Finite Group Databasebeta

← #13180430

Subgroups — #13180430

Order 1728.

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

normal subgroups only

This group has 286 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
1#11724365C211yesmaximal, characteristic, complemented
2C4×#157⋊D3C211yesmaximal, characteristic
3C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
4AGL(1,𝔽9)⋊Dic3C212yesmaximal, complemented
5#11724104C212yesmaximal
6GO+(4,𝔽2)⋊Q8—31nomaximal
7C2×C3⋊PSU(3,𝔽2)—21no—
8C2×#157⋊D3C2211yescharacteristic
9C2×C3×AGL(1,𝔽9)C2212yes—
10C4×#1113—21no—
11C4⋊(C3⋊#157)—21no—
12C3×C4×#157C2211yescharacteristic
13#88946C2212yes—
14#88940C2211yescharacteristic
15#88939—21no—
16C4⋊PSU(3,𝔽2)—31no—
17C4×GO+(4,𝔽2)—31no—
18C4⋊AGL(1,𝔽9)D311yescharacteristic, complemented
19AGL(1,𝔽9)⋊C4—32no—
20GO+(4,𝔽2)⋊C4—32no—
21C2×#1113—21no—
22C2×C3⋊#157—22no—
23C2×C3×#157D411yescharacteristic
24C2×C3×#157C2311yescharacteristic
25C3⋊PSU(3,𝔽2)—42no—
26#157⋊D3—21no—
27#157⋊D3Q812yescomplemented
28C3×AGL(1,𝔽9)—22no—
29C3×C4×C3⋊D3—41no—
30C3×C4×C3⋊D3D411yescharacteristic
31#6294—21no—
32#6294—41no—
33#4842—91nomaximal
34C2×PSU(3,𝔽2)—61no—
35C2×GO+(4,𝔽2)—31no—
36C2×AGL(1,𝔽9)C2×D312yes—
37C4×D32—61no—
38C4×D32—121no—
39C4⋊#157—61no—
40C4×#157C2×D311yescharacteristic
41#3777—32no—
42#3773—31no—
43D32⋊C4—61no—
44C2×C3×C3⋊D3—41no—
45C2×C3×C3⋊D3C2×D411yescharacteristic
46#1113—22no—
47#1113—41no—
48C3⋊#157—42no—
49C3×#157—21no—
50C3×#157C2×Q811yescharacteristic
51C3×#157#4111yescharacteristic
52C33×C4SD1611yescharacteristic
53C3×C3⋊Dic3—41no—
54C3×C3⋊Dic3SD1611yescharacteristic
55C4×D12—91no—
56#869—91no—
57C3×C4⋊C8—91no—
58#821—92no—
59#817—92no—
60C2×D32—61no—
61C2×D32—121no—
62C2×#157—62no—
63C2×#157D1211yescharacteristic
64C2×#157C22×D311yescharacteristic
65PSU(3,𝔽2)—122no—
66GO+(4,𝔽2)—32no—
67GO+(4,𝔽2)—61no—
68AGL(1,𝔽9)—22no—
69C4×C3⋊D3—41no—
70C4×C3⋊D3D1211yescharacteristic
71C3×C4×D3—121no—
72C3×C4×D3—121no—
73C3×C4×D3—121no—
74C3×C4×D3—241no—
75#624—61no—
76#624—121no—
77D3×Dic3—121no—
78D3×Dic3—121no—
79D3×Dic3—121no—
80#476—271noSylow
81C2×C33C2×SD1611yescharacteristic
82C3×C3⋊D3—42no—
83C3×C3⋊D3C22⋊Q811yescharacteristic
84C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
85C2×D12—91no—
86C2×C4×D3—181no—
87C2×C4×D3—361no—
88C2×Dic6—181no—
89C2×C3×C8—92no—
90C3×C42—91no—
91(C2×D3)⋊C4—181no—
92C4⋊Dic3—91no—
93C4⋊Dic3—92no—
94C4⋊Dic3—181no—
95C2×C3⋊D3—41no—
96C2×C3⋊D3C2×D1211yescharacteristic
97C2×C3×D3—121no—
98C2×C3×D3—121no—
99C2×C3×D3—121no—
100C2×C3×D3—241no—
101D32—62no—
102D32—122no—
103D32—122no—
104D32—121no—
105#157—21no—
106#157—122no—
107#157#23711yescharacteristic
108#157D3×Q811yescharacteristic
109C32×C4—41no—
110C32×C4—81no—
111C32×C4#20411yescharacteristic
112C3⋊Dic3—41no—
113C3⋊Dic3#20411yescharacteristic
114C3×Dic3—121no—
115C3×Dic3—121no—
116C3×Dic3—121no—
117C3×Dic3—241no—
118C4⋊Q8—271no—
119C4×D4—271no—
120#106—272no—
121C4⋊C8—91no—
122#102—272no—
123C33#47611yescharacteristic, complemented, Sylow
124C22×D3—181no—
125C22×D3—361no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—361no—
130C2×Dic3—181no—
131C2×Dic3—182no—
132C2×Dic3—361no—
133D12—92no—
134D12—181no—
135C4×D3—121no—
136C4×D3—121no—
137C4×D3—121no—
138C4×D3—241no—
139C4×D3—361no—
140C4×D3—361no—
141C4×D3—361no—
142Dic6—362no—
143C3×C8—182no—
144C2×C32—41no—
145C2×C32—81no—
146C2×C32C2×#20411yescharacteristic
147C3⋊D3—42no—
148C3⋊D3#89711yescharacteristic
149C3⋊D3#90811yescharacteristic
150C3×D3—121no—
151C3×D3—121no—
152C3×D3—122no—
153C3×D3—122no—
154C3×D3—242no—
155C2×Q8—541no—
156C2×D4—271no—
157C22×C4—541no—
158C2×C8—92no—
159C4⋊C4—272no—
160C4⋊C4—271no—
161C4⋊C4—541no—
162C22⋊C4—541no—
163C42—91no—
164C22×C3—91no—
165C22×C3—361no—
166C2×D3—121no—
167C2×D3—121no—
168C2×D3—121no—
169C2×D3—182no—
170C2×D3—241no—
171C2×D3—362no—
172C2×D3—361no—
173C2×D3—362no—
174C3×C4—41no—
175C3×C4—81no—
176C3×C4—91no—
177C3×C4—91no—
178C3×C4—91no—
179C3×C4—181no—
180C3×C4—361no—
181C3×C4AΓL(1,𝔽9)11yescharacteristic
182Dic3—121no—
183Dic3—121no—
184Dic3—121no—
185Dic3—241no—
186Dic3—362no—
187C32—41no—
188C32—81no—
189C32#484211yescharacteristic, complemented
190C23—541no—
191Q8—1082no—
192D4—272no—
193D4—541no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—361no—
198C2×C4—541no—
199C2×C4—542no—
200C2×C4—1081no—
201C8—182no—
202C2×C3—41no—
203C2×C3—81no—
204C2×C3—91no—
205C2×C3—91no—
206C2×C3—362no—
207C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
208D3—121no—
209D3—121no—
210D3—122no—
211D3—122no—
212D3—242no—
213C22—91no—
214C22—361no—
215C22—542no—
216C22—1081no—
217C4—91no—
218C4—91no—
219C4—91no—
220C4—181no—
221C4—361no—
222C4—1082no—
223C4AGL(1,𝔽9)⋊D311yescharacteristic
224C3—41no—
225C3—81no—
226C3GO+(4,𝔽2)⋊Q811yescharacteristic, complemented
227C2—91no—
228C2—91no—
229C2—362no—
230C2C2×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#476
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
1. #11724365C2semidirect
3. C3×C4⋊AGL(1,𝔽9)C2semidirect
4. AGL(1,𝔽9)⋊Dic3C2semidirect
18. C4⋊AGL(1,𝔽9)D3semidirect
27. #157⋊D3Q8semidirect
123. C33#476semidirect
189. C32#4842semidirect
226. C3GO+(4,𝔽2)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#117243652
C4×#157⋊D32
C3×C4⋊AGL(1,𝔽9)2
AGL(1,𝔽9)⋊Dic32
#117241042
GO+(4,𝔽2)⋊Q83
#48429