Finite Group Databasebeta

← #3568⋊C3 #85663

Subgroups — #3568⋊C3 #85663

Order 384.

193 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 17 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#5464C211yesmaximal, characteristic, complemented
2#3568C311yesmaximal, characteristic, complemented, Sylow
3#126⋊C3—42nomaximal
4#120⋊C3—42nomaximal
5C2×C42⋊C3—41nomaximal
6#563—31no—
7H(3,𝔽4)C2×C311yescharacteristic, complemented
8#556—32no—
9#551—32no—
10#542—32no—
11#538—32no—
12#529—31no—
13C22⋊A4—42no—
14C42⋊C3—41no—
15C42⋊C3—42no—
16C4⋊Q8—32no—
17#126—32no—
18#126—32no—
19#126—32no—
20#126—32no—
21#126A412yescomplemented
22#125—31no—
23#124—32no—
24#124—32no—
25#124—31no—
26#124—32no—
27#123—32no—
28#123—32no—
29#123—32no—
30C22⋊Q8—31no—
31C22⋊Q8—32no—
32C22⋊Q8—32no—
33C22⋊Q8—32no—
34#121—31no—
35#121—32no—
36#120—32no—
37#120A412yescomplemented
38C4×Q8—31no—
39C4×Q8—32no—
40C4×D4—32no—
41C4×D4—31no—
42C4×D4—32no—
43#117—32no—
44#117—32no—
45C2×C22⋊C4—32no—
46C2×C42A411yescharacteristic, complemented
47C2×A4—161no—
48C24C2×A412yes—
49C2×Q8—32no—
50C2×Q8—31no—
51C2×D4—31no—
52C2×D4—32no—
53C2×D4—32no—
54C22×C4—32no—
55C22×C4—32no—
56C22×C4—31no—
57C4⋊C4—32no—
58C4⋊C4—32no—
59C4⋊C4—32no—
60C4⋊C4—32no—
61C4⋊C4—31no—
62C4⋊C4—32no—
63C4⋊C4—31no—
64C4⋊C4—32no—
65C22⋊C4—32no—
66C22⋊C4—32no—
67C22⋊C4—32no—
68C22⋊C4—32no—
69C22⋊C4—32no—
70C22⋊C4—32no—
71C22⋊C4—32no—
72C22⋊C4—32no—
73C22⋊C4—32no—
74C42—31no—
75C42—32no—
76C42—32no—
77C42C2×A411yescharacteristic
78C42C2×A412yes—
79A4—161no—
80A4—324no—
81C23—32no—
82C23—122no—
83C23C22⋊A411yescharacteristic
84Q8—122no—
85Q8—121no—
86D4—122no—
87D4—121no—
88D4—122no—
89C2×C4—31no—
90C2×C4—32no—
91C2×C4—32no—
92C2×C4—31no—
93C2×C4—32no—
94C2×C4—122no—
95C2×C4—121no—
96C2×C4—122no—
97C2×C3—641no—
98C22—61no—
99C22—62no—
100C22—62no—
101C22—62no—
102C22—84no—
103C22C2×C22⋊A411yescharacteristic
104C4—61no—
105C4—62no—
106C4—62no—
107C4—61no—
108C4—62no—
109C3—641noSylow
110C2—31no—
111C2—41no—
112C2—122no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterH(3,𝔽4)C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3568C3
LCSG ▷ H(3,𝔽4)
UCSC1

Complements

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

NH ≅ G/Ntype
1. #5464C2semidirect
2. #3568C3semidirect
7. H(3,𝔽4)C2×C3semidirect
21. #126A4semidirect
37. #120A4semidirect
46. C2×C42A4semidirect

Maximal subgroups

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

subgroupindex
#54642
#35683
#126⋊C34
#120⋊C34
C2×C42⋊C34