Finite Group Databasebeta

← #10589802⋊C3 #13153206

Subgroups — #10589802⋊C3 #13153206

Order 1728.

688 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 202 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×#125C211yesmaximal, characteristic, complemented
2#63458⋊C3C214yesmaximal, complemented
3#63457⋊C3C214yesmaximal, complemented
4#63456⋊C3C214yesmaximal, complemented
5#63437⋊C3C212yesmaximal, complemented
6C3×#5700—31nomaximal
7#10589802C311yesmaximal, characteristic, complemented
8C2×C4⋊AGL(1,ℤ/9)C2214yes—
9C2×C4⋊AGL(1,ℤ/9)C2212yes—
10C2×C4×AGL(1,ℤ/9)C2214yes—
11C2×3−1+2⋊Q8C2212yes—
12C4⋊C4×3−1+2C2212yes—
13C4⋊C4×3−1+2C2214yes—
14C42×3−1+2C2211yescharacteristic
15(C2×AGL(1,ℤ/9))⋊C4C2214yes—
16(C2×AGL(1,ℤ/9))⋊C4C2214yes—
17(C2×AGL(1,ℤ/9))⋊C4C2214yes—
18(3−1+2⋊C4)⋊C4C2214yes—
19C32×#125D311yescharacteristic
20C3×#896—34no—
21C3×#895—34no—
22C3×#894—34no—
23C3×#875—32no—
24C9×#125—21no—
25C9×#125C2×C311yescharacteristic, complemented
26#63458C2×C314yescomplemented
27#63457C2×C314yescomplemented
28#63456C2×C314yescomplemented
29#63437C2×C312yescomplemented
30C22×AGL(1,ℤ/9)C2314yes—
31C2×C4×3−1+2C2312yes—
32C2×C4×3−1+2C2314yes—
33C2×C4×3−1+2C2311yescharacteristic
34C2×3−1+2⋊C4C2314yes—
35C4⋊AGL(1,ℤ/9)—42no—
36C4⋊AGL(1,ℤ/9)—44no—
37C4×AGL(1,ℤ/9)—44no—
383−1+2⋊Q8—42no—
39C3×#562—91no—
40#5700—31no—
41C2×C3×D12—34no—
42C2×C3×D12—32no—
43C2×C3×C4×D3—34no—
44C2×C3×Dic6—32no—
45C32×C4⋊C4C2×D312yes—
46C32×C4⋊C4C2×D314yes—
47C32×C42C2×D311yescharacteristic
48C3×(C2×D3)⋊C4—34no—
49C3×(C2×D3)⋊C4—34no—
50C3×(C2×D3)⋊C4—34no—
51C3×Dic3⋊C4—34no—
52C2×D36C22×C312yes—
53C2×D36C22×C314yes—
54C2×C4×D9C22×C314yes—
55C2×Dic18C22×C312yes—
56C9×C4⋊C4—22no—
57C9×C4⋊C4—24no—
58C9×C4⋊C4C22×C312yes—
59C9×C4⋊C4C22×C314yes—
60C42×C9—21no—
61C42×C9C22×C311yescharacteristic
62(C2×D9)⋊C4C22×C314yes—
63(C2×D9)⋊C4C22×C314yes—
64(C2×D9)⋊C4C22×C314yes—
65Dic9⋊C4C22×C314yes—
66C22×3−1+2C2411yescharacteristic
67C2×AGL(1,ℤ/9)—24no—
68C2×AGL(1,ℤ/9)—24no—
69C2×AGL(1,ℤ/9)—24no—
70C4×3−1+2—22no—
71C4×3−1+2—24no—
72C4×3−1+2—21no—
733−1+2⋊C4—24no—
74C3×#125—31no—
75C3×#125C3×D311yescharacteristic
76C3×#124—92no—
77C3×#123—94no—
78C3×C22⋊Q8—94no—
79C3×#121—94no—
80#896—34no—
81#895—34no—
82#894—34no—
83#875—32no—
84C22×C3×D3—34no—
85C2×C32×C4C22×D314yes—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D312yes—
88C2×C3×Dic3—34no—
89C3×D12—122no—
90C3×D12—124no—
91C3×C4×D3—124no—
92C3×Dic6—122no—
93C22×D9C23×C314yes—
94C2×C4×C9—21no—
95C2×C4×C9—22no—
96C2×C4×C9—24no—
97C2×C4×C9C23×C314yes—
98C2×C4×C9C23×C311yescharacteristic
99C2×C4×C9C23×C312yes—
100C2×Dic9C23×C314yes—
101D36—42no—
102D36—44no—
103C4×D9—44no—
104Dic18—42no—
105#562—91noSylow
106C2×3−1+2H(5,𝔽2)12yes—
107C2×3−1+22−1+411yescharacteristic
108AGL(1,ℤ/9)—44no—
109C2×C3×Q8—92no—
110C2×C3×D4—94no—
111C2×C3×D4—92no—
112C22×C3×C4—94no—
113C2×D12—32no—
114C2×D12—34no—
115C2×C4×D3—34no—
116C2×Dic6—32no—
117C3×C4⋊C4—34no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—94no—
120C3×C4⋊C4C2×C3×D312yes—
121C3×C4⋊C4C2×C3×D314yes—
122C3×C22⋊C4—94no—
123C3×C22⋊C4—94no—
124C3×C22⋊C4—94no—
125C3×C42—31no—
126C3×C42C2×C3×D311yescharacteristic
127(C2×D3)⋊C4—34no—
128(C2×D3)⋊C4—34no—
129(C2×D3)⋊C4—34no—
130Dic3⋊C4—34no—
131C22×C32C23×D311yescharacteristic
132C2×C3×D3—64no—
133C2×C3×D3—64no—
134C2×C3×D3—64no—
135C32×C4—21no—
136C32×C4—22no—
137C32×C4—24no—
138C3×Dic3—64no—
139C22×C9—21no—
140C22×C9C24×C311yescharacteristic
141C2×D9—24no—
142C2×D9—24no—
143C2×D9—24no—
144C4×C9—22no—
145C4×C9—22no—
146C4×C9—24no—
147C4×C9—21no—
148C4×C9—42no—
149C4×C9—44no—
150Dic9—24no—
151#125AGL(1,ℤ/9)11yescharacteristic, complemented
152#124—92no—
153#123—94no—
154C22⋊Q8—94no—
155#121—94no—
1563−1+2#56211yescharacteristic, complemented, Sylow
157C23×C3—94no—
158C22×D3—34no—
159C3×Q8—362no—
160C3×D4—364no—
161C3×D4—362no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—34no—
165C2×C3×C4—94no—
166C2×C3×C4—364no—
167C2×C3×C4C22×C3×D314yes—
168C2×C3×C4C22×C3×D311yescharacteristic
169C2×C3×C4C22×C3×D312yes—
170C2×Dic3—34no—
171D12—122no—
172D12—124no—
173C4×D3—124no—
174Dic6—122no—
175C2×C32#100711yescharacteristic
176C2×C32#100912yes—
177C3×D3—124no—
178C2×C9—21no—
179C2×C9—22no—
180C2×C9C3×H(5,𝔽2)12yes—
181C2×C9C3×2−1+411yescharacteristic
182D9—44no—
183C2×Q8—92no—
184C2×D4—94no—
185C2×D4—92no—
186C22×C4—94no—
187C4⋊C4—94no—
188C4⋊C4C2×AGL(1,ℤ/9)14yes—
189C4⋊C4C2×AGL(1,ℤ/9)12yes—
190C22⋊C4—94no—
191C22⋊C4—94no—
192C22⋊C4—94no—
193C42C2×AGL(1,ℤ/9)11yescharacteristic
194C22×C3—31no—
195C22×C3—184no—
196C22×C3—184no—
197C22×C3—184no—
198C22×C3C23×C3×D311yescharacteristic
199C2×D3—64no—
200C2×D3—64no—
201C2×D3—64no—
202C3×C4—21no—
203C3×C4—22no—
204C3×C4—24no—
205C3×C4—64no—
206C3×C4—62no—
207C3×C4—61no—
208C3×C4—184no—
209Dic3—64no—
210C32#570011yescharacteristic
211C9—21no—
212C9C3×#56211yescharacteristic, complemented
213C23—94no—
214Q8—362no—
215D4—364no—
216D4—362no—
217C2×C4—94no—
218C2×C4—364no—
219C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
220C2×C4C22×AGL(1,ℤ/9)12yes—
221C2×C4C22×AGL(1,ℤ/9)14yes—
222C2×C3—31no—
223C2×C3—32no—
224C2×C3—364no—
225C2×C3C3×#100711yescharacteristic
226C2×C3C3×#100912yes—
227D3—124no—
228C22—184no—
229C22—184no—
230C22—184no—
231C22C23×AGL(1,ℤ/9)11yescharacteristic
232C4—21no—
233C4—24no—
234C4—22no—
235C4—184no—
236C3—31no—
237C3C3×#570011yescharacteristic
238C2—364no—
239C2#1172263011yescharacteristic
240C2#1172264112yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#125C2
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#562
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#125C2semidirect
2. #63458⋊C3C2semidirect
3. #63457⋊C3C2semidirect
4. #63456⋊C3C2semidirect
5. #63437⋊C3C2semidirect
7. #10589802C3semidirect
25. C9×#125C2×C3semidirect
26. #63458C2×C3semidirect
27. #63457C2×C3semidirect
28. #63456C2×C3semidirect
29. #63437C2×C3semidirect
151. #125AGL(1,ℤ/9)semidirect
156. 3−1+2#562semidirect
212. C9C3×#562semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63458⋊C32
#63457⋊C32
#63456⋊C32
#63437⋊C32
C3×#57003
#105898023