Finite Group Databasebeta

← #13180160

Subgroups — #13180160

Order 1728.

246 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 46 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
1#11724153C211yesmaximal, characteristic
2#11724125C211yesmaximal, characteristic
3#11724105C211yesmaximal, characteristic
4#11724102C211yesmaximal, characteristic
5#11724099C211yesmaximal, characteristic
6C3×PSU(3,𝔽2)⋊C4C211yesmaximal, characteristic
7Dic3×AGL(1,𝔽9)C211yesmaximal, characteristic
8#10596404—31nomaximal
9C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
10C2×C3×PSU(3,𝔽2)C2211yescharacteristic
11C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
12C2×C3×AGL(1,𝔽9)C2211yescharacteristic
13#88957—21no—
14#88945—21no—
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—
23PSU(3,𝔽2)⋊C4—31no—
24PSU(3,𝔽2)⋊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)—21no—
33C3×PSU(3,𝔽2)—21no—
34C3×PSU(3,𝔽2)—21no—
35C3⋊AGL(1,𝔽9)—21no—
36C3×AGL(1,𝔽9)—21no—
37#6294—21no—
38#6293—41no—
39Dic3×C3⋊D3D411yescharacteristic
40C3×#627—41no—
41C3×#624—21no—
42#4792—91nomaximal
43C2×PSU(3,𝔽2)—31no—
44C2×PSU(3,𝔽2)C2×D311yescharacteristic
45C2×AGL(1,𝔽9)—31no—
46C2×AGL(1,𝔽9)C2×D311yescharacteristic
47#3797—61no—
48D3×Dic6—121no—
49C4⋊#157—61no—
50C4×#157—31no—
51#3777—61no—
52#3773—31no—
53#3773—61no—
54#3773C2×D311yescharacteristic
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C3⋊#157—41no—
57C3⋊#157Q1612yes—
58C3×#157—41no—
59C3×#157#4111yescharacteristic
60C3×#157#4111yescharacteristic
61C3⋊(C3⋊Dic3)SD1611yescharacteristic
62C3×C3⋊Dic3—41no—
63C32×Dic3—41no—
64C32×Dic3SD1611yescharacteristic
65#944—91no—
66#888—91no—
67C3×#103—91no—
68#835—91no—
69#816—91no—
70C8×Dic3—91no—
71#810—91no—
72C2×#157—21no—
73C2×#157—31no—
74C2×#157—61no—
75C2×#157C22×D311yescharacteristic
76PSU(3,𝔽2)—21no—
77PSU(3,𝔽2)—21no—
78PSU(3,𝔽2)—61no—
79PSU(3,𝔽2)—61no—
80AGL(1,𝔽9)—21no—
81AGL(1,𝔽9)—61no—
82C4×C3⋊D3—31no—
83#634—121no—
84C3×C4×D3—121no—
85C3×Dic6—121no—
86C3×Dic6—121no—
87#627—41no—
88#627—121no—
89#627—121no—
90#627—241no—
91#624—21no—
92#624—61no—
93D3×Dic3—121no—
94D3×Dic3—121no—
95#487—271noSylow
96C2×C33C2×SD1611yescharacteristic
97C3×C3⋊D3#12411yescharacteristic
98C3×C3⋊D3C2×Q1611yescharacteristic
99C2×C3×Q8—91no—
100D3×Q8—361no—
101C2×Dic6—91no—
102C2×Dic6—181no—
103C2×C3×C8—91no—
104C3×C4⋊C4—91no—
105C4⋊Dic3—91no—
106Dic3⋊C4—181no—
107Dic3⋊C4—181no—
108Dic3⋊C4—181no—
109C4×Dic3—91no—
110C2×C3⋊C8—91no—
111C2×C3⋊D3D3×D411yescharacteristic
112C2×C3×D3—121no—
113#157—32no—
114#157—41no—
115#157—121no—
116#157#23511yescharacteristic
117#157#23711yescharacteristic
118C32×C4—31no—
119C32×C4—121no—
120C3⋊Dic3—31no—
121C3⋊Dic3—41no—
122C3⋊Dic3—121no—
123C3⋊Dic3—241no—
124C3×Dic3—41no—
125C3×Dic3—41no—
126C3×Dic3—41no—
127C3×Dic3—81no—
128C3×Dic3—121no—
129C3×Dic3—121no—
130C3×Dic3—241no—
131C4⋊Q8—271no—
132C4⋊Q8—271no—
133#103—91no—
134#103—271no—
135#103—271no—
136#103—271no—
137C4×C8—271no—
138C33#48711yescharacteristic, complemented, Sylow
139C3×Q8—181no—
140C3×Q8—181no—
141C3×Q8—361no—
142C2×C3×C4—91no—
143C2×C3×C4—181no—
144C2×C3×C4—181no—
145C2×Dic3—91no—
146C2×Dic3—91no—
147C2×Dic3—181no—
148C2×Dic3—181no—
149C4×D3—121no—
150C4×D3—361no—
151C4×D3—361no—
152Dic6—121no—
153Dic6—121no—
154Dic6—181no—
155Dic6—181no—
156Dic6—361no—
157Dic6—361no—
158Dic6—361no—
159Dic6—721no—
160C3×C8—181no—
161C3⋊C8—181no—
162C2×C32—41no—
163C2×C32—81no—
164C2×C32D3×SD1611yescharacteristic
165C3⋊D3#88511yescharacteristic
166C3⋊D3D3×Q1611yescharacteristic
167C3×D3—121no—
168C3×D3—121no—
169C2×Q8—91no—
170C2×Q8—271no—
171C2×Q8—541no—
172C2×C8—91no—
173C2×C8—271no—
174C4⋊C4—91no—
175C4⋊C4—271no—
176C4⋊C4—541no—
177C4⋊C4—541no—
178C4⋊C4—541no—
179C42—271no—
180C22×C3—91no—
181C2×D3—121no—
182C3×C4—91no—
183C3×C4—91no—
184C3×C4—121no—
185C3×C4—121no—
186C3×C4—121no—
187C3×C4—241no—
188C3×C4—361no—
189C3×C4—361no—
190Dic3—41no—
191Dic3—92no—
192Dic3—91no—
193Dic3—121no—
194Dic3—121no—
195Dic3—241no—
196Dic3—361no—
197Dic3—361no—
198Dic3—721no—
199Dic3AΓL(1,𝔽9)11yescharacteristic
200C32—41no—
201C32—81no—
202C32#479211yescharacteristic, complemented
203Q8—181no—
204Q8—181no—
205Q8—361no—
206Q8—541no—
207Q8—541no—
208Q8—1081no—
209C2×C4—91no—
210C2×C4—181no—
211C2×C4—181no—
212C2×C4—271no—
213C2×C4—271no—
214C2×C4—541no—
215C2×C4—541no—
216C8—181no—
217C8—541no—
218C2×C3—41no—
219C2×C3—81no—
220C2×C3—91no—
221C2×C3—91no—
222C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
223D3—121no—
224D3—121no—
225C22—91no—
226C4—31no—
227C4—91no—
228C4—91no—
229C4—121no—
230C4—272no—
231C4—271no—
232C4—361no—
233C4—361no—
234C4—1081no—
235C3—41no—
236C3—81no—
237C3#1059640411yescharacteristic, complemented
238C2—91no—
239C2—91no—
240C2D3×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#487
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
138. C33#487semidirect
202. C32#4792semidirect
237. C3#10596404semidirect

Maximal subgroups

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

subgroupindex
#117241532
#117241252
#117241052
#117241022
#117240992
C3×PSU(3,𝔽2)⋊C42
Dic3×AGL(1,𝔽9)2
#105964043
#47929