Finite Group Databasebeta

← #10588205⋊C3 #13135569

Subgroups — #10588205⋊C3 #13135569

Order 1728.

280 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 86 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×C4⋊C4×3−1+2C211yesmaximal, characteristic
2#63389⋊C3C212yesmaximal
3C3×#4528—31nomaximal
4#10588205C311yesmaximal, characteristic, complemented
5C22×C4×3−1+2C411yescharacteristic
6C22×C4×3−1+2C411yescharacteristic
7C22×C4×3−1+2C2211yescharacteristic
8C4⋊C4×3−1+2—22no—
9C2×3−1+2⋊C8—22no—
10C2×C32×C4⋊C4D311yescharacteristic
11C3×#830—32no—
12C2×C9×C4⋊C4—21no—
13C2×C9×C4⋊C4C2×C311yescharacteristic
14#63389C2×C312yes—
15C23×3−1+2C2×C411yescharacteristic
16C2×C4×3−1+2—21no—
17C2×C4×3−1+2—21no—
18C2×C4×3−1+2—21no—
19C2×C4×3−1+2—41no—
20C2×C4×3−1+2—41no—
21C2×C4×3−1+2C812yescomplemented
22C2×C4×3−1+2D412yes—
233−1+2⋊C8—42no—
24C3×#324—91no—
25#4528—31no—
26C22×C32×C4Dic311yescharacteristic
27C22×C32×C4Dic311yescharacteristic
28C22×C32×C4C2×D311yescharacteristic
29C32×C4⋊C4—22no—
30C2×C3×C3⋊C8—62no—
31C22×C4×C9—21no—
32C22×C4×C9—21no—
33C22×C4×C9—21no—
34C22×C4×C9C3×C411yescharacteristic
35C22×C4×C9C3×C411yescharacteristic
36C22×C4×C9C22×C311yescharacteristic
37C9×C4⋊C4—24no—
38C9×C4⋊C4—22no—
39C2×C9⋊C8—22no—
40C22×3−1+2—21no—
41C22×3−1+2—21no—
42C22×3−1+2C22⋊C411yescharacteristic
43C22×3−1+2C2×C811yescharacteristic
44C22×3−1+2M1611yescharacteristic
45C4×3−1+2—22no—
46C4×3−1+2—22no—
47C4×3−1+2—41no—
48C2×C3×C4⋊C4—31no—
49C2×C3×C4⋊C4C3×D311yescharacteristic
50C3×C22⋊C8—92no—
51#830—32no—
52C23×C32C2×Dic311yescharacteristic
53C2×C32×C4—21no—
54C2×C32×C4—21no—
55C2×C32×C4—21no—
56C2×C32×C4—41no—
57C2×C32×C4—41no—
58C2×C32×C4C3⋊C812yes—
59C2×C32×C4#6712yes—
60C23×C9—21no—
61C23×C9C2×C3×C411yescharacteristic
62C3×C3⋊C8—122no—
63C2×C4×C9—22no—
64C2×C4×C9—22no—
65C2×C4×C9—21no—
66C2×C4×C9—21no—
67C2×C4×C9—22no—
68C2×C4×C9—21no—
69C2×C4×C9—22no—
70C2×C4×C9—22no—
71C2×C4×C9—41no—
72C2×C4×C9—42no—
73C2×C4×C9—42no—
74C2×C4×C9—41no—
75C2×C4×C9C3×C812yescomplemented
76C2×C4×C9C3×D412yes—
77C9⋊C8—42no—
78#324—91noSylow
79C2×3−1+2—21no—
80C2×3−1+2—21no—
81C2×3−1+2C22⋊C811yescharacteristic
82C2×3−1+2#9911yescharacteristic
83C2×3−1+2#10111yescharacteristic
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4C3×Dic311yescharacteristic
88C22×C3×C4C3×Dic311yescharacteristic
89C22×C3×C4C2×C3×D311yescharacteristic
90C2×C3×C8—182no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—62no—
93C2×C3⋊C8—62no—
94C22×C32—21no—
95C22×C32—21no—
96C22×C32C2×C3⋊C811yescharacteristic
97C22×C32#20811yescharacteristic
98C22×C32#21711yescharacteristic
99C32×C4—22no—
100C32×C4—22no—
101C32×C4—41no—
102C22×C9—22no—
103C22×C9—22no—
104C22×C9—21no—
105C22×C9—21no—
106C22×C9—21no—
107C22×C9—21no—
108C22×C9—21no—
109C22×C9C3×C22⋊C411yescharacteristic
110C22×C9C2×C3×C811yescharacteristic
111C22×C9C3×M1611yescharacteristic
112C4×C9—24no—
113C4×C9—22no—
114C4×C9—22no—
115C4×C9—42no—
116C4×C9—42no—
117C4×C9—41no—
118C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
119C22⋊C8—92no—
1203−1+2#32411yescharacteristic, complemented, Sylow
121C23×C3—31no—
122C23×C3C2×C3×Dic311yescharacteristic
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—61no—
133C2×C3×C4—121no—
134C2×C3×C4—121no—
135C2×C3×C4C3×C3⋊C812yes—
136C2×C3×C4C3×#6712yes—
137C3×C8—362no—
138C3⋊C8—122no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32#83011yescharacteristic
142C2×C32#83411yescharacteristic
143C2×C32#83611yescharacteristic
144C2×C9—22no—
145C2×C9—22no—
146C2×C9—21no—
147C2×C9—21no—
148C2×C9—21no—
149C2×C9—21no—
150C2×C9—21no—
151C2×C9C3×C22⋊C811yescharacteristic
152C2×C9C3×#9911yescharacteristic
153C2×C9C3×#10111yescharacteristic
154C22×C43−1+2⋊C411yescharacteristic
155C22×C43−1+2⋊C411yescharacteristic
156C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
157C2×C8—182no—
158C4⋊C4—22no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—61no—
165C22×C3—61no—
166C22×C3C2×C3×C3⋊C811yescharacteristic
167C22×C3C3×#20811yescharacteristic
168C22×C3C3×#21711yescharacteristic
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—41no—
172C3×C4—62no—
173C3×C4—62no—
174C3×C4—121no—
175C32#452811yescharacteristic
176C9—21no—
177C9C3×#32411yescharacteristic, complemented
178C23C2×3−1+2⋊C411yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C43−1+2⋊C812yescomplemented
185C2×C4#611⋊C312yes—
186C8—362no—
187C2×C3—21no—
188C2×C3—21no—
189C2×C3—31no—
190C2×C3—31no—
191C2×C3—31no—
192C2×C3—61no—
193C2×C3—61no—
194C2×C3C3×#83011yescharacteristic
195C2×C3C3×#83411yescharacteristic
196C2×C3C3×#83611yescharacteristic
197C22—21no—
198C22—21no—
199C22C2×3−1+2⋊C811yescharacteristic
200C22#3667⋊C311yescharacteristic
201C22#3676⋊C311yescharacteristic
202C4—22no—
203C4—22no—
204C4—41no—
205C3—31no—
206C3C3×#452811yescharacteristic
207C2—21no—
208C2—21no—
209C2#63389⋊C311yescharacteristic
210C2#63393⋊C311yescharacteristic
211C2#63394⋊C311yescharacteristic
subgroupG/H
centerC22#3676⋊C3
commutatorC22×C9C2×C3×C8
FrattiniC22×C3×C4C2×C3×D3
FittingC2×C4⋊C4×3−1+2C2
socleC22×C3C3×#217
prime radicalC23×C32C2×Dic3
cosocleC22×C4×C9C22×C3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#324
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #10588205C3semidirect
21. C2×C4×3−1+2C8semidirect
75. C2×C4×C9C3×C8semidirect
120. 3−1+2#324semidirect
177. C9C3×#324semidirect
184. C2×C43−1+2⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
#63389⋊C32
C3×#45283
#105882053