Finite Group Databasebeta

← #10588399⋊C3 #13141623

Subgroups — #10588399⋊C3 #13141623

Order 1728.

355 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×3−1+2⋊C4C211yesmaximal, characteristic
23−1+2×C22⋊C8C211yesmaximal, characteristic
3#63389⋊C3C211yesmaximal, characteristic
4#63375⋊C3C212yesmaximal, complemented
5(3−1+2⋊C4)⋊C8C212yesmaximal, complemented
6C3×#4719—31nomaximal
7#10588399C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C22×3−1+2⋊C4C411yescharacteristic
10C22×3−1+2⋊C4C411yescharacteristic
11C2×C8×3−1+2C2212yes—
12C4×3−1+2⋊C4C411yescharacteristic
13C4×3−1+2⋊C4C411yescharacteristic
14C4×3−1+2⋊C4C2211yescharacteristic
15C4×3−1+2⋊C4C2211yescharacteristic
16C2×3−1+2⋊C8C2212yes—
17C2×C3×C4×Dic3—31no—
18C32×C22⋊C8D311yescharacteristic
19C3×#830—31no—
20C3×#815—32no—
21C3×Dic3⋊C8—32no—
22C2×C4×Dic9C2×C311yescharacteristic
23C9×C22⋊C8—21no—
24C9×C22⋊C8C2×C311yescharacteristic
25#63389C2×C311yescharacteristic
26#63375C2×C312yescomplemented
27Dic9⋊C8C2×C312yescomplemented
28C23×3−1+2C2×C411yescharacteristic
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2C2×C411yescharacteristic
31C2×C4×3−1+2C2311yescharacteristic
32C2×3−1+2⋊C4—22no—
33C2×3−1+2⋊C4—22no—
34C2×3−1+2⋊C4C2×C411yescharacteristic
35C2×3−1+2⋊C4C2×C411yescharacteristic
36C2×3−1+2⋊C4C2×C411yescharacteristic
37C2×3−1+2⋊C4C2×C411yescharacteristic
38C8×3−1+2—22no—
393−1+2⋊C8—22no—
40C3×#432—91no—
41#4719—31no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C3×Dic3—31no—
44C22×C3×Dic3—31no—
45C2×C32×C8C2×D312yes—
46C3×C4×Dic3—31no—
47C3×C4×Dic3—31no—
48C3×C4×Dic3—31no—
49C3×C4×Dic3—31no—
50C2×C3×C3⋊C8—32no—
51C22×C4×C9—21no—
52C22×C4×C9C22×C311yescharacteristic
53C22×Dic9C3×C411yescharacteristic
54C22×Dic9C3×C411yescharacteristic
55C2×C8×C9—22no—
56C2×C8×C9C22×C312yes—
57C4×Dic9C3×C411yescharacteristic
58C4×Dic9C3×C411yescharacteristic
59C4×Dic9C22×C311yescharacteristic
60C4×Dic9C22×C311yescharacteristic
61C2×C9⋊C8C22×C312yes—
62C22×3−1+2—21no—
63C22×3−1+2—21no—
64C22×3−1+2M1612yes—
65C22×3−1+2C22×C411yescharacteristic
66C4×3−1+2—21no—
67C4×3−1+2#4112yes—
683−1+2⋊C4—21no—
693−1+2⋊C4—21no—
703−1+2⋊C4—21no—
713−1+2⋊C4M1612yes—
72C2×C3×C42—91no—
73C2×C4×Dic3—31no—
74C3×C4⋊C8—92no—
75C3×C22⋊C8—31no—
76C3×C22⋊C8—91no—
77C3×C22⋊C8C3×D311yescharacteristic
78C3×#97—92no—
79#830—31no—
80#815—32no—
81Dic3⋊C8—32no—
82C23×C32C4×D311yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4C4×D311yescharacteristic
85C2×C32×C4C22×D311yescharacteristic
86C2×C3×Dic3—31no—
87C2×C3×Dic3—31no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—31no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C23×C9—21no—
93C23×C9C2×C3×C411yescharacteristic
94C32×C8—22no—
95C3×C3⋊C8—62no—
96C2×C4×C9—22no—
97C2×C4×C9—21no—
98C2×C4×C9—21no—
99C2×C4×C9—42no—
100C2×C4×C9C2×C3×C411yescharacteristic
101C2×C4×C9C23×C311yescharacteristic
102C2×Dic9—22no—
103C2×Dic9—22no—
104C2×Dic9C2×C3×C411yescharacteristic
105C2×Dic9C2×C3×C411yescharacteristic
106C2×Dic9C2×C3×C411yescharacteristic
107C2×Dic9C2×C3×C411yescharacteristic
108C8×C9—22no—
109C8×C9—42no—
110C9⋊C8—22no—
111#432—91noSylow
112C2×3−1+2—22no—
113C2×3−1+2#11711yescharacteristic
114C2×3−1+2C2×M1611yescharacteristic
115C2×3−1+2C2×M1611yescharacteristic
116C22×C3×C4—31no—
117C22×C3×C4—91no—
118C22×C3×C4—91no—
119C22×C3×C4C2×C3×D311yescharacteristic
120C22×Dic3—31no—
121C22×Dic3—31no—
122C2×C3×C8—32no—
123C2×C3×C8—92no—
124C2×C3×C8C2×C3×D312yes—
125C3×C42—91no—
126C3×C42—91no—
127C3×C42—91no—
128C3×C42—91no—
129C4×Dic3—31no—
130C4×Dic3—31no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C2×C3⋊C8—32no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32ΓL(1,𝔽25)12yes—
137C22×C32C2×C4×D311yescharacteristic
138C32×C4—21no—
139C32×C4#23712yes—
140C3×Dic3—32no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143C3×Dic3—61no—
144C22×C9—21no—
145C22×C9—21no—
146C22×C9—22no—
147C22×C9—21no—
148C22×C9—41no—
149C22×C9—41no—
150C22×C9C3×M1612yes—
151C22×C9C22×C3×C411yescharacteristic
152C4×C9—21no—
153C4×C9—22no—
154C4×C9—41no—
155C4×C9C3×#4112yes—
156Dic9—21no—
157Dic9—21no—
158Dic9—21no—
159Dic9C3×M1612yes—
160C2×C42—91no—
161C4⋊C8—92no—
162C22⋊C8—91no—
163C22⋊C8AGL(1,ℤ/9)11yescharacteristic
164#97—92no—
1653−1+2#43211yescharacteristic, complemented, Sylow
166C23×C3—31no—
167C23×C3C3×C4×D311yescharacteristic
168C2×C3×C4—22no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—62no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4C3×C4×D311yescharacteristic
179C2×C3×C4C22×C3×D311yescharacteristic
180C2×Dic3—31no—
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—31no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C3×C8—22no—
187C3×C8—62no—
188C3×C8—182no—
189C3⋊C8—62no—
190C2×C32—22no—
191C2×C32#87711yescharacteristic
192C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
193C2×C32D3×M1611yescharacteristic
194C2×C9—22no—
195C2×C9—21no—
196C2×C9—21no—
197C2×C9—21no—
198C2×C9—42no—
199C2×C9C3×#11711yescharacteristic
200C2×C9C2×C3×M1611yescharacteristic
201C2×C9C2×C3×M1611yescharacteristic
202C22×C4—91no—
203C22×C4—91no—
204C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
205C2×C8—92no—
206C2×C8C2×AGL(1,ℤ/9)12yes—
207C42—91no—
208C42—91no—
209C42—91no—
210C42—91no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—31no—
214C22×C3—32no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3C3×ΓL(1,𝔽25)12yes—
218C22×C3C2×C3×C4×D311yescharacteristic
219C3×C4—21no—
220C3×C4—32no—
221C3×C4—61no—
222C3×C4—92no—
223C3×C4—181no—
224C3×C4—181no—
225C3×C4—181no—
226C3×C4C3×#23712yes—
227Dic3—32no—
228Dic3—61no—
229Dic3—61no—
230Dic3—61no—
231C32#471911yescharacteristic
232C9—21no—
233C9C3×#43211yescharacteristic, complemented
234C23C4×AGL(1,ℤ/9)11yescharacteristic
235C2×C4—22no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4C4×AGL(1,ℤ/9)11yescharacteristic
243C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
244C8—22no—
245C8—182no—
246C2×C3—22no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—62no—
251C2×C3C3×#87711yescharacteristic
252C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
253C2×C3C3×D3×M1611yescharacteristic
254C22—21no—
255C22—21no—
256C22#3663⋊C312yes—
257C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
258C4—21no—
259C4—92no—
260C4—181no—
261C4—181no—
262C4—181no—
263C4#3699⋊C312yes—
264C3—31no—
265C3C3×#471911yescharacteristic
266C2—22no—
267C2#63439⋊C311yescharacteristic
268C2C2×#3663⋊C311yescharacteristic
269C2M16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2×C4C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC23×C32C4×D3
cosocleC2×C4×C9C23×C3
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#432
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63375⋊C3C2semidirect
5. (3−1+2⋊C4)⋊C8C2semidirect
7. #10588399C3semidirect
26. #63375C2×C3semidirect
27. Dic9⋊C8C2×C3semidirect
165. 3−1+2#432semidirect
233. C9C3×#432semidirect

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⋊C42
3−1+2×C22⋊C82
#63389⋊C32
#63375⋊C32
(3−1+2⋊C4)⋊C82
C3×#47193
#105883993