Finite Group Databasebeta

← #13147099

Subgroups — #13147099

Order 1728.

206 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C4×C9⋊SL(2,𝔽3)C211yesmaximal, characteristic
2Dic9⋊SL(2,𝔽3)C211yesmaximal, characteristic
3Dic9⋊SL(2,𝔽3)C211yesmaximal, characteristic
4#5569⋊C3—31nomaximal
5#10589671C311yesmaximal, characteristic, complemented
6C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
7(3−1+2⋊C4)⋊C4—41nomaximal
8C3×C4×SL(2,𝔽3)D311yescharacteristic
9Dic3×SL(2,𝔽3)—31no—
10Dic3×SL(2,𝔽3)—31no—
11C4×Q8×C9C2×C311yescharacteristic
12Q8×Dic9C2×C311yescharacteristic
13Q8×Dic9C2×C311yescharacteristic
14#63452—31no—
15#63450—31no—
16#63432—31no—
17C4×Dic18—31no—
18C4×Q8⋊C9—21no—
19C2×C4×3−1+2—41no—
20C2×3−1+2⋊C4—41no—
21C2×3−1+2⋊C4—41no—
22C9⋊SL(2,𝔽3)Q811yescharacteristic, complemented
23#5569—31no—
24#557⋊C3—91no—
25C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
26C3×Dic3⋊C4—121no—
27C2×Q8×C9C22×C311yescharacteristic
28C2×Dic18—31no—
29C2×Q8⋊C9—21no—
30C9×C4⋊C4—31no—
31C42×C9—31no—
32C4⋊Dic9—31no—
33C4⋊Dic9—31no—
34C4⋊Dic9—31no—
35Dic9⋊C4—32no—
36Dic9⋊C4—31no—
37Dic9⋊C4A411yescharacteristic
38C4×Dic9—31no—
39C4×Dic9—31no—
40C22×3−1+2—41no—
41C4×3−1+2—81no—
423−1+2⋊C4—42no—
433−1+2⋊C4—81no—
44C3×C4×Q8C3×D311yescharacteristic
45Q8×Dic3—31no—
46Q8×Dic3—31no—
47#890—91no—
48#888—91no—
49#870—91no—
50C4×Dic6—91no—
51C4×SL(2,𝔽3)—31no—
52C4×SL(2,𝔽3)—91no—
53C4×SL(2,𝔽3)—91no—
54C2×C32×C4—41no—
55C2×C3×Dic3—121no—
56C2×C3×Dic3—121no—
57C3×SL(2,𝔽3)Dic611yescharacteristic
58Q8×C9—31no—
59Q8×C9C3×Q811yescharacteristic
60C2×C4×C9—31no—
61C2×C4×C9—31no—
62C2×C4×C9—81no—
63C2×C4×C9C2×A411yescharacteristic
64C2×Dic9—31no—
65C2×Dic9—31no—
66C2×Dic9C2×A411yescharacteristic
67C2×Dic9C2×A411yescharacteristic
68Dic18—62no—
69Q8⋊C9—21no—
70#557—91noSylow
71C2×3−1+2—41no—
72C2×3−1+2—41no—
73C2×3−1+2—41no—
74C2×C3×Q8C2×C3×D311yescharacteristic
75C2×Dic6—91no—
76C2×SL(2,𝔽3)—31no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—361no—
79C3×C42—31no—
80C4⋊Dic3—91no—
81C4⋊Dic3—91no—
82C4⋊Dic3—91no—
83Dic3⋊C4—31no—
84Dic3⋊C4—92no—
85Dic3⋊C4—91no—
86C4×Dic3—91no—
87C4×Dic3—91no—
88C22×C32—41no—
89C32×C4—81no—
90C3×Dic3—122no—
91C3×Dic3—241no—
92C22×C9—81no—
93C22×C9C22×A411yescharacteristic
94C4×C9—21no—
95C4×C9—31no—
96C4×C9—31no—
97C4×C9—61no—
98C4×C9—82no—
99Dic9—21no—
100Dic9—61no—
101Dic9—61no—
102Dic9#41⋊C312yes—
103C4⋊Q8—271no—
104#125—271no—
105#125—271no—
106C4×Q8—91no—
107C4×Q8—91no—
108C4×Q8—271no—
109C4×Q8AGL(1,ℤ/9)11yescharacteristic
1103−1+2—41noSylow
111C3×Q8—31no—
112C3×Q8C3×Dic611yescharacteristic
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—121no—
116C2×C3×C4—361no—
117C2×C3×C4—361no—
118C2×C3×C4D3×A411yescharacteristic
119C2×Dic3—31no—
120C2×Dic3—31no—
121C2×Dic3—91no—
122C2×Dic3—91no—
123Dic6—182no—
124SL(2,𝔽3)—31no—
125C2×C32—41no—
126C2×C32—41no—
127C2×C32—41no—
128C2×C9—81no—
129C2×C9—81no—
130C2×C9—81no—
131C2×C9Q8×A411yescharacteristic
132C2×C9C2×#41⋊C311yescharacteristic
133C2×C92−1+4⋊C311yescharacteristic
134C2×Q8—271no—
135C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
136C4⋊C4—31no—
137C4⋊C4—91no—
138C4⋊C4—271no—
139C4⋊C4—271no—
140C4⋊C4—271no—
141C4⋊C4—272no—
142C4⋊C4—271no—
143C42—31no—
144C42—271no—
145C42—271no—
146C22×C3—121no—
147C22×C3C2×D3×A411yescharacteristic
148C3×C4—21no—
149C3×C4—31no—
150C3×C4—31no—
151C3×C4—61no—
152C3×C4—241no—
153C3×C4—362no—
154C3×C4—721no—
155Dic3—32no—
156Dic3—61no—
157Dic3—181no—
158Dic3—181no—
159C32—41no—
160C9—81no—
161C9#557⋊C311yescharacteristic, complemented
162Q8—31no—
163Q8—542no—
164Q83−1+2⋊Q811yescharacteristic
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—271no—
170C2×C4—271no—
171C2×C4D9⋊A411yescharacteristic
172C2×C3—121no—
173C2×C3—121no—
174C2×C3—121no—
175C2×C3A4×Dic611yescharacteristic
176C2×C3#1007⋊C311yescharacteristic
177C2×C3D3×#41⋊C311yescharacteristic
178C22C2×D9⋊A411yescharacteristic
179C4—21no—
180C4—31no—
181C4—31no—
182C4—61no—
183C4—92no—
184C4—181no—
185C4—541no—
186C4—541no—
187C3—121no—
188C3#5569⋊C311yescharacteristic
189C2Dic18⋊A411yescharacteristic
190C2#1172240811yescharacteristic
191C2D9⋊(#41⋊C3)11yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
prime radicalC2×C3×SL(2,𝔽3)C2×D3
hypercenterC2×C4D9⋊A4
hypocenterQ8×C9C3×Q8
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. #10589671C3semidirect
22. C9⋊SL(2,𝔽3)Q8semidirect
161. C9#557⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C9⋊SL(2,𝔽3)2
Dic9⋊SL(2,𝔽3)2
#5569⋊C33
#105896713
(3−1+2⋊C4)⋊C44