Finite Group Databasebeta

← #10590937

Subgroups — #10590937

Order 576.

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

normal subgroups only

This group has 414 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×D32C211yesmaximal, characteristic
2C2×#3743C211yesmaximal, characteristic, complemented
3C2×#3712C211yesmaximal, characteristic, complemented
4C3×(C2×D3)⋊C8C212yesmaximal, complemented
5(C2×D3)⋊(C3⋊C8)C212yesmaximal, complemented
6#4726—32nomaximal
7C22×D32C411yescharacteristic
8C2×C4×C3⋊D3C2211yescharacteristic
9C2×C3×C4×D3C2212yes—
10C2×#624C411yescharacteristic
11C2×D3×Dic3C412yes—
12C4×D32—22no—
13C4×D32—22no—
14C2×C32×C8C2211yescharacteristic, complemented
15C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
16#3743—22no—
17C2×C3×C3⋊C8C2212yescomplemented
18#3712—22no—
19C22×C4×D3—32no—
20C2×ΓL(1,𝔽25)—32no—
21C2×ΓL(1,𝔽25)—32no—
22C2×ΓL(1,𝔽25)—61no—
23C3×C22⋊C8—32no—
24#830—32no—
25(C2×D3)⋊C8—32no—
26(C2×D3)⋊C8D312yescomplemented
27C22×C3⋊D3C2×C411yescharacteristic
28C22×C3×D3C2×C412yes—
29C2×D32—22no—
30C2×D32—24no—
31C2×C32×C4C2311yescharacteristic
32C2×C3⋊Dic3C2×C411yescharacteristic
33C4×C3⋊D3D412yes—
34C4×C3⋊D3D412yes—
35C2×C3×Dic3C2×C412yes—
36C3×C4×D3—22no—
37C3×C4×D3—22no—
38#624—22no—
39D3×Dic3—24no—
40C32×C8—21no—
41C3⋊(C3⋊C8)—21no—
42C3×C3⋊C8—22no—
43#407—91noSylow
44C23×D3—32no—
45C22×C3×C4—32no—
46C22×Dic3—32no—
47C2×C4×D3—32no—
48C2×C4×D3—32no—
49C2×C4×D3—32no—
50C2×C4×D3—61no—
51C2×C4×D3—64no—
52C2×C4×D3—64no—
53C2×C4×D3C2×D312yes—
54C2×C3×C8—21no—
55C2×C3×C8—32no—
56C2×C3×C8C2×D312yescomplemented
57C2×C3⋊C8—32no—
58C2×C3⋊C8—61no—
59C2×C3⋊C8C2×D312yescomplemented
60ΓL(1,𝔽25)—64no—
61ΓL(1,𝔽25)—64no—
62ΓL(1,𝔽25)—64no—
63C22×C32C22×C411yescharacteristic
64C2×C3⋊D3—22no—
65C2×C3⋊D3C22⋊C412yes—
66C2×C3×D3—22no—
67C2×C3×D3—22no—
68C2×C3×D3M1614yescomplemented
69D32—28no—
70C32×C4C2×D411yescharacteristic
71C32×C4C2×D411yescharacteristic
72C3⋊Dic3C22⋊C412yes—
73C3×Dic3—22no—
74C23×C4—91no—
75C2×M16—91no—
76C2×M16—91no—
77C22⋊C8—32no—
78C22⋊C8—92no—
79C23×C3—32no—
80C22×D3—32no—
81C22×D3—61no—
82C22×D3—64no—
83C22×D3—64no—
84C22×D3—64no—
85C22×D3C4×D312yes—
86C2×C3×C4—21no—
87C2×C3×C4—32no—
88C2×C3×C4—62no—
89C2×C3×C4—62no—
90C2×C3×C4C22×D312yes—
91C2×Dic3—32no—
92C2×Dic3—61no—
93C2×Dic3—64no—
94C2×Dic3C4×D312yes—
95C4×D3—22no—
96C4×D3—22no—
97C4×D3—34no—
98C4×D3—34no—
99C4×D3—62no—
100C4×D3—62no—
101C4×D3—64no—
102C4×D3—64no—
103C3×C8—22no—
104C3×C8—22no—
105C3×C8—62no—
106C3⋊C8—22no—
107C3⋊C8—62no—
108C3⋊C8—62no—
109C2×C32C2×C22⋊C411yescharacteristic
110C2×C32C2×M1612yes—
111C3⋊D3—22no—
112C3×D3—24no—
113C24—91no—
114C22×C4—32no—
115C22×C4—91no—
116C22×C4—92no—
117C22×C4—91no—
118C22×C4—182no—
119C22×C4—182no—
120M16—182no—
121M16—182no—
122C2×C8—32no—
123C2×C8—91no—
124C2×C8D3211yescharacteristic, complemented
125C22×C3—21no—
126C22×C3—34no—
127C22×C3—62no—
128C22×C3—62no—
129C22×C3C2×C4×D312yes—
130C2×D3—22no—
131C2×D3—22no—
132C2×D3—34no—
133C2×D3—62no—
134C2×D3—616no—
135C2×D3—62no—
136C2×D3—62no—
137C2×D3—122no—
138C2×D3ΓL(1,𝔽25)14yescomplemented
139C3×C4—21no—
140C3×C4—21no—
141C3×C4—62no—
142C3×C4D3×D412yes—
143C3×C4D3×D412yes—
144Dic3—22no—
145Dic3—34no—
146Dic3—62no—
147C32#40711yescharacteristic, complemented, Sylow
148C23—32no—
149C23—91no—
150C23—184no—
151C23—182no—
152C2×C4—32no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—91no—
156C2×C4—92no—
157C2×C4—92no—
158C2×C4—182no—
159C2×C4—184no—
160C2×C4C2×D3211yescharacteristic
161C8—21no—
162C8—62no—
163C8—181no—
164C2×C3—22no—
165C2×C3—21no—
166C2×C3—64no—
167C2×C3D3×C22⋊C412yes—
168C2×C3C2×ΓL(1,𝔽25)12yes—
169C2×C3D3×M1612yes—
170D3—24no—
171D3—64no—
172D3—122no—
173C22—34no—
174C22—62no—
175C22—62no—
176C22—92no—
177C22—188no—
178C22—182no—
179C22C4×D3211yescharacteristic
180C4—62no—
181C4—92no—
182C4#380211yescharacteristic
183C4#381111yescharacteristic
184C3—21no—
185C3#472612yescomplemented
186C2—64no—
187C2—182no—
188C2D3×ΓL(1,𝔽25)12yes—
189C2#6392111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D32
commutator, socleC22×C32C22×C4
FittingC2×C32×C8C22
prime radicalC22×D32C4
cosocleC2×C32×C4C23
hypercenterC2×C8D32
hypocenterC32#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743C2semidirect
3. C2×#3712C2semidirect
4. C3×(C2×D3)⋊C8C2semidirect
5. (C2×D3)⋊(C3⋊C8)C2semidirect
14. C2×C32×C8C22semidirect
15. C2×C3⋊(C3⋊C8)C22semidirect
17. C2×C3×C3⋊C8C22semidirect
26. (C2×D3)⋊C8D3semidirect
56. C2×C3×C8C2×D3semidirect
59. C2×C3⋊C8C2×D3semidirect
68. C2×C3×D3M16semidirect
124. C2×C8D32semidirect
138. C2×D3ΓL(1,𝔽25)semidirect
147. C32#407semidirect
185. C3#4726semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D322
C2×#37432
C2×#37122
C3×(C2×D3)⋊C82
(C2×D3)⋊(C3⋊C8)2
#47263