Finite Group Databasebeta

← #13180158

Subgroups — #13180158

Order 1728.

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

normal subgroups only

This group has 278 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
1#11724150C211yesmaximal, characteristic
2D3×#3773C211yesmaximal, characteristic
3#11724105C211yesmaximal, characteristic, complemented
4#11724103C211yesmaximal, characteristic, complemented
5#11724099C211yesmaximal, characteristic, complemented
6C3×AGL(1,𝔽9)⋊C4C211yesmaximal, characteristic, complemented
7(C2×D3)⋊AGL(1,𝔽9)C211yesmaximal, characteristic
8#10596415—31nomaximal
9C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
10C2×D3×#157C2211yescharacteristic
11C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
12C2×C3×AGL(1,𝔽9)C2211yescharacteristic
13D3×#624—21no—
14#88946—21no—
15C3×#3777C2211yescharacteristic
16#88940C2211yescharacteristic
17#88934—21no—
18C3×#3773C2211yescharacteristic
19(C2×D3)⋊#157—21no—
20C22⋊PSU(3,𝔽2)—31no—
21C2×#3773—31no—
22C22⋊AGL(1,𝔽9)—31no—
23AGL(1,𝔽9)⋊C4—31no—
24AGL(1,𝔽9)⋊C4D311yescharacteristic, complemented
25PSU(3,𝔽2)⋊C4—31no—
26PSU(3,𝔽2)⋊C4—31no—
27C2×D3×C3⋊D3D411yescharacteristic
28C2×C3⋊#157—21no—
29C2×C3⋊#157D411yescharacteristic
30C2×C3×#157—21no—
31C2×C3×#157C2311yescharacteristic
32C3⋊PSU(3,𝔽2)—21no—
33C3⋊PSU(3,𝔽2)—21no—
34D3×#157—21no—
35D3×#157—21no—
36C3⋊AGL(1,𝔽9)—21no—
37C3×AGL(1,𝔽9)—21no—
38#6294—21no—
39#6289—41no—
40C3×#624—21no—
41C3×D3×Dic3—41no—
42#4862—91nomaximal
43C22×#157—31no—
44C2×PSU(3,𝔽2)—31no—
45C2×AGL(1,𝔽9)—31no—
46C2×AGL(1,𝔽9)C2×D311yescharacteristic
47C2×#624—61no—
48C4×D32—121no—
49C22⋊#157—61no—
50#3777—61no—
51#3777C2×D311yescharacteristic
52#3773—31no—
53#3773—61no—
54#3773C2×D311yescharacteristic
55C2×C3⋊(C3⋊D3)SD1611yescharacteristic
56C2×C3×C3⋊D3C2×D411yescharacteristic
57C2×C32×D3SD1611yescharacteristic
58D3×C3⋊D3—21no—
59D3×C3⋊D3—21no—
60C3⋊#157—21no—
61C3⋊#157—41no—
62C3×#157—41no—
63C3×#157#4111yescharacteristic
64C3×#157#4111yescharacteristic
65C3×C3⋊Dic3—41no—
66C32×Dic3—41no—
67#897—91no—
68D3×C4⋊C4—91no—
69C3×#106—91no—
70(C2×D3)⋊C8—91no—
71#816—91no—
72#810—91no—
73#808—91no—
74C22×C3⋊D3—31no—
75C2×D32—121no—
76C2×#157—21no—
77C2×#157—31no—
78C2×#157—61no—
79C2×#157—61no—
80C2×#157—61no—
81C2×#157C22×D311yescharacteristic
82PSU(3,𝔽2)—61no—
83PSU(3,𝔽2)—61no—
84AGL(1,𝔽9)—21no—
85AGL(1,𝔽9)—61no—
86C4×C3⋊D3—121no—
87C2×C3×Dic3—121no—
88C3×C4×D3—121no—
89C3×C4×D3—121no—
90#624—21no—
91#624—61no—
92#624—121no—
93#624—121no—
94#624—241no—
95D3×Dic3—41no—
96D3×Dic3—121no—
97#483—271noSylow
98C2×C33C2×SD1611yescharacteristic
99C3⋊(C3⋊D3)—21no—
100C3×C3⋊D3#12311yescharacteristic
101C3×C3⋊D3#13711yescharacteristic
102C32×D3—21no—
103C2×C4×D3—91no—
104C2×C4×D3—181no—
105C2×C4×D3—361no—
106C2×Dic6—91no—
107C2×C3×C8—91no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110(C2×D3)⋊C4—181no—
111C4⋊Dic3—91no—
112C4⋊Dic3—181no—
113Dic3⋊C4—181no—
114C2×C3⋊C8—91no—
115C22×C32—31no—
116C2×C3⋊D3—31no—
117C2×C3⋊D3—61no—
118C2×C3⋊D3—61no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—241no—
121C2×C3⋊D3D3×D411yescharacteristic
122C2×C3×D3—41no—
123C2×C3×D3—121no—
124D32—122no—
125D32—122no—
126#157—41no—
127#157—61no—
128#157—121no—
129#157#23511yescharacteristic
130#157#23911yescharacteristic
131C32×C4—121no—
132C3⋊Dic3—41no—
133C3×Dic3—41no—
134C3×Dic3—41no—
135C3×Dic3—81no—
136C3×Dic3—121no—
137C3×Dic3—121no—
138C3×Dic3—241no—
139C22⋊Q8—271no—
140C2×C4⋊C4—271no—
141#106—91no—
142#106—271no—
143#103—271no—
144#103—271no—
145C22⋊C8—271no—
146C33#48311yescharacteristic, complemented, Sylow
147C22×D3—91no—
148C22×D3—361no—
149C2×C3×C4—91no—
150C2×C3×C4—181no—
151C2×C3×C4—181no—
152C2×C3×C4—361no—
153C2×Dic3—91no—
154C2×Dic3—121no—
155C2×Dic3—181no—
156C2×Dic3—181no—
157C4×D3—121no—
158C4×D3—121no—
159C4×D3—181no—
160C4×D3—181no—
161C4×D3—361no—
162C4×D3—361no—
163C4×D3—361no—
164C4×D3—361no—
165C4×D3—721no—
166Dic6—181no—
167Dic6—181no—
168C3×C8—181no—
169C3⋊C8—181no—
170C2×C32—41no—
171C2×C32—61no—
172C2×C32—81no—
173C2×C32D3×SD1611yescharacteristic
174C3⋊D3—61no—
175C3⋊D3—122no—
176C3⋊D3—242no—
177C3⋊D3#89411yescharacteristic
178C3⋊D3#91511yescharacteristic
179C3×D3—42no—
180C3×D3—121no—
181C3×D3—121no—
182C2×Q8—271no—
183C22×C4—271no—
184C22×C4—541no—
185C2×C8—91no—
186C2×C8—271no—
187C4⋊C4—91no—
188C4⋊C4—91no—
189C4⋊C4—271no—
190C4⋊C4—541no—
191C4⋊C4—541no—
192C22⋊C4—541no—
193C22×C3—91no—
194C22×C3—121no—
195C2×D3—91no—
196C2×D3—121no—
197C2×D3—181no—
198C2×D3—181no—
199C2×D3—361no—
200C2×D3—362no—
201C2×D3—362no—
202C2×D3—721no—
203C2×D3AΓL(1,𝔽9)11yescharacteristic
204C3×C4—91no—
205C3×C4—91no—
206C3×C4—121no—
207C3×C4—121no—
208C3×C4—241no—
209C3×C4—361no—
210C3×C4—361no—
211Dic3—41no—
212Dic3—121no—
213Dic3—121no—
214Dic3—181no—
215Dic3—241no—
216Dic3—361no—
217C32—41no—
218C32—81no—
219C32#486211yescharacteristic, complemented
220C23—271no—
221Q8—541no—
222Q8—541no—
223C2×C4—91no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4—271no—
227C2×C4—361no—
228C2×C4—541no—
229C2×C4—541no—
230C2×C4—541no—
231C2×C4—541no—
232C2×C4—1081no—
233C8—181no—
234C8—541no—
235C2×C3—41no—
236C2×C3—81no—
237C2×C3—91no—
238C2×C3—91no—
239C2×C3—122no—
240C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
241D3—21no—
242D3—121no—
243D3—121no—
244D3—181no—
245D3—362no—
246D3—722no—
247C22—31no—
248C22—91no—
249C22—271no—
250C22—541no—
251C22—541no—
252C4—91no—
253C4—91no—
254C4—121no—
255C4—361no—
256C4—361no—
257C4—541no—
258C4—1081no—
259C3—41no—
260C3—81no—
261C3#1059641511yescharacteristic, complemented
262C2—61no—
263C2—91no—
264C2—91no—
265C2—541no—
266C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radicalC2×D3×C3⋊D3D4
hypocenterC33#483
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. #11724105C2semidirect
4. #11724103C2semidirect
5. #11724099C2semidirect
6. C3×AGL(1,𝔽9)⋊C4C2semidirect
24. AGL(1,𝔽9)⋊C4D3semidirect
146. C33#483semidirect
219. C32#4862semidirect
261. C3#10596415semidirect

Maximal subgroups

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

subgroupindex
#117241502
D3×#37732
#117241052
#117241032
#117240992
C3×AGL(1,𝔽9)⋊C42
(C2×D3)⋊AGL(1,𝔽9)2
#105964153
#48629