Finite Group Databasebeta

← 2−1+4×AGL(1,ℤ/9) #13164961

Subgroups — 2−1+4×AGL(1,ℤ/9) #13164961

Order 1728.

2455 conjugacy classes of subgroups; only the 958 normal subgroups are shown.

normal subgroups only

This group has 2455 conjugacy classes of subgroups — too many to draw. Only its 958 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
13−1+2×2−1+4C211yesmaximal, characteristic, complemented
2#11722642C2110yesmaximal, complemented
3#41×AGL(1,ℤ/9)C2110yesmaximal, complemented
4#11722630C215yesmaximal, complemented
5C2×Q8×AGL(1,ℤ/9)C215yesmaximal, complemented
6C3×D3×2−1+4—31nomaximal
7D9×2−1+4C311yesmaximal, characteristic, complemented
8#41×3−1+2C22110yescomplemented
9C2×Q8×3−1+2C2215yescomplemented
10#88724C22110yescomplemented
11Q8×AGL(1,ℤ/9)C22110yes—
12Q8×AGL(1,ℤ/9)C22120yescomplemented
13#3699⋊C3C22130yescomplemented
14D4×AGL(1,ℤ/9)C22110yes—
15#3697⋊C3C22130yescomplemented
16C2×C4×AGL(1,ℤ/9)C22115yes—
17C2×3−1+2⋊Q8C22115yescomplemented
18C32×2−1+4D311yescharacteristic
19C3×#1010—310no—
20C3×D3×#41—310no—
21C3×#1007—35no—
22C2×C3×D3×Q8—35no—
23C9×2−1+4—21no—
24C9×2−1+4C2×C311yescharacteristic, complemented
25#63716C2×C3110yescomplemented
26#41×D9C2×C3110yescomplemented
27#63713C2×C315yescomplemented
28C2×Q8×D9C2×C315yescomplemented
29C22×AGL(1,ℤ/9)C2315yes—
30Q8×3−1+2C23110yes—
31D4×3−1+2C23110yes—
32C2×C4×3−1+2C23115yes—
33#611⋊C3C23120yes—
34C2×3−1+2⋊C4C23115yes—
35C4⋊AGL(1,ℤ/9)C23110yes—
36C4×AGL(1,ℤ/9)C23110yes—
37C4×AGL(1,ℤ/9)C23130yes—
383−1+2⋊Q8C23130yes—
39C2×C3×2−1+4—91no—
40D3×2−1+4—31no—
41C32×#41C2×D3110yes—
42C2×C32×Q8C2×D315yes—
43C3×#239—310no—
44C3×D3×Q8—310no—
45C3×D3×Q8—320no—
46C3×#237—330no—
47C3×D3×D4—310no—
48C3×#235—330no—
49C2×C3×C4×D3—315no—
50C2×C3×Dic6—315no—
51C9×#41—210no—
52C9×#41C22×C3110yescomplemented
53C2×Q8×C9—25no—
54C2×Q8×C9C22×C315yescomplemented
55#3701C22×C3110yescomplemented
56Q8×D9C22×C3110yes—
57Q8×D9C22×C3120yescomplemented
58#3699C22×C3130yescomplemented
59D4×D9C22×C3110yes—
60#3697C22×C3130yescomplemented
61C2×C4×D9C22×C3115yes—
62C2×Dic18C22×C3115yescomplemented
63C22×3−1+2C2415yes—
64C2×AGL(1,ℤ/9)—210no—
65C2×AGL(1,ℤ/9)C2415yes—
66C2×AGL(1,ℤ/9)C2411yescharacteristic
67C4×3−1+2C24110yes—
683−1+2⋊C4C24110yes—
69C3×2−1+4—31no—
70C3×2−1+4—95no—
71C3×2−1+4—910no—
72C3×2−1+4C3×D311yescharacteristic
73C2×C3×#41—910no—
74C22×C3×Q8—95no—
75#1010—310no—
76D3×#41—310no—
77#1007—35no—
78C2×D3×Q8—35no—
79C22×C3×D3—35no—
80C32×Q8C22×D3110yes—
81C32×D4C22×D3110yes—
82C2×C32×C4C22×D3115yes—
83C3×#67—320no—
84C2×C3×Dic3—315no—
85C3×D12—310no—
86C3×C4×D3—310no—
87C3×C4×D3—330no—
88C3×Dic6—330no—
89C22×D9C23×C315yes—
90Q8×C9—210no—
91Q8×C9C23×C3110yes—
92D4×C9—210no—
93D4×C9C23×C3110yes—
94C2×C4×C9—215no—
95C2×C4×C9C23×C3115yes—
96#611C23×C3120yes—
97C2×Dic9C23×C3115yes—
98D36C23×C3110yes—
99C4×D9C23×C3110yes—
100C4×D9C23×C3130yes—
101Dic18C23×C3130yes—
102C2×2−1+4—91noSylow
103C2×3−1+2—25no—
104C2×3−1+2C2511yescharacteristic
105AGL(1,ℤ/9)—25no—
106AGL(1,ℤ/9)2−1+412yesdirect product factor
107C3×#41—310no—
108C3×#41—930no—
109C3×#41—930no—
110C3×#41—910no—
111C3×#41C2×C3×D3110yes—
112C2×C3×Q8—35no—
113C2×C3×Q8—915no—
114C2×C3×Q8—910no—
115C2×C3×Q8—920no—
116C2×C3×Q8C2×C3×D315yes—
117C2×C3×D4—910no—
118C22×C3×C4—915no—
119#239—310no—
120D3×Q8—310no—
121D3×Q8—320no—
122#237—330no—
123D3×D4—310no—
124#235—330no—
125C2×C4×D3—315no—
126C2×Dic6—315no—
127C22×C32C23×D315yes—
128C2×C3×D3—31no—
129C2×C3×D3—35no—
130C2×C3×D3—610no—
131C32×C4C23×D3110yes—
132C3×Dic3—310no—
133C22×C9—25no—
134C22×C9C24×C315yes—
135C2×D9—210no—
136C2×D9C24×C315yes—
137C2×D9C24×C311yescharacteristic
138C4×C9—210no—
139C4×C9C24×C3110yes—
140Dic9C24×C3110yes—
1412−1+4—910no—
1422−1+4—95no—
1432−1+4AGL(1,ℤ/9)11yescharacteristic, direct product factor
144C2×#41—910no—
145C22×Q8—95no—
1463−1+2C2×2−1+411yescharacteristic, complemented, Sylow
147C23×C3—95no—
148C22×D3—35no—
149C3×Q8—310no—
150C3×Q8—930no—
151C3×Q8C22×C3×D3110yes—
152C3×D4—310no—
153C3×D4—910no—
154C3×D4—920no—
155C3×D4C22×C3×D3110yes—
156C2×C3×C4—315no—
157C2×C3×C4—910no—
158C2×C3×C4—930no—
159C2×C3×C4—915no—
160C2×C3×C4C22×C3×D3115yes—
161#67—320no—
162C2×Dic3—315no—
163D12—310no—
164C4×D3—310no—
165C4×D3—330no—
166Dic6—330no—
167C2×C32—25no—
168C2×C32C24×D311yescharacteristic
169C3×D3—32no—
170C3×D3—65no—
171C2×C9—25no—
172C2×C9—21no—
173C2×C9—45no—
174C2×C9C25×C311yescharacteristic
175D9—25no—
176D9C3×2−1+412yescomplemented
177#41—910no—
178#41—930no—
179#41—930no—
180#41C2×AGL(1,ℤ/9)110yescomplemented
181C2×Q8—910no—
182C2×Q8—920no—
183C2×Q8—915no—
184C2×Q8C2×AGL(1,ℤ/9)15yescomplemented
185C2×D4—910no—
186C22×C4—915no—
187C22×C3—35no—
188C22×C3—91no—
189C22×C3—95no—
190C22×C3—1810no—
191C22×C3C23×C3×D315yes—
192C2×D3—31no—
193C2×D3—35no—
194C2×D3—610no—
195C3×C4—310no—
196C3×C4—910no—
197C3×C4C23×C3×D3110yes—
198Dic3—310no—
199C32D3×2−1+411yescharacteristic
200C9—21no—
201C9C2×C3×2−1+411yescharacteristic, complemented
202C23—95no—
203Q8—930no—
204Q8C22×AGL(1,ℤ/9)110yes—
205D4—920no—
206D4—910no—
207D4C22×AGL(1,ℤ/9)110yes—
208C2×C4—915no—
209C2×C4—910no—
210C2×C4—930no—
211C2×C4C22×AGL(1,ℤ/9)115yes—
212C2×C3—25no—
213C2×C3—31no—
214C2×C3—65no—
215C2×C3—92no—
216C2×C3—185no—
217C2×C3C24×C3×D311yescharacteristic
218D3—32no—
219D3—65no—
220C22—91no—
221C22—95no—
222C22—1810no—
223C22C23×AGL(1,ℤ/9)15yes—
224C4—910no—
225C4C23×AGL(1,ℤ/9)110yes—
226C3—31no—
227C3C3×D3×2−1+411yescharacteristic
228C2—25no—
229C2—92no—
230C2—185no—
231C2C24×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C24×AGL(1,ℤ/9)
commutator, cosocleC2×C9C25×C3
Frattini, socleC2×C3C24×C3×D3
Fitting3−1+2×2−1+4C2
hypercenter2−1+4AGL(1,ℤ/9)
hypocenterC9C2×C3×2−1+4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. 3−1+2×2−1+4C2semidirect
2. #11722642C2semidirect
3. #41×AGL(1,ℤ/9)C2semidirect
4. #11722630C2semidirect
5. C2×Q8×AGL(1,ℤ/9)C2semidirect
7. D9×2−1+4C3semidirect
8. #41×3−1+2C22semidirect
9. C2×Q8×3−1+2C22semidirect
10. #88724C22semidirect
12. Q8×AGL(1,ℤ/9)C22semidirect
13. #3699⋊C3C22semidirect
15. #3697⋊C3C22semidirect
17. C2×3−1+2⋊Q8C22semidirect
24. C9×2−1+4C2×C3semidirect
25. #63716C2×C3semidirect
26. #41×D9C2×C3semidirect
27. #63713C2×C3semidirect
28. C2×Q8×D9C2×C3semidirect
52. C9×#41C22×C3semidirect
54. C2×Q8×C9C22×C3semidirect
55. #3701C22×C3semidirect
57. Q8×D9C22×C3semidirect
58. #3699C22×C3semidirect
60. #3697C22×C3semidirect
62. C2×Dic18C22×C3semidirect
106. AGL(1,ℤ/9)2−1+4direct (N × H)
143. 2−1+4AGL(1,ℤ/9)direct (N × H)
146. 3−1+2C2×2−1+4semidirect
176. D9C3×2−1+4semidirect
180. #41C2×AGL(1,ℤ/9)semidirect
184. C2×Q8C2×AGL(1,ℤ/9)semidirect
201. C9C2×C3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
3−1+2×2−1+42
#117226422
#41×AGL(1,ℤ/9)2
#117226302
C2×Q8×AGL(1,ℤ/9)2
C3×D3×2−1+43
D9×2−1+43