Finite Group Databasebeta

← #13146217

Subgroups — #13146217

Order 1728.

259 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 41 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×#88615C211yesmaximal, characteristic
2C9⋊#859C211yesmaximal, characteristic, complemented
3C9×#859C211yesmaximal, characteristic, complemented
4#10593200—31nomaximal
5#10588483—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7#88615C412yescomplemented
8Dic9⋊Dic3—41nomaximal
9C2×C3⋊(#41⋊C3)—31no—
10C3⋊#859—31no—
11C3×#859D311yescharacteristic
12C2×#3701D311yescharacteristic
13C2×#3663—31no—
14C4⋊C4×D9—31no—
15#63421—61no—
16C9×#103—31no—
17#63395—31no—
18#63380—31no—
19#63369—31no—
20C2×C3⋊Dic9—41no—
21C2×C9×Dic3—41no—
22C2×C3×Dic9—41no—
23C9×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24#5426—91no—
25#4803—91no—
26C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
27C3⋊(#41⋊C3)—32no—
28Dic3⋊Dic3—121no—
29C2×Q8×C9C2×D311yescharacteristic
30#3701—32no—
31#3701Dic312yescomplemented
32C2×D36—31no—
33C2×C4×D9—31no—
34C2×C4×D9—61no—
35C2×Q8⋊C9—21no—
36C2×C8×C9—31no—
37C9×C4⋊C4—31no—
38C4⋊Dic9—31no—
39Dic9⋊C4—61no—
40C2×C9⋊C8—31no—
41#3663—62no—
42C22×C3×C9—41no—
43C3⋊Dic9—81no—
44C9×Dic3—81no—
45C3×Dic9—42no—
46C2×#239—31no—
47C2×#41⋊C3—91no—
48C2×ΓL(1,𝔽25)—91no—
49D3×C4⋊C4—91no—
50#859—91no—
51#859—181no—
52#859D911yescharacteristic, complemented
53C3×#103—31no—
54#835—91no—
55#821—91no—
56#809—91no—
57C2×C3⋊Dic3—121no—
58C2×C3×Dic3—41no—
59C2×C3×Dic3—121no—
60C3×SL(2,𝔽3)C4×D311yescharacteristic
61C22×D9—31no—
62Q8×C9—31no—
63Q8×C9C2×Dic311yescharacteristic
64C2×C4×C9—31no—
65C2×C4×C9—61no—
66C2×Dic9—61no—
67C2×Dic9—241no—
68C2×Dic9S411yescharacteristic
69D36—32no—
70D36—61no—
71C4×D9—32no—
72C4×D9—32no—
73C4×D9—122no—
74Q8⋊C9—21no—
75C8×C9—61no—
76C9⋊C8—61no—
77#417—271noSylow
78C2×C3×C9—41no—
79C2×C3×C9—41no—
80C2×C3×C9—41no—
81C2×C3×Q8D3211yescharacteristic
82#239—32no—
83#239—92no—
84C2×D12—91no—
85C2×C4×D3—91no—
86C2×C4×D3—181no—
87#41⋊C3—92no—
88C2×SL(2,𝔽3)—21no—
89C2×SL(2,𝔽3)C2×D911yescharacteristic
90C2×C3×C8—31no—
91C3×C4⋊C4—31no—
92C4⋊Dic3—91no—
93C4⋊Dic3—361no—
94Dic3⋊C4—181no—
95C2×C3⋊C8—91no—
96ΓL(1,𝔽25)—182no—
97C22×C32—41no—
98C3⋊Dic3—241no—
99C3×Dic3—81no—
100C3×Dic3—122no—
101C22×C9—81no—
102C22×C9C2×S411yescharacteristic
103C2×D9—32no—
104C2×D9—61no—
105C2×D9—61no—
106C4×C9—31no—
107C4×C9—31no—
108C4×C9—121no—
109Dic9—121no—
110Dic9—242no—
111Dic9SL(2,ℤ/4)12yes—
112C2×#41—91no—
113C2×M16—271no—
114C2×C4⋊C4—271no—
115#103—31no—
116#103—271no—
117#102—271no—
118#102—271no—
119C3×C9—41noSylow
120C22×D3—91no—
121C3×Q8—31no—
122C3×Q8D3×Dic311yescharacteristic
123C2×C3×C4—31no—
124C2×C3×C4—61no—
125C2×C3×C4—361no—
126C2×Dic3—31no—
127C2×Dic3—41no—
128C2×Dic3—181no—
129C2×Dic3—361no—
130C2×Dic3—721no—
131D12—92no—
132D12—181no—
133C4×D3—92no—
134C4×D3—92no—
135C4×D3—362no—
136SL(2,𝔽3)—21no—
137SL(2,𝔽3)C4×D911yescharacteristic, complemented
138C3×C8—61no—
139C3⋊C8—181no—
140C2×C32—41no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C9—81no—
144C2×C9—81no—
145C2×C9—81no—
146C2×C9#98411yescharacteristic
147C2×C9#98611yescharacteristic
148C2×C9C2×SL(2,ℤ/4)11yescharacteristic
149D9—62no—
150#41—92no—
151#41—272no—
152C2×Q8D3×D911yescharacteristic
153C2×D4—271no—
154C22×C4—271no—
155C22×C4—541no—
156M16—542no—
157C2×C8—31no—
158C2×C8—271no—
159C4⋊C4—31no—
160C4⋊C4—271no—
161C4⋊C4—541no—
162C22×C3—41no—
163C22×C3—81no—
164C22×C3D3×S411yescharacteristic
165C2×D3—92no—
166C2×D3—181no—
167C2×D3—181no—
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—121no—
171C3×C4—362no—
172Dic3—32no—
173Dic3—81no—
174Dic3—361no—
175Dic3—721no—
176Dic3—722no—
177C32—41no—
178C9—81no—
179C9#542611yescharacteristic, complemented
180C23—271no—
181Q8—31no—
182Q8Dic3×D911yescharacteristic
183D4—272no—
184D4—541no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—91no—
188C2×C4—272no—
189C2×C4—272no—
190C2×C4—541no—
191C2×C4—1082no—
192C8—61no—
193C8—541no—
194C2×C3—41no—
195C2×C3—41no—
196C2×C3—41no—
197C2×C3—81no—
198C2×C3—81no—
199C2×C3—81no—
200C2×C3#6419611yescharacteristic
201C2×C3#6419911yescharacteristic
202C2×C3D3×SL(2,ℤ/4)11yescharacteristic
203D3—182no—
204C22—272no—
205C22—541no—
206C22—541no—
207C22D9×S411yescharacteristic
208C4—31no—
209C4—31no—
210C4—92no—
211C4—121no—
212C4—1081no—
213C3—41no—
214C3—81no—
215C3#1059320011yescharacteristic
216C2—542no—
217C2#1172223511yescharacteristic
218C2#1172223611yescharacteristic
219C2D9×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×SL(2,𝔽3)C2×C4
Frattini, socleC22×C3D3×S4
FittingC2×Q8×C9C2×D3
prime radicalC2×#88615C2
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. C9⋊#859C2semidirect
3. C9×#859C2semidirect
7. #88615C4semidirect
23. C9×SL(2,𝔽3)C2×C4semidirect
31. #3701Dic3semidirect
52. #859D9semidirect
137. SL(2,𝔽3)C4×D9semidirect
179. C9#5426semidirect

Maximal subgroups

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

subgroupindex
C2×#886152
C9⋊#8592
C9×#8592
#105932003
#105884833
Dic9⋊Dic34