Finite Group Databasebeta

← #10588376⋊C3 #13141483

Subgroups — #10588376⋊C3 #13141483

Order 1728.

376 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#11721394C211yesmaximal, characteristic
2#63431⋊C3C211yesmaximal, characteristic, complemented
3C4×C8×3−1+2C211yesmaximal, characteristic, complemented
4#63380⋊C3C214yesmaximal, complemented
5C3×#4694—31nomaximal
6#10588376C311yesmaximal, characteristic, complemented
7C2×C4⋊AGL(1,ℤ/9)—21no—
8C2×C4⋊AGL(1,ℤ/9)—21no—
9C2×C4⋊AGL(1,ℤ/9)C2212yes—
10C2×3−1+2⋊Q8—21no—
11C2×C8×3−1+2C2212yes—
12C42×3−1+2C2211yescharacteristic
13C4⋊(3−1+2⋊C4)—21no—
14C4⋊(3−1+2⋊C4)C2212yes—
15C3×#874—31no—
16C3×#869—31no—
17C32×C4×C8D311yescharacteristic
18C3×#821—34no—
19#63436C2×C311yescharacteristic
20#63431C2×C311yescharacteristic, complemented
21C4×C8×C9—21no—
22C4×C8×C9C2×C311yescharacteristic, complemented
23#63380C2×C314yescomplemented
24C22×AGL(1,ℤ/9)—22no—
25C2×C4×3−1+2D411yescharacteristic
26C2×C4×3−1+2D411yescharacteristic
27C2×C4×3−1+2C2311yescharacteristic
28C2×3−1+2⋊C4—22no—
29C4⋊AGL(1,ℤ/9)—24no—
30C4⋊AGL(1,ℤ/9)—42no—
31C4⋊AGL(1,ℤ/9)—42no—
323−1+2⋊Q8—42no—
33C8×3−1+2—22no—
34C3×#486—91no—
35#4694—31no—
36C2×C3×D12—32no—
37C2×C3×D12—61no—
38C2×C3×D12—61no—
39C2×C3×Dic6—61no—
40C2×C32×C8C2×D312yes—
41C32×C42C2×D311yescharacteristic
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—61no—
44C2×D36—21no—
45C2×D36—21no—
46C2×D36C22×C312yes—
47C2×Dic18—21no—
48C2×C8×C9—22no—
49C2×C8×C9C22×C312yes—
50C42×C9—21no—
51C42×C9C22×C311yescharacteristic
52C4⋊Dic9—21no—
53C4⋊Dic9C22×C312yes—
54C22×3−1+2C2×D411yescharacteristic
55C2×AGL(1,ℤ/9)—42no—
56C2×AGL(1,ℤ/9)—42no—
57C2×AGL(1,ℤ/9)—42no—
58C4×3−1+2D812yes—
59C4×3−1+2SD1612yes—
60C4×3−1+2#4112yes—
613−1+2⋊C4—42no—
62C3×C4⋊Q8—91no—
63C3×#127—91no—
64#874—31no—
65#869—31no—
66C3×#102—94no—
67C3×C4×C8—31no—
68C3×C4×C8C3×D311yescharacteristic
69#821—34no—
70C22×C3×D3—62no—
71C2×C32×C4D1211yescharacteristic
72C2×C32×C4D1211yescharacteristic
73C2×C32×C4C22×D311yescharacteristic
74C2×C3×Dic3—62no—
75C3×D12—64no—
76C3×D12—122no—
77C3×D12—122no—
78C3×Dic6—122no—
79C22×D9—22no—
80C32×C8—22no—
81C2×C4×C9—21no—
82C2×C4×C9—21no—
83C2×C4×C9—21no—
84C2×C4×C9C3×D411yescharacteristic
85C2×C4×C9C3×D411yescharacteristic
86C2×C4×C9C23×C311yescharacteristic
87C2×Dic9—22no—
88D36—24no—
89D36—42no—
90D36—42no—
91Dic18—42no—
92C8×C9—24no—
93C8×C9—22no—
94#486—91noSylow
95C2×3−1+2#12411yescharacteristic
96C2×3−1+2C2×D811yescharacteristic
97C2×3−1+2C2×SD1611yescharacteristic
98AGL(1,ℤ/9)—82no—
99C2×C3×Q8—181no—
100C2×C3×D4—92no—
101C2×C3×D4—181no—
102C2×C3×D4—181no—
103C2×D12—32no—
104C2×D12—61no—
105C2×D12—61no—
106C2×Dic6—61no—
107C2×C3×C8—32no—
108C2×C3×C8C2×C3×D312yes—
109C3×C4⋊C4—92no—
110C3×C4⋊C4—181no—
111C3×C42—31no—
112C3×C42C2×C3×D311yescharacteristic
113C4⋊Dic3—32no—
114C4⋊Dic3—61no—
115C22×C32C2×D1211yescharacteristic
116C2×C3×D3—122no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C32×C4#20412yes—
120C32×C4D2412yes—
121C32×C4#23512yes—
122C3×Dic3—122no—
123C22×C9—21no—
124C22×C9C2×C3×D411yescharacteristic
125C2×D9—42no—
126C2×D9—42no—
127C2×D9—42no—
128C4×C9—22no—
129C4×C9—22no—
130C4×C9—22no—
131C4×C9C3×D812yes—
132C4×C9C3×SD1612yes—
133C4×C9C3×#4112yes—
134Dic9—42no—
135C4⋊Q8—91no—
136#127—91no—
137#102—94no—
138C4×C8AGL(1,ℤ/9)11yescharacteristic, complemented
1393−1+2#48611yescharacteristic, complemented, Sylow
140C23×C3—182no—
141C22×D3—62no—
142C3×Q8—362no—
143C3×D4—184no—
144C3×D4—362no—
145C3×D4—362no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—182no—
150C2×C3×C4C3×D1211yescharacteristic
151C2×C3×C4C3×D1211yescharacteristic
152C2×C3×C4C22×C3×D311yescharacteristic
153C2×Dic3—62no—
154D12—64no—
155D12—122no—
156D12—122no—
157Dic6—122no—
158C3×C8—22no—
159C3×C8—62no—
160C2×C32#87511yescharacteristic
161C2×C32C2×#20411yescharacteristic
162C2×C32C2×D2411yescharacteristic
163C3×D3—242no—
164C2×C9—21no—
165C2×C9—21no—
166C2×C9—21no—
167C2×C9C3×#12411yescharacteristic
168C2×C9C2×C3×D811yescharacteristic
169C2×C9C2×C3×SD1611yescharacteristic
170D9—82no—
171C2×Q8—181no—
172C2×D4—92no—
173C2×D4—181no—
174C2×D4—181no—
175C2×C8C2×AGL(1,ℤ/9)12yes—
176C4⋊C4—92no—
177C4⋊C4—181no—
178C42C2×AGL(1,ℤ/9)11yescharacteristic
179C22×C3—31no—
180C22×C3—362no—
181C22×C3—362no—
182C22×C3—362no—
183C22×C3C2×C3×D1211yescharacteristic
184C2×D3—122no—
185C2×D3—122no—
186C2×D3—122no—
187C3×C4—32no—
188C3×C4—32no—
189C3×C4—32no—
190C3×C4—362no—
191C3×C4C3×#20412yes—
192C3×C4C3×D2412yes—
193C3×C4C3×#23512yes—
194Dic3—122no—
195C32#469411yescharacteristic
196C9—21no—
197C9C3×#48611yescharacteristic, complemented
198C23—182no—
199Q8—362no—
200D4—184no—
201D4—362no—
202D4—362no—
203C2×C4—182no—
204C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
205C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
206C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
207C8—22no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—31no—
211C2×C3—722no—
212C2×C3C3×#87511yescharacteristic
213C2×C3C2×C3×#20411yescharacteristic
214C2×C3C2×C3×D2411yescharacteristic
215D3—242no—
216C22—362no—
217C22—362no—
218C22—362no—
219C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
220C4—362no—
221C4#3664⋊C312yes—
222C4D72⋊C312yes—
223C4#3697⋊C312yes—
224C3—31no—
225C3C3×#469411yescharacteristic
226C2—722no—
227C2#63437⋊C311yescharacteristic
228C2C2×#3664⋊C311yescharacteristic
229C2C2×D72⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×C8×3−1+2C2
socleC22×C3C2×C3×D12
prime radical#11721394C2
hypercenterC4×C8AGL(1,ℤ/9)
hypocenterC9C3×#486
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #63431⋊C3C2semidirect
3. C4×C8×3−1+2C2semidirect
4. #63380⋊C3C2semidirect
6. #10588376C3semidirect
20. #63431C2×C3semidirect
22. C4×C8×C9C2×C3semidirect
23. #63380C2×C3semidirect
138. C4×C8AGL(1,ℤ/9)semidirect
139. 3−1+2#486semidirect
197. C9C3×#486semidirect

Maximal subgroups

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

subgroupindex
#117213942
#63431⋊C32
C4×C8×3−1+22
#63380⋊C32
C3×#46943
#105883763