Finite Group Databasebeta

← #10588769⋊C3 #13143065

Subgroups — #10588769⋊C3 #13143065

Order 1728.

358 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
3#63369⋊C3C212yesmaximal
4#63367⋊C3C212yesmaximal, complemented
5C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
6C3×#5084—31nomaximal
7#10588769C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—21no—
9C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
10C2×C4×AGL(1,ℤ/9)—21no—
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2C2212yes—
13C42×3−1+2C2211yescharacteristic
14(C2×AGL(1,ℤ/9))⋊C4—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C2×3−1+2⋊C8C2212yes—
17C32×C4⋊Q8D311yescharacteristic
18C3×C4×D12—31no—
19C3×#809—32no—
20C3×#808—32no—
21C3×C4⋊(C3⋊C8)—31no—
22C9×C4⋊Q8—21no—
23C9×C4⋊Q8C2×C311yescharacteristic, complemented
24C4×D36C2×C311yescharacteristic
25#63369C2×C312yes—
26#63367C2×C312yescomplemented
27C4⋊(C9⋊C8)C2×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)—21no—
36C4⋊AGL(1,ℤ/9)Q812yescomplemented
37C4×AGL(1,ℤ/9)—42no—
383−1+2⋊C8—22no—
39C3×#476—91no—
40#5084—31no—
41C2×C32×Q8—21no—
42C2×C3×D12—31no—
43C2×C3×C4×D3—61no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4C2×D312yes—
46C32×C42C2×D311yescharacteristic
47C3×(C2×D3)⋊C4—61no—
48C3×C4⋊Dic3—31no—
49C2×C3×C3⋊C8—32no—
50C2×Q8×C9—22no—
51C2×Q8×C9—21no—
52C2×D36C22×C311yescharacteristic
53C2×C4×D9—21no—
54C9×C4⋊C4—22no—
55C9×C4⋊C4—22no—
56C9×C4⋊C4—21no—
57C9×C4⋊C4C22×C312yes—
58C42×C9—21no—
59C42×C9C22×C311yescharacteristic
60(C2×D9)⋊C4—21no—
61C4⋊Dic9C22×C311yescharacteristic
62C2×C9⋊C8C22×C312yes—
63C22×3−1+2C2×D411yescharacteristic
64C2×AGL(1,ℤ/9)—22no—
65C2×AGL(1,ℤ/9)—41no—
66C2×AGL(1,ℤ/9)—41no—
67C4×3−1+2—21no—
68C4×3−1+2—42no—
69C4×3−1+2SD1612yes—
70C4×3−1+2C2×Q811yescharacteristic
71C4×3−1+2#4111yescharacteristic
723−1+2⋊C4—41no—
73C3×C4⋊Q8—31no—
74C3×C4⋊Q8C3×D311yescharacteristic
75C3×C4×D4—91no—
76C4×D12—31no—
77C3×#106—92no—
78C3×C4⋊C8—91no—
79C3×#102—92no—
80#809—32no—
81#808—32no—
82C4⋊(C3⋊C8)—31no—
83C22×C3×D3—61no—
84C32×Q8—42no—
85C2×C32×C4—22no—
86C2×C32×C4#6711yescharacteristic
87C2×C32×C4#6711yescharacteristic
88C2×C32×C4C22×D311yescharacteristic
89C2×C3×Dic3—61no—
90C3×D12—32no—
91C3×D12—61no—
92C3×C4×D3—122no—
93C22×D9—21no—
94C3×C3⋊C8—62no—
95Q8×C9—44no—
96Q8×C9—42no—
97C2×C4×C9—24no—
98C2×C4×C9—22no—
99C2×C4×C9—21no—
100C2×C4×C9—21no—
101C2×C4×C9—21no—
102C2×C4×C9C3×D411yescharacteristic
103C2×C4×C9C3×D411yescharacteristic
104C2×C4×C9C23×C311yescharacteristic
105C2×Dic9—21no—
106D36—21no—
107D36C3×Q812yescomplemented
108C4×D9—42no—
109C9⋊C8—22no—
110#476—91noSylow
111C2×3−1+2C22⋊Q811yescharacteristic
112C2×3−1+2C2×SD1611yescharacteristic
113C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
114AGL(1,ℤ/9)—42no—
115C2×C3×Q8—21no—
116C2×C3×Q8—61no—
117C2×C3×D4—91no—
118C22×C3×C4—181no—
119C2×D12—31no—
120C2×C4×D3—61no—
121C2×C3×C8—92no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—91no—
126C3×C4⋊C4C2×C3×D312yes—
127C3×C22⋊C4—181no—
128C3×C42—31no—
129C3×C42C2×C3×D311yescharacteristic
130(C2×D3)⋊C4—61no—
131C4⋊Dic3—31no—
132C2×C3⋊C8—32no—
133C22×C32C2×#6711yescharacteristic
134C2×C3×D3—62no—
135C2×C3×D3—121no—
136C2×C3×D3—121no—
137C32×C4—21no—
138C32×C4—42no—
139C32×C4#21512yes—
140C32×C4D3×Q811yescharacteristic
141C32×C4#23911yescharacteristic
142C3×Dic3—121no—
143C22×C9—21no—
144C22×C9C2×C3×D411yescharacteristic
145C2×D9—22no—
146C2×D9—41no—
147C2×D9—41no—
148C4×C9—22no—
149C4×C9—21no—
150C4×C9—22no—
151C4×C9—21no—
152C4×C9—21no—
153C4×C9—44no—
154C4×C9—42no—
155C4×C9C3×SD1612yes—
156C4×C9C2×C3×Q811yescharacteristic
157C4×C9C3×#4111yescharacteristic
158Dic9—41no—
159C4⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
160C4×D4—91no—
161#106—92no—
162C4⋊C8—91no—
163#102—92no—
1643−1+2#47611yescharacteristic, complemented, Sylow
165C23×C3—181no—
166C22×D3—61no—
167C3×Q8—42no—
168C3×Q8—122no—
169C3×D4—92no—
170C3×D4—181no—
171C2×C3×C4—22no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—62no—
176C2×C3×C4—181no—
177C2×C3×C4—362no—
178C2×C3×C4C3×#6711yescharacteristic
179C2×C3×C4C3×#6711yescharacteristic
180C2×C3×C4C22×C3×D311yescharacteristic
181C2×Dic3—61no—
182D12—32no—
183D12—61no—
184C4×D3—122no—
185C3×C8—182no—
186C3⋊C8—62no—
187C2×C32#93211yescharacteristic
188C2×C32C2×#21511yescharacteristic
189C2×C32#94611yescharacteristic
190C3×D3—122no—
191C2×C9—21no—
192C2×C9—21no—
193C2×C9—21no—
194C2×C9C3×C22⋊Q811yescharacteristic
195C2×C9C2×C3×SD1611yescharacteristic
196C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
197D9—42no—
198C2×Q8—21no—
199C2×D4—91no—
200C22×C4—181no—
201C2×C8—92no—
202C4⋊C4—21no—
203C4⋊C4—91no—
204C4⋊C4C2×AGL(1,ℤ/9)12yes—
205C22⋊C4—181no—
206C42C2×AGL(1,ℤ/9)11yescharacteristic
207C22×C3—31no—
208C22×C3—182no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3C2×C3×#6711yescharacteristic
212C2×D3—62no—
213C2×D3—121no—
214C2×D3—121no—
215C3×C4—21no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—32no—
219C3×C4—42no—
220C3×C4—61no—
221C3×C4—122no—
222C3×C4—361no—
223C3×C4C3×#21512yes—
224C3×C4C3×D3×Q811yescharacteristic
225C3×C4C3×#23911yescharacteristic
226Dic3—121no—
227C32#508411yescharacteristic
228C9—21no—
229C9C3×#47611yescharacteristic, complemented
230C23—181no—
231Q8—42no—
232D4—92no—
233D4—181no—
234C2×C4—22no—
235C2×C4—181no—
236C2×C4—362no—
237C2×C4#611⋊C311yescharacteristic
238C2×C4#611⋊C311yescharacteristic
239C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
240C8—182no—
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—31no—
244C2×C3—362no—
245C2×C3C3×#93211yescharacteristic
246C2×C3C2×C3×#21511yescharacteristic
247C2×C3C3×#94611yescharacteristic
248D3—122no—
249C22—182no—
250C22—361no—
251C22—361no—
252C22C2×#611⋊C311yescharacteristic
253C4—21no—
254C4—42no—
255C4—361no—
256C4#3675⋊C312yes—
257C4Q8×AGL(1,ℤ/9)11yescharacteristic
258C4#8872411yescharacteristic
259C3—31no—
260C3C3×#508411yescharacteristic
261C2—362no—
262C2#63495⋊C311yescharacteristic
263C2C2×#3675⋊C311yescharacteristic
264C2#63508⋊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×#476
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. #63367⋊C3C2semidirect
5. C4⋊(3−1+2⋊C8)C2semidirect
7. #10588769C3semidirect
23. C9×C4⋊Q8C2×C3semidirect
26. #63367C2×C3semidirect
27. C4⋊(C9⋊C8)C2×C3semidirect
36. C4⋊AGL(1,ℤ/9)Q8semidirect
107. D36C3×Q8semidirect
159. C4⋊Q8AGL(1,ℤ/9)semidirect
164. 3−1+2#476semidirect
229. C9C3×#476semidirect

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
C4×C4⋊AGL(1,ℤ/9)2
#63369⋊C32
#63367⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50843
#105887693