Finite Group Databasebeta

← #10591159

Subgroups — #10591159

Order 576.

498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#3712C212yesmaximal, complemented
3C3×#830C212yesmaximal, complemented
4#63557C212yesmaximal, complemented
5#4726—32nomaximal
6C23×C3⋊D3C411yescharacteristic
7C22×C32×C4C2211yescharacteristic
8C22×C3⋊Dic3C411yescharacteristic
9C2×C4×C3⋊D3—24no—
10C2×C4×C3⋊D3C411yescharacteristic
11C2×C4×C3⋊D3C411yescharacteristic
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C3⋊C8C2214yescomplemented
15#3712—24no—
16C22×C4×D3—32no—
17C22×C4×D3—61no—
18C2×ΓL(1,𝔽25)—34no—
19C3×C22⋊C8—32no—
20#830D312yescomplemented
21(C2×D3)⋊C8—34no—
22C23×C32C2×C411yescharacteristic
23C22×C3⋊D3—22no—
24C22×C3⋊D3—22no—
25C22×C3⋊D3—22no—
26C22×C3⋊D3C2×C411yescharacteristic
27C22×C3⋊D3C2×C411yescharacteristic
28C2×C32×C4—22no—
29C2×C32×C4C2×C411yescharacteristic
30C2×C32×C4C2311yescharacteristic
31C2×C3⋊Dic3—22no—
32C2×C3⋊Dic3C2×C411yescharacteristic
33C2×C3⋊Dic3C2×C411yescharacteristic
34C4×C3⋊D3—22no—
35C4×C3⋊D3—22no—
36C4×C3⋊D3—22no—
37C4×C3⋊D3D414yes—
38C3×C3⋊C8—24no—
39#407—91noSylow
40C23×D3—32no—
41C23×D3—61no—
42C22×C3×C4—21no—
43C22×C3×C4C2×D312yes—
44C22×Dic3—32no—
45C22×Dic3—61no—
46C2×C4×D3—32no—
47C2×C4×D3—32no—
48C2×C4×D3—32no—
49C2×C4×D3—32no—
50C2×C4×D3—61no—
51C2×C4×D3—61no—
52C2×C4×D3—61no—
53C2×C4×D3—61no—
54C2×C4×D3—68no—
55C2×C4×D3—68no—
56C2×C3×C8—34no—
57C2×C3⋊C8C2×D314yescomplemented
58ΓL(1,𝔽25)—68no—
59C22×C32—21no—
60C22×C32—21no—
61C22×C32M1612yescomplemented
62C22×C32C22×C411yescharacteristic
63C2×C3⋊D3—28no—
64C2×C3⋊D3—21no—
65C2×C3⋊D3—21no—
66C2×C3⋊D3—21no—
67C2×C3⋊D3—21no—
68C2×C3⋊D3C22⋊C412yes—
69C2×C3⋊D3M1612yescomplemented
70C32×C4—21no—
71C32×C4C2×D412yes—
72C3⋊Dic3—21no—
73C3⋊Dic3C22⋊C412yes—
74C23×C4—91no—
75C2×M16—92no—
76C22⋊C8—32no—
77C22⋊C8—92no—
78C23×C3—21no—
79C23×C3C4×D312yes—
80C22×D3—32no—
81C22×D3—32no—
82C22×D3—61no—
83C22×D3—61no—
84C22×D3—64no—
85C22×D3—64no—
86C22×D3—64no—
87C22×D3—64no—
88C22×D3—64no—
89C22×D3—64no—
90C2×C3×C4—24no—
91C2×C3×C4—24no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4C4×D312yes—
95C2×C3×C4C22×D312yes—
96C2×Dic3—32no—
97C2×Dic3—32no—
98C2×Dic3—61no—
99C2×Dic3—61no—
100C2×Dic3—64no—
101C2×Dic3—64no—
102C4×D3—38no—
103C4×D3—64no—
104C4×D3—64no—
105C4×D3—64no—
106C4×D3—64no—
107C4×D3—64no—
108C4×D3—64no—
109C4×D3—64no—
110C3×C8—64no—
111C3⋊C8—24no—
112C2×C32—22no—
113C2×C32C2×C22⋊C411yescharacteristic
114C2×C32C2×M1611yescharacteristic
115C2×C32C2×M1611yescharacteristic
116C3⋊D3—22no—
117C3⋊D3—22no—
118C24—91no—
119C22×C4—91no—
120C22×C4—91no—
121C22×C4—91no—
122C22×C4—91no—
123C22×C4—91no—
124C22×C4—184no—
125C22×C4D3211yescharacteristic
126M16—184no—
127C2×C8—34no—
128C22×C3—22no—
129C22×C3—22no—
130C22×C3—22no—
131C22×C3—22no—
132C22×C3—22no—
133C22×C3—21no—
134C22×C3ΓL(1,𝔽25)14yescomplemented
135C22×C3C2×C4×D312yes—
136C2×D3—34no—
137C2×D3—34no—
138C2×D3—62no—
139C2×D3—62no—
140C2×D3—62no—
141C2×D3—616no—
142C2×D3—616no—
143C2×D3—62no—
144C2×D3—62no—
145C2×D3—62no—
146C2×D3—62no—
147C2×D3—62no—
148C2×D3—62no—
149C2×D3—62no—
150C3×C4—22no—
151C3×C4—22no—
152C3×C4—22no—
153C3×C4D3×D414yes—
154Dic3—34no—
155Dic3—62no—
156Dic3—62no—
157Dic3—62no—
158C32#40711yescharacteristic, complemented, Sylow
159C23—91no—
160C23—91no—
161C23—182no—
162C23—182no—
163C23—182no—
164C23#62411yescharacteristic
165C2×C4—22no—
166C2×C4—91no—
167C2×C4—94no—
168C2×C4—91no—
169C2×C4—182no—
170C2×C4—182no—
171C2×C4—182no—
172C2×C4—182no—
173C2×C4#62411yescharacteristic
174C2×C4C2×D3211yescharacteristic
175C8—64no—
176C2×C3—24no—
177C2×C3—24no—
178C2×C3—21no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3D3×C22⋊C412yes—
182C2×C3C2×ΓL(1,𝔽25)12yes—
183C2×C3D3×M1612yes—
184D3—64no—
185D3—64no—
186D3—64no—
187D3—64no—
188C22—21no—
189C22—21no—
190C22—92no—
191C22—92no—
192C22—188no—
193C22—181no—
194C22—181no—
195C22—181no—
196C22—181no—
197C22#371212yescomplemented
198C22C2×#62411yescharacteristic
199C4—21no—
200C4—92no—
201C4—181no—
202C4#381112yes—
203C3—21no—
204C3#472612yescomplemented
205C2—22no—
206C2—182no—
207C2—182no—
208C2#6381811yescharacteristic
209C2C2×#371211yescharacteristic
210C2#6397411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D32
commutator, socleC22×C32C22×C4
FittingC22×C32×C4C22
prime radicalC23×C3⋊D3C4
cosocleC2×C32×C4C23
hypercenterC22×C4D32
hypocenterC32#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3712C2semidirect
3. C3×#830C2semidirect
4. #63557C2semidirect
14. C2×C3×C3⋊C8C22semidirect
20. #830D3semidirect
57. C2×C3⋊C8C2×D3semidirect
61. C22×C32M16semidirect
69. C2×C3⋊D3M16semidirect
134. C22×C3ΓL(1,𝔽25)semidirect
158. C32#407semidirect
197. C22#3712semidirect
204. C3#4726semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D32
C2×#37122
C3×#8302
#635572
#47263