋊Dic3—62no—
140Dic3⋊C4—64no—
141C22×C32C2×D1211yescharacteristic
142C22×C32#23511yescharacteristic
143C22×C32#23712yes—
144C22×C32D3×Q812yes—
145C22×C32C2×#6711yescharacteristic
146C32×C4—22no—
147C32×C4—22no—
148C32×C4—41no—
149C3×Dic3—64no—
150C3×Dic3—122no—
151C22×C9—21no—
152C22×C9—21no—
153C22×C9—22no—
154C22×C9—22no—
155C22×C9—21no—
156C22×C9C2×C3×D411yescharacteristic
157C22×C9C2×C3×D411yescharacteristic
158C22×C9C2×C3×Q812yes—
159C22×C9C3×#4111yescharacteristic
160C22×C9C3×#4112yes—
161C4×C9—24no—
162C4×C9—22no—
163C4×C9—22no—
164C4×C9—42no—
165C4×C9—42no—
166C4×C9—41no—
167Dic9—24no—
168Dic9—42no—
169C2×C4⋊C4—92no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
172#95—91no—
173#95—92no—
1743−1+2#39811yescharacteristic, complemented, Sylow
175C23×C3—31no—
176C23×C3C22×C3×D311yescharacteristic
177C2×C3×C4—21no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—62no—
187C2×C3×C4—61no—
188C2×C3×C4—62no—
189C2×C3×C4—94no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4C3×D1212yes—
196C2×C3×C4C3×#6712yes—
197C2×Dic3—34no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×C32#88912yes—
204C2×C32#89712yes—
205C2×C32#93011yescharacteristic
206C2×C32#93511yescharacteristic
207C2×C32#94411yescharacteristic
208C2×C9—22no—
209C2×C9—22no—
210C2×C9—21no—
211C2×C9—21no—
212C2×C9—21no—
213C2×C9C3×#12111yescharacteristic
214C2×C9C3×C22⋊Q812yes—
215C2×C9C3×#12311yescharacteristic
216C2×C9C3×#12512yes—
217C2×C9C3×C4⋊Q811yescharacteristic
218C22×C4—92no—
219C22×C4—92no—
220C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
221C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
222C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—184no—
225C4⋊C4—182no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—31no—
231C22×C3C2×C3×D1211yescharacteristic
232C22×C3C3×#23511yescharacteristic
233C22×C3C3×#23712yes—
234C22×C3C3×D3×Q812yes—
235C22×C3C2×C3×#6711yescharacteristic
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—41no—
239C3×C4—62no—
240C3×C4—62no—
241C3×C4—121no—
242C3×C4—184no—
243C3×C4—362no—
244Dic3—64no—
245Dic3—122no—
246C32#498111yescharacteristic
247C9—21no—
248C9C3×#39811yescharacteristic, complemented
249C23C22×AGL(1,ℤ/9)11yescharacteristic
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—94no—
256C2×C4—182no—
257C2×C4—182no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C4—182no—
261C2×C4C4⋊AGL(1,ℤ/9)12yes—
262C2×C4#611⋊C312yes—
263C2×C3—32no—
264C2×C3—32no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3C3×#88912yes—
269C2×C3C3×#89712yes—
270C2×C3C3×#93011yescharacteristic
271C2×C3C3×#93511yescharacteristic
272C2×C3C3×#94411yescharacteristic
273C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
274C22#3697⋊C311yescharacteristic
275C22#3699⋊C312yes—
276C22Q8×AGL(1,ℤ/9)12yes—
277C22C2×#611⋊C311yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—184no—
282C4—362no—
283C3—31no—
284C3C3×#498111yescharacteristic
285C2#63451⋊C312yes—
286C2#63459⋊C312yes—
287C2#63492⋊C311yescharacteristic
288C2#63497⋊C311yescharacteristic
289C2#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. #10588661C3semidirect
174. 3−1+2#398semidirect
248. 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×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49813
#105886613