Finite Group Databasebeta

← C2×#11724103 #13180426

Subgroups — C2×#11724103 #13180426

Order 1728.

330 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×C3⋊AGL(1,𝔽9)C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×#88940C211yesmaximal, characteristic
4#11724103C214yesmaximal, direct product factor
5C2×AGL(1,𝔽9)⋊C4—31nomaximal
6C22×C3×#157—21no—
7C22×C3×#157C2211yescharacteristic
8C2×C3⋊AGL(1,𝔽9)C414yescomplemented
9C2×C3⋊AGL(1,𝔽9)C2212yes—
10C2×#6294—21no—
11C3×#3777—21no—
12C3×#3777C2212yes—
13#88940—21no—
14#88940C2212yes—
15C22×AGL(1,𝔽9)—31no—
16C2×#3777D311yescharacteristic
17C2×#3773—31no—
18AGL(1,𝔽9)⋊C4—34no—
19C22×C3×C3⋊D3D411yescharacteristic
20C2×C3×#157—22no—
21C2×C3×#157—42no—
22C2×C3×#157C2×C412yes—
23C2×C3×#157D411yescharacteristic
24C2×C3×#157Q812yes—
25C2×C3×#157C2311yescharacteristic
26C3⋊AGL(1,𝔽9)C2×C414yescomplemented
27C2×C3×C3⋊Dic3—41no—
28#6294—22no—
29#6294—41no—
30C2×#808—91nomaximal
31C22×#157—21no—
32C22×#157C2×D311yescharacteristic
33C2×AGL(1,𝔽9)—34no—
34C2×AGL(1,𝔽9)—32no—
35C2×#624—61no—
36C2×D3×Dic3—121no—
37#3777—21no—
38#3777C2×D312yes—
39#3773—32no—
40#3773—61no—
41C22×C33SD1611yescharacteristic
42C2×C3×C3⋊D3C4⋊C412yes—
43C2×C3×C3⋊D3SD1611yescharacteristic
44C2×C3×C3⋊D3SD1612yes—
45C2×C3×C3⋊D3C2×D411yescharacteristic
46C3×#157—42no—
47C3×#157C4⋊C412yes—
48C3×#157C22×C411yescharacteristic
49C3×#157C2×Q811yescharacteristic
50C3×C3⋊Dic3—42no—
51C2×C3×C4⋊C4—91no—
52C2×C4⋊Dic3—91no—
53C22×C3⋊C8—91no—
54#808—94no—
55C22×C3⋊D3#6711yescharacteristic
56C22×C3×D3—121no—
57C2×#157—22no—
58C2×#157—42no—
59C2×#157Dic612yes—
60C2×#157C4×D312yes—
61C2×#157#6711yescharacteristic
62C2×#157C22×D311yescharacteristic
63AGL(1,𝔽9)—34no—
64C2×C3⋊Dic3—41no—
65C2×C3×Dic3—121no—
66C2×C3×Dic3—121no—
67C2×C3×Dic3—241no—
68#624—62no—
69#624—121no—
70D3×Dic3—122no—
71D3×Dic3—122no—
72C2×#106—271noSylow
73C2×C33#10612yescomplemented
74C2×C33C2×SD1611yescharacteristic
75C3×C3⋊D3#10612yescomplemented
76C3×C3⋊D3C2×C4⋊C411yescharacteristic
77C3×C3⋊D3C2×SD1611yescharacteristic
78C22×C3×C4—91no—
79C22×C3×C4—181no—
80C22×Dic3—181no—
81C2×C4×D3—361no—
82C3×C4⋊C4—92no—
83C3×C4⋊C4—181no—
84C4⋊Dic3—92no—
85C4⋊Dic3—181no—
86C2×C3⋊C8—94no—
87C2×C3⋊C8—92no—
88C22×C32—41no—
89C22×C32—81no—
90C22×C32#21511yescharacteristic
91C2×C3⋊D3Dic3⋊C412yes—
92C2×C3⋊D3#21412yes—
93C2×C3⋊D3#21511yescharacteristic
94C2×C3⋊D3C2×#6711yescharacteristic
95C2×C3×D3—122no—
96C2×C3×D3—121no—
97C2×C3×D3—122no—
98C2×C3×D3—121no—
99#157—42no—
100#157Dic3⋊C412yes—
101#157C2×Dic611yescharacteristic
102#157C2×C4×D311yescharacteristic
103C3⋊Dic3—42no—
104C3×Dic3—122no—
105C3×Dic3—122no—
106C3×Dic3—242no—
107C22×C8—271no—
108C2×C4⋊C4—91no—
109C2×C4⋊C4—271no—
110#106—274no—
111C33C2×#10611yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—121no—
114C2×C3×C4—92no—
115C2×C3×C4—91no—
116C2×C3×C4—92no—
117C2×C3×C4—91no—
118C2×C3×C4—182no—
119C2×C3×C4—361no—
120C2×C3×C4—362no—
121C2×Dic3—121no—
122C2×Dic3—121no—
123C2×Dic3—182no—
124C2×Dic3—241no—
125C2×Dic3—361no—
126C4×D3—362no—
127C4×D3—362no—
128C3⋊C8—94no—
129C2×C32—41no—
130C2×C32—42no—
131C2×C32—82no—
132C2×C32—81no—
133C2×C32#80812yescomplemented
134C2×C32C2×#21511yescharacteristic
135C3⋊D3#80812yescomplemented
136C3⋊D3C2×Dic3⋊C411yescharacteristic
137C3⋊D3C2×#21411yescharacteristic
138C3×D3—122no—
139C3×D3—121no—
140C3×D3—121no—
141C22×C4—91no—
142C22×C4—181no—
143C22×C4—541no—
144C2×C8—272no—
145C2×C8—274no—
146C4⋊C4—92no—
147C4⋊C4—181no—
148C4⋊C4—272no—
149C4⋊C4—541no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3—91no—
153C22×C3—91no—
154C22×C3—92no—
155C22×C3—92no—
156C22×C3AΓL(1,𝔽9)11yescharacteristic
157C2×D3—121no—
158C2×D3—121no—
159C2×D3—122no—
160C2×D3—122no—
161C3×C4—91no—
162C3×C4—92no—
163C3×C4—91no—
164C3×C4—362no—
165C3×C4—362no—
166Dic3—122no—
167Dic3—122no—
168Dic3—242no—
169C32—41no—
170C32—81no—
171C32C2×#80811yescharacteristic, complemented
172C23—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—92no—
176C2×C4—92no—
177C2×C4—182no—
178C2×C4—362no—
179C2×C4—361no—
180C2×C4—542no—
181C2×C4—1081no—
182C8—274no—
183C2×C3—42no—
184C2×C3—41no—
185C2×C3—82no—
186C2×C3—81no—
187C2×C3—91no—
188C2×C3—91no—
189C2×C3—92no—
190C2×C3AGL(1,𝔽9)⋊C412yescomplemented
191C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
192D3—122no—
193D3—121no—
194D3—121no—
195C22—92no—
196C22—92no—
197C22—91no—
198C22—91no—
199C22#8909211yescharacteristic
200C4—91no—
201C4—92no—
202C4—91no—
203C4—362no—
204C4—362no—
205C3—41no—
206C3—81no—
207C3C2×AGL(1,𝔽9)⋊C411yescharacteristic, complemented
208C2—91no—
209C2—91no—
210C2—92no—
211C2#1172410312yesdirect product factor
212C2C2×#8909211yescharacteristic
subgroupG/H
center, hypercenterC22#89092
commutatorC3×#157C22×C4
FrattiniC2C2×#89092
Fitting, socleC22×C33SD16
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×#157C23
hypocenterC33C2×#106
LCSG ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11724103C2direct (N × H)
8. C2×C3⋊AGL(1,𝔽9)C4semidirect
26. C3⋊AGL(1,𝔽9)C2×C4semidirect
73. C2×C33#106semidirect
75. C3×C3⋊D3#106semidirect
111. C33C2×#106semidirect
133. C2×C32#808semidirect
135. C3⋊D3#808semidirect
171. C32C2×#808semidirect
190. C2×C3AGL(1,𝔽9)⋊C4semidirect
207. C3C2×AGL(1,𝔽9)⋊C4semidirect
211. C2#11724103direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊AGL(1,𝔽9)2
C2×C3×#37772
C2×#889402
#117241032
C2×AGL(1,𝔽9)⋊C43
C2×#8089