Finite Group Databasebeta

← #10588341⋊C3 #13141380

Subgroups — #10588341⋊C3 #13141380

Order 1728.

488 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
2C2×C4⋊(3−1+2⋊C4)C212yesmaximal
3C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic
43−1+2×#95C211yesmaximal, characteristic
53−1+2⋊#95C212yesmaximal
6C3×#4661—31nomaximal
7#10588341C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2212yes—
10C22×3−1+2⋊C4C2211yescharacteristic
11C22×3−1+2⋊C4C2212yes—
12C22×3−1+2⋊C4C2211yescharacteristic
13C4⋊(3−1+2⋊C4)—22no—
14C4⋊(3−1+2⋊C4)—24no—
15(3−1+2⋊C4)⋊C4—22no—
16C2×C3×C4⋊Dic3—32no—
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—31no—
19C32×#95D311yescharacteristic
20C3×C3⋊#95—32no—
21C2×C4⋊Dic9C2×C311yescharacteristic
22C2×C4⋊Dic9C2×C312yes—
23C2×Dic9⋊C4C2×C311yescharacteristic
24C9×#95—21no—
25C9×#95C2×C311yescharacteristic
26C9⋊#95C2×C312yes—
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+2D412yes—
32C2×C4×3−1+2Q814yes—
33C2×3−1+2⋊C4—21no—
34C2×3−1+2⋊C4—21no—
35C2×3−1+2⋊C4—21no—
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⋊C4D412yes—
41C3×#398—91no—
42#4661—31no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3×Dic3—31no—
46C22×C3×Dic3—31no—
47C22×C3×Dic3—32no—
48C3×C4⋊Dic3—64no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—62no—
51C22×C4×C9—22no—
52C22×C4×C9—21no—
53C22×C4×C9C22×C311yescharacteristic
54C22×C4×C9C22×C312yes—
55C22×Dic9C22×C311yescharacteristic
56C22×Dic9C22×C312yes—
57C22×Dic9C22×C311yescharacteristic
58C4⋊Dic9—22no—
59C4⋊Dic9—24no—
60Dic9⋊C4—22no—
61C22×3−1+2C2×D411yescharacteristic
62C22×3−1+2C2×D411yescharacteristic
63C22×3−1+2C2×Q812yes—
64C22×3−1+2#4111yescharacteristic
65C22×3−1+2#4112yes—
66C4×3−1+2—24no—
67C4×3−1+2—22no—
683−1+2⋊C4—22no—
693−1+2⋊C4—42no—
703−1+2⋊C4—41no—
71C2×C3×C4⋊C4—91no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C4⋊C4—92no—
74C2×C4⋊Dic3—32no—
75C2×C4⋊Dic3—31no—
76C2×Dic3⋊C4—31no—
77C3×#95—31no—
78C3×#95—92no—
79C3×#95C3×D311yescharacteristic
80C3⋊#95—32no—
81C23×C32C22×D311yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4Dic614yes—
86C2×C32×C4D1212yes—
87C2×C3×Dic3—32no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C23×C9—21no—
96C23×C9C23×C311yescharacteristic
97C2×C4×C9—22no—
98C2×C4×C9—22no—
99C2×C4×C9—22no—
100C2×C4×C9—24no—
101C2×C4×C9—22no—
102C2×C4×C9—42no—
103C2×C4×C9—42no—
104C2×C4×C9—42no—
105C2×C4×C9C3×D412yes—
106C2×C4×C9C3×Q814yes—
107C2×Dic9—21no—
108C2×Dic9—21no—
109C2×Dic9—21no—
110C2×Dic9—22no—
111C2×Dic9—22no—
112C2×Dic9—22no—
113C2×Dic9—22no—
114C2×Dic9C3×D412yes—
115#398—91noSylow
116C2×3−1+2#12111yescharacteristic
117C2×3−1+2C22⋊Q812yes—
118C2×3−1+2#12311yescharacteristic
119C2×3−1+2#12512yes—
120C2×3−1+2C4⋊Q811yescharacteristic
121C22×C3×C4—32no—
122C22×C3×C4—31no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—92no—
126C22×C3×C4C2×C3×D312yes—
127C22×C3×C4C2×C3×D311yescharacteristic
128C22×Dic3—31no—
129C22×Dic3—31no—
130C22×Dic3—32no—
131C3×C4⋊C4—182no—
132C3×C4⋊C4—182no—
133C3×C4⋊C4—184no—
134C4⋊Dic3—64no—
135C4⋊Dic3—62no—
136Dic3⋊C4—62no—
137C22×C32C2×Dic612yes—
138C22×C32C2×D1211yescharacteristic
139C22×C32D3×D411yescharacteristic
140C22×C32#23712yes—
141C22×C32#23911yescharacteristic
142C32×C4—24no—
143C32×C4—22no—
144C3×Dic3—62no—
145C3×Dic3—121no—
146C3×Dic3—122no—
147C22×C9—22no—
148C22×C9—21no—
149C22×C9—21no—
150C22×C9—22no—
151C22×C9—21no—
152C22×C9C2×C3×D411yescharacteristic
153C22×C9C2×C3×D411yescharacteristic
154C22×C9C2×C3×Q812yes—
155C22×C9C3×#4112yes—
156C22×C9C3×#4111yescharacteristic
157C4×C9—24no—
158C4×C9—22no—
159C4×C9—44no—
160C4×C9—42no—
161Dic9—22no—
162Dic9—42no—
163Dic9—41no—
164C2×C4⋊C4—91no—
165C2×C4⋊C4—92no—
166C2×C4⋊C4—91no—
167#95—92no—
168#95AGL(1,ℤ/9)11yescharacteristic
1693−1+2#39811yescharacteristic, complemented, Sylow
170C23×C3—31no—
171C23×C3C22×C3×D311yescharacteristic
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—34no—
176C2×C3×C4—32no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—92no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—182no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×C3×C4C3×Dic614yes—
189C2×C3×C4C3×D1212yes—
190C2×Dic3—32no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×C32#86911yescharacteristic
199C2×C32#87812yes—
200C2×C32#88611yescharacteristic
201C2×C32#89012yes—
202C2×C32#89511yescharacteristic
203C2×C9—22no—
204C2×C9—21no—
205C2×C9—22no—
206C2×C9—21no—
207C2×C9—21no—
208C2×C9C3×#12111yescharacteristic
209C2×C9C3×C22⋊Q812yes—
210C2×C9C3×#12311yescharacteristic
211C2×C9C3×#12512yes—
212C2×C9C3×C4⋊Q811yescharacteristic
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—92no—
216C22×C4C2×AGL(1,ℤ/9)12yes—
217C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
218C4⋊C4—184no—
219C4⋊C4—182no—
220C4⋊C4—182no—
221C22×C3—32no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—32no—
225C22×C3—31no—
226C22×C3C2×C3×Dic612yes—
227C22×C3C2×C3×D1211yescharacteristic
228C22×C3C3×D3×D411yescharacteristic
229C22×C3C3×#23712yes—
230C22×C3C3×#23911yescharacteristic
231C3×C4—24no—
232C3×C4—22no—
233C3×C4—64no—
234C3×C4—62no—
235C3×C4—182no—
236C3×C4—362no—
237C3×C4—361no—
238Dic3—62no—
239Dic3—121no—
240Dic3—122no—
241C32#466111yescharacteristic
242C9—21no—
243C9C3×#39811yescharacteristic, complemented
244C23C22×AGL(1,ℤ/9)11yescharacteristic
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—92no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4—182no—
252C2×C4—181no—
253C2×C4—182no—
254C2×C4—182no—
255C2×C4—182no—
256C2×C43−1+2⋊Q814yes—
257C2×C4C4⋊AGL(1,ℤ/9)12yes—
258C2×C3—32no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—32no—
262C2×C3—31no—
263C2×C3C3×#86911yescharacteristic
264C2×C3C3×#87812yes—
265C2×C3C3×#88611yescharacteristic
266C2×C3C3×#89012yes—
267C2×C3C3×#89511yescharacteristic
268C22C2×3−1+2⋊Q812yes—
269C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
270C22D4×AGL(1,ℤ/9)11yescharacteristic
271C22#3699⋊C312yes—
272C22#8872411yescharacteristic
273C4—24no—
274C4—22no—
275C4—182no—
276C4—362no—
277C4—361no—
278C3—31no—
279C3C3×#466111yescharacteristic
280C2#63431⋊C311yescharacteristic
281C2#63440⋊C312yes—
282C2#63448⋊C311yescharacteristic
283C2#63452⋊C312yes—
284C2#63457⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×C32C22×D3
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#398
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
7. #10588341C3semidirect
169. 3−1+2#398semidirect
243. C9C3×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
3−1+2×#952
3−1+2⋊#952
C3×#46613
#105883413