Finite Group Databasebeta

← #13180437

Subgroups — #13180437

Order 1728.

310 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 51 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×AGL(1,𝔽9)⋊D3C212yesmaximal
2#11724365C211yesmaximal, characteristic, complemented
3C4×#157⋊D3C211yesmaximal, characteristic
4C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
5#11724105C212yesmaximal, complemented
6#10596409—31nomaximal
7C2×C3⋊PSU(3,𝔽2)C2212yes—
8C2×#157⋊D3C2211yescharacteristic
9C2×C3×AGL(1,𝔽9)C2212yes—
10AGL(1,𝔽9)⋊D3—24no—
11C4×#1113—21no—
12C4⋊(C3⋊#157)—21no—
13C3×C4×#157C2211yescharacteristic
14#88946—21no—
15#88940C2211yescharacteristic
16#88939—21no—
17C2×AΓL(1,𝔽9)—32no—
18C4⋊PSU(3,𝔽2)—31no—
19C4×GO+(4,𝔽2)—31no—
20C4⋊AGL(1,𝔽9)D311yescharacteristic, complemented
21PSU(3,𝔽2)⋊C4—32no—
22C2×#1113—21no—
23C2×C3⋊#157—22no—
24C2×C3×#157D411yescharacteristic
25C2×C3×#157C2311yescharacteristic
26C3⋊PSU(3,𝔽2)—22no—
27C3⋊PSU(3,𝔽2)—22no—
28#157⋊D3—21no—
29#157⋊D3D412yes—
30C3×AGL(1,𝔽9)—22no—
31C3×C4×C3⋊D3—41no—
32C3×C4×C3⋊D3D411yescharacteristic
33#6294—21no—
34#6294—41no—
35#4841—91nomaximal
36C2×PSU(3,𝔽2)—32no—
37C2×GO+(4,𝔽2)—31no—
38C2×AGL(1,𝔽9)C2×D312yes—
39AΓL(1,𝔽9)—64no—
40C4×D32—61no—
41C4×D32—121no—
42C4⋊#157—61no—
43C4×#157C2×D311yescharacteristic
44#3777—61no—
45#3773—31no—
46D32⋊C4—61no—
47C2×C3×C3⋊D3—41no—
48C2×C3×C3⋊D3C2×D411yescharacteristic
49#1113—22no—
50#1113—41no—
51C3⋊#157—42no—
52C3×#157—21no—
53C3×#157C2×D411yescharacteristic
54C3×#157#4111yescharacteristic
55C33×C4SD1611yescharacteristic, complemented
56C3×C3⋊Dic3—41no—
57C3×C3⋊Dic3SD1611yescharacteristic, complemented
58C2×#204—92no—
59C4×D12—91no—
60#869—91no—
61C3×C4⋊C8—91no—
62#816—92no—
63C2×D32—61no—
64C2×D32—121no—
65C2×#157—62no—
66C2×#157D1211yescharacteristic
67C2×#157C22×D311yescharacteristic
68PSU(3,𝔽2)—62no—
69PSU(3,𝔽2)—62no—
70GO+(4,𝔽2)—32no—
71GO+(4,𝔽2)—61no—
72AGL(1,𝔽9)—22no—
73C4×C3⋊D3—41no—
74C4×C3⋊D3D1211yescharacteristic
75C3×C4×D3—121no—
76C3×C4×D3—121no—
77C3×C4×D3—121no—
78C3×C4×D3—241no—
79#624—61no—
80#624—121no—
81D3×Dic3—121no—
82D3×Dic3—121no—
83D3×Dic3—121no—
84#461—271noSylow
85C2×C33C2×SD1611yescharacteristic
86C3×C3⋊D3—42no—
87C3×C3⋊D3#12111yescharacteristic
88C3×C3⋊D3#13711yescharacteristic
89C2×D12—91no—
90C2×C4×D3—181no—
91C2×C4×D3—361no—
92C2×Dic6—92no—
93C2×C3×C8—92no—
94C3×C42—91no—
95(C2×D3)⋊C4—181no—
96C4⋊Dic3—91no—
97C4⋊Dic3—181no—
98C4⋊Dic3—181no—
99#204—184no—
100C2×C3⋊D3—41no—
101C2×C3⋊D3C2×D1211yescharacteristic
102C2×C3×D3—121no—
103C2×C3×D3—121no—
104C2×C3×D3—121no—
105C2×C3×D3—241no—
106D32—62no—
107D32—122no—
108D32—122no—
109D32—121no—
110#157—21no—
111#157—122no—
112#157D3×D411yescharacteristic
113#157#23911yescharacteristic
114C32×C4—41no—
115C32×C4—81no—
116C32×C4#20411yescharacteristic, complemented
117C3⋊Dic3—41no—
118C3⋊Dic3#20411yescharacteristic, complemented
119C3×Dic3—121no—
120C3×Dic3—121no—
121C3×Dic3—121no—
122C3×Dic3—241no—
123C2×SD16—272no—
124C4⋊Q8—271no—
125C4×D4—271no—
126C4⋊C8—91no—
127#103—272no—
128C33#46111yescharacteristic, complemented, Sylow
129C22×D3—181no—
130C22×D3—361no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—361no—
135C2×Dic3—181no—
136C2×Dic3—182no—
137C2×Dic3—361no—
138D12—92no—
139D12—181no—
140C4×D3—121no—
141C4×D3—121no—
142C4×D3—121no—
143C4×D3—241no—
144C4×D3—361no—
145C4×D3—361no—
146C4×D3—361no—
147Dic6—182no—
148Dic6—182no—
149C3×C8—182no—
150C2×C32—41no—
151C2×C32—81no—
152C2×C32C2×#20411yescharacteristic
153C3⋊D3—42no—
154C3⋊D3#89511yescharacteristic
155C3⋊D3#90911yescharacteristic
156C3×D3—121no—
157C3×D3—121no—
158C3×D3—122no—
159C3×D3—122no—
160C3×D3—242no—
161C2×Q8—272no—
162C2×D4—271no—
163C22×C4—541no—
164SD16—544no—
165C2×C8—92no—
166C4⋊C4—271no—
167C4⋊C4—541no—
168C4⋊C4—541no—
169C22⋊C4—541no—
170C42—91no—
171C22×C3—91no—
172C22×C3—361no—
173C2×D3—121no—
174C2×D3—121no—
175C2×D3—121no—
176C2×D3—182no—
177C2×D3—241no—
178C2×D3—362no—
179C2×D3—361no—
180C2×D3—362no—
181C3×C4—41no—
182C3×C4—81no—
183C3×C4—91no—
184C3×C4—91no—
185C3×C4—91no—
186C3×C4—181no—
187C3×C4—361no—
188C3×C4AΓL(1,𝔽9)11yescharacteristic, complemented
189Dic3—121no—
190Dic3—121no—
191Dic3—121no—
192Dic3—241no—
193Dic3—362no—
194C32—41no—
195C32—81no—
196C32#484111yescharacteristic, complemented
197C23—541no—
198Q8—542no—
199Q8—542no—
200D4—272no—
201D4—541no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—361no—
206C2×C4—541no—
207C2×C4—542no—
208C2×C4—1081no—
209C8—182no—
210C2×C3—41no—
211C2×C3—81no—
212C2×C3—91no—
213C2×C3—91no—
214C2×C3—362no—
215C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
216D3—121no—
217D3—121no—
218D3—122no—
219D3—122no—
220D3—242no—
221C22—91no—
222C22—361no—
223C22—542no—
224C22—1081no—
225C4—91no—
226C4—91no—
227C4—91no—
228C4—181no—
229C4—361no—
230C4—1082no—
231C4AGL(1,𝔽9)⋊D311yescharacteristic, complemented
232C3—41no—
233C3—81no—
234C3#1059640911yescharacteristic, complemented
235C2—91no—
236C2—91no—
237C2—362no—
238C2C2×AGL(1,𝔽9)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽9)⋊D3
commutator, cosocleC2×C3×#157C23
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radicalC2×#157⋊D3C22
hypercenterC4AGL(1,𝔽9)⋊D3
hypocenterC33#461
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11724365C2semidirect
4. C3×C4⋊AGL(1,𝔽9)C2semidirect
5. #11724105C2semidirect
20. C4⋊AGL(1,𝔽9)D3semidirect
55. C33×C4SD16semidirect
57. C3×C3⋊Dic3SD16semidirect
116. C32×C4#204semidirect
118. C3⋊Dic3#204semidirect
128. C33#461semidirect
188. C3×C4AΓL(1,𝔽9)semidirect
196. C32#4841semidirect
231. C4AGL(1,𝔽9)⋊D3semidirect
234. C3#10596409semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,𝔽9)⋊D32
#117243652
C4×#157⋊D32
C3×C4⋊AGL(1,𝔽9)2
#117241052
#105964093
#48419