Finite Group Databasebeta

← #11714103

Subgroups — #11714103

Order 768.

593 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#5407C214yesmaximal, complemented
2#85346C212yesmaximal, complemented
3C4×#131⋊C3C211yesmaximal, characteristic, complemented
4#33190—31nomaximal, Sylow
5C2×#986C2214yes—
6C2×#131⋊C3C2212yes—
7C4×#41⋊C3C2211yescharacteristic
8#5407—28no—
9C4⋊GL(2,𝔽3)—24no—
10#4695—41nomaximal
11C2×#493—31no—
12C2×#472—34no—
13#3038—32no—
14C4×#131D311yescharacteristic, complemented
15#1978—34no—
16#1965—32no—
17#1848—31no—
18C2×#41⋊C3C2311yescharacteristic
19#986—28no—
20C2×GL(2,𝔽3)—24no—
21C2×D24—44no—
22#874—42no—
23#131⋊C3D414yescomplemented
24C4×SL(2,𝔽3)D412yes—
25C3×C4×C8—41no—
26C2×AGL(1,ℤ/8)—34no—
27C22×D8—34no—
28C2×#131C2×D312yes—
29C2×#127—32no—
30C4×#41C2×D311yescharacteristic
31#493—62no—
32#493—61no—
33#493—61no—
34#472—68no—
35#472—68no—
36#460—64no—
37#459—64no—
38C2×#108—34no—
39#423—32no—
40#418—34no—
41C2×#100—38no—
42#405—32no—
43C4×M16—31no—
44C2×C4×C8—31no—
45C2×D12—44no—
46C2×D12—84no—
47#41⋊C3C2×D412yes—
48C2×SL(2,𝔽3)C2×D411yescharacteristic
49GL(2,𝔽3)—44no—
50C2×C3×C8—42no—
51C3×C42—41no—
52D24—88no—
53C2×#41C22×D311yescharacteristic
54C22×D4—34no—
55C22×D4—64no—
56AGL(1,ℤ/8)—68no—
57AGL(1,ℤ/8)—68no—
58C2×SD16—64no—
59C2×D8—68no—
60C2×D8—68no—
61C2×D8—64no—
62C2×D8—64no—
63C2×D8—64no—
64#131—34no—
65#131D1214yescomplemented
66C2×M16—32no—
67C2×M16—34no—
68C22×C8—32no—
69#127—64no—
70#127—62no—
71#127—62no—
72C4×Q8D1212yes—
73C4×D4—32no—
74#117—31no—
75C2×C42—31no—
76#108—68no—
77C4⋊C8—64no—
78#102—68no—
79#100—616no—
80#97—61no—
81C4×C8—32no—
82C4×C8—31no—
83C4×C8S411yescharacteristic
84C22×D3—84no—
85C2×C3×C4—42no—
86C2×C3×C4—41no—
87D12—88no—
88D12—168no—
89SL(2,𝔽3)#12711yescharacteristic
90C3×C8—44no—
91C24—64no—
92#41—32no—
93#41C2×D1212yes—
94C2×Q8C2×D1211yescharacteristic
95C2×D4—31no—
96C2×D4—68no—
97C2×D4—64no—
98C2×D4—68no—
99C2×D4—64no—
100C2×D4—124no—
101C2×D4—124no—
102C2×D4—248no—
103C22×C4—31no—
104C22×C4—32no—
105SD16—124no—
106SD16—124no—
107D8—128no—
108D8—128no—
109D8—128no—
110M16—34no—
111M16—62no—
112M16—68no—
113C2×C8—32no—
114C2×C8—34no—
115C2×C8—34no—
116C2×C8—64no—
117C2×C8C2×S412yes—
118C4⋊C4—32no—
119C22⋊C4—61no—
120C42—32no—
121C42—31no—
122C42C2×S411yescharacteristic
123C22×C3—41no—
124C2×D3—164no—
125C2×D3—168no—
126C3×C4—42no—
127C3×C4—44no—
128C23—31no—
129C23—128no—
130C23—124no—
131C23—248no—
132Q8—31no—
133Q8#87411yescharacteristic
134D4—32no—
135D4—61no—
136D4—124no—
137D4—124no—
138D4—128no—
139D4—248no—
140D4—248no—
141C2×C4—31no—
142C2×C4—32no—
143C2×C4—32no—
144C2×C4—32no—
145C2×C4—64no—
146C2×C4C4⋊S412yes—
147C2×C4C22×S411yescharacteristic
148C8—34no—
149C8—124no—
150C8C4⋊S414yes—
151C2×C3—41no—
152C2×C3—42no—
153D3—324no—
154C22—32no—
155C22—62no—
156C22—244no—
157C22—248no—
158C22—488no—
159C22C2×C4⋊S411yescharacteristic
160C4—31no—
161C4—31no—
162C4—62no—
163C4#540714yescomplemented
164C4C2×C4⋊S412yes—
165C3—41noSylow
166C2—62no—
167C2—484no—
168C2#8528911yescharacteristic
169C2C2×#540712yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×#131D3
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#5407C2semidirect
2. #85346C2semidirect
3. C4×#131⋊C3C2semidirect
14. C4×#131D3semidirect
23. #131⋊C3D4semidirect
65. #131D12semidirect
163. C4#5407semidirect

Maximal subgroups

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

subgroupindex
C2×#54072
#853462
C4×#131⋊C32
#331903
#46954