Finite Group Databasebeta

← #11791470

Subgroups — #11791470

Order 1152.

546 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 80 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×#64231C211yesmaximal, characteristic
2C2×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3C2×GO+(4,𝔽2)⋊C4C214yesmaximal, complemented
4C4×#3777C211yesmaximal, characteristic, complemented
5C22×GO+(4,𝔽2)—21no—
6C22×GO+(4,𝔽2)C2212yes—
7C22×AGL(1,𝔽9)C2212yes—
8C2×#3802—21no—
9C2×C4×#157—21no—
10C2×C4×#157C2211yescharacteristic
11C2×#3777C2212yes—
12#64231—22no—
13#64231C414yescomplemented
14C4⋊AGL(1,𝔽9)—22no—
15GO+(4,𝔽2)⋊C4—28no—
16#63747—21no—
17C22×D32—22no—
18C22×#157—22no—
19C22×#157D411yescharacteristic
20C22×#157C2311yescharacteristic
21C2×GO+(4,𝔽2)—24no—
22C2×GO+(4,𝔽2)—24no—
23C2×GO+(4,𝔽2)—24no—
24C2×GO+(4,𝔽2)—22no—
25C2×GO+(4,𝔽2)—44no—
26C2×GO+(4,𝔽2)C2×C414yes—
27C2×AGL(1,𝔽9)—24no—
28C2×AGL(1,𝔽9)—22no—
29C2×C4×C3⋊D3D411yescharacteristic
30C2×C3×D12—41no—
31C2×#625—41no—
32#3802—24no—
33#3802—42no—
34C4×#157—42no—
35C4×#157C2×C412yes—
36C4×#157D412yes—
37#3777—22no—
38#3777D414yescomplemented
39#1982—91nomaximal, Sylow
40C2×D3×D4—121no—
41C22×C3⋊D3C2×D411yescharacteristic
42C22×C3×D3—42no—
43C2×D32—24no—
44C2×D32—42no—
45C2×D32—44no—
46C2×D32—44no—
47C2×#157—22no—
48C2×#157—24no—
49C2×#157—42no—
50C2×#157—42no—
51C2×#157C22⋊C412yes—
52C2×#157C22×C411yescharacteristic
53C2×#157C2×D412yes—
54C2×#157C2×D411yescharacteristic
55C2×#157#4112yes—
56GO+(4,𝔽2)—28no—
57GO+(4,𝔽2)—48no—
58GO+(4,𝔽2)—42no—
59GO+(4,𝔽2)—42no—
60AGL(1,𝔽9)—42no—
61C2×C32×C4SD1611yescharacteristic
62C2×C3⋊Dic3SD1611yescharacteristic
63C4×C3⋊D3C22⋊C412yes—
64C4×C3⋊D3D812yes—
65C3×D12—44no—
66#625—44no—
67C2×#127—91no—
68C2×C4⋊C8—91no—
69C2×#102—94no—
70C4×C4⋊C4—91no—
71C23×D3—122no—
72C2×C3×D4—121no—
73C2×#67—122no—
74D3×D4—124no—
75D3×D4—124no—
76C2×D12—41no—
77C2×C4×D3—121no—
78C22×C32C2×SD1611yescharacteristic
79C2×C3⋊D3C2×C22⋊C411yescharacteristic
80C2×C3⋊D3#12112yes—
81C2×C3⋊D3C2×D811yescharacteristic
82C2×C3⋊D3AGL(1,ℤ/8)12yes—
83C2×C3×D3—44no—
84C2×C3×D3—84no—
85D32—44no—
86D32—44no—
87D32—84no—
88#157—22no—
89#157—44no—
90#157C4×D412yes—
91#157#12411yescharacteristic
92#157#12711yescharacteristic
93C32×C4#10212yescomplemented
94C3⋊Dic3#10212yescomplemented
95C22×D4—92no—
96C22×D4—181no—
97C22×D4—181no—
98C22×C8—92no—
99#127—94no—
100#127—182no—
101C2×C4⋊C4—92no—
102C2×C42—91no—
103C2×C42—181no—
104C4⋊C8—182no—
105#102—188no—
106#95—181no—
107C23×C3—122no—
108C22×D3—42no—
109C22×D3—121no—
110C22×D3—124no—
111C22×D3—124no—
112C22×D3—244no—
113C22×D3—244no—
114C3×D4—124no—
115C2×C3×C4—41no—
116#67—128no—
117C2×Dic3—121no—
118D12—44no—
119C4×D3—122no—
120C4×D3—122no—
121C2×C32C2×#10211yescharacteristic
122C2×C32#46012yes—
123C3⋊D3#39011yescharacteristic
124C3⋊D3#41811yescharacteristic
125C3⋊D3#45912yes—
126C3×D3—84no—
127C24—182no—
128C2×D4—94no—
129C2×D4—121no—
130C2×D4—184no—
131C2×D4—184no—
132C2×D4—184no—
133C2×D4—184no—
134C2×D4—182no—
135C2×D4—362no—
136C2×D4—361no—
137C2×D4—364no—
138C22×C4—91no—
139C22×C4—91no—
140C22×C4—91no—
141C22×C4—182no—
142C2×C8—184no—
143C2×C8—182no—
144C4⋊C4—94no—
145C4⋊C4—182no—
146C42—92no—
147C42—92no—
148C42—362no—
149C22×C3—41no—
150C22×C3—124no—
151C22×C3—244no—
152C2×D3—44no—
153C2×D3—84no—
154C2×D3—121no—
155C2×D3—121no—
156C2×D3—122no—
157C2×D3—122no—
158C2×D3—244no—
159C2×D3—244no—
160C2×D3—248no—
161C3×C4—42no—
162Dic3—122no—
163C32#198211yescharacteristic, complemented, Sylow
164C23—91no—
165C23—122no—
166C23—184no—
167C23—362no—
168C23—364no—
169C23—364no—
170D4—124no—
171D4—188no—
172D4—364no—
173D4—362no—
174D4—362no—
175D4—368no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—92no—
180C2×C4—91no—
181C2×C4—92no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—182no—
185C2×C4—184no—
186C2×C4—362no—
187C2×C4—362no—
188C2×C4AΓL(1,𝔽9)11yescharacteristic
189C8—362no—
190C2×C3—42no—
191C2×C3—41no—
192C2×C3—244no—
193D3—84no—
194D3—121no—
195D3—121no—
196D3—122no—
197C22—92no—
198C22—92no—
199C22—91no—
200C22—91no—
201C22—124no—
202C22—244no—
203C22—364no—
204C22—364no—
205C22—724no—
206C22C2×AΓL(1,𝔽9)11yescharacteristic
207C4—91no—
208C4—92no—
209C4—91no—
210C4—92no—
211C4—182no—
212C4—364no—
213C4GO+(4,𝔽2)⋊C412yescomplemented
214C3—41no—
215C2—91no—
216C2—91no—
217C2—92no—
218C2—244no—
219C2#1059641012yes—
220C2C2×GO+(4,𝔽2)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC22C2×AΓL(1,𝔽9)
commutatorC2×#157C22×C4
FittingC2×C32×C4SD16
socleC22×C32C2×SD16
prime radicalC2×#64231C2
cosocleC22×#157C23
hypercenterC2×C4AΓL(1,𝔽9)
hypocenterC32#1982
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊AGL(1,𝔽9)C2semidirect
3. C2×GO+(4,𝔽2)⋊C4C2semidirect
4. C4×#3777C2semidirect
13. #64231C4semidirect
38. #3777D4semidirect
93. C32×C4#102semidirect
94. C3⋊Dic3#102semidirect
163. C32#1982semidirect
213. C4GO+(4,𝔽2)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#642312
C2×C4⋊AGL(1,𝔽9)2
C2×GO+(4,𝔽2)⋊C42
C4×#37772
#19829