Finite Group Databasebeta

← #13165126

Subgroups — #13165126

Order 1728.

238 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 39 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#11724344C211yesmaximal, characteristic
2C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3#11722804C211yesmaximal, characteristic, complemented
4#10593155—31nomaximal
5C2×#157⋊D3—21no—
6C2×#157⋊D3C412yes—
7C2×C3×AGL(1,𝔽9)—21no—
8C4⋊#1113—21no—
9#88988—21no—
10C3×C4×#157C2211yescharacteristic
11#64231—31no—
12C4⋊AGL(1,𝔽9)D311yescharacteristic, complemented
13#63744—31no—
14C2×#1113—22no—
15C2×C3×#157C2×C411yescharacteristic
16C2×C3×#157D411yescharacteristic
17#157⋊D3—42no—
18#157⋊D3—41no—
19#157⋊D3—41no—
20C3×AGL(1,𝔽9)—41no—
21C3×C4⋊(C3⋊D3)—41no—
22C3×C4×C3⋊D3D411yescharacteristic
23#6295—41no—
24C3⋊#622—41no—
25#4485—91nomaximal
26C2×GO+(4,𝔽2)—32no—
27C2×GO+(4,𝔽2)—61no—
28C2×AGL(1,𝔽9)—21no—
29#3802—61no—
30D3×D12—121no—
31C4×#157C2×D311yescharacteristic
32#3787—61no—
33C2×C3×C3⋊D3—42no—
34C2×C3×C3⋊D3C22⋊C411yescharacteristic
35#1113—42no—
36#1113—42no—
37C3×#157—21no—
38C3×#157D811yescharacteristic
39C3×#157SD1611yescharacteristic
40C33×C4SD1611yescharacteristic
41C3×C3⋊Dic3D811yescharacteristic
42#874—91no—
43C3×C4⋊C8—91no—
44C4⋊(C3⋊C8)—91no—
45C2×D32—62no—
46C2×D32—122no—
47C2×#157C4×D311yescharacteristic
48C2×#157#6711yescharacteristic
49GO+(4,𝔽2)—122no—
50GO+(4,𝔽2)—121no—
51GO+(4,𝔽2)—121no—
52AGL(1,𝔽9)—41no—
53C4⋊(C3⋊D3)—41no—
54C4×C3⋊D3D1211yescharacteristic
55C3×D12—121no—
56C3×D12—121no—
57C3×D12—241no—
58C3×C4×D3—121no—
59#626—122no—
60#625—121no—
61#622—121no—
62#331—271noSylow
63C2×C33#10211yescharacteristic
64C3×C3⋊D3—82no—
65C3×C3⋊D3#10011yescharacteristic
66C3×C3⋊D3#10211yescharacteristic
67D3×D4—361no—
68C2×D12—92no—
69C2×D12—181no—
70C2×D12—181no—
71C2×C3×C8—181no—
72C3×C42—91no—
73C2×C3⋊C8—181no—
74C2×C3⋊D3—42no—
75C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
76C2×C3×D3—121no—
77C2×C3×D3—122no—
78C2×C3×D3—122no—
79C2×C3×D3—242no—
80D32—122no—
81D32—122no—
82D32—242no—
83D32—242no—
84#157—21no—
85#157#21311yescharacteristic
86#157#21511yescharacteristic
87C32×C4—41no—
88C32×C4—81no—
89C32×C4#20411yescharacteristic
90C3⋊Dic3D2411yescharacteristic
91C3×Dic3—121no—
92#127—271no—
93C4⋊C8—91no—
94C4⋊C8—271no—
95C33#33111yescharacteristic, complemented, Sylow
96C22×D3—182no—
97C22×D3—362no—
98C3×D4—361no—
99C2×C3×C4—91no—
100C2×C3×C4—91no—
101C2×C3×C4—91no—
102#67—362no—
103D12—121no—
104D12—121no—
105D12—241no—
106D12—361no—
107D12—361no—
108D12—361no—
109D12—362no—
110C4×D3—121no—
111C3×C8—361no—
112C3⋊C8—361no—
113C2×C32—41no—
114C2×C32—81no—
115C2×C32#82111yescharacteristic
116C3⋊D3—82no—
117C3⋊D3#80911yescharacteristic
118C3⋊D3#82311yescharacteristic
119C3×D3—121no—
120C3×D3—121no—
121C3×D3—242no—
122C3×D3—242no—
123C3×D3—482no—
124C2×D4—272no—
125C2×D4—541no—
126C2×D4—541no—
127C2×C8—181no—
128C2×C8—541no—
129C42—91no—
130C22×C3—91no—
131C22×C3—362no—
132C2×D3—121no—
133C2×D3—122no—
134C2×D3—122no—
135C2×D3—242no—
136C2×D3—362no—
137C2×D3—362no—
138C2×D3—722no—
139C2×D3—722no—
140C3×C4—41no—
141C3×C4—81no—
142C3×C4—91no—
143C3×C4—91no—
144C3×C4—91no—
145C3×C4—181no—
146C3×C4AΓL(1,𝔽9)11yescharacteristic
147Dic3—121no—
148C32—41no—
149C32—81no—
150C32#448511yescharacteristic, complemented
151C23—542no—
152D4—361no—
153D4—1081no—
154D4—1081no—
155D4—1081no—
156D4—1082no—
157C2×C4—91no—
158C2×C4—91no—
159C2×C4—91no—
160C8—361no—
161C8—1081no—
162C2×C3—41no—
163C2×C3—81no—
164C2×C3—91no—
165C2×C3—91no—
166C2×C3—722no—
167C2×C3GO+(4,𝔽2)⋊C411yescharacteristic
168D3—121no—
169D3—121no—
170D3—242no—
171D3—242no—
172D3—482no—
173C22—91no—
174C22—362no—
175C22—1082no—
176C22—1082no—
177C4—91no—
178C4—91no—
179C4—91no—
180C4—181no—
181C4AGL(1,𝔽9)⋊D311yescharacteristic
182C3—41no—
183C3—81no—
184C3#1059315511yescharacteristic, complemented
185C2—91no—
186C2—91no—
187C2—722no—
188C2#1172410411yescharacteristic
subgroupG/H
centerC2#11724104
commutatorC2×C3×#157C2×C4
Frattini, hypercenterC4AGL(1,𝔽9)⋊D3
FittingC33×C4SD16
socleC2×C33#102
prime radical#11724344C2
cosocleC3×C4×#157C22
hypocenterC33#331
LCSG ▷ C2×C3×#157 ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊AGL(1,𝔽9)C2semidirect
3. #11722804C2semidirect
12. C4⋊AGL(1,𝔽9)D3semidirect
95. C33#331semidirect
150. C32#4485semidirect
184. C3#10593155semidirect

Maximal subgroups

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

subgroupindex
#117243442
C3×C4⋊AGL(1,𝔽9)2
#117228042
#105931553
#44859