Finite Group Databasebeta

← #3459⋊C3 #85651

Subgroups — #3459⋊C3 #85651

Order 384.

250 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 23 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
1#543⋊C3C211yesmaximal, characteristic, complemented
2#3459C311yesmaximal, characteristic, complemented, Sylow
3C2×#41⋊C3—42nomaximal
4C2×C4×A4—41nomaximal
5C4×SL(2,𝔽3)—42nomaximal
6#543C2×C311yescharacteristic
7#539—32no—
8#538—32no—
9#534—31no—
10#529—32no—
11#525—32no—
12C2×C4×D4—31no—
13C22×A4—41no—
14#41⋊C3—44no—
15C2×SL(2,𝔽3)—42no—
16C2×SL(2,𝔽3)—42no—
17C4×A4—42no—
18C2×#41A412yescomplemented
19C22×D4—31no—
20C23×C4—31no—
21C23×C4A411yescharacteristic
22#126—34no—
23#125—32no—
24#124—32no—
25#123—34no—
26#123—34no—
27C22⋊Q8—32no—
28C22⋊Q8—32no—
29C22⋊Q8—32no—
30#121—32no—
31#121—32no—
32#121—32no—
33#120—34no—
34C4×Q8A412yescomplemented
35C4×D4—34no—
36C4×D4—34no—
37C4×D4—32no—
38#117—32no—
39C2×C4⋊C4—31no—
40C2×C22⋊C4—31no—
41C2×C22⋊C4—31no—
42C2×C42—31no—
43C2×A4—41no—
44C2×A4—41no—
45C2×A4—41no—
46C2×C3×C4—161no—
47SL(2,𝔽3)—42no—
48SL(2,𝔽3)—42no—
49C24—31no—
50C24C2×A411yescharacteristic
51#41—44no—
52C2×Q8C2×A412yes—
53C2×Q8C2×A412yes—
54C2×D4—34no—
55C2×D4—32no—
56C2×D4—32no—
57C2×D4—62no—
58C2×D4—61no—
59C2×D4—61no—
60C22×C4—31no—
61C22×C4—31no—
62C22×C4—31no—
63C22×C4—32no—
64C22×C4—31no—
65C22×C4—32no—
66C22×C4—31no—
67C22×C4—42no—
68C22×C4—122no—
69C4⋊C4—32no—
70C4⋊C4—34no—
71C4⋊C4—32no—
72C4⋊C4—32no—
73C4⋊C4—32no—
74C22⋊C4—34no—
75C22⋊C4—34no—
76C22⋊C4—34no—
77C22⋊C4—34no—
78C42—32no—
79C42—32no—
80C22×C3—161no—
81A4—41no—
82C3×C4—162no—
83C23—31no—
84C23—31no—
85C23—32no—
86C23—41no—
87C23—41no—
88C23—61no—
89C23—61no—
90C23—121no—
91C23—121no—
92C23#41⋊C311yescharacteristic, complemented
93Q8—42no—
94Q8—42no—
95D4—64no—
96D4—64no—
97D4—122no—
98D4—122no—
99C2×C4—32no—
100C2×C4—32no—
101C2×C4—31no—
102C2×C4—32no—
103C2×C4—61no—
104C2×C4—61no—
105C2×C4—62no—
106C2×C4—64no—
107C2×C4—121no—
108C2×C4—122no—
109C2×C4—121no—
110C2×C4—121no—
111C2×C4C22⋊A411yescharacteristic
112C2×C3—161no—
113C2×C3—161no—
114C2×C3—161no—
115C22—31no—
116C22—31no—
117C22—41no—
118C22—61no—
119C22—61no—
120C22—62no—
121C22—62no—
122C22—62no—
123C22—121no—
124C22—122no—
125C22—122no—
126C22C2×C22⋊A411yescharacteristic
127C4—62no—
128C4—61no—
129C4—62no—
130C4—62no—
131C4#99712yes—
132C3—161noSylow
133C2—61no—
134C2—61no—
135C2—122no—
136C2#594811yescharacteristic
137C2C2×#99711yescharacteristic
138C2#595011yescharacteristic
subgroupG/H
center, hypercenterC2×C4C22⋊A4
commutator, cosocle, hypocenter#543C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3459C3
LCSG ▷ #543
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #543⋊C3C2semidirect
2. #3459C3semidirect
18. C2×#41A4semidirect
34. C4×Q8A4semidirect
92. C23#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
#543⋊C32
#34593
C2×#41⋊C34
C2×C4×A44
C4×SL(2,𝔽3)4