Finite Group Databasebeta

← #10588431⋊C3 #13141703

Subgroups — #10588431⋊C3 #13141703

Order 1728.

469 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 114 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×#88724C211yesmaximal, characteristic
2M16×AGL(1,ℤ/9)C212yesmaximal, complemented
33−1+2×#101C211yesmaximal, characteristic, complemented
4#63394⋊C3C211yesmaximal, characteristic, complemented
5#63383⋊C3C212yesmaximal, complemented
6C3×#4751—31nomaximal
7#10588431C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9#88724—22no—
10#88724—22no—
11C2×C4⋊AGL(1,ℤ/9)C412yes—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×C4×AGL(1,ℤ/9)C412yes—
14C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
15M16×3−1+2C2212yes—
16#3667⋊C3C2212yes—
17#3663⋊C3—22no—
18C8×AGL(1,ℤ/9)—22no—
19C2×C3×#239—31no—
20C3×D3×M16—32no—
21C2×#3701C2×C311yescharacteristic
22C32×#101D311yescharacteristic
23C3×#836—31no—
24C3×#823—32no—
25M16×D9C2×C312yescomplemented
26C9×#101—21no—
27C9×#101C2×C311yescharacteristic, complemented
28#63394C2×C311yescharacteristic, complemented
29#63383C2×C312yescomplemented
30C22×AGL(1,ℤ/9)C2×C412yes—
31C22×AGL(1,ℤ/9)C2×C411yescharacteristic
32Q8×3−1+2—22no—
33C2×C4×3−1+2C2×C412yes—
34C2×C4×3−1+2C2311yescharacteristic
35C2×3−1+2⋊C4C2×C411yescharacteristic
36C4⋊AGL(1,ℤ/9)—24no—
37C4⋊AGL(1,ℤ/9)—22no—
38C4×AGL(1,ℤ/9)—22no—
39C4×AGL(1,ℤ/9)—22no—
40C4×AGL(1,ℤ/9)D412yes—
41C4×AGL(1,ℤ/9)D412yes—
42C8×3−1+2—22no—
433−1+2⋊C8—22no—
44C3×#413—91no—
45#4751—31no—
46C2×C32×Q8C2×D311yescharacteristic
47C3×#239—62no—
48C3×#239—62no—
49C2×C3×D12—32no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—32no—
53C32×M16C2×D312yes—
54C3×#208—32no—
55C3×ΓL(1,𝔽25)—62no—
56C3×D3×C8—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C22×C311yescharacteristic
59#3701—22no—
60#3701—22no—
61C2×D36C3×C412yes—
62C2×D36C22×C311yescharacteristic
63C2×C4×D9C3×C412yes—
64C2×C4×D9C22×C311yescharacteristic
65C9×M16—22no—
66C9×M16C22×C312yes—
67#3667C22×C312yes—
68#3663—22no—
69C8×D9—22no—
70C22×3−1+2C22×C411yescharacteristic
71C2×AGL(1,ℤ/9)—22no—
72C2×AGL(1,ℤ/9)—41no—
73C2×AGL(1,ℤ/9)—42no—
74C2×AGL(1,ℤ/9)C22⋊C412yes—
75C4×3−1+2—22no—
76C4×3−1+2C2×D412yes—
773−1+2⋊C4C22⋊C411yescharacteristic
783−1+2⋊C4C22⋊C411yescharacteristic
79C2×C3×#41—91no—
80C2×#239—31no—
81C2×C3×M16—92no—
82D3×M16—32no—
83C3×#101—31no—
84C3×#101—91no—
85C3×#101C3×D311yescharacteristic
86C3×#100—92no—
87#836—31no—
88#823—32no—
89C22×C3×D3—31no—
90C22×C3×D3—32no—
91C32×Q8—22no—
92C2×C32×C4C4×D312yes—
93C2×C32×C4C22×D311yescharacteristic
94C2×C3×Dic3—31no—
95C3×D12—64no—
96C3×D12—62no—
97C3×C4×D3—32no—
98C3×C4×D3—32no—
99C3×C4×D3—62no—
100C3×C4×D3—62no—
101C22×D9C2×C3×C412yes—
102C22×D9C2×C3×C411yescharacteristic
103C32×C8—22no—
104C3×C3⋊C8—62no—
105Q8×C9—22no—
106Q8×C9—42no—
107C2×C4×C9—22no—
108C2×C4×C9—21no—
109C2×C4×C9C2×C3×C412yes—
110C2×C4×C9C23×C311yescharacteristic
111C2×Dic9C2×C3×C411yescharacteristic
112D36—24no—
113D36—22no—
114C4×D9—22no—
115C4×D9—22no—
116C4×D9C3×D412yes—
117C4×D9C3×D412yes—
118C8×C9—22no—
119C8×C9—42no—
120C9⋊C8—22no—
121#413—91noSylow
122C2×3−1+2—21no—
123C2×3−1+2C2×C22⋊C411yescharacteristic
124AGL(1,ℤ/9)—22no—
125AGL(1,ℤ/9)—42no—
126C3×#41—182no—
127C3×#41—182no—
128C2×C3×Q8—31no—
129C2×C3×Q8C2×C3×D311yescharacteristic
130C2×C3×D4—91no—
131C2×C3×D4—92no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134#239—62no—
135#239—62no—
136C2×D12—32no—
137C2×D12—31no—
138C2×C4×D3—31no—
139C2×C4×D3—32no—
140C3×M16—32no—
141C3×M16—92no—
142C3×M16—182no—
143C3×M16C2×C3×D312yes—
144C2×C3×C8—182no—
145#208—32no—
146ΓL(1,𝔽25)—62no—
147D3×C8—62no—
148C22×C32C2×C4×D311yescharacteristic
149C2×C3×D3—32no—
150C2×C3×D3—62no—
151C2×C3×D3—121no—
152C2×C3×D3—122no—
153C32×C4—22no—
154C32×C4D3×D412yes—
155C3×Dic3—31no—
156C3×Dic3—31no—
157C22×C9—21no—
158C22×C9C22×C3×C411yescharacteristic
159C2×D9—22no—
160C2×D9—41no—
161C2×D9—42no—
162C2×D9C3×C22⋊C412yes—
163C4×C9—22no—
164C4×C9—22no—
165C4×C9—42no—
166C4×C9C2×C3×D412yes—
167Dic9C3×C22⋊C411yescharacteristic
168Dic9C3×C22⋊C411yescharacteristic
169C2×#41—91no—
170C2×M16—92no—
171#101—91no—
172#101AGL(1,ℤ/9)11yescharacteristic, complemented
173#100—92no—
1743−1+2#41311yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C23×C3—92no—
177C22×D3—31no—
178C22×D3—32no—
179C3×Q8—22no—
180C3×Q8—62no—
181C3×D4—182no—
182C3×D4—184no—
183C2×C3×C4—32no—
184C2×C3×C4—31no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—91no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4C3×C4×D312yes—
191C2×C3×C4C22×C3×D311yescharacteristic
192C2×Dic3—31no—
193D12—64no—
194D12—62no—
195C4×D3—32no—
196C4×D3—32no—
197C4×D3—62no—
198C4×D3—62no—
199C3×C8—22no—
200C3×C8—62no—
201C3×C8—182no—
202C3⋊C8—62no—
203C2×C32—21no—
204C2×C32D3×C22⋊C411yescharacteristic
205C3×D3—62no—
206C3×D3—122no—
207C2×C9—21no—
208C2×C9—21no—
209C2×C9—41no—
210C2×C9C2×C3×C22⋊C411yescharacteristic
211D9—22no—
212D9—42no—
213#41—182no—
214#41—182no—
215C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
216C2×D4—92no—
217C2×D4—91no—
218C22×C4—91no—
219C22×C4—92no—
220M16—92no—
221M16—182no—
222M16C2×AGL(1,ℤ/9)12yes—
223C2×C8—182no—
224C22×C3—31no—
225C22×C3—92no—
226C22×C3—182no—
227C22×C3—361no—
228C22×C3—362no—
229C22×C3C2×C3×C4×D311yescharacteristic
230C2×D3—32no—
231C2×D3—62no—
232C2×D3—121no—
233C2×D3—122no—
234C3×C4—22no—
235C3×C4—32no—
236C3×C4—62no—
237C3×C4—91no—
238C3×C4—91no—
239C3×C4C3×D3×D412yes—
240Dic3—31no—
241Dic3—31no—
242C32#475111yescharacteristic
243C9—21no—
244C9C3×#41311yescharacteristic, complemented
245C23—91no—
246C23—92no—
247Q8—22no—
248D4—184no—
249D4—182no—
250C2×C4—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—182no—
254C2×C4—182no—
255C2×C4C4×AGL(1,ℤ/9)12yes—
256C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
257C8—22no—
258C8—182no—
259C2×C3—21no—
260C2×C3—31no—
261C2×C3—61no—
262C2×C3—182no—
263C2×C3—362no—
264C2×C3C3×D3×C22⋊C411yescharacteristic
265D3—62no—
266D3—122no—
267C22—92no—
268C22—182no—
269C22—361no—
270C22—362no—
271C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
272C4—22no—
273C4—91no—
274C4—91no—
275C4D4×AGL(1,ℤ/9)12yes—
276C3—31no—
277C3C3×#475111yescharacteristic
278C2—21no—
279C2—182no—
280C2—362no—
281C2C22⋊C4×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C22⋊C4×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#101C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC2×#88724C2
cosocleC2×C4×C9C23×C3
hypercenter#101AGL(1,ℤ/9)
hypocenterC9C3×#413
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. M16×AGL(1,ℤ/9)C2semidirect
3. 3−1+2×#101C2semidirect
4. #63394⋊C3C2semidirect
5. #63383⋊C3C2semidirect
7. #10588431C3semidirect
25. M16×D9C2×C3semidirect
27. C9×#101C2×C3semidirect
28. #63394C2×C3semidirect
29. #63383C2×C3semidirect
172. #101AGL(1,ℤ/9)semidirect
174. 3−1+2#413semidirect
244. C9C3×#413semidirect

Maximal subgroups

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

subgroupindex
C2×#887242
M16×AGL(1,ℤ/9)2
3−1+2×#1012
#63394⋊C32
#63383⋊C32
C3×#47513
#105884313