Finite Group Databasebeta

← #10588432⋊C3 #13141704

Subgroups — #10588432⋊C3 #13141704

Order 1728.

445 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 90 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#11722630C211yesmaximal, characteristic
2#63506⋊C3C211yesmaximal, characteristic, complemented
3#63471⋊C3C212yesmaximal, complemented
43−1+2×#101C211yesmaximal, characteristic, complemented
5#63384⋊C3C212yesmaximal, complemented
6C3×#4752—31nomaximal
7#10588432C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9#88724—22no—
10Q8×AGL(1,ℤ/9)—22no—
11#3697⋊C3—22no—
12#3697⋊C3C2212yes—
13C2×3−1+2⋊Q8C2211yescharacteristic
14M16×3−1+2C2212yes—
15(3−1+2⋊C4)⋊C4—22no—
16C4×3−1+2⋊C4C2211yescharacteristic
17#3664⋊C3—22no—
18Dic36⋊C3—22no—
19C3×#1007—31no—
20C3×#944—31no—
21C3×#909—32no—
22#63713C2×C311yescharacteristic
23C32×#101D311yescharacteristic
24C3×#825—32no—
25#63506C2×C311yescharacteristic, complemented
26#63471C2×C312yescomplemented
27C9×#101—21no—
28C9×#101C2×C311yescharacteristic, complemented
29#63384C2×C312yescomplemented
30Q8×3−1+2—22no—
31C2×C4×3−1+2D412yes—
32C2×C4×3−1+2C2311yescharacteristic
33#611⋊C3—22no—
34C2×3−1+2⋊C4—21no—
35C2×3−1+2⋊C4—21no—
36C4⋊AGL(1,ℤ/9)—22no—
37C4⋊AGL(1,ℤ/9)D412yes—
38C4×AGL(1,ℤ/9)—24no—
39C4×AGL(1,ℤ/9)—22no—
403−1+2⋊Q8—22no—
413−1+2⋊Q8—22no—
423−1+2⋊Q8D412yes—
43C8×3−1+2—22no—
44C3×#456—91no—
45#4752—31no—
46C2×C32×Q8C2×D311yescharacteristic
47C3×#239—62no—
48C3×D3×Q8—62no—
49C3×#235—32no—
50C3×#235—62no—
51C2×C3×Dic6—31no—
52C32×M16C2×D312yes—
53C3×Dic3⋊C4—62no—
54C3×C4×Dic3—31no—
55C3×Dic12—62no—
56C3×#204—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C22×C311yescharacteristic
59#3701—22no—
60Q8×D9—22no—
61#3697—22no—
62#3697C22×C312yes—
63C2×Dic18C22×C311yescharacteristic
64C9×M16—22no—
65C9×M16C22×C312yes—
66Dic9⋊C4—22no—
67C4×Dic9C22×C311yescharacteristic
68#3664—22no—
69Dic36—22no—
70C22×3−1+2C2×D411yescharacteristic
71C2×AGL(1,ℤ/9)—22no—
72C4×3−1+2—22no—
73C4×3−1+2C2×D412yes—
743−1+2⋊C4—22no—
753−1+2⋊C4—22no—
763−1+2⋊C4—41no—
77C3×2−1+4—91no—
78#1007—31no—
79C3×#137—92no—
80C3×C4⋊Q8—91no—
81#944—31no—
82#909—32no—
83C3×#104—92no—
84C3×#101—31no—
85C3×#101C3×D311yescharacteristic
86#825—32no—
87C32×Q8—22no—
88C2×C32×C4D1212yes—
89C2×C32×C4C22×D311yescharacteristic
90C3×#67—62no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C3×D12—32no—
94C3×D12—62no—
95C3×C4×D3—64no—
96C3×C4×D3—62no—
97C3×Dic6—32no—
98C3×Dic6—62no—
99C3×Dic6—62no—
100C32×C8—22no—
101Q8×C9—22no—
102Q8×C9—42no—
103C2×C4×C9—22no—
104C2×C4×C9—21no—
105C2×C4×C9C3×D412yes—
106C2×C4×C9C23×C311yescharacteristic
107#611—22no—
108C2×Dic9—21no—
109C2×Dic9—21no—
110D36—22no—
111D36C3×D412yes—
112C4×D9—24no—
113C4×D9—22no—
114Dic18—22no—
115Dic18—22no—
116Dic18C3×D412yes—
117C8×C9—22no—
118C8×C9—42no—
119#456—91noSylow
120C2×3−1+2—21no—
121C2×3−1+2#12011yescharacteristic
122AGL(1,ℤ/9)—42no—
123C3×#41—92no—
124C3×#41—182no—
125C3×#41—182no—
126C2×C3×Q8—31no—
127C2×C3×Q8—91no—
128C2×C3×Q8—182no—
129C2×C3×Q8C2×C3×D311yescharacteristic
130#239—62no—
131D3×Q8—62no—
132#235—32no—
133#235—62no—
134C2×Dic6—31no—
135C3×Q16—182no—
136C3×SD16—182no—
137C3×M16—32no—
138C3×M16C2×C3×D312yes—
139C3×C4⋊C4—182no—
140C3×C42—91no—
141Dic3⋊C4—62no—
142C4×Dic3—31no—
143Dic12—62no—
144#204—62no—
145C22×C32C2×D1211yescharacteristic
146C2×C3×D3—62no—
147C32×C4—22no—
148C32×C4D3×D412yes—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—121no—
152C22×C9—21no—
153C22×C9C2×C3×D411yescharacteristic
154C2×D9—22no—
155C4×C9—22no—
156C4×C9—22no—
157C4×C9—42no—
158C4×C9C2×C3×D412yes—
159Dic9—22no—
160Dic9—22no—
161Dic9—41no—
1622−1+4—91no—
163#137—92no—
164C4⋊Q8—91no—
165#104—92no—
166#101AGL(1,ℤ/9)11yescharacteristic, complemented
1673−1+2#45611yescharacteristic, complemented, Sylow
168C3×Q8—22no—
169C3×Q8—62no—
170C3×Q8—92no—
171C3×Q8—182no—
172C3×Q8—182no—
173C3×D4—92no—
174C3×D4—182no—
175C3×D4—182no—
176C2×C3×C4—32no—
177C2×C3×C4—31no—
178C2×C3×C4—182no—
179C2×C3×C4—184no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4C3×D1212yes—
183C2×C3×C4C22×C3×D311yescharacteristic
184#67—62no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187D12—32no—
188D12—62no—
189C4×D3—64no—
190C4×D3—62no—
191Dic6—32no—
192Dic6—62no—
193Dic6—62no—
194C3×C8—22no—
195C3×C8—62no—
196C2×C32—21no—
197C2×C32#88211yescharacteristic
198C3×D3—122no—
199C2×C9—21no—
200C2×C9—21no—
201C2×C9—41no—
202C2×C9C3×#12011yescharacteristic
203D9—42no—
204#41—92no—
205#41—182no—
206#41—182no—
207C2×Q8—91no—
208C2×Q8—182no—
209C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
210Q16—182no—
211SD16—182no—
212M16C2×AGL(1,ℤ/9)12yes—
213C4⋊C4—182no—
214C42—91no—
215C22×C3—31no—
216C22×C3—182no—
217C22×C3C2×C3×D1211yescharacteristic
218C2×D3—62no—
219C3×C4—22no—
220C3×C4—32no—
221C3×C4—62no—
222C3×C4—182no—
223C3×C4—182no—
224C3×C4—361no—
225C3×C4C3×D3×D412yes—
226Dic3—62no—
227Dic3—62no—
228Dic3—121no—
229C32#475211yescharacteristic
230C9—21no—
231C9C3×#45611yescharacteristic, complemented
232Q8—22no—
233Q8—92no—
234Q8—182no—
235Q8—182no—
236D4—92no—
237D4—182no—
238D4—182no—
239C2×C4—184no—
240C2×C4—181no—
241C2×C4—182no—
242C2×C4—181no—
243C2×C4C4⋊AGL(1,ℤ/9)12yes—
244C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
245C8—22no—
246C2×C3—21no—
247C2×C3—31no—
248C2×C3—61no—
249C2×C3—362no—
250C2×C3C3×#88211yescharacteristic
251D3—122no—
252C22—182no—
253C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
254C4—22no—
255C4—182no—
256C4—182no—
257C4—361no—
258C4D4×AGL(1,ℤ/9)12yes—
259C3—31no—
260C3C3×#475211yescharacteristic
261C2—21no—
262C2—362no—
263C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#101C2
socleC2×C3C3×#882
prime radical#11722630C2
hypercenter#101AGL(1,ℤ/9)
hypocenterC9C3×#456
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #63506⋊C3C2semidirect
3. #63471⋊C3C2semidirect
4. 3−1+2×#101C2semidirect
5. #63384⋊C3C2semidirect
7. #10588432C3semidirect
25. #63506C2×C3semidirect
26. #63471C2×C3semidirect
28. C9×#101C2×C3semidirect
29. #63384C2×C3semidirect
166. #101AGL(1,ℤ/9)semidirect
167. 3−1+2#456semidirect
231. C9C3×#456semidirect

Maximal subgroups

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

subgroupindex
#117226302
#63506⋊C32
#63471⋊C32
3−1+2×#1012
#63384⋊C32
C3×#47523
#105884323