Finite Group Databasebeta

← Dic9×GL(2,𝔽3) #13146227

Subgroups — Dic9×GL(2,𝔽3) #13146227

Order 1728.

286 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 43 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×C9×GL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic9C211yesmaximal, characteristic, complemented
3C9⋊#859C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)—31nomaximal
5SD16×Dic9—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7C2×D3×Dic9—41nomaximal
8C9×GL(2,𝔽3)C412yescomplemented
9C2×C3×GL(2,𝔽3)D311yescharacteristic
10Dic3×SL(2,𝔽3)—31no—
11C3⋊#859—31no—
12C2×C9×SD16—31no—
13Q8×Dic9D311yescharacteristic, complemented
14D4×Dic9—31no—
15#63421—61no—
16#63395—31no—
17#63392—31no—
18#63377—31no—
19C8×Dic9—31no—
20C22×D3×C9—41no—
21C2×C3⋊Dic9—41no—
22C2×C3×Dic9—41no—
23C9×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24D3×Dic9—44no—
25C4×GL(2,𝔽3)—91no—
26Dic3×SD16—91no—
27C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
28C2×D3×Dic3—121no—
29C3×GL(2,𝔽3)Dic312yes—
30C2×Q8×C9C2×D311yescharacteristic
31C2×D4×C9—31no—
32C22×Dic9—61no—
33C2×Q8⋊C9—21no—
34C9×SD16—32no—
35C9×SD16—32no—
36C2×C8×C9—31no—
37#3676—61no—
38C4⋊Dic9—31no—
39C4⋊Dic9—31no—
40C4×Dic9—31no—
41C2×C9⋊C8—31no—
42C22×C3×C9—41no—
43C2×D3×C9—42no—
44C2×D3×C9—42no—
45C2×D3×C9—42no—
46C3⋊Dic9—42no—
47C3×Dic9—42no—
48C2×GL(2,𝔽3)D911yescharacteristic
49C2×C3×SD16—31no—
50Q8×Dic3—31no—
51D4×Dic3—91no—
52C4×SL(2,𝔽3)—91no—
53#859—91no—
54#859—181no—
55#835—91no—
56#832—91no—
57#817—91no—
58C8×Dic3—91no—
59C22×C3×D3—41no—
60C2×C3⋊Dic3—121no—
61C2×C3×Dic3—121no—
62C3×SL(2,𝔽3)C2×Dic311yescharacteristic
63D3×Dic3—124no—
64C23×C9—61no—
65Q8×C9—31no—
66Q8×C9C4×D311yescharacteristic, complemented
67D4×C9—32no—
68D4×C9—61no—
69C2×C4×C9—31no—
70C2×Dic9—31no—
71C2×Dic9—61no—
72C2×Dic9—122no—
73C2×Dic9—241no—
74C2×Dic9S411yescharacteristic
75Q8⋊C9—21no—
76C8×C9—32no—
77C9⋊C8—61no—
78C4×SD16—271noSylow
79C2×C3×C9—41no—
80C2×C3×C9—41no—
81C2×C3×C9—41no—
82D3×C9—44no—
83C2×C3×Q8D3211yescharacteristic
84C2×C3×D4—31no—
85C22×Dic3—181no—
86C2×C4×D3—361no—
87C2×SL(2,𝔽3)—21no—
88C2×SL(2,𝔽3)C2×D911yescharacteristic
89GL(2,𝔽3)Dic912yesdirect product factor
90C3×SD16—32no—
91C3×SD16—32no—
92C2×C3×C8—31no—
93#217—181no—
94C4⋊Dic3—91no—
95C4⋊Dic3—91no—
96C4×Dic3—91no—
97C2×C3⋊C8—91no—
98C22×C32—41no—
99C2×C3×D3—42no—
100C2×C3×D3—42no—
101C2×C3×D3—42no—
102C3⋊Dic3—122no—
103C3×Dic3—122no—
104C22×C9—62no—
105C22×C9—81no—
106C22×C9—121no—
107C22×C9—121no—
108C22×C9C2×S411yescharacteristic
109C4×C9—31no—
110C4×C9—31no—
111Dic9—61no—
112Dic9—121no—
113Dic9—242no—
114Dic9GL(2,𝔽3)12yesdirect product factor
115C2×SD16—31no—
116C4×Q8—91no—
117C4×D4—271no—
118#106—271no—
119#103—271no—
120#102—271no—
121C4×C8—271no—
122C3×C9—41noSylow
123C23×C3—61no—
124C22×D3—41no—
125C3×Q8—31no—
126C3×Q8D3×Dic311yescharacteristic
127C3×D4—32no—
128C3×D4—61no—
129C2×C3×C4—31no—
130C2×C3×C4—361no—
131C2×Dic3—31no—
132C2×Dic3—91no—
133C2×Dic3—181no—
134C2×Dic3—361no—
135C2×Dic3—362no—
136C2×Dic3—721no—
137C4×D3—364no—
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)C2×Dic911yescharacteristic, complemented
140C3×C8—32no—
141C3⋊C8—181no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—41no—
145C3×D3—44no—
146C2×C9—81no—
147C2×C9—81no—
148C2×C9—81no—
149C2×C9—122no—
150C2×C9C4×S411yescharacteristic
151C2×C9C2×GL(2,𝔽3)11yescharacteristic
152C2×C9#98511yescharacteristic
153C2×Q8D3×D911yescharacteristic
154C2×D4—31no—
155C22×C4—541no—
156SD16—32no—
157SD16—32no—
158C2×C8—31no—
159C2×C8—271no—
160C4⋊C4—271no—
161C4⋊C4—271no—
162C22⋊C4—541no—
163C42—271no—
164C22×C3—41no—
165C22×C3—62no—
166C22×C3—81no—
167C22×C3—121no—
168C22×C3—121no—
169C22×C3D3×S411yescharacteristic
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C3×C4—31no—
174C3×C4—31no—
175C3×C4—362no—
176Dic3—32no—
177Dic3—181no—
178Dic3—362no—
179Dic3—361no—
180Dic3—722no—
181C32—41no—
182C9—81no—
183C9C4×GL(2,𝔽3)11yescharacteristic, complemented
184C23—61no—
185Q8—31no—
186Q8D3×Dic911yescharacteristic, complemented
187D4—32no—
188D4—61no—
189C2×C4—31no—
190C2×C4—91no—
191C2×C4—271no—
192C2×C4—541no—
193C2×C4—1082no—
194C8—32no—
195C8—541no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3—41no—
199C2×C3—81no—
200C2×C3—81no—
201C2×C3—81no—
202C2×C3—122no—
203C2×C3#6419711yescharacteristic
204C2×C3D3×GL(2,𝔽3)11yescharacteristic
205C2×C3Dic3×S411yescharacteristic
206D3—44no—
207C22—62no—
208C22—121no—
209C22—121no—
210C22D9×S411yescharacteristic
211C4—31no—
212C4—31no—
213C4—92no—
214C4—541no—
215C4—1081no—
216C3—41no—
217C3—81no—
218C3Dic3×GL(2,𝔽3)11yescharacteristic
219C2—122no—
220C2#1172223111yescharacteristic
221C2D9×GL(2,𝔽3)11yescharacteristic
222C2S4×Dic911yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×SL(2,𝔽3)C2×C4
Frattini, socleC22×C3D3×S4
FittingC2×Q8×C9C2×D3
prime radicalC2×C3×GL(2,𝔽3)D3
cosocleC2×C9×SL(2,𝔽3)C22
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×Dic9C2semidirect
3. C9⋊#859C2semidirect
8. C9×GL(2,𝔽3)C4semidirect
13. Q8×Dic9D3semidirect
23. C9×SL(2,𝔽3)C2×C4semidirect
66. Q8×C9C4×D3semidirect
89. GL(2,𝔽3)Dic9direct (N × H)
114. Dic9GL(2,𝔽3)direct (N × H)
139. SL(2,𝔽3)C2×Dic9semidirect
183. C9C4×GL(2,𝔽3)semidirect
186. Q8D3×Dic9semidirect

Maximal subgroups

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

subgroupindex
C2×C9×GL(2,𝔽3)2
SL(2,𝔽3)×Dic92
C9⋊#8592
Dic3×GL(2,𝔽3)3
SD16×Dic93
C2×D3×Dic94