Finite Group Databasebeta

← #10588153⋊C3 #13135355

Subgroups — #10588153⋊C3 #13135355

Order 1728.

305 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 50 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#63499⋊C3C211yesmaximal, characteristic
23−1+2×#100C211yesmaximal, characteristic, complemented
3#63393⋊C3C211yesmaximal, characteristic, complemented
4C3×#4475—31nomaximal
5#10588153C311yesmaximal, characteristic, complemented
6C23×AGL(1,ℤ/9)C411yescharacteristic, complemented
7C2×D4×3−1+2C2211yescharacteristic
8C2×#611⋊C3—21no—
9M16×3−1+2—21no—
10#3676⋊C3—21no—
11#3676⋊C3C411yescharacteristic, complemented
12(C2×AGL(1,ℤ/9))⋊C4—21no—
13C3×#937—31no—
14C32×#100D311yescharacteristic
15C3×#834—31no—
16#63499C2×C311yescharacteristic
17C9×#100—21no—
18C9×#100C2×C311yescharacteristic, complemented
19#63393C2×C311yescharacteristic, complemented
20C23×3−1+2C2×C411yescharacteristic
21C23×3−1+2D411yescharacteristic
22C22×AGL(1,ℤ/9)—21no—
23C22×AGL(1,ℤ/9)—42no—
24C22×AGL(1,ℤ/9)—41no—
25D4×3−1+2—41no—
26C2×C4×3−1+2D411yescharacteristic
27#611⋊C3—42no—
28C2×3−1+2⋊C4—21no—
29C2×3−1+2⋊C4—41no—
30C8×3−1+2—41no—
31C3×AΣL(1,𝔽16)—91no—
32#4475—31no—
33C23×C3×D3—31no—
34C2×C32×D4C2×D311yescharacteristic
35C2×C3×#67—61no—
36C23×D9C3×C411yescharacteristic, complemented
37C32×M16—21no—
38C3×#217—31no—
39C3×#217—61no—
40C3×(C2×D3)⋊C4—61no—
41C2×D4×C9—21no—
42C2×D4×C9C22×C311yescharacteristic
43C2×#611—21no—
44C9×M16—22no—
45C9×M16—21no—
46#3676—21no—
47#3676C3×C411yescharacteristic, complemented
48(C2×D9)⋊C4—21no—
49C22×3−1+2—21no—
50C22×3−1+2—21no—
51C22×3−1+2—41no—
52C22×3−1+2—41no—
53C22×3−1+2C22⋊C411yescharacteristic
54C2×AGL(1,ℤ/9)—22no—
55C2×AGL(1,ℤ/9)—41no—
56C2×AGL(1,ℤ/9)—42no—
57C2×AGL(1,ℤ/9)—41no—
58C2×AGL(1,ℤ/9)—42no—
59C4×3−1+2—21no—
603−1+2⋊C4—41no—
613−1+2⋊C4—81no—
62C3×#120—91no—
63#937—31no—
64C3×#100—31no—
65C3×#100C3×D311yescharacteristic
66C3×#99—91no—
67#834—31no—
68C23×C32C4×D311yescharacteristic
69C23×C32D1211yescharacteristic
70C22×C3×D3—61no—
71C22×C3×D3—122no—
72C22×C3×D3—121no—
73C32×D4—41no—
74C2×C32×C4#6711yescharacteristic
75C3×#67—122no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—121no—
78C23×C9—21no—
79C23×C9—21no—
80C23×C9C2×C3×C411yescharacteristic
81C23×C9C3×D411yescharacteristic
82C22×D9—21no—
83C22×D9—42no—
84C22×D9—41no—
85C32×C8—41no—
86D4×C9—41no—
87D4×C9—42no—
88C2×C4×C9—21no—
89C2×C4×C9C3×D411yescharacteristic
90#611—42no—
91C2×Dic9—21no—
92C2×Dic9—41no—
93C8×C9—42no—
94C8×C9—41no—
95AΣL(1,𝔽16)—91noSylow
96C2×3−1+2—21no—
97C2×3−1+2—41no—
98C2×3−1+2—41no—
99C2×3−1+2#9911yescharacteristic
100AGL(1,ℤ/9)—42no—
101C24×C3—91no—
102C23×D3—31no—
103C2×C3×D4—31no—
104C2×C3×D4—181no—
105C2×C3×D4C2×C3×D311yescharacteristic
106C2×#67—61no—
107C3×M16—21no—
108C3×M16—61no—
109C3×C22⋊C4—91no—
110C3×C22⋊C4—181no—
111C3×C22⋊C4—181no—
112#217—31no—
113#217—61no—
114(C2×D3)⋊C4—61no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—41no—
118C22×C32—41no—
119C22×C32(C2×D3)⋊C411yescharacteristic
120C2×C3×D3—62no—
121C2×C3×D3—121no—
122C2×C3×D3—122no—
123C2×C3×D3—121no—
124C2×C3×D3—122no—
125C32×C4—21no—
126C3×Dic3—121no—
127C3×Dic3—241no—
128C22×C9—21no—
129C22×C9—21no—
130C22×C9—21no—
131C22×C9—41no—
132C22×C9—41no—
133C22×C9—41no—
134C22×C9—41no—
135C22×C9—81no—
136C22×C9—81no—
137C22×C9C3×C22⋊C411yescharacteristic
138C2×D9—22no—
139C2×D9—41no—
140C2×D9—42no—
141C2×D9—41no—
142C2×D9—42no—
143C4×C9—22no—
144C4×C9—21no—
145Dic9—41no—
146Dic9—81no—
147#120—91no—
148#100AGL(1,ℤ/9)11yescharacteristic, complemented
149#99—91no—
1503−1+2AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—181no—
154C23×C3—361no—
155C23×C3—362no—
156C23×C3C3×C4×D311yescharacteristic
157C23×C3C3×D1211yescharacteristic
158C22×D3—61no—
159C22×D3—122no—
160C22×D3—121no—
161C3×D4—41no—
162C3×D4—121no—
163C3×D4—362no—
164C2×C3×C4—31no—
165C2×C3×C4—181no—
166C2×C3×C4—361no—
167C2×C3×C4C3×#6711yescharacteristic
168#67—122no—
169C2×Dic3—61no—
170C2×Dic3—121no—
171C3×C8—41no—
172C3×C8—121no—
173C2×C32—21no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32#80611yescharacteristic
177C3×D3—122no—
178C2×C9—21no—
179C2×C9—21no—
180C2×C9—41no—
181C2×C9—41no—
182C2×C9—41no—
183C2×C9—81no—
184C2×C9—81no—
185C2×C9C3×#9911yescharacteristic
186D9—42no—
187C24—91no—
188C2×D4—181no—
189C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
190M16—21no—
191C22⋊C4—91no—
192C22⋊C4—181no—
193C22⋊C4—181no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—31no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—61no—
200C22×C3—61no—
201C22×C3—121no—
202C22×C3—121no—
203C22×C3—182no—
204C22×C3—361no—
205C22×C3—361no—
206C22×C3—362no—
207C22×C3—362no—
208C22×C3C3×(C2×D3)⋊C411yescharacteristic
209C2×D3—62no—
210C2×D3—121no—
211C2×D3—122no—
212C2×D3—121no—
213C2×D3—122no—
214C3×C4—21no—
215C3×C4—61no—
216C3×C4—361no—
217C3×C4—721no—
218Dic3—121no—
219Dic3—241no—
220C32#447511yescharacteristic
221C9—21no—
222C9C3×AΣL(1,𝔽16)11yescharacteristic, complemented
223C23—181no—
224C23—361no—
225C23—362no—
226C23C4×AGL(1,ℤ/9)11yescharacteristic
227C23C4⋊AGL(1,ℤ/9)11yescharacteristic
228D4—41no—
229D4—362no—
230C2×C4—181no—
231C2×C4—361no—
232C2×C4#611⋊C311yescharacteristic
233C8—41no—
234C2×C3—21no—
235C2×C3—31no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—61no—
239C2×C3—121no—
240C2×C3—121no—
241C2×C3—362no—
242C2×C3C3×#80611yescharacteristic
243D3—122no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—182no—
249C22—361no—
250C22—362no—
251C22—362no—
252C22—361no—
253C22(C2×AGL(1,ℤ/9))⋊C411yescharacteristic
254C4—21no—
255C4—361no—
256C4—721no—
257C3—31no—
258C3C3×#447511yescharacteristic
259C2—21no—
260C2—41no—
261C2—41no—
262C2—362no—
263C2#63365⋊C311yescharacteristic
subgroupG/H
centerC2#63365⋊C3
commutatorC23×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#100C2
socleC2×C3C3×#806
prime radical#63499⋊C3C2
cosocleC2×D4×C9C22×C3
hypercenter#100AGL(1,ℤ/9)
hypocenterC9C3×AΣL(1,𝔽16)
LCSG ▷ C23×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#100C2semidirect
3. #63393⋊C3C2semidirect
5. #10588153C3semidirect
6. C23×AGL(1,ℤ/9)C4semidirect
11. #3676⋊C3C4semidirect
18. C9×#100C2×C3semidirect
19. #63393C2×C3semidirect
36. C23×D9C3×C4semidirect
47. #3676C3×C4semidirect
148. #100AGL(1,ℤ/9)semidirect
150. 3−1+2AΣL(1,𝔽16)semidirect
222. C9C3×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#63499⋊C32
3−1+2×#1002
#63393⋊C32
C3×#44753
#105881533