Finite Group Databasebeta

← #73073

Subgroups — #73073

Order 384.

161 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 24 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#5923C211yesmaximal, characteristic
2C4×#41⋊C3C211yesmaximal, characteristic, complemented
3#5423C211yesmaximal, characteristic, complemented
4#1906—31nomaximal, Sylow
5C2×#41⋊C3C2211yescharacteristic
6#986C411yescharacteristic, complemented
7#985—21no—
8#984C411yescharacteristic, complemented
9#983—21no—
10C4×SL(2,𝔽3)—21no—
11GU(2,𝔽3)—21no—
12#805—41nomaximal
13#575—31no—
14C4×#41D311yescharacteristic, complemented
15#421—31no—
16C2×#104—31no—
17#410—31no—
18#344—31no—
19#339—31no—
20#235—41no—
21#41⋊C3C2×C411yescharacteristic
22#41⋊C3D411yescharacteristic
23C2×SL(2,𝔽3)D411yescharacteristic
24GL(2,𝔽3)—21no—
25#226—21no—
26C3×C42—41no—
27#208—41no—
28C2×#41—31no—
29C2×#41C2×D311yescharacteristic
30#137—31no—
31#137—31no—
32#137—61no—
33AGL(1,ℤ/8)—31no—
34AGL(1,ℤ/8)—31no—
35AGL(1,ℤ/8)—61no—
36#135—61no—
37#135—61no—
38C2×M16—31no—
39C2×M16—31no—
40C4×Q8—21no—
41C4×D4—61no—
42#117—31no—
43#117—31no—
44C2×C42—31no—
45#104—61no—
46#104—61no—
47#104—61no—
48#104—61no—
49#99—62no—
50C2×C3×C4—41no—
51C2×C3×C4—81no—
52#67—81no—
53D12—41no—
54C4×D3—81no—
55Dic6—41no—
56SL(2,𝔽3)C22⋊C411yescharacteristic
57C3⋊C8—81no—
58#41—31no—
59#41—31no—
60#41—61no—
61#41—61no—
62#41—61no—
63#41—61no—
64#41C4×D311yescharacteristic
65#41D1211yescharacteristic
66C2×Q8—31no—
67C2×Q8#6711yescharacteristic
68C2×D4—31no—
69C2×D4—31no—
70C2×D4—61no—
71C22×C4—31no—
72C22×C4—61no—
73C22×C4—61no—
74Q16—61no—
75Q16—61no—
76SD16—61no—
77SD16—61no—
78SD16—61no—
79SD16—61no—
80D8—61no—
81D8—61no—
82M16—31no—
83M16—31no—
84M16—61no—
85M16—61no—
86M16—61no—
87C2×C8—61no—
88C2×C8—61no—
89C4⋊C4—61no—
90C4⋊C4—61no—
91C22⋊C4—32no—
92C22⋊C4—61no—
93C42—31no—
94C42—61no—
95C42—61no—
96C42S411yescharacteristic
97C22×C3—41no—
98C2×D3—81no—
99C3×C4—41no—
100C3×C4—41no—
101C3×C4—82no—
102Dic3—81no—
103C23—31no—
104C23—61no—
105Q8—31no—
106Q8—61no—
107Q8—61no—
108Q8(C2×D3)⋊C411yescharacteristic
109D4—31no—
110D4—31no—
111D4—61no—
112D4—61no—
113D4—61no—
114D4—121no—
115D4—121no—
116C2×C4—21no—
117C2×C4—31no—
118C2×C4—31no—
119C2×C4—31no—
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—122no—
125C2×C4—121no—
126C2×C4—121no—
127C2×C4—121no—
128C2×C4C2×S411yescharacteristic
129C8—61no—
130C8—61no—
131C8—121no—
132C2×C3—41no—
133C2×C3—81no—
134D3—161no—
135C22—31no—
136C22—31no—
137C22—121no—
138C22—121no—
139C22—241no—
140C22GL(2,ℤ/4)11yescharacteristic
141C4—22no—
142C4—31no—
143C4—31no—
144C4—121no—
145C4—121no—
146C4—241no—
147C4C4×S411yescharacteristic
148C4C4⋊S411yescharacteristic
149C3—41noSylow
150C2—21no—
151C2—61no—
152C2—61no—
153C2—241no—
154C2(C2×S4)⋊C411yescharacteristic
subgroupG/H
centerC4C4⋊S4
commutator#41⋊C3C2×C4
FrattiniC2×C4C2×S4
FittingC4×#41D3
socleC2(C2×S4)⋊C4
prime radical#5923C2
cosocleC2×#41⋊C3C22
hypercenterC42S4
hypocenterSL(2,𝔽3)C22⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#41⋊C3C2semidirect
3. #5423C2semidirect
6. #986C4semidirect
8. #984C4semidirect
14. C4×#41D3semidirect

Maximal subgroups

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

subgroupindex
#59232
C4×#41⋊C32
#54232
#19063
#8054