Finite Group Databasebeta

← #10589522⋊C3 #13146899

Subgroups — #10589522⋊C3 #13146899

Order 1728.

314 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 55 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×#6204C211yesmaximal, characteristic
2#63421⋊C3C212yesmaximal, complemented
3C3×#5225—31nomaximal
4#10589522C311yesmaximal, characteristic, complemented
5C2×#6204C412yescomplemented
6C2×#6204C2211yescharacteristic
7#3676⋊C3—41nomaximal
8C22×C32×Q8D311yescharacteristic
9C3×#926—31no—
10C2×C3×#208—31no—
11C22×Q8⋊C9—21no—
12C22×Q8⋊C9C2×C311yescharacteristic
13C3×#835—32no—
14C3×#835—32no—
15#63421C2×C312yescomplemented
16C23×3−1+2—41no—
17C2×3−1+2⋊C4—42no—
18#6204C2×C411yescharacteristic
19C3×#419—91no—
20#5225—31no—
21C2×C32×Q8—32no—
22C2×C32×Q8—31no—
23C2×C32×Q8Dic312yes—
24C2×C32×Q8C2×D311yescharacteristic
25C22×C32×C4—31no—
26C3×#217—61no—
27C3×C4⋊Dic3—32no—
28C3×C4×Dic3—61no—
29C3×#208—62no—
30C2×C3×C3⋊C8—32no—
31C2×Q8⋊C9—22no—
32C2×Q8⋊C9—21no—
33C2×Q8⋊C9C3×C412yescomplemented
34C2×Q8⋊C9C22×C311yescharacteristic
35#3676—41no—
36C22×3−1+2—41no—
37C22×3−1+2—42no—
38C22×3−1+2—82no—
393−1+2⋊C4—82no—
40C22×C3×Q8—31no—
41C22×C3×Q8C3×D311yescharacteristic
42C2×C3×M16—91no—
43C3×#117—91no—
44#926—31no—
45C2×#208—31no—
46C3×#103—92no—
47C3×#103—92no—
48#835—32no—
49#835—32no—
50C23×C32S411yescharacteristic
51C32×Q8—32no—
52C32×Q8—31no—
53C32×Q8—61no—
54C32×Q8C2×Dic311yescharacteristic
55C2×C32×C4—32no—
56C2×C32×C4—32no—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C3×Dic3—62no—
60C23×C9—41no—
61C23×C9—81no—
62C3×C3⋊C8—62no—
63C2×Dic9—42no—
64Q8⋊C9—21no—
65Q8⋊C9C2×C3×C411yescharacteristic
66#419—91noSylow
67C2×3−1+2—42no—
68C2×3−1+2—41no—
69C2×3−1+2—82no—
70C2×C3×Q8—31no—
71C2×C3×Q8—32no—
72C2×C3×Q8—31no—
73C2×C3×Q8—32no—
74C2×C3×Q8—32no—
75C2×C3×Q8—31no—
76C2×C3×Q8—61no—
77C2×C3×Q8—62no—
78C2×C3×Q8C3×Dic312yes—
79C2×C3×Q8C2×C3×D311yescharacteristic
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—61no—
83C3×M16—182no—
84C2×C3×C8—92no—
85C3×C4⋊C4—92no—
86C3×C22⋊C4—181no—
87C3×C42—181no—
88#217—61no—
89C4⋊Dic3—32no—
90C4×Dic3—61no—
91#208—62no—
92C2×C3⋊C8—32no—
93C22×C32—22no—
94C22×C32SL(2,ℤ/4)12yes—
95C22×C32C2×S411yescharacteristic
96C32×C4—31no—
97C32×C4—31no—
98C32×C4—32no—
99C3×Dic3—122no—
100C22×C9—41no—
101C22×C9—42no—
102C22×C9—82no—
103C22×C9—84no—
104C22×C9—81no—
105C22×C9—82no—
106Dic9—82no—
107C22×Q8AGL(1,ℤ/9)11yescharacteristic
108C2×M16—91no—
109#117—91no—
110#103—92no—
111#103—92no—
1123−1+2—41noSylow
113C23×C3—31no—
114C23×C3C3×S411yescharacteristic
115C3×Q8—32no—
116C3×Q8—31no—
117C3×Q8—32no—
118C3×Q8—31no—
119C3×Q8—31no—
120C3×Q8—62no—
121C3×Q8—61no—
122C3×Q8—61no—
123C3×Q8—61no—
124C3×Q8—62no—
125C3×Q8C2×C3×Dic311yescharacteristic
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—32no—
129C2×C3×C4—32no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—32no—
133C2×C3×C4—32no—
134C2×C3×C4—62no—
135C2×C3×C4—62no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—182no—
139C2×Dic3—62no—
140C3×C8—182no—
141C3⋊C8—62no—
142C2×C32—22no—
143C2×C32#98312yes—
144C2×C32C2×SL(2,ℤ/4)11yescharacteristic
145C2×C9—41no—
146C2×C9—42no—
147C2×C9—82no—
148C2×C9—84no—
149C2×C9—81no—
150C2×C9—82no—
151C2×Q8—31no—
152C2×Q8—32no—
153C2×Q83−1+2⋊C412yescomplemented
154C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
155C22×C4—31no—
156M16—182no—
157C2×C8—92no—
158C4⋊C4—92no—
159C22⋊C4—181no—
160C42—181no—
161C22×C3—22no—
162C22×C3—32no—
163C22×C3—31no—
164C22×C3—62no—
165C22×C3C3×SL(2,ℤ/4)12yes—
166C22×C3C2×C3×S411yescharacteristic
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—32no—
170C3×C4—31no—
171C3×C4—31no—
172C3×C4—32no—
173C3×C4—61no—
174C3×C4—61no—
175C3×C4—62no—
176C3×C4—362no—
177Dic3—122no—
178C32#542011yescharacteristic
179C9—41no—
180C9—81no—
181C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
182Q8—32no—
183Q8—31no—
184Q8—61no—
185Q8C2×3−1+2⋊C411yescharacteristic
186C2×C4—32no—
187C2×C4—32no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—182no—
191C8—182no—
192C2×C3—22no—
193C2×C3—32no—
194C2×C3—31no—
195C2×C3—62no—
196C2×C3C3×#98312yes—
197C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
198C22—22no—
199C22#1094⋊C412yes—
200C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
201C4—31no—
202C4—31no—
203C4—32no—
204C4—362no—
205C3—31no—
206C3C3×#542011yescharacteristic
207C2—22no—
208C2#63689⋊C312yes—
209C2C2×#1094⋊C411yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, hypocenterQ8⋊C9C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
prime radicalC23×C32S4
cosocleC2×Q8⋊C9C22×C3
hypercenterC23((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #63421⋊C3C2semidirect
4. #10589522C3semidirect
5. C2×#6204C4semidirect
15. #63421C2×C3semidirect
33. C2×Q8⋊C9C3×C4semidirect
153. C2×Q83−1+2⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#62042
#63421⋊C32
C3×#52253
#105895223
#3676⋊C34