Finite Group Databasebeta

← #3529⋊C3 #85657

Subgroups — #3529⋊C3 #85657

Order 384.

238 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 22 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#5465C211yesmaximal, characteristic, complemented
2#3529C311yesmaximal, characteristic, complemented, Sylow
3C2×#41⋊C3—41nomaximal
4C2×C4×A4—42nomaximal
5C4×SL(2,𝔽3)—42nomaximal
6#563—31no—
7#562—32no—
8H(3,𝔽4)C2×C311yescharacteristic
9#560—32no—
10#532—31no—
11#529—32no—
12#525—32no—
13C22×A4—42no—
14#41⋊C3—42no—
15C2×SL(2,𝔽3)—42no—
16C2×SL(2,𝔽3)—41no—
17C4×A4—44no—
18C2×#41—31no—
19C2×#41A411yescharacteristic, complemented
20C23×C4A412yescomplemented
21C4⋊Q8—31no—
22#127—31no—
23#126—34no—
24#125—32no—
25#124—31no—
26#124—32no—
27#124—31no—
28#123—32no—
29#123—32no—
30#123—34no—
31C22⋊Q8—34no—
32C22⋊Q8—32no—
33#121—34no—
34#121—32no—
35#121—34no—
36#120—32no—
37#120—32no—
38C4×Q8A412yescomplemented
39C4×D4—32no—
40C4×D4—34no—
41#117—32no—
42C2×C42—31no—
43C2×A4—42no—
44C2×A4—44no—
45C2×C3×C4—161no—
46SL(2,𝔽3)—42no—
47SL(2,𝔽3)—41no—
48C24C2×A412yes—
49#41—42no—
50#41—122no—
51C2×Q8—31no—
52C2×Q8C2×A412yes—
53C2×Q8C2×A411yescharacteristic
54C2×D4—31no—
55C2×D4—32no—
56C2×D4—34no—
57C2×D4—31no—
58C2×D4—32no—
59C22×C4—32no—
60C22×C4—31no—
61C22×C4—32no—
62C22×C4—32no—
63C22×C4—44no—
64C4⋊C4—32no—
65C4⋊C4—34no—
66C4⋊C4—34no—
67C4⋊C4—32no—
68C22⋊C4—32no—
69C22⋊C4—32no—
70C22⋊C4—34no—
71C22⋊C4—34no—
72C22⋊C4—34no—
73C42—31no—
74C42—31no—
75C42—32no—
76C22×C3—161no—
77A4—42no—
78C3×C4—162no—
79C23—32no—
80C23—31no—
81C23—42no—
82C23—44no—
83Q8—41no—
84Q8—42no—
85Q8—121no—
86D4—121no—
87D4—124no—
88D4—122no—
89D4—122no—
90D4—121no—
91C2×C4—32no—
92C2×C4—32no—
93C2×C4—31no—
94C2×C4—32no—
95C2×C4—64no—
96C2×C4—62no—
97C2×C4—122no—
98C2×C4—122no—
99C2×C4C22⋊A411yescharacteristic
100C2×C3—161no—
101C2×C3—162no—
102C22—42no—
103C22—62no—
104C22—61no—
105C22—62no—
106C22—64no—
107C22—122no—
108C22C2×C22⋊A411yescharacteristic
109C4—62no—
110C4—61no—
111C4—62no—
112C4—62no—
113C4#99712yes—
114C3—161noSylow
115C2—122no—
116C2—121no—
117C2C2×#99711yescharacteristic
118C2#595012yes—
subgroupG/H
center, hypercenterC2×C4C22⋊A4
commutator, cosocle, hypocenterH(3,𝔽4)C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3529C3
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #5465C2semidirect
2. #3529C3semidirect
19. C2×#41A4semidirect
20. C23×C4A4semidirect
38. C4×Q8A4semidirect

Maximal subgroups

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

subgroupindex
#54652
#35293
C2×#41⋊C34
C2×C4×A44
C4×SL(2,𝔽3)4