Finite Group Databasebeta

← C2×#458⋊C3 #73258

Subgroups — C2×#458⋊C3 #73258

Order 384.

179 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 20 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×#997C211yesmaximal, characteristic
2#458⋊C3C212yesmaximal, direct product factor
3C2×#458C311yesmaximal, characteristic, complemented, Sylow
4#997C2211yescharacteristic
5C2×C4×A4—41nomaximal
6C2×H(5,𝔽2)C2×C311yescharacteristic
7C2×#123—31no—
8#458—32no—
9#458C2×C312yescomplemented
10C2×#99—31no—
11C22×A4—41no—
12C22×A4—81no—
13C2×SL(2,𝔽3)—81no—
14C4×A4—42no—
15H(5,𝔽2)—31no—
16H(5,𝔽2)—61no—
17H(5,𝔽2)—61no—
18H(5,𝔽2)C22×C311yescharacteristic
19C2×#41—61no—
20C22×D4—31no—
21C22×D4—61no—
22C23×C4A411yescharacteristic, complemented
23#123—32no—
24#123—32no—
25#123—61no—
26#123—61no—
27C2×C4⋊C4—61no—
28C2×C22⋊C4—31no—
29C2×C22⋊C4—61no—
30#99—32no—
31#99—32no—
32C2×A4—42no—
33C2×A4—41no—
34C2×A4—81no—
35C2×A4—82no—
36C2×C3×C4—161no—
37SL(2,𝔽3)—81no—
38C24—21no—
39C24—31no—
40C24C2×A411yescharacteristic
41#41—61no—
42#41—62no—
43#41—61no—
44#41—61no—
45#41—62no—
46#41—61no—
47C2×Q8—21no—
48C2×D4—31no—
49C2×D4—31no—
50C2×D4—31no—
51C2×D4—31no—
52C2×D4—61no—
53C2×D4—62no—
54C2×D4—62no—
55C2×D4—62no—
56C2×D4—61no—
57C2×D4—61no—
58C2×D4—62no—
59C2×D4—61no—
60C2×D4—61no—
61C22×C4—31no—
62C22×C4—61no—
63C22×C4—61no—
64C22×C4—61no—
65C22×C4—61no—
66C22×C4C2×A412yescomplemented
67C4⋊C4—62no—
68C4⋊C4—62no—
69C22⋊C4—32no—
70C22⋊C4—61no—
71C22⋊C4—62no—
72C22⋊C4—62no—
73C22×C3—161no—
74A4—41no—
75A4—81no—
76C3×C4—162no—
77C23—21no—
78C23—31no—
79C23—31no—
80C23—31no—
81C23—31no—
82C23—42no—
83C23—61no—
84C23—61no—
85C23—61no—
86C23—82no—
87C23—122no—
88C23C22×A411yescharacteristic
89Q8—21no—
90Q8—61no—
91D4—62no—
92D4—62no—
93D4—62no—
94D4—62no—
95D4—61no—
96D4—61no—
97D4—62no—
98C2×C4—31no—
99C2×C4—31no—
100C2×C4—41no—
101C2×C4—62no—
102C2×C4—61no—
103C2×C4—62no—
104C2×C4—61no—
105C2×C4—61no—
106C2×C4—61no—
107C2×C4—62no—
108C2×C4—121no—
109C2×C4—122no—
110C2×C3—162no—
111C2×C3—161no—
112C22—31no—
113C22—31no—
114C22—41no—
115C22—62no—
116C22—61no—
117C22—61no—
118C22—81no—
119C22—121no—
120C22—121no—
121C22—121no—
122C22—122no—
123C22—241no—
124C22—241no—
125C22#120⋊C311yescharacteristic
126C4—42no—
127C4—61no—
128C4—61no—
129C4—122no—
130C3—161noSylow
131C2—61no—
132C2—61no—
133C2—121no—
134C2—121no—
135C2#458⋊C312yesdirect product factor
136C2C2×#120⋊C311yescharacteristic
subgroupG/H
center, socle, hypercenterC22#120⋊C3
commutator, cosocle, hypocenterH(5,𝔽2)C22×C3
FrattiniC23C22×A4
FittingC2×#458C3
prime radicalC2×#997C2
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #458⋊C3C2direct (N × H)
3. C2×#458C3semidirect
9. #458C2×C3semidirect
22. C23×C4A4semidirect
66. C22×C4C2×A4semidirect
135. C2#458⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#9972
#458⋊C32
C2×#4583
C2×C4×A44