Finite Group Databasebeta

← #10588647⋊C3 #13142635

Subgroups — #10588647⋊C3 #13142635

Order 1728.

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

normal subgroups only

This group has 358 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
2#63486⋊C3C211yesmaximal, characteristic
3#63389⋊C3C211yesmaximal, characteristic
4#63368⋊C3C212yesmaximal, complemented
5#63367⋊C3C212yesmaximal, complemented
6C3×#4971—31nomaximal
7#10588647C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2—21no—
9C22×C4×3−1+2C2211yescharacteristic
10C2×3−1+2⋊Q8C2211yescharacteristic
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2C2212yes—
13#3676⋊C3—21no—
14C4⋊(3−1+2⋊C4)C2211yescharacteristic
15(3−1+2⋊C4)⋊C4—21no—
16C2×3−1+2⋊C8C2212yes—
17C2×C32×C4⋊C4D311yescharacteristic
18C3×#924—31no—
19C3×#830—31no—
20C3×#810—32no—
21C3×#808—32no—
22C2×C9×C4⋊C4—21no—
23C2×C9×C4⋊C4C2×C311yescharacteristic
24#63486C2×C311yescharacteristic
25#63389C2×C311yescharacteristic
26#63368C2×C312yescomplemented
27#63367C2×C312yescomplemented
28C23×3−1+2D411yescharacteristic
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2—42no—
32C2×C4×3−1+2D411yescharacteristic
33C2×C4×3−1+2C2311yescharacteristic
34C2×3−1+2⋊C4—21no—
35C2×3−1+2⋊C4—21no—
363−1+2⋊Q8—22no—
373−1+2⋊C8—22no—
38C3×#483—91no—
39#4971—31no—
40C22×C32×C4—21no—
41C22×C32×C4C2×D311yescharacteristic
42C2×C3×Dic6—31no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×D312yes—
45C3×#217—61no—
46C3×C4⋊Dic3—31no—
47C3×Dic3⋊C4—61no—
48C2×C3×C3⋊C8—32no—
49C22×C4×C9—21no—
50C22×C4×C9—22no—
51C22×C4×C9—21no—
52C22×C4×C9C22×C311yescharacteristic
53C2×Dic18C22×C311yescharacteristic
54C9×C4⋊C4—22no—
55C9×C4⋊C4—22no—
56C9×C4⋊C4—21no—
57C9×C4⋊C4C22×C312yes—
58#3676—21no—
59C4⋊Dic9C22×C311yescharacteristic
60Dic9⋊C4—21no—
61C2×C9⋊C8C22×C312yes—
62C22×3−1+2—21no—
63C22×3−1+2—21no—
64C22×3−1+2SD1612yes—
65C22×3−1+2C2×D411yescharacteristic
66C4×3−1+2—21no—
67C4×3−1+2—42no—
68C4×3−1+2#4112yes—
693−1+2⋊C4—41no—
703−1+2⋊C4—41no—
71C3×C22⋊Q8—91no—
72C2×C3×C4⋊C4—31no—
73C2×C3×C4⋊C4C3×D311yescharacteristic
74#924—31no—
75C3×#106—92no—
76C3×#103—92no—
77C3×C22⋊C8—91no—
78#830—31no—
79#810—32no—
80#808—32no—
81C23×C32#6711yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—42no—
85C2×C32×C4#6711yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C3×Dic6—62no—
90C23×C9—21no—
91C23×C9C3×D411yescharacteristic
92C3×C3⋊C8—62no—
93C2×C4×C9—24no—
94C2×C4×C9—22no—
95C2×C4×C9—24no—
96C2×C4×C9—22no—
97C2×C4×C9—21no—
98C2×C4×C9—21no—
99C2×C4×C9—44no—
100C2×C4×C9—42no—
101C2×C4×C9C3×D411yescharacteristic
102C2×C4×C9C23×C311yescharacteristic
103C2×Dic9—21no—
104C2×Dic9—21no—
105Dic18—22no—
106C9⋊C8—22no—
107#483—91noSylow
108C2×3−1+2—22no—
109C2×3−1+2#12311yescharacteristic
110C2×3−1+2C2×SD1611yescharacteristic
111C2×3−1+2#13711yescharacteristic
112C2×C3×Q8—91no—
113C22×C3×C4—21no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C22×C3×C4C2×C3×D311yescharacteristic
117C2×Dic6—31no—
118C2×C3×C8—92no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—181no—
124C3×C4⋊C4C2×C3×D312yes—
125C3×C22⋊C4—181no—
126#217—61no—
127C4⋊Dic3—31no—
128Dic3⋊C4—61no—
129C2×C3⋊C8—32no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32#21412yes—
133C22×C32C2×#6711yescharacteristic
134C32×C4—21no—
135C32×C4—42no—
136C32×C4#23512yes—
137C3×Dic3—121no—
138C3×Dic3—121no—
139C22×C9—22no—
140C22×C9—22no—
141C22×C9—21no—
142C22×C9—21no—
143C22×C9—22no—
144C22×C9—21no—
145C22×C9C3×SD1612yes—
146C22×C9C2×C3×D411yescharacteristic
147C4×C9—22no—
148C4×C9—21no—
149C4×C9—22no—
150C4×C9—44no—
151C4×C9—42no—
152C4×C9C3×#4112yes—
153Dic9—41no—
154Dic9—41no—
155C22⋊Q8—91no—
156C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
157#106—92no—
158#103—92no—
159C22⋊C8—91no—
1603−1+2#48311yescharacteristic, complemented, Sylow
161C23×C3—31no—
162C23×C3C3×#6711yescharacteristic
163C3×Q8—182no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—42no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—122no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4C3×#6711yescharacteristic
175C2×C3×C4C22×C3×D311yescharacteristic
176C2×Dic3—61no—
177C2×Dic3—61no—
178Dic6—62no—
179C3×C8—182no—
180C3⋊C8—62no—
181C2×C32—22no—
182C2×C32#92911yescharacteristic
183C2×C32C2×#21411yescharacteristic
184C2×C32#94211yescharacteristic
185C2×C9—24no—
186C2×C9—22no—
187C2×C9—21no—
188C2×C9—21no—
189C2×C9—21no—
190C2×C9C3×#12311yescharacteristic
191C2×C9C2×C3×SD1611yescharacteristic
192C2×C9C3×#13711yescharacteristic
193C2×Q8—91no—
194C22×C4—21no—
195C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
196C2×C8—92no—
197C4⋊C4—21no—
198C4⋊C4—91no—
199C4⋊C4—181no—
200C4⋊C4C2×AGL(1,ℤ/9)12yes—
201C22⋊C4—181no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—31no—
205C22×C3—32no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3C3×#21412yes—
209C22×C3C2×C3×#6711yescharacteristic
210C3×C4—21no—
211C3×C4—32no—
212C3×C4—42no—
213C3×C4—61no—
214C3×C4—122no—
215C3×C4—361no—
216C3×C4—361no—
217C3×C4C3×#23512yes—
218Dic3—121no—
219Dic3—121no—
220C32#497111yescharacteristic
221C9—21no—
222C9C3×#48311yescharacteristic, complemented
223C23#611⋊C311yescharacteristic
224Q8—182no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—181no—
229C2×C4—181no—
230C2×C4#611⋊C311yescharacteristic
231C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
232C8—182no—
233C2×C3—22no—
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—62no—
238C2×C3C3×#92911yescharacteristic
239C2×C3C2×C3×#21411yescharacteristic
240C2×C3C3×#94211yescharacteristic
241C22—21no—
242C22—21no—
243C22#3672⋊C312yes—
244C22C2×#611⋊C311yescharacteristic
245C4—21no—
246C4—42no—
247C4—361no—
248C4—361no—
249C4#3697⋊C312yes—
250C3—31no—
251C3C3×#497111yescharacteristic
252C2—22no—
253C2#63491⋊C311yescharacteristic
254C2C2×#3672⋊C311yescharacteristic
255C2#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 radicalC23×C32#67
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#483
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
4. #63368⋊C3C2semidirect
5. #63367⋊C3C2semidirect
7. #10588647C3semidirect
26. #63368C2×C3semidirect
27. #63367C2×C3semidirect
160. 3−1+2#483semidirect
222. C9C3×#483semidirect

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
#63486⋊C32
#63389⋊C32
#63368⋊C32
#63367⋊C32
C3×#49713
#105886473