Finite Group Databasebeta

← (D9⋊SL(2,𝔽3))⋊C4 #13147109

Subgroups — (D9⋊SL(2,𝔽3))⋊C4 #13147109

Order 1728.

269 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 48 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
4#5572⋊C3—31nomaximal
5#10589674C311yesmaximal, characteristic, complemented
6C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
7D9⋊SL(2,𝔽3)C412yescomplemented
8(C2×AGL(1,ℤ/9))⋊C4—41nomaximal
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
15#63454—31no—
16#63449—31no—
17#63434—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—
23C9⋊SL(2,𝔽3)C2×C411yescharacteristic
24#5572—31no—
25#519⋊C3—91no—
26C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
27D3×SL(2,𝔽3)—32no—
28C3×(C2×D3)⋊C4—121no—
29C2×Q8×C9C22×C311yescharacteristic
30Q8×D9—32no—
31Q8×D9C3×C412yescomplemented
32C2×C4×D9—31no—
33C2×Dic18—31no—
34C2×Q8⋊C9—21no—
35C9×C4⋊C4—31no—
36C42×C9—31no—
37(C2×D9)⋊C4—31no—
38(C2×D9)⋊C4A411yescharacteristic
39C4⋊Dic9—31no—
40Dic9⋊C4—31no—
41Dic9⋊C4—31no—
42C4×Dic9—32no—
43C4×Dic9—31no—
44C22×3−1+2—41no—
45C2×AGL(1,ℤ/9)—42no—
46C2×AGL(1,ℤ/9)—81no—
47C2×AGL(1,ℤ/9)—81no—
48C4×3−1+2—81no—
493−1+2⋊C4—81no—
50C2×D3×Q8—31no—
51C22×SL(2,𝔽3)—91no—
52C3×C4×Q8C3×D311yescharacteristic
53Q8×Dic3—31no—
54#892—91no—
55#887—91no—
56#872—91no—
57C4×Dic6—91no—
58C4×SL(2,𝔽3)—31no—
59C4×SL(2,𝔽3)—91no—
60C22×C3×D3—121no—
61C2×C32×C4—41no—
62C2×C3×Dic3—121no—
63C3×SL(2,𝔽3)C4×D311yescharacteristic
64C22×D9C2×A411yescharacteristic
65Q8×C9—31no—
66Q8×C9C2×C3×C411yescharacteristic
67C2×C4×C9—31no—
68C2×C4×C9—31no—
69C2×C4×C9—81no—
70C2×C4×C9C2×A411yescharacteristic
71C2×Dic9—31no—
72C2×Dic9—31no—
73C2×Dic9C2×A411yescharacteristic
74C4×D9—32no—
75C4×D9—32no—
76Dic18—32no—
77Dic18—32no—
78Q8⋊C9—21no—
79#519—91noSylow
80C2×3−1+2—41no—
81C2×3−1+2—41no—
82C2×3−1+2—41no—
83AGL(1,ℤ/9)—82no—
84C2×C3×Q8C2×C3×D311yescharacteristic
85D3×Q8—32no—
86D3×Q8—92no—
87C2×C4×D3—91no—
88C2×Dic6—91no—
89C2×SL(2,𝔽3)—31no—
90C2×SL(2,𝔽3)—92no—
91C3×C4⋊C4—31no—
92C3×C22⋊C4—361no—
93C3×C42—31no—
94(C2×D3)⋊C4—31no—
95(C2×D3)⋊C4—91no—
96C4⋊Dic3—91no—
97Dic3⋊C4—91no—
98Dic3⋊C4—91no—
99C4×Dic3—91no—
100C4×Dic3—92no—
101C22×C32—41no—
102C2×C3×D3—122no—
103C2×C3×D3—241no—
104C2×C3×D3—241no—
105C32×C4—81no—
106C3×Dic3—241no—
107C22×C9—81no—
108C22×C9C22×A411yescharacteristic
109C2×D9—21no—
110C2×D9—21no—
111C2×D9C4×A412yes—
112C4×C9—21no—
113C4×C9—31no—
114C4×C9—31no—
115C4×C9—61no—
116C4×C9—82no—
117Dic9—21no—
118Dic9—32no—
119Dic9—61no—
120C22×Q8—91no—
121C4×Q8—91no—
122C4×Q8—271no—
123C4×Q8—271no—
124C4×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
125#117—271no—
126#117—271no—
1273−1+2—41noSylow
128C23×C3—361no—
129C22×D3—31no—
130C3×Q8—31no—
131C3×Q8C3×C4×D311yescharacteristic
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—121no—
135C2×C3×C4—361no—
136C2×C3×C4D3×A411yescharacteristic
137C2×Dic3—31no—
138C2×Dic3—91no—
139C2×Dic3—91no—
140C4×D3—92no—
141C4×D3—92no—
142Dic6—92no—
143Dic6—92no—
144SL(2,𝔽3)—31no—
145C2×C32—41no—
146C2×C32—41no—
147C2×C32—41no—
148C3×D3—242no—
149C2×C9—81no—
150C2×C9—81no—
151C2×C9—81no—
152C2×C9C2×C4×A411yescharacteristic
153C2×C92−1+4⋊C311yescharacteristic
154C2×C92−1+4⋊C311yescharacteristic
155D9—22no—
156C2×Q8—92no—
157C2×Q8—271no—
158C2×Q8—272no—
159C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
160C22×C4—271no—
161C4⋊C4—31no—
162C4⋊C4—271no—
163C4⋊C4—271no—
164C4⋊C4—271no—
165C22⋊C4—91no—
166C22⋊C4—271no—
167C42—31no—
168C42—271no—
169C42—272no—
170C22×C3—121no—
171C22×C3—362no—
172C22×C3—721no—
173C22×C3—721no—
174C22×C3C2×D3×A411yescharacteristic
175C2×D3—32no—
176C2×D3—61no—
177C2×D3—61no—
178C3×C4—21no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—61no—
182C3×C4—241no—
183C3×C4—721no—
184Dic3—61no—
185Dic3—92no—
186Dic3—181no—
187C32—41no—
188C9—81no—
189C9#519⋊C311yescharacteristic, complemented
190C23—91no—
191Q8—31no—
192Q8—272no—
193Q8—272no—
194Q8C4×AGL(1,ℤ/9)11yescharacteristic
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—91no—
198C2×C4—272no—
199C2×C4—272no—
200C2×C4—271no—
201C2×C4—271no—
202C2×C4D9⋊A411yescharacteristic
203C2×C3—121no—
204C2×C3—121no—
205C2×C3—121no—
206C2×C3—722no—
207C2×C3C4×D3×A411yescharacteristic
208C2×C3#1007⋊C311yescharacteristic
209C2×C3#1010⋊C311yescharacteristic
210D3—62no—
211C22—92no—
212C22—181no—
213C22—181no—
214C22C2×D9⋊A411yescharacteristic
215C4—21no—
216C4—31no—
217C4—31no—
218C4—61no—
219C4—181no—
220C4—272no—
221C4—541no—
222C3—121no—
223C3#5572⋊C311yescharacteristic
224C2—182no—
225C2C4×D9⋊A411yescharacteristic
226C2#1172240811yescharacteristic
227C2#1172241411yescharacteristic
subgroupG/H
centerC22C2×D9⋊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
hypercenterC2×C4D9⋊A4
LCSG ▷ Q8×C9
UCSC1 ◁ C22 ◁ 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. #10589674C3semidirect
7. D9⋊SL(2,𝔽3)C4semidirect
13. C4×Q8×C9C2×C3semidirect
14. Q8×Dic9C2×C3semidirect
31. Q8×D9C3×C4semidirect
124. C4×Q8AGL(1,ℤ/9)semidirect
189. C9#519⋊C3semidirect

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
#5572⋊C33
#105896743
(C2×AGL(1,ℤ/9))⋊C44