Finite Group Databasebeta

← #72252779

Subgroups — #72252779

Order 2304.

551 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 162 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#11786220C214yesmaximal
2#11786219C211yesmaximal, characteristic
3#11786216C211yesmaximal, characteristic
4C2×C8×AGL(1,𝔽9)C211yesmaximal, characteristic
5C2×C4×AGL(1,𝔽9)C2212yes—
6C2×AGL(1,𝔽9)⋊C4—28no—
7C2×#63768—21no—
8C2×C8×#157C2211yescharacteristic
9#10593553C2212yes—
10C2×#63745—21no—
11#10593413C2212yes—
12#10593387—21no—
13C8×AGL(1,𝔽9)C414yescomplemented
14#10590685—21no—
15C22×AGL(1,𝔽9)D414yes—
16C2×C4⋊#157—22no—
17C2×C4×#157C2311yescharacteristic
18C2×#3787D411yescharacteristic
19C2×#3777—24no—
20C2×#3773—24no—
21C4×AGL(1,𝔽9)C2×C414yes—
22C4×AGL(1,𝔽9)Q814yes—
23AGL(1,𝔽9)⋊C4—416no—
24C2×C8×C3⋊D3D411yescharacteristic
25#63900—22no—
26#63768—42no—
27C8×#157C2×C412yes—
28C8×#157Q812yes—
29#63745—42no—
30#63742—42no—
31C3×#817—41no—
32#63574—41no—
33#8813—91nomaximal, Sylow
34D3×#106—121no—
35C22×#157—42no—
36C22×#157C2×D412yes—
37C2×AGL(1,𝔽9)C4⋊C418yes—
38C2×AGL(1,𝔽9)SD16116yes—
39C2×C4×C3⋊D3C2×D411yescharacteristic
40C2×#624—42no—
41C2×#622SD1612yes—
42C4⋊#157—42no—
43C4⋊#157—42no—
44C4×#157C22×C411yescharacteristic
45C4×#157C2×Q811yescharacteristic
46C4×#157C2×Q812yes—
47#3787C4⋊C412yes—
48#3787SD1612yes—
49#3777—48no—
50#3773—48no—
51C2×C32×C8SD1611yescharacteristic
52C2×C3⋊(C3⋊C8)SD1611yescharacteristic
53C8×C3⋊D3C4⋊C412yes—
54C8×C3⋊D3SD1612yes—
55C3×C4⋊Dic3—42no—
56#3724—42no—
57#1844—91no—
58#1844—91no—
59#1844—94no—
60C2×C82—91no—
61C2×D3×C8—121no—
62D3×C4⋊C4—122no—
63C3×#106—121no—
64#817—41no—
65#808—122no—
66C22×C3⋊D3#12711yescharacteristic
67C2×#157—84no—
68C2×#157—82no—
69C2×#157C2×C4⋊C412yes—
70C2×#157C4⋊Q812yes—
71C2×#157C2×SD1618yes—
72AGL(1,𝔽9)#106116yescomplemented
73C2×C32×C4C2×SD1611yescharacteristic
74C2×C3⋊Dic3C2×SD1611yescharacteristic
75C4×C3⋊D3C2×C4⋊C411yescharacteristic
76C4×C3⋊D3C4⋊Q811yescharacteristic
77C4×C3⋊D3C2×SD1611yescharacteristic
78C4×C3⋊D3C2×SD1611yescharacteristic
79C2×C3×Dic3—82no—
80#624—84no—
81#622#10614yescomplemented
82C32×C8#10612yescomplemented
83C3⋊(C3⋊C8)#10612yescomplemented
84C2×#106—181no—
85C2×#106—181no—
86C2×#106—188no—
87C2×#106—181no—
88C2×#106—181no—
89C2×C4×C8—92no—
90C2×C4×C8—91no—
91#384—92no—
92#384—92no—
93C82—94no—
94C2×C4×D3—121no—
95C2×C4×D3—242no—
96C2×C3×C8—41no—
97C3×C4⋊C4—122no—
98C4⋊Dic3—42no—
99Dic3⋊C4—242no—
100C2×C3⋊C8—121no—
101D3×C8—122no—
102D3×C8—122no—
103C22×C32#49211yescharacteristic
104C2×C3⋊D3#38411yescharacteristic
105C2×C3⋊D3#49214yes—
106C2×C3⋊D3#49211yescharacteristic
107#157—162no—
108#157C2×#10614yes—
109#157#49814yes—
110C32×C4C2×#10611yescharacteristic
111C32×C4#49811yescharacteristic
112C3⋊Dic3C2×#10611yescharacteristic
113C3⋊Dic3#49811yescharacteristic
114C3×Dic3—162no—
115C22×C8—91no—
116C22×C8—94no—
117C22×C8—91no—
118C2×C4⋊C4—182no—
119C2×C4⋊C4—182no—
120C2×C4⋊C4—184no—
121C2×C4⋊C4—184no—
122C2×C42—91no—
123#106—121no—
124#106—361no—
125#106—362no—
126#106—362no—
127#106—3616no—
128#106—362no—
129C4×C8—92no—
130C4×C8—92no—
131C4×C8—94no—
132C4×C8—94no—
133C22×D3—121no—
134C2×C3×C4—41no—
135C2×C3×C4—242no—
136C2×Dic3—82no—
137C2×Dic3—121no—
138C4×D3—121no—
139C4×D3—121no—
140C4×D3—121no—
141C4×D3—121no—
142C4×D3—484no—
143C3×C8—42no—
144C3⋊C8—122no—
145C2×C32#170512yes—
146C2×C32#184411yescharacteristic
147C3⋊D3#170512yes—
148C3⋊D3#184412yes—
149C22×C4—92no—
150C22×C4—91no—
151C22×C4—362no—
152C22×C4—362no—
153C2×C8—98no—
154C2×C8—916no—
155C2×C8—92no—
156C2×C8—92no—
157C2×C8—92no—
158C2×C8—91no—
159C2×C8—92no—
160C2×C8—92no—
161C2×C8AΓL(1,𝔽9)11yescharacteristic
162C4⋊C4—122no—
163C4⋊C4—362no—
164C4⋊C4—368no—
165C4⋊C4—368no—
166C4⋊C4—362no—
167C4⋊C4—362no—
168C42—91no—
169C42—92no—
170C42—91no—
171C22×C3—41no—
172C2×D3—121no—
173C2×D3—121no—
174C2×D3—124no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4—482no—
178Dic3—121no—
179Dic3—121no—
180Dic3—162no—
181C32#881311yescharacteristic, complemented, Sylow
182C23—91no—
183C2×C4—92no—
184C2×C4—92no—
185C2×C4—98no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—242no—
192C2×C4—724no—
193C2×C4—724no—
194C2×C4—722no—
195C2×C4C2×AΓL(1,𝔽9)11yescharacteristic
196C8—94no—
197C8—916no—
198C8—92no—
199C8AGL(1,𝔽9)⋊C412yescomplemented
200C2×C3—42no—
201C2×C3—41no—
202D3—122no—
203D3—122no—
204C22—94no—
205C22—91no—
206C22—91no—
207C22#1059640811yescharacteristic
208C4—94no—
209C4—94no—
210C4—91no—
211C4—91no—
212C4—482no—
213C4—1442no—
214C4AGL(1,𝔽9)⋊Q811yescharacteristic
215C4C2×AGL(1,𝔽9)⋊C411yescharacteristic
216C3—41no—
217C2—92no—
218C2—92no—
219C2#1178243012yes—
220C2#1178622011yescharacteristic
subgroupG/H
centerC22#10596408
commutatorC4×#157C22×C4
FrattiniC2×C4C2×AΓL(1,𝔽9)
FittingC2×C32×C8SD16
socleC22×C32#492
prime radicalC22×C3⋊D3#127
cosocleC2×C4×#157C23
hypercenterC2×C8AΓL(1,𝔽9)
hypocenterC32#8813
LCSG ▷ C4×#157 ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
13. C8×AGL(1,𝔽9)C4semidirect
72. AGL(1,𝔽9)#106semidirect
81. #622#106semidirect
82. C32×C8#106semidirect
83. C3⋊(C3⋊C8)#106semidirect
181. C32#8813semidirect
199. C8AGL(1,𝔽9)⋊C4semidirect

Maximal subgroups

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

subgroupindex
#117862202
#117862192
#117862162
C2×C8×AGL(1,𝔽9)2
#88139