Finite Group Databasebeta

← #3549⋊C3 #85660

Subgroups — #3549⋊C3 #85660

Order 384.

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

normal subgroups only

This group has 243 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#3549C311yesmaximal, characteristic, complemented, Sylow
3#127⋊C3—42nomaximal
4#120⋊C3—42nomaximal
5C2×C42⋊C3—41nomaximal
6H(3,𝔽4)—31no—
7H(3,𝔽4)C2×C311yescharacteristic, complemented
8#555—32no—
9D42—32no—
10#535—32no—
11#535—32no—
12#532—31no—
13C22⋊A4—42no—
14C42⋊C3—41no—
15C42⋊C3—42no—
16C2×#41—31no—
17C2×#41—31no—
18C22×D4—32no—
19C22×D4—32no—
20C4⋊Q8—31no—
21#127—31no—
22#127—32no—
23#127A412yescomplemented
24#124—32no—
25#124—34no—
26#124—31no—
27#124—31no—
28#124—32no—
29#121—34no—
30#121—34no—
31#121—34no—
32#120—32no—
33#120—34no—
34#120—32no—
35#120—32no—
36#120A412yescomplemented
37C4×D4—34no—
38#117—32no—
39#117—32no—
40C2×C42A411yescharacteristic, complemented
41C2×A4—161no—
42C24—32no—
43C24C2×A412yes—
44#41—122no—
45#41—122no—
46C2×Q8—31no—
47C2×Q8—31no—
48C2×D4—34no—
49C2×D4—32no—
50C2×D4—31no—
51C2×D4—31no—
52C2×D4—34no—
53C2×D4—32no—
54C2×D4—32no—
55C2×D4—32no—
56C2×D4—124no—
57C2×D4—124no—
58C22×C4—32no—
59C22×C4—32no—
60C22×C4—31no—
61C4⋊C4—34no—
62C22⋊C4—34no—
63C22⋊C4—32no—
64C22⋊C4—34no—
65C22⋊C4—34no—
66C22⋊C4—32no—
67C42—31no—
68C42—32no—
69C42C2×A411yescharacteristic
70C42C2×A412yes—
71A4—161no—
72A4—324no—
73C23—32no—
74C23—32no—
75C23—122no—
76C23—122no—
77C23—124no—
78C23C22⋊A411yescharacteristic
79Q8—121no—
80Q8—121no—
81D4—68no—
82D4—122no—
83D4—124no—
84D4—122no—
85D4—122no—
86D4—122no—
87D4—121no—
88D4—121no—
89C2×C4—31no—
90C2×C4—32no—
91C2×C4—31no—
92C2×C4—32no—
93C2×C4—64no—
94C2×C4—64no—
95C2×C4—62no—
96C2×C3—641no—
97C22—61no—
98C22—62no—
99C22—62no—
100C22—64no—
101C22—64no—
102C22—84no—
103C22—244no—
104C22C2×C22⋊A411yescharacteristic
105C4—32no—
106C4—62no—
107C4—62no—
108C4—61no—
109C3—641noSylow
110C2—31no—
111C2—41no—
112C2—122no—
113C2—122no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterH(3,𝔽4)C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3549C3
LCSG ▷ H(3,𝔽4)
UCSC1

Complements

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

NH ≅ G/Ntype
1. #5464C2semidirect
2. #3549C3semidirect
7. H(3,𝔽4)C2×C3semidirect
23. #127A4semidirect
36. #120A4semidirect
40. C2×C42A4semidirect

Maximal subgroups

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

subgroupindex
#54642
#35493
#127⋊C34
#120⋊C34
C2×C42⋊C34