Finite Group Databasebeta

← #10588774⋊C3 #13143066

Subgroups — #10588774⋊C3 #13143066

Order 1728.

334 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 92 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×C4⋊Q8C211yesmaximal, characteristic, complemented
2#63437⋊C3C211yesmaximal, characteristic
3#63369⋊C3C212yesmaximal
4#63368⋊C3C212yesmaximal, complemented
5#63362⋊C3C211yesmaximal, characteristic, complemented
6C3×#5089—31nomaximal
7#10588774C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—21no—
9C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
10C2×3−1+2⋊Q8C2211yescharacteristic
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2C2212yes—
13C42×3−1+2C2211yescharacteristic
14(C2×AGL(1,ℤ/9))⋊C4—21no—
15(C2×AGL(1,ℤ/9))⋊C4—21no—
16C2×3−1+2⋊C8C2212yes—
17C32×C4⋊Q8D311yescharacteristic
18C3×#875—31no—
19C3×#810—32no—
20C3×#809—32no—
21C3×#803—31no—
22C9×C4⋊Q8—21no—
23C9×C4⋊Q8C2×C311yescharacteristic, complemented
24#63437C2×C311yescharacteristic
25#63369C2×C312yes—
26#63368C2×C312yescomplemented
27#63362C2×C311yescharacteristic, complemented
28C22×AGL(1,ℤ/9)—21no—
29Q8×3−1+2—42no—
30C2×C4×3−1+2—22no—
31C2×C4×3−1+2D411yescharacteristic
32C2×C4×3−1+2D411yescharacteristic
33C2×C4×3−1+2C2311yescharacteristic
34C2×3−1+2⋊C4—21no—
35C4⋊AGL(1,ℤ/9)—22no—
363−1+2⋊Q8—22no—
373−1+2⋊C8—22no—
38C3×#489—91no—
39#5089—31no—
40C2×C32×Q8—21no—
41C2×C3×D12—31no—
42C2×C3×Dic6—31no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×D312yes—
45C32×C42C2×D311yescharacteristic
46C3×(C2×D3)⋊C4—61no—
47C3×(C2×D3)⋊C4—61no—
48C2×C3×C3⋊C8—32no—
49C2×Q8×C9—22no—
50C2×Q8×C9—21no—
51C2×D36C22×C311yescharacteristic
52C2×Dic18C22×C311yescharacteristic
53C9×C4⋊C4—22no—
54C9×C4⋊C4—22no—
55C9×C4⋊C4—21no—
56C9×C4⋊C4C22×C312yes—
57C42×C9—21no—
58C42×C9C22×C311yescharacteristic
59(C2×D9)⋊C4—21no—
60(C2×D9)⋊C4—21no—
61C2×C9⋊C8C22×C312yes—
62C22×3−1+2C2×D411yescharacteristic
63C2×AGL(1,ℤ/9)—41no—
64C2×AGL(1,ℤ/9)—41no—
65C2×AGL(1,ℤ/9)—41no—
66C4×3−1+2—21no—
67C4×3−1+2—21no—
68C4×3−1+2—42no—
69C4×3−1+2#4112yes—
703−1+2⋊C4—41no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8C3×D311yescharacteristic
73C3×#124—91no—
74#875—31no—
75C3×#103—92no—
76C3×#102—92no—
77C3×#97—91no—
78#810—32no—
79#809—32no—
80#803—31no—
81C22×C3×D3—61no—
82C32×Q8—42no—
83C2×C32×C4—22no—
84C2×C32×C4#6711yescharacteristic
85C2×C32×C4#6711yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3×Dic3—61no—
88C3×D12—62no—
89C3×Dic6—62no—
90C22×D9—21no—
91C3×C3⋊C8—62no—
92Q8×C9—44no—
93Q8×C9—42no—
94C2×C4×C9—24no—
95C2×C4×C9—22no—
96C2×C4×C9—21no—
97C2×C4×C9—21no—
98C2×C4×C9—21no—
99C2×C4×C9C3×D411yescharacteristic
100C2×C4×C9C3×D411yescharacteristic
101C2×C4×C9C23×C311yescharacteristic
102C2×Dic9—21no—
103D36—22no—
104Dic18—22no—
105C9⋊C8—22no—
106#489—91noSylow
107C2×3−1+2#12411yescharacteristic
108C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
109C2×3−1+2#13711yescharacteristic
110AGL(1,ℤ/9)—81no—
111C2×C3×Q8—21no—
112C2×C3×Q8—61no—
113C2×C3×Q8—91no—
114C2×C3×D4—91no—
115C2×D12—31no—
116C2×Dic6—31no—
117C2×C3×C8—92no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4C2×C3×D312yes—
122C3×C22⋊C4—181no—
123C3×C22⋊C4—181no—
124C3×C42—31no—
125C3×C42C2×C3×D311yescharacteristic
126(C2×D3)⋊C4—61no—
127(C2×D3)⋊C4—61no—
128C2×C3⋊C8—32no—
129C22×C32C2×#6711yescharacteristic
130C2×C3×D3—121no—
131C2×C3×D3—121no—
132C2×C3×D3—121no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—42no—
136C32×C4#23912yes—
137C3×Dic3—121no—
138C22×C9—21no—
139C22×C9C2×C3×D411yescharacteristic
140C2×D9—41no—
141C2×D9—41no—
142C2×D9—41no—
143C4×C9—22no—
144C4×C9—22no—
145C4×C9—21no—
146C4×C9—21no—
147C4×C9—22no—
148C4×C9—44no—
149C4×C9—42no—
150C4×C9C3×#4112yes—
151Dic9—41no—
152C4⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
153#124—91no—
154#103—92no—
155#102—92no—
156#97—91no—
1573−1+2#48911yescharacteristic, complemented, Sylow
158C23×C3—181no—
159C22×D3—61no—
160C3×Q8—42no—
161C3×Q8—122no—
162C3×Q8—182no—
163C3×D4—182no—
164C2×C3×C4—22no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—62no—
169C2×C3×C4—181no—
170C2×C3×C4C3×#6711yescharacteristic
171C2×C3×C4C3×#6711yescharacteristic
172C2×C3×C4C22×C3×D311yescharacteristic
173C2×Dic3—61no—
174D12—62no—
175Dic6—62no—
176C3×C8—182no—
177C3⋊C8—62no—
178C2×C32#93211yescharacteristic
179C2×C32#94211yescharacteristic
180C2×C32#94711yescharacteristic
181C3×D3—241no—
182C2×C9—21no—
183C2×C9—21no—
184C2×C9—21no—
185C2×C9C3×#12411yescharacteristic
186C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
187C2×C9C3×#13711yescharacteristic
188D9—81no—
189C2×Q8—21no—
190C2×Q8—91no—
191C2×D4—91no—
192C2×C8—92no—
193C4⋊C4—21no—
194C4⋊C4C2×AGL(1,ℤ/9)12yes—
195C22⋊C4—181no—
196C22⋊C4—181no—
197C42C2×AGL(1,ℤ/9)11yescharacteristic
198C22×C3—31no—
199C22×C3—361no—
200C22×C3—361no—
201C22×C3—361no—
202C22×C3C2×C3×#6711yescharacteristic
203C2×D3—121no—
204C2×D3—121no—
205C2×D3—121no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—32no—
209C3×C4—42no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4—122no—
213C3×C4—361no—
214C3×C4C3×#23912yes—
215Dic3—121no—
216C32#508911yescharacteristic
217C9—21no—
218C9C3×#48911yescharacteristic, complemented
219C23—181no—
220Q8—42no—
221Q8—182no—
222D4—182no—
223C2×C4—22no—
224C2×C4—181no—
225C2×C4#611⋊C311yescharacteristic
226C2×C4#611⋊C311yescharacteristic
227C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
228C8—182no—
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—721no—
233C2×C3C3×#93211yescharacteristic
234C2×C3C3×#94211yescharacteristic
235C2×C3C3×#94711yescharacteristic
236D3—241no—
237C22—361no—
238C22—361no—
239C22—361no—
240C22C2×#611⋊C311yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—361no—
245C4#8872412yes—
246C3—31no—
247C3C3×#508911yescharacteristic
248C2—721no—
249C2#63495⋊C311yescharacteristic
250C2#63505⋊C311yescharacteristic
251C2#63509⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C4⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenterC4⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#489
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. 3−1+2×C4⋊Q8C2semidirect
4. #63368⋊C3C2semidirect
5. #63362⋊C3C2semidirect
7. #10588774C3semidirect
23. C9×C4⋊Q8C2×C3semidirect
26. #63368C2×C3semidirect
27. #63362C2×C3semidirect
152. C4⋊Q8AGL(1,ℤ/9)semidirect
157. 3−1+2#489semidirect
218. C9C3×#489semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C4⋊Q82
#63437⋊C32
#63369⋊C32
#63368⋊C32
#63362⋊C32
C3×#50893
#105887743