Finite Group Databasebeta

← #107×AGL(1,ℤ/9) #13142240

Subgroups — #107×AGL(1,ℤ/9) #13142240

Order 1728.

408 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 140 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×C8×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C4⋊C4×AGL(1,ℤ/9)C212yesmaximal
33−1+2×#107C211yesmaximal, characteristic, complemented
4#63378⋊C3C211yesmaximal, characteristic, complemented
5#63366⋊C3C212yesmaximal, complemented
6C3×D3×#107—31nomaximal
7D9×#107C311yesmaximal, characteristic, complemented
8C2×C4×AGL(1,ℤ/9)—22no—
9C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
10C2×C8×3−1+2C2211yescharacteristic
11C4⋊C4×3−1+2C2212yes—
12C4⋊(3−1+2⋊C4)C2212yes—
13(3−1+2⋊C4)⋊C4—22no—
14C2×3−1+2⋊C8C2211yescharacteristic
15C8×AGL(1,ℤ/9)C414yescomplemented
16C2×C3×D3×C8—31no—
17C3×D3×C4⋊C4—32no—
18C32×#107D311yescharacteristic
19C3×#818—31no—
20C3×#807—32no—
21C2×C8×D9C2×C311yescharacteristic
22C4⋊C4×D9C2×C312yes—
23C9×#107—21no—
24C9×#107C2×C311yescharacteristic, complemented
25#63378C2×C311yescharacteristic, complemented
26#63366C2×C312yescomplemented
27C22×AGL(1,ℤ/9)D411yescharacteristic
28C2×C4×3−1+2—22no—
29C2×C4×3−1+2C2311yescharacteristic
30C2×3−1+2⋊C4—22no—
31C2×3−1+2⋊C4D411yescharacteristic
32C4×AGL(1,ℤ/9)—44no—
33C4×AGL(1,ℤ/9)C2×C412yes—
34C4×AGL(1,ℤ/9)Q812yes—
35C8×3−1+2C2×C412yescomplemented
363−1+2⋊C8C2×C412yescomplemented
37C2×C3×#107—91no—
38D3×#107—31no—
39C2×C3×C4×D3—31no—
40C2×C3×C4×D3—62no—
41C2×C32×C8C2×D311yescharacteristic
42C32×C4⋊C4C2×D312yes—
43C3×C4⋊Dic3—32no—
44C3×Dic3⋊C4—62no—
45C2×C3×C3⋊C8—31no—
46C3×D3×C8—34no—
47C2×C4×D9—22no—
48C2×C4×D9C22×C311yescharacteristic
49C2×C8×C9—21no—
50C2×C8×C9C22×C311yescharacteristic
51C9×C4⋊C4—22no—
52C9×C4⋊C4C22×C312yes—
53C4⋊Dic9C22×C312yes—
54Dic9⋊C4—22no—
55C2×C9⋊C8C22×C311yescharacteristic
56C8×D9C3×C414yescomplemented
57C22×3−1+2C2×D411yescharacteristic
58C2×AGL(1,ℤ/9)C4⋊C412yes—
59C2×AGL(1,ℤ/9)D812yes—
60C2×AGL(1,ℤ/9)Q1612yes—
61C4×3−1+2—42no—
62C4×3−1+2C22×C411yescharacteristic
63C4×3−1+2C2×Q811yescharacteristic
643−1+2⋊C4—42no—
653−1+2⋊C4C4⋊C412yes—
66C22×C3×C8—91no—
67C2×C3×C4⋊C4—92no—
68C2×D3×C8—31no—
69D3×C4⋊C4—32no—
70C3×#107—31no—
71C3×#107—92no—
72C3×#107—91no—
73C3×#107C3×D311yescharacteristic
74#818—31no—
75#807—32no—
76C22×C3×D3—31no—
77C2×C32×C4—22no—
78C2×C32×C4C22×D311yescharacteristic
79C2×C3×Dic3—31no—
80C2×C3×Dic3—62no—
81C3×C4×D3—32no—
82C3×C4×D3—32no—
83C3×C4×D3—124no—
84C22×D9C3×D411yescharacteristic
85C32×C8C4×D312yes—
86C3×C3⋊C8—32no—
87C2×C4×C9—22no—
88C2×C4×C9—21no—
89C2×C4×C9—42no—
90C2×C4×C9C23×C311yescharacteristic
91C2×Dic9—22no—
92C2×Dic9C3×D411yescharacteristic
93C4×D9—44no—
94C4×D9C2×C3×C412yes—
95C4×D9C3×Q812yes—
96C8×C9—22no—
97C8×C9C2×C3×C412yescomplemented
98C9⋊C8C2×C3×C412yescomplemented
99C2×#107—91noSylow
100C2×3−1+2C2×C4⋊C411yescharacteristic
101C2×3−1+2C2×D811yescharacteristic
102C2×3−1+2C2×Q1611yescharacteristic
103AGL(1,ℤ/9)#10714yesdirect product factor
104C22×C3×C4—91no—
105C22×C3×C4—182no—
106C2×C4×D3—31no—
107C2×C4×D3—62no—
108C2×C3×C8—31no—
109C2×C3×C8—94no—
110C2×C3×C8—91no—
111C2×C3×C8C2×C3×D311yescharacteristic
112C3×C4⋊C4—32no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4—182no—
115C3×C4⋊C4C2×C3×D312yes—
116C4⋊Dic3—32no—
117Dic3⋊C4—62no—
118C2×C3⋊C8—31no—
119D3×C8—34no—
120C22×C32D3×D411yescharacteristic
121C2×C3×D3—32no—
122C2×C3×D3—32no—
123C2×C3×D3—32no—
124C32×C4—42no—
125C32×C4C2×C4×D311yescharacteristic
126C32×C4D3×Q811yescharacteristic
127C3×Dic3—32no—
128C3×Dic3—122no—
129C22×C9—21no—
130C22×C9C2×C3×D411yescharacteristic
131C2×D9C3×C4⋊C412yes—
132C2×D9C3×D812yes—
133C2×D9C3×Q1612yes—
134C4×C9—21no—
135C4×C9—21no—
136C4×C9—42no—
137C4×C9—82no—
138C4×C9C22×C3×C411yescharacteristic
139C4×C9C2×C3×Q811yescharacteristic
140Dic9—42no—
141Dic9C3×C4⋊C412yes—
142C22×C8—91no—
143C2×C4⋊C4—92no—
144#107—91no—
145#107—92no—
146#107AGL(1,ℤ/9)11yescharacteristic, direct product factor
1473−1+2C2×#10711yescharacteristic, complemented, Sylow
148C23×C3—91no—
149C22×D3—31no—
150C2×C3×C4—22no—
151C2×C3×C4—31no—
152C2×C3×C4—62no—
153C2×C3×C4—92no—
154C2×C3×C4—92no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×C3×C4—364no—
158C2×C3×C4C22×C3×D311yescharacteristic
159C2×Dic3—31no—
160C2×Dic3—62no—
161C4×D3—32no—
162C4×D3—32no—
163C4×D3—124no—
164C3×C8—32no—
165C3×C8—92no—
166C3×C8C3×C4×D312yes—
167C3⋊C8—32no—
168C2×C32D3×C4⋊C411yescharacteristic
169C2×C32D3×D811yescharacteristic
170C2×C32D3×Q1611yescharacteristic
171C3×D3—34no—
172C2×C9—21no—
173C2×C9—21no—
174C2×C9—21no—
175C2×C9C2×C3×C4⋊C411yescharacteristic
176C2×C9C2×C3×D811yescharacteristic
177C2×C9C2×C3×Q1611yescharacteristic
178D9C3×#10714yescomplemented
179C22×C4—91no—
180C22×C4—182no—
181C2×C8—91no—
182C2×C8—94no—
183C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
184C4⋊C4—92no—
185C4⋊C4—182no—
186C4⋊C4C2×AGL(1,ℤ/9)12yes—
187C22×C3—31no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—92no—
191C22×C3C3×D3×D411yescharacteristic
192C2×D3—32no—
193C2×D3—32no—
194C2×D3—32no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—42no—
198C3×C4—92no—
199C3×C4—122no—
200C3×C4—362no—
201C3×C4C2×C3×C4×D311yescharacteristic
202C3×C4C3×D3×Q811yescharacteristic
203Dic3—32no—
204Dic3—122no—
205C32D3×#10711yescharacteristic
206C9—21no—
207C9C2×C3×#10711yescharacteristic, complemented
208C23—91no—
209C2×C4—22no—
210C2×C4—91no—
211C2×C4—92no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C4—364no—
215C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
216C8—92no—
217C8C4×AGL(1,ℤ/9)12yescomplemented
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—94no—
222C2×C3C3×D3×C4⋊C411yescharacteristic
223C2×C3C3×D3×D811yescharacteristic
224C2×C3C3×D3×Q1611yescharacteristic
225D3—34no—
226C22—92no—
227C22—92no—
228C22—92no—
229C22D4×AGL(1,ℤ/9)11yescharacteristic
230C4—42no—
231C4—92no—
232C4—362no—
233C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
234C4Q8×AGL(1,ℤ/9)11yescharacteristic
235C3—31no—
236C3C3×D3×#10711yescharacteristic
237C2—94no—
238C2C4⋊C4×AGL(1,ℤ/9)11yescharacteristic
239C2D8×AGL(1,ℤ/9)11yescharacteristic
240C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×AGL(1,ℤ/9)D4
cosocleC2×C4×C9C23×C3
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C2×C3×#107
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. 3−1+2×#107C2semidirect
4. #63378⋊C3C2semidirect
5. #63366⋊C3C2semidirect
7. D9×#107C3semidirect
15. C8×AGL(1,ℤ/9)C4semidirect
24. C9×#107C2×C3semidirect
25. #63378C2×C3semidirect
26. #63366C2×C3semidirect
35. C8×3−1+2C2×C4semidirect
36. 3−1+2⋊C8C2×C4semidirect
56. C8×D9C3×C4semidirect
97. C8×C9C2×C3×C4semidirect
98. C9⋊C8C2×C3×C4semidirect
103. AGL(1,ℤ/9)#107direct (N × H)
146. #107AGL(1,ℤ/9)direct (N × H)
147. 3−1+2C2×#107semidirect
178. D9C3×#107semidirect
207. C9C2×C3×#107semidirect
217. C8C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×C8×AGL(1,ℤ/9)2
C4⋊C4×AGL(1,ℤ/9)2
3−1+2×#1072
#63378⋊C32
#63366⋊C32
C3×D3×#1073
D9×#1073