Finite Group Databasebeta

← #11774783

Subgroups — #11774783

Order 1152.

266 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 62 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#10596542C211yesmaximal, characteristic
2C2×C4×AGL(1,𝔽9)C211yesmaximal, characteristic
3#10593328C211yesmaximal, characteristic
4PSU(3,𝔽2)⋊C8C214yesmaximal, complemented
5C22×AGL(1,𝔽9)—21no—
6#64286—21no—
7C2×C4×#157C2211yescharacteristic
8C22⋊PSU(3,𝔽2)—21no—
9C22⋊PSU(3,𝔽2)C412yes—
10C4⋊PSU(3,𝔽2)C411yescharacteristic
11C4⋊PSU(3,𝔽2)C411yescharacteristic
12C4×PSU(3,𝔽2)—21no—
13C4×PSU(3,𝔽2)C2212yes—
14#64243—21no—
15C4×AGL(1,𝔽9)—21no—
16C4×AGL(1,𝔽9)C2212yes—
17#63818—21no—
18#63732—21no—
19#63732C2212yes—
20C22×#157C2×C411yescharacteristic
21C22×#157D411yescharacteristic
22C2×PSU(3,𝔽2)—21no—
23C2×PSU(3,𝔽2)C2×C412yes—
24C2×AGL(1,𝔽9)—24no—
25C2×AGL(1,𝔽9)—22no—
26C2×C4×C3⋊D3D411yescharacteristic
27C22⋊#157—22no—
28C4⋊#157—21no—
29C4⋊#157—21no—
30C4×#157—22no—
31C4×#157C2×C411yescharacteristic
32C4×#157D411yescharacteristic
33C4×#157D411yescharacteristic
34C4×#157C2311yescharacteristic
35#3777—22no—
36#3777—21no—
37#3777—22no—
38#3777C2×C412yes—
39C3×#208—41no—
40#3712—41no—
41#3710—22no—
42#1493—91nomaximal, Sylow
43D3×M16—121no—
44C22×C3⋊D3C22⋊C411yescharacteristic
45C2×#157—22no—
46C2×#157—22no—
47C2×#157—22no—
48C2×#157—22no—
49C2×#157C22⋊C411yescharacteristic
50C2×#157C22⋊C411yescharacteristic
51C2×#157C22×C411yescharacteristic
52C2×#157C2×D411yescharacteristic
53PSU(3,𝔽2)—24no—
54PSU(3,𝔽2)—42no—
55AGL(1,𝔽9)—24no—
56C2×C32×C4SD1611yescharacteristic
57C2×C3⋊Dic3Q1611yescharacteristic
58C4×C3⋊D3C22⋊C411yescharacteristic
59C4×C3⋊D3SD1611yescharacteristic
60C4×C3⋊D3Q1611yescharacteristic
61C4×C3⋊D3C2×D411yescharacteristic
62C3×C3⋊C8—42no—
63#533—91no—
64#423—91no—
65C2×C4×C8—91no—
66Q8⋊C8—94no—
67C2×C4×D3—121no—
68C3×M16—121no—
69#208—41no—
70ΓL(1,𝔽25)—122no—
71D3×C8—122no—
72C22×C32#10311yescharacteristic
73C2×C3⋊D3—21no—
74C2×C3⋊D3—21no—
75C2×C3⋊D3#10311yescharacteristic
76C2×C3⋊D3C2×C22⋊C411yescharacteristic
77#157—21no—
78#157—21no—
79#157—42no—
80#157—42no—
81#157#10412yes—
82#157#13112yes—
83C32×C4#10311yescharacteristic
84C32×C4C2×SD1611yescharacteristic
85C3⋊Dic3#10311yescharacteristic
86C3⋊Dic3C2×Q1611yescharacteristic
87C2×M16—181no—
88C22×C8—181no—
89C4⋊Q8—91no—
90C4⋊Q8—91no—
91#125—181no—
92C22⋊Q8—92no—
93C22⋊Q8—181no—
94C4×Q8—92no—
95C4×Q8—181no—
96#117—181no—
97C2×C42—91no—
98C4⋊C8—92no—
99C4⋊C8—181no—
100C4×C8—92no—
101C4×C8—181no—
102C22×D3—121no—
103C2×C3×C4—41no—
104C2×Dic3—121no—
105C4×D3—121no—
106C4×D3—121no—
107C4×D3—121no—
108C4×D3—121no—
109C3×C8—122no—
110C3⋊C8—42no—
111C2×C32—21no—
112C2×C32C2×#10311yescharacteristic
113C3⋊D3—21no—
114C3⋊D3#40811yescharacteristic
115C3⋊D3C2×#10411yescharacteristic
116C2×Q8—92no—
117C2×Q8—181no—
118C22×C4—91no—
119C22×C4—91no—
120C22×C4—91no—
121M16—121no—
122M16—361no—
123C2×C8—182no—
124C2×C8—182no—
125C2×C8—184no—
126C4⋊C4—92no—
127C4⋊C4—181no—
128C4⋊C4—181no—
129C4⋊C4—182no—
130C4⋊C4—182no—
131C4⋊C4—181no—
132C22⋊C4—182no—
133C42—91no—
134C42—91no—
135C42—91no—
136C42—91no—
137C42—182no—
138C22×C3—41no—
139C2×D3—121no—
140C2×D3—121no—
141C2×D3—241no—
142C2×D3—241no—
143C3×C4—41no—
144C3×C4—41no—
145Dic3—121no—
146Dic3—121no—
147C32#149311yescharacteristic, complemented, Sylow
148C23—91no—
149Q8—184no—
150Q8—362no—
151C2×C4—91no—
152C2×C4—91no—
153C2×C4—91no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4—91no—
157C2×C4—91no—
158C2×C4—91no—
159C2×C4—91no—
160C2×C4—182no—
161C2×C4—182no—
162C2×C4—182no—
163C2×C4—182no—
164C2×C4AΓL(1,𝔽9)11yescharacteristic
165C8—122no—
166C8—184no—
167C2×C3—41no—
168C2×C3—81no—
169D3—121no—
170D3—121no—
171D3—241no—
172C22—91no—
173C22—91no—
174C22—181no—
175C22—181no—
176C22PSU(3,𝔽2)⋊C411yescharacteristic
177C4—92no—
178C4—92no—
179C4—91no—
180C4—91no—
181C4—181no—
182C4—181no—
183C4—362no—
184C4—362no—
185C4PSU(3,𝔽2)⋊C411yescharacteristic
186C4C2×AΓL(1,𝔽9)11yescharacteristic
187C3—41no—
188C2—21no—
189C2—91no—
190C2—91no—
191C2—181no—
192C2C2×PSU(3,𝔽2)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC4C2×AΓL(1,𝔽9)
commutatorC2×#157C22×C4
FittingC2×C32×C4SD16
socleC2×C32C2×#103
prime radicalC22×C3⋊D3C22⋊C4
cosocleC4×#157C23
hypercenterC2×C4AΓL(1,𝔽9)
hypocenterC32#1493
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. PSU(3,𝔽2)⋊C8C2semidirect
147. C32#1493semidirect

Maximal subgroups

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

subgroupindex
#105965422
C2×C4×AGL(1,𝔽9)2
#105933282
PSU(3,𝔽2)⋊C82
#14939