Finite Group Databasebeta

← #10588650⋊C3 #13142639

Subgroups — #10588650⋊C3 #13142639

Order 1728.

388 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 102 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×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic
3#63389⋊C3C211yesmaximal, characteristic, complemented
4#63369⋊C3C212yesmaximal, complemented
5#63366⋊C3C212yesmaximal, complemented
6C3×#4967—31nomaximal
7#10588650C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2—21no—
9C22×C4×3−1+2C2211yescharacteristic
10C2×#611⋊C3—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C4⋊C4×3−1+2—21no—
13C4⋊C4×3−1+2C2212yes—
14(C2×AGL(1,ℤ/9))⋊C4—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C2×3−1+2⋊C8C2212yes—
17C2×C32×C4⋊C4D311yescharacteristic
18C3×#930—31no—
19C3×#830—31no—
20C3×#809—32no—
21C3×#807—32no—
22C2×C9×C4⋊C4—21no—
23C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
24#63492C2×C311yescharacteristic
25#63389C2×C311yescharacteristic, complemented
26#63369C2×C312yescomplemented
27#63366C2×C312yescomplemented
28C23×3−1+2D411yescharacteristic
29C22×AGL(1,ℤ/9)—21no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2—42no—
33C2×C4×3−1+2D411yescharacteristic
34C2×C4×3−1+2C2311yescharacteristic
35#611⋊C3—42no—
36C2×3−1+2⋊C4—21no—
37C4⋊AGL(1,ℤ/9)—22no—
383−1+2⋊C8—22no—
39C3×#480—91no—
40#4967—31no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C2×C3×#67—61no—
44C2×C3×D12—31no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4C2×D312yes—
47C3×(C2×D3)⋊C4—61no—
48C3×C4⋊Dic3—31no—
49C2×C3×C3⋊C8—32no—
50C22×C4×C9—21no—
51C22×C4×C9—22no—
52C22×C4×C9—21no—
53C22×C4×C9C22×C311yescharacteristic
54C2×#611—21no—
55C2×D36C22×C311yescharacteristic
56C9×C4⋊C4—22no—
57C9×C4⋊C4—22no—
58C9×C4⋊C4—21no—
59C9×C4⋊C4C22×C312yes—
60(C2×D9)⋊C4—21no—
61C4⋊Dic9C22×C311yescharacteristic
62C2×C9⋊C8C22×C312yes—
63C22×3−1+2—21no—
64C22×3−1+2—21no—
65C22×3−1+2D812yes—
66C22×3−1+2C2×D411yescharacteristic
67C2×AGL(1,ℤ/9)—41no—
68C2×AGL(1,ℤ/9)—41no—
69C2×AGL(1,ℤ/9)—41no—
70C4×3−1+2—21no—
71C4×3−1+2—42no—
72C4×3−1+2#4112yes—
733−1+2⋊C4—41no—
74C3×#121—91no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4C3×D311yescharacteristic
77#930—31no—
78C3×#107—92no—
79C3×#102—92no—
80C3×C22⋊C8—91no—
81#830—31no—
82#809—32no—
83#807—32no—
84C23×C32#6711yescharacteristic
85C22×C3×D3—61no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—42no—
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4C22×D311yescharacteristic
91C3×#67—122no—
92C2×C3×Dic3—61no—
93C3×D12—62no—
94C23×C9—21no—
95C23×C9C3×D411yescharacteristic
96C22×D9—21no—
97C3×C3⋊C8—62no—
98C2×C4×C9—24no—
99C2×C4×C9—22no—
100C2×C4×C9—24no—
101C2×C4×C9—22no—
102C2×C4×C9—21no—
103C2×C4×C9—21no—
104C2×C4×C9—44no—
105C2×C4×C9—42no—
106C2×C4×C9C3×D411yescharacteristic
107C2×C4×C9C23×C311yescharacteristic
108#611—42no—
109C2×Dic9—21no—
110D36—22no—
111C9⋊C8—22no—
112#480—91noSylow
113C2×3−1+2—22no—
114C2×3−1+2#12311yescharacteristic
115C2×3−1+2C2×D811yescharacteristic
116C2×3−1+2#13711yescharacteristic
117AGL(1,ℤ/9)—81no—
118C2×C3×D4—91no—
119C2×C3×D4—181no—
120C22×C3×C4—21no—
121C22×C3×C4—31no—
122C22×C3×C4—61no—
123C22×C3×C4C2×C3×D311yescharacteristic
124C2×#67—61no—
125C2×D12—31no—
126C2×C3×C8—92no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4C2×C3×D312yes—
132C3×C22⋊C4—181no—
133(C2×D3)⋊C4—61no—
134C4⋊Dic3—31no—
135C2×C3⋊C8—32no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32#21312yes—
139C22×C32C2×#6711yescharacteristic
140C2×C3×D3—121no—
141C2×C3×D3—121no—
142C2×C3×D3—121no—
143C32×C4—21no—
144C32×C4—42no—
145C32×C4#23512yes—
146C3×Dic3—121no—
147C22×C9—22no—
148C22×C9—22no—
149C22×C9—21no—
150C22×C9—21no—
151C22×C9—22no—
152C22×C9—21no—
153C22×C9C3×D812yes—
154C22×C9C2×C3×D411yescharacteristic
155C2×D9—41no—
156C2×D9—41no—
157C2×D9—41no—
158C4×C9—22no—
159C4×C9—21no—
160C4×C9—22no—
161C4×C9—44no—
162C4×C9—42no—
163C4×C9C3×#4112yes—
164Dic9—41no—
165#121—91no—
166C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
167#107—92no—
168#102—92no—
169C22⋊C8—91no—
1703−1+2#48011yescharacteristic, complemented, Sylow
171C23×C3—31no—
172C23×C3—181no—
173C23×C3C3×#6711yescharacteristic
174C22×D3—61no—
175C3×D4—182no—
176C3×D4—362no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—42no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—122no—
185C2×C3×C4—181no—
186C2×C3×C4C3×#6711yescharacteristic
187C2×C3×C4C22×C3×D311yescharacteristic
188#67—122no—
189C2×Dic3—61no—
190D12—62no—
191C3×C8—182no—
192C3⋊C8—62no—
193C2×C32—22no—
194C2×C32#92911yescharacteristic
195C2×C32C2×#21311yescharacteristic
196C2×C32#94211yescharacteristic
197C3×D3—241no—
198C2×C9—24no—
199C2×C9—22no—
200C2×C9—21no—
201C2×C9—21no—
202C2×C9—21no—
203C2×C9C3×#12311yescharacteristic
204C2×C9C2×C3×D811yescharacteristic
205C2×C9C3×#13711yescharacteristic
206D9—81no—
207C2×D4—91no—
208C2×D4—181no—
209C22×C4—21no—
210C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
211C2×C8—92no—
212C4⋊C4—21no—
213C4⋊C4—91no—
214C4⋊C4C2×AGL(1,ℤ/9)12yes—
215C22⋊C4—181no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—31no—
219C22×C3—32no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—361no—
223C22×C3—361no—
224C22×C3—361no—
225C22×C3C3×#21312yes—
226C22×C3C2×C3×#6711yescharacteristic
227C2×D3—121no—
228C2×D3—121no—
229C2×D3—121no—
230C3×C4—21no—
231C3×C4—32no—
232C3×C4—42no—
233C3×C4—61no—
234C3×C4—122no—
235C3×C4—361no—
236C3×C4C3×#23512yes—
237Dic3—121no—
238C32#496711yescharacteristic
239C9—21no—
240C9C3×#48011yescharacteristic, complemented
241C23—181no—
242C23#611⋊C311yescharacteristic
243D4—182no—
244D4—362no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—181no—
249C2×C4#611⋊C311yescharacteristic
250C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
251C8—182no—
252C2×C3—22no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—62no—
257C2×C3—721no—
258C2×C3C3×#92911yescharacteristic
259C2×C3C2×C3×#21311yescharacteristic
260C2×C3C3×#94211yescharacteristic
261D3—241no—
262C22—21no—
263C22—21no—
264C22—361no—
265C22—361no—
266C22—361no—
267C22#3673⋊C312yes—
268C22C2×#611⋊C311yescharacteristic
269C4—21no—
270C4—42no—
271C4—361no—
272C4#3697⋊C312yes—
273C3—31no—
274C3C3×#496711yescharacteristic
275C2—22no—
276C2—721no—
277C2#63491⋊C311yescharacteristic
278C2C2×#3673⋊C311yescharacteristic
279C2#63505⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
socleC22×C3C2×C3×#67
prime radical#63492⋊C3C2
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#480
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
3. #63389⋊C3C2semidirect
4. #63369⋊C3C2semidirect
5. #63366⋊C3C2semidirect
7. #10588650C3semidirect
23. C2×C9×C4⋊C4C2×C3semidirect
25. #63389C2×C3semidirect
26. #63369C2×C3semidirect
27. #63366C2×C3semidirect
166. C2×C4⋊C4AGL(1,ℤ/9)semidirect
170. 3−1+2#480semidirect
240. C9C3×#480semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
#63492⋊C32
#63389⋊C32
#63369⋊C32
#63366⋊C32
C3×#49673
#105886503