Finite Group Databasebeta

← #13180153

Subgroups — #13180153

Order 1728.

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

normal subgroups only

This group has 286 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
1#11724154C211yesmaximal, characteristic, complemented
2Dic3×GO+(4,𝔽2)C211yesmaximal, characteristic
3AGL(1,𝔽9)⋊Dic3C211yesmaximal, characteristic, complemented
4#11724103C211yesmaximal, characteristic, complemented
5GO+(4,𝔽2)⋊Dic3C211yesmaximal, characteristic
6C3×GO+(4,𝔽2)⋊C4C211yesmaximal, characteristic
7Dic3⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
8GO+(4,𝔽2)⋊Q8—31nomaximal
9C2×C3⋊PSU(3,𝔽2)—21no—
10C2×C3×GO+(4,𝔽2)C2211yescharacteristic
11C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
12C2×C3×AGL(1,𝔽9)C2211yescharacteristic
13D32×Dic3—21no—
14#88946C2211yescharacteristic
15C3×#3777C2211yescharacteristic
16#88940C2211yescharacteristic
17D32⋊Dic3—21no—
18Dic3⋊#157—21no—
19Dic3×#157C2211yescharacteristic
20C4⋊PSU(3,𝔽2)—31no—
21C4×GO+(4,𝔽2)—31no—
22C4⋊AGL(1,𝔽9)—31no—
23AGL(1,𝔽9)⋊C4—31no—
24AGL(1,𝔽9)⋊C4—31no—
25GO+(4,𝔽2)⋊C4—31no—
26GO+(4,𝔽2)⋊C4D311yescharacteristic
27C2×C3×D32—21no—
28C2×C3⋊#157—21no—
29C2×C3⋊#157D411yescharacteristic
30C2×C3×#157—21no—
31C2×C3×#157C2311yescharacteristic
32C3⋊PSU(3,𝔽2)—42no—
33C3×GO+(4,𝔽2)—21no—
34C3×GO+(4,𝔽2)Q812yescomplemented
35C3⋊AGL(1,𝔽9)—21no—
36C3×AGL(1,𝔽9)—21no—
37#6294—21no—
38Dic3×C3⋊D3D411yescharacteristic
39D3×C3⋊Dic3—41no—
40C3×D3×Dic3—41no—
41#4761—91nomaximal
42C2×PSU(3,𝔽2)—61no—
43C2×GO+(4,𝔽2)C2×D311yescharacteristic
44C2×AGL(1,𝔽9)—31no—
45C2×AGL(1,𝔽9)C2×D311yescharacteristic
46C2×D3×Dic3—121no—
47C4×D32—61no—
48C4⋊#157—61no—
49C4×#157—31no—
50#3777—31no—
51#3777C2×D311yescharacteristic
52#3773—31no—
53D32⋊C4—61no—
54C2×C3×C3⋊D3C2×D411yescharacteristic
55C2×C32×D3—41no—
56C3×D32—22no—
57C3×D32—41no—
58C3⋊#157—21no—
59C3⋊#157—41no—
60C3×#157—41no—
61C3×#157C2×Q811yescharacteristic
62C3×#157#4111yescharacteristic
63C3⋊(C3⋊Dic3)SD1611yescharacteristic
64C3×C3⋊Dic3—41no—
65C32×Dic3SD1611yescharacteristic
66D4×Dic3—91no—
67#888—91no—
68C3×#102—91no—
69#832—91no—
70#817—91no—
71Dic3⋊C8—91no—
72#808—91no—
73C22×C3×D3—121no—
74C2×D32—21no—
75C2×#157—21no—
76C2×#157—31no—
77C2×#157—61no—
78C2×#157C22×D311yescharacteristic
79PSU(3,𝔽2)—122no—
80GO+(4,𝔽2)—21no—
81GO+(4,𝔽2)Dic612yescomplemented
82AGL(1,𝔽9)—21no—
83AGL(1,𝔽9)—61no—
84C2×C3⋊Dic3—121no—
85C4×C3⋊D3—31no—
86C2×C3×Dic3—121no—
87C3×C4×D3—121no—
88#624—61no—
89D3×Dic3—41no—
90D3×Dic3—121no—
91D3×Dic3—121no—
92D3×Dic3—121no—
93D3×Dic3—121no—
94D3×Dic3—241no—
95#476—271noSylow
96C2×C33C2×SD1611yescharacteristic
97C3×C3⋊D3C22⋊Q811yescharacteristic
98C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
99C32×D3—42no—
100C2×C3×D4—91no—
101C22×Dic3—181no—
102C2×C4×D3—361no—
103C2×Dic6—181no—
104C2×C3×C8—91no—
105C3×C4⋊C4—91no—
106#217—181no—
107C4⋊Dic3—91no—
108C4⋊Dic3—91no—
109Dic3⋊C4—181no—
110C4×Dic3—91no—
111C2×C3⋊C8—91no—
112C22×C32—121no—
113C2×C3⋊D3D3×D411yescharacteristic
114C2×C3×D3—41no—
115C2×C3×D3—41no—
116C2×C3×D3—81no—
117C2×C3×D3—121no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120D32—22no—
121D32—41no—
122#157—41no—
123#157—61no—
124#157—121no—
125#157C2×Dic611yescharacteristic
126#157#23711yescharacteristic
127C32×C4—31no—
128C3⋊Dic3—31no—
129C3⋊Dic3—41no—
130C3⋊Dic3—121no—
131C3⋊Dic3—241no—
132C3×Dic3—41no—
133C3×Dic3—121no—
134C3×Dic3—121no—
135C3×Dic3—241no—
136C4⋊Q8—271no—
137C4×D4—271no—
138#106—271no—
139#106—271no—
140C4⋊C8—271no—
141#102—91no—
142#102—271no—
143C33#47611yescharacteristic, complemented, Sylow
144C23×C3—181no—
145C22×D3—121no—
146C3×D4—92no—
147C3×D4—181no—
148C2×C3×C4—91no—
149C2×C3×C4—181no—
150C2×C3×C4—361no—
151C2×Dic3—91no—
152C2×Dic3—91no—
153C2×Dic3—121no—
154C2×Dic3—181no—
155C2×Dic3—181no—
156C2×Dic3—361no—
157C2×Dic3—361no—
158C2×Dic3—721no—
159C4×D3—121no—
160C4×D3—361no—
161C4×D3—361no—
162C4×D3—361no—
163Dic6—362no—
164C3×C8—181no—
165C3⋊C8—181no—
166C2×C32—41no—
167C2×C32—81no—
168C2×C32—122no—
169C2×C32D3×SD1611yescharacteristic
170C3⋊D3#87811yescharacteristic
171C3⋊D3#91111yescharacteristic
172C3×D3—42no—
173C3×D3—42no—
174C3×D3—82no—
175C3×D3—121no—
176C3×D3—121no—
177C2×Q8—541no—
178C2×D4—91no—
179C22×C4—541no—
180C2×C8—91no—
181C2×C8—271no—
182C4⋊C4—91no—
183C4⋊C4—271no—
184C4⋊C4—271no—
185C4⋊C4—541no—
186C22⋊C4—541no—
187C42—271no—
188C22×C3—91no—
189C22×C3—121no—
190C22×C3—121no—
191C22×C3—182no—
192C22×C3—241no—
193C22×C3—361no—
194C2×D3—41no—
195C2×D3—121no—
196C2×D3—122no—
197C2×D3—122no—
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—121no—
201C3×C4—361no—
202C3×C4—361no—
203Dic3—91no—
204Dic3—121no—
205Dic3—121no—
206Dic3—181no—
207Dic3—241no—
208Dic3—361no—
209Dic3—361no—
210Dic3—721no—
211Dic3AΓL(1,𝔽9)11yescharacteristic
212C32—41no—
213C32—81no—
214C32#476111yescharacteristic, complemented
215C23—181no—
216Q8—1082no—
217D4—92no—
218D4—181no—
219C2×C4—91no—
220C2×C4—181no—
221C2×C4—271no—
222C2×C4—271no—
223C2×C4—361no—
224C2×C4—541no—
225C2×C4—541no—
226C2×C4—1081no—
227C8—181no—
228C8—541no—
229C2×C3—41no—
230C2×C3—81no—
231C2×C3—91no—
232C2×C3—91no—
233C2×C3—122no—
234C2×C3—122no—
235C2×C3—242no—
236C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
237D3—42no—
238D3—121no—
239D3—121no—
240C22—91no—
241C22—121no—
242C22—182no—
243C22—361no—
244C4—31no—
245C4—91no—
246C4—91no—
247C4—271no—
248C4—361no—
249C4—361no—
250C4—541no—
251C4—1081no—
252C3—41no—
253C3—81no—
254C3GO+(4,𝔽2)⋊Q811yescharacteristic, complemented
255C2—91no—
256C2—91no—
257C2—122no—
258C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radicalC2×C3×GO+(4,𝔽2)C22
hypocenterC33#476
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #11724154C2semidirect
3. AGL(1,𝔽9)⋊Dic3C2semidirect
4. #11724103C2semidirect
7. Dic3⋊AGL(1,𝔽9)C2semidirect
34. C3×GO+(4,𝔽2)Q8semidirect
81. GO+(4,𝔽2)Dic6semidirect
143. C33#476semidirect
214. C32#4761semidirect
254. C3GO+(4,𝔽2)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#117241542
Dic3×GO+(4,𝔽2)2
AGL(1,𝔽9)⋊Dic32
#117241032
GO+(4,𝔽2)⋊Dic32
C3×GO+(4,𝔽2)⋊C42
Dic3⋊AGL(1,𝔽9)2
GO+(4,𝔽2)⋊Q83
#47619