Finite Group Databasebeta

← C4×D9⋊SL(2,𝔽3) #13147105

Subgroups — C4×D9⋊SL(2,𝔽3) #13147105

Order 1728.

335 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 68 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×D9⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C4×C9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3Dic9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)—31nomaximal
5C4×Q8×D9C311yesmaximal, characteristic, complemented
6C2×C4×AGL(1,ℤ/9)—41nomaximal
7C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
8D9⋊SL(2,𝔽3)C412yesdirect product factor
9C2×D3×SL(2,𝔽3)—31no—
10C3×C4×SL(2,𝔽3)D311yescharacteristic
11Dic3×SL(2,𝔽3)—31no—
12C2×Q8×D9C2×C311yescharacteristic
13C4×Q8×C9C2×C311yescharacteristic, complemented
14Q8×Dic9C2×C311yescharacteristic, complemented
15C4⋊C4×D9—31no—
16#63449—31no—
17C42×D9—31no—
18C4×Dic18—31no—
19C4×Q8⋊C9—21no—
20C22×AGL(1,ℤ/9)—41no—
21C2×C4×3−1+2—41no—
22C2×3−1+2⋊C4—41no—
23C4×AGL(1,ℤ/9)—44no—
24C9⋊SL(2,𝔽3)C2×C411yescharacteristic, complemented
25C4×D3×Q8—31no—
26C2×C4×SL(2,𝔽3)—91no—
27C2×C3×C4×D3—121no—
28C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
29D3×SL(2,𝔽3)—32no—
30C2×Q8×C9C22×C311yescharacteristic
31Q8×D9—32no—
32Q8×D9C3×C412yescomplemented
33C2×C4×D9—31no—
34C2×C4×D9—31no—
35C2×C4×D9A411yescharacteristic
36C2×Dic18—31no—
37C2×Q8⋊C9—21no—
38C9×C4⋊C4—31no—
39C42×C9—31no—
40C4⋊Dic9—31no—
41Dic9⋊C4—32no—
42C4×Dic9—31no—
43C4×Dic9—31no—
44C4×Dic9—31no—
45C22×3−1+2—41no—
46C2×AGL(1,ℤ/9)—42no—
47C2×AGL(1,ℤ/9)—42no—
48C2×AGL(1,ℤ/9)—42no—
49C4×3−1+2—42no—
503−1+2⋊C4—42no—
51C2×D3×Q8—31no—
52C22×SL(2,𝔽3)—91no—
53C3×C4×Q8C3×D311yescharacteristic
54Q8×Dic3—31no—
55D3×C4⋊C4—91no—
56#887—91no—
57C42×D3—91no—
58C4×Dic6—91no—
59C4×SL(2,𝔽3)—31no—
60C4×SL(2,𝔽3)—91no—
61C22×C3×D3—121no—
62C2×C32×C4—41no—
63C2×C3×Dic3—121no—
64C3×C4×D3—124no—
65C3×SL(2,𝔽3)C4×D311yescharacteristic
66C22×D9C2×A411yescharacteristic
67Q8×C9—31no—
68Q8×C9C2×C3×C411yescharacteristic, complemented
69C2×C4×C9—31no—
70C2×C4×C9—31no—
71C2×C4×C9—81no—
72C2×C4×C9C2×A411yescharacteristic
73C2×Dic9—31no—
74C2×Dic9—31no—
75C2×Dic9C2×A411yescharacteristic
76C4×D9—32no—
77C4×D9—32no—
78C4×D9—62no—
79C4×D9SL(2,𝔽3)14yescomplemented
80Dic18—32no—
81Dic18—32no—
82Q8⋊C9—21no—
83C2×C4×Q8—91noSylow
84C2×3−1+2—41no—
85C2×3−1+2—41no—
86C2×3−1+2—41no—
87AGL(1,ℤ/9)—44no—
88C2×C3×Q8C2×C3×D311yescharacteristic
89C22×C3×C4—361no—
90D3×Q8—32no—
91D3×Q8—92no—
92C2×C4×D3—31no—
93C2×C4×D3—91no—
94C2×C4×D3—91no—
95C2×Dic6—91no—
96C2×SL(2,𝔽3)—31no—
97C2×SL(2,𝔽3)—92no—
98C3×C4⋊C4—31no—
99C3×C42—31no—
100C4⋊Dic3—91no—
101Dic3⋊C4—92no—
102C4×Dic3—91no—
103C4×Dic3—91no—
104C4×Dic3—91no—
105C22×C32—41no—
106C2×C3×D3—122no—
107C2×C3×D3—122no—
108C2×C3×D3—122no—
109C32×C4—42no—
110C3×Dic3—122no—
111C22×C9—81no—
112C22×C9C22×A411yescharacteristic
113C2×D9C4×A412yes—
114C2×D9C2×SL(2,𝔽3)12yes—
115C2×D9#41⋊C312yes—
116C4×C9—31no—
117C4×C9—31no—
118C4×C9—61no—
119C4×C9—82no—
120C4×C9C2×SL(2,𝔽3)12yescomplemented
121Dic9—32no—
122Dic9—61no—
123Dic9C2×SL(2,𝔽3)12yescomplemented
124C22×Q8—91no—
125C4×Q8—91no—
126C4×Q8—271no—
127C4×Q8—271no—
128C4×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
129C2×C4⋊C4—271no—
130C2×C42—271no—
1313−1+2—41noSylow
132C23×C3—361no—
133C22×D3—31no—
134C3×Q8—31no—
135C3×Q8C3×C4×D311yescharacteristic
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—121no—
139C2×C3×C4—364no—
140C2×C3×C4—361no—
141C2×C3×C4D3×A411yescharacteristic
142C2×Dic3—31no—
143C2×Dic3—91no—
144C2×Dic3—91no—
145C4×D3—34no—
146C4×D3—92no—
147C4×D3—92no—
148C4×D3—182no—
149Dic6—92no—
150Dic6—92no—
151SL(2,𝔽3)—31no—
152C2×C32—41no—
153C2×C32—41no—
154C2×C32—41no—
155C3×D3—124no—
156C2×C9—81no—
157C2×C9—81no—
158C2×C9—81no—
159C2×C9C2×C4×A411yescharacteristic
160C2×C9C22×SL(2,𝔽3)11yescharacteristic
161C2×C9C2×#41⋊C311yescharacteristic
162D9C4×SL(2,𝔽3)14yescomplemented
163C2×Q8—92no—
164C2×Q8—271no—
165C2×Q8—272no—
166C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
167C22×C4—91no—
168C22×C4—271no—
169C22×C4—271no—
170C4⋊C4—31no—
171C4⋊C4—271no—
172C4⋊C4—272no—
173C42—31no—
174C42—271no—
175C42—271no—
176C42—271no—
177C22×C3—121no—
178C22×C3—362no—
179C22×C3—362no—
180C22×C3—362no—
181C22×C3C2×D3×A411yescharacteristic
182C2×D3—32no—
183C2×D3—32no—
184C2×D3—32no—
185C3×C4—31no—
186C3×C4—31no—
187C3×C4—61no—
188C3×C4—122no—
189C3×C4—362no—
190C3×C4D3×SL(2,𝔽3)12yes—
191Dic3—32no—
192Dic3—92no—
193Dic3—181no—
194C32—41no—
195C9—81no—
196C9C2×C4×SL(2,𝔽3)11yescharacteristic, complemented
197C23—91no—
198Q8—31no—
199Q8—272no—
200Q8—272no—
201Q8C4×AGL(1,ℤ/9)11yescharacteristic, complemented
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—91no—
205C2×C4—94no—
206C2×C4—272no—
207C2×C4—272no—
208C2×C4—271no—
209C2×C4—271no—
210C2×C4—542no—
211C2×C4D9⋊A411yescharacteristic
212C2×C3—121no—
213C2×C3—121no—
214C2×C3—121no—
215C2×C3—364no—
216C2×C3C4×D3×A411yescharacteristic
217C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
218C2×C3D3×#41⋊C311yescharacteristic
219D3—34no—
220C22—92no—
221C22—92no—
222C22—92no—
223C22C2×D9⋊A411yescharacteristic
224C4—31no—
225C4—31no—
226C4—61no—
227C4—92no—
228C4—272no—
229C4—541no—
230C4D9⋊SL(2,𝔽3)12yesdirect product factor
231C3—121no—
232C3C4×D3×SL(2,𝔽3)11yescharacteristic
233C2—94no—
234C2C4×D9⋊A411yescharacteristic
235C2C2×D9⋊SL(2,𝔽3)11yescharacteristic
236C2D9⋊(#41⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4D9⋊A4
commutator, hypocenterQ8×C9C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
prime radicalC2×D9⋊SL(2,𝔽3)C2
cosocleC2×Q8×C9C22×C3
LCSG ▷ Q8×C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C9⋊SL(2,𝔽3)C2semidirect
3. Dic9⋊SL(2,𝔽3)C2semidirect
5. C4×Q8×D9C3semidirect
8. D9⋊SL(2,𝔽3)C4direct (N × H)
13. C4×Q8×C9C2×C3semidirect
14. Q8×Dic9C2×C3semidirect
24. C9⋊SL(2,𝔽3)C2×C4semidirect
32. Q8×D9C3×C4semidirect
68. Q8×C9C2×C3×C4semidirect
79. C4×D9SL(2,𝔽3)semidirect
120. C4×C9C2×SL(2,𝔽3)semidirect
123. Dic9C2×SL(2,𝔽3)semidirect
128. C4×Q8AGL(1,ℤ/9)semidirect
162. D9C4×SL(2,𝔽3)semidirect
196. C9C2×C4×SL(2,𝔽3)semidirect
201. Q8C4×AGL(1,ℤ/9)semidirect
230. C4D9⋊SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D9⋊SL(2,𝔽3)2
C4×C9⋊SL(2,𝔽3)2
Dic9⋊SL(2,𝔽3)2
C4×D3×SL(2,𝔽3)3
C4×Q8×D93
C2×C4×AGL(1,ℤ/9)4