Finite Group Databasebeta

← #13180151

Subgroups — #13180151

Order 1728.

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

normal subgroups only

This group has 242 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
1Dic3×PSU(3,𝔽2)C211yesmaximal, characteristic
2#11724125C211yesmaximal, characteristic
3AGL(1,𝔽9)⋊Dic3C211yesmaximal, characteristic
4#11724102C211yesmaximal, characteristic
5#11724100C211yesmaximal, characteristic
6C3×PSU(3,𝔽2)⋊C4C211yesmaximal, characteristic
7Dic3⋊AGL(1,𝔽9)C211yesmaximal, characteristic
8PSU(3,𝔽2)⋊Q8—31nomaximal
9C2×C3×PSU(3,𝔽2)C2211yescharacteristic
10C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
11C2×C3×AGL(1,𝔽9)C2211yescharacteristic
12#88957—21no—
13#88946—21no—
14#88946C2211yescharacteristic
15#88940C2211yescharacteristic
16#88934—21no—
17C3×#3773C2211yescharacteristic
18Dic3×#157—21no—
19Dic3×#157C2211yescharacteristic
20C4×PSU(3,𝔽2)—31no—
21#64222—31no—
22C4⋊AGL(1,𝔽9)—31no—
23AGL(1,𝔽9)⋊C4—31no—
24AGL(1,𝔽9)⋊C4—31no—
25PSU(3,𝔽2)⋊C4—31no—
26PSU(3,𝔽2)⋊C4D311yescharacteristic
27C2×C3⋊#157—21no—
28C2×C3⋊#157D411yescharacteristic
29C2×C3×#157—21no—
30C2×C3×#157C2311yescharacteristic
31C3×PSU(3,𝔽2)—21no—
32C3×PSU(3,𝔽2)Q812yescomplemented
33C3⋊AGL(1,𝔽9)—21no—
34C3×AGL(1,𝔽9)—21no—
35#6294—21no—
36#6293—41no—
37Dic3×C3⋊D3D411yescharacteristic
38C3×#627—41no—
39C3×#624—21no—
40#4791—91nomaximal
41C2×PSU(3,𝔽2)C2×D311yescharacteristic
42C2×AGL(1,𝔽9)—31no—
43C2×AGL(1,𝔽9)C2×D311yescharacteristic
44#3797—61no—
45D3×Dic6—121no—
46C4×#157—31no—
47C4×#157—61no—
48#3777—31no—
49#3777—61no—
50#3773—31no—
51#3773—61no—
52#3773C2×D311yescharacteristic
53C2×C3×C3⋊D3C2×D411yescharacteristic
54C3⋊#157—21no—
55C3⋊#157—41no—
56C3×#157—22no—
57C3×#157C2×Q811yescharacteristic
58C3×#157#4111yescharacteristic
59C3⋊(C3⋊Dic3)SD1611yescharacteristic
60C3×C3⋊Dic3—41no—
61C32×Dic3—41no—
62C32×Dic3SD1611yescharacteristic
63Q8×Dic3—91no—
64#888—91no—
65C3×#103—91no—
66#835—91no—
67#817—91no—
68Dic3⋊C8—91no—
69#808—91no—
70C2×#157—21no—
71C2×#157—31no—
72C2×#157—61no—
73C2×#157C22×D311yescharacteristic
74PSU(3,𝔽2)—21no—
75PSU(3,𝔽2)Dic612yescomplemented
76AGL(1,𝔽9)—21no—
77AGL(1,𝔽9)—61no—
78C4×C3⋊D3—31no—
79#634—121no—
80C3×C4×D3—121no—
81C3×Dic6—121no—
82C3×Dic6—121no—
83#627—41no—
84#627—121no—
85#627—121no—
86#627—241no—
87#624—21no—
88#624—61no—
89D3×Dic3—121no—
90D3×Dic3—121no—
91#475—271noSylow
92C2×C33C2×SD1611yescharacteristic
93C3×C3⋊D3C22⋊Q811yescharacteristic
94C3×C3⋊D3#13711yescharacteristic
95C2×C3×Q8—91no—
96D3×Q8—361no—
97C2×Dic6—181no—
98C2×C3×C8—91no—
99C3×C4⋊C4—91no—
100C4⋊Dic3—91no—
101C4⋊Dic3—91no—
102C4⋊Dic3—181no—
103Dic3⋊C4—181no—
104C4×Dic3—91no—
105C4×Dic3—181no—
106C2×C3⋊C8—91no—
107C2×C3⋊D3D3×D411yescharacteristic
108C2×C3×D3—121no—
109#157—22no—
110#157—61no—
111#157—121no—
112#157C2×Dic611yescharacteristic
113#157#23711yescharacteristic
114C32×C4—31no—
115C32×C4—121no—
116C3⋊Dic3—31no—
117C3⋊Dic3—41no—
118C3⋊Dic3—121no—
119C3⋊Dic3—241no—
120C3×Dic3—41no—
121C3×Dic3—41no—
122C3×Dic3—41no—
123C3×Dic3—81no—
124C3×Dic3—121no—
125C3×Dic3—121no—
126C3×Dic3—241no—
127C4⋊Q8—271no—
128C4×Q8—271no—
129#106—271no—
130#106—271no—
131C4⋊C8—271no—
132#103—91no—
133#103—271no—
134C33#47511yescharacteristic, complemented, Sylow
135C3×Q8—92no—
136C3×Q8—181no—
137C3×Q8—361no—
138C2×C3×C4—91no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×Dic3—91no—
142C2×Dic3—91no—
143C2×Dic3—181no—
144C2×Dic3—181no—
145C4×D3—121no—
146C4×D3—361no—
147C4×D3—361no—
148Dic6—121no—
149Dic6—121no—
150Dic6—361no—
151Dic6—361no—
152Dic6—361no—
153Dic6—721no—
154C3×C8—181no—
155C3⋊C8—181no—
156C2×C32—41no—
157C2×C32—81no—
158C2×C32D3×SD1611yescharacteristic
159C3⋊D3#87811yescharacteristic
160C3⋊D3#91811yescharacteristic
161C3×D3—121no—
162C3×D3—121no—
163C2×Q8—91no—
164C2×Q8—541no—
165C2×C8—91no—
166C2×C8—271no—
167C4⋊C4—91no—
168C4⋊C4—271no—
169C4⋊C4—271no—
170C4⋊C4—541no—
171C4⋊C4—541no—
172C42—271no—
173C42—541no—
174C22×C3—91no—
175C2×D3—121no—
176C3×C4—91no—
177C3×C4—91no—
178C3×C4—121no—
179C3×C4—121no—
180C3×C4—121no—
181C3×C4—182no—
182C3×C4—241no—
183C3×C4—361no—
184Dic3—41no—
185Dic3—91no—
186Dic3—121no—
187Dic3—121no—
188Dic3—181no—
189Dic3—241no—
190Dic3—361no—
191Dic3—361no—
192Dic3—721no—
193Dic3AΓL(1,𝔽9)11yescharacteristic
194C32—41no—
195C32—81no—
196C32#479111yescharacteristic, complemented
197Q8—92no—
198Q8—181no—
199Q8—361no—
200Q8—1081no—
201C2×C4—91no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C4—271no—
205C2×C4—271no—
206C2×C4—541no—
207C2×C4—541no—
208C8—181no—
209C8—541no—
210C2×C3—41no—
211C2×C3—81no—
212C2×C3—91no—
213C2×C3—91no—
214C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
215D3—121no—
216D3—121no—
217C22—91no—
218C4—31no—
219C4—91no—
220C4—91no—
221C4—121no—
222C4—182no—
223C4—271no—
224C4—361no—
225C4—541no—
226C4—1081no—
227C3—41no—
228C3—81no—
229C3PSU(3,𝔽2)⋊Q811yescharacteristic, complemented
230C2—91no—
231C2—91no—
232C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radicalC2×C3×C3⋊D3C2×D4
hypocenterC33#475
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
32. C3×PSU(3,𝔽2)Q8semidirect
75. PSU(3,𝔽2)Dic6semidirect
134. C33#475semidirect
196. C32#4791semidirect
229. C3PSU(3,𝔽2)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
Dic3×PSU(3,𝔽2)2
#117241252
AGL(1,𝔽9)⋊Dic32
#117241022
#117241002
C3×PSU(3,𝔽2)⋊C42
Dic3⋊AGL(1,𝔽9)2
PSU(3,𝔽2)⋊Q83
#47919