Finite Group Databasebeta

← #2880⋊C3 #73228

Subgroups — #2880⋊C3 #73228

Order 384.

199 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 38 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×#41⋊C3C211yesmaximal, characteristic
2C2×#131⋊C3C212yesmaximal, complemented
3#2880C311yesmaximal, characteristic, complemented, Sylow
4C2×#41⋊C3—22no—
5C2×#41⋊C3C411yescharacteristic
6C2×#41⋊C3C2211yescharacteristic
7C22×SL(2,𝔽3)C411yescharacteristic
8#131⋊C3—22no—
9C3×C22⋊C8—41nomaximal
10C22×#41C2×C311yescharacteristic
11C2×#131C2×C312yescomplemented
12#408—32no—
13#407—31no—
14C2×C22⋊C8—31no—
15C22×C3×C4—41no—
16#41⋊C3—21no—
17#41⋊C3D412yes—
18C2×SL(2,𝔽3)—21no—
19C2×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C3×C8—42no—
21C2×#41—22no—
22C2×#41—31no—
23C2×#41—31no—
24C2×#41—62no—
25C2×#41C3×C411yescharacteristic
26C2×#41C22×C311yescharacteristic
27C22×Q8C3×C411yescharacteristic
28C22×D4—31no—
29C23×C4—31no—
30#131—22no—
31#131—62no—
32C2×M16—32no—
33C22×C8—32no—
34C22⋊C8—32no—
35C22⋊C8—31no—
36C22⋊C8—32no—
37C22⋊C8A411yescharacteristic
38C23×C3—41no—
39C2×C3×C4—41no—
40C2×C3×C4—41no—
41C2×C3×C4—82no—
42SL(2,𝔽3)C22⋊C411yescharacteristic
43C3×C8—82no—
44C24—31no—
45#41—21no—
46#41—32no—
47#41—62no—
48#41—62no—
49#41—62no—
50#41—62no—
51#41—61no—
52#41C3×D412yes—
53C2×Q8—21no—
54C2×Q8—31no—
55C2×Q8—61no—
56C2×Q8C2×C3×C411yescharacteristic
57C2×D4—31no—
58C2×D4—31no—
59C2×D4—32no—
60C2×D4—62no—
61C2×D4—62no—
62C22×C4—31no—
63C22×C4—31no—
64C22×C4—31no—
65C22×C4—31no—
66C22×C4—31no—
67C22×C4—62no—
68C22×C4—62no—
69C22×C4C2×A411yescharacteristic
70M16—62no—
71M16—62no—
72C2×C8—32no—
73C2×C8—62no—
74C2×C8—62no—
75C2×C8C2×A412yes—
76C22×C3—42no—
77C22×C3—41no—
78C22×C3—81no—
79C22×C3—81no—
80C3×C4—42no—
81C3×C4—81no—
82C23—31no—
83C23—31no—
84C23—62no—
85C23—62no—
86C23—121no—
87C23C4×A411yescharacteristic
88Q8—31no—
89Q8—61no—
90Q8—61no—
91Q8C3×C22⋊C411yescharacteristic
92D4—32no—
93D4—32no—
94D4—61no—
95D4—61no—
96D4—64no—
97C2×C4—22no—
98C2×C4—31no—
99C2×C4—32no—
100C2×C4—32no—
101C2×C4—31no—
102C2×C4—61no—
103C2×C4—61no—
104C2×C4—62no—
105C2×C4—61no—
106C2×C4—61no—
107C2×C4—62no—
108C2×C4C4×A411yescharacteristic
109C2×C4C22×A411yescharacteristic
110C8—22no—
111C8—62no—
112C2×C3—41no—
113C2×C3—41no—
114C2×C3—41no—
115C2×C3—82no—
116C22—21no—
117C22—21no—
118C22—32no—
119C22—61no—
120C22—61no—
121C22—61no—
122C22—61no—
123C22—61no—
124C22—124no—
125C22#131⋊C312yescomplemented
126C22C2×C4×A411yescharacteristic
127C4—21no—
128C4—31no—
129C4—31no—
130C4—61no—
131C4D4×A412yes—
132C3—41noSylow
133C2—22no—
134C2—62no—
135C2—121no—
136C2A4×C22⋊C411yescharacteristic
137C2C2×#131⋊C311yescharacteristic
138C2#568⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×A4
commutatorC2×Q8C2×C3×C4
Fitting#2880C3
socleC22C2×C4×A4
prime radicalC22×#41⋊C3C2
cosocleC2×#41C22×C3
hypercenterC22⋊C8A4
hypocenterQ8C3×C22⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#131⋊C3C2semidirect
3. #2880C3semidirect
11. C2×#131C2×C3semidirect
125. C22#131⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
C2×#131⋊C32
#28803
C3×C22⋊C84