Finite Group Databasebeta

← #11714104

Subgroups — #11714104

Order 768.

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

normal subgroups only

This group has 521 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×#5403C214yesmaximal
2#85336C212yesmaximal
3C4×#131⋊C3C211yesmaximal, characteristic
4#33191—31nomaximal, Sylow
5C2×#984C2214yes—
6C2×#131⋊C3C2212yes—
7C4×#41⋊C3C2211yescharacteristic
8#5403—28no—
9C4⋊#226—24no—
10#4699—41nomaximal
11C2×#494—31no—
12C2×#473—34no—
13#3039—32no—
14C4×#131D311yescharacteristic
15#1979—34no—
16#1966—32no—
17#1848—31no—
18C2×#41⋊C3C2311yescharacteristic
19#984—28no—
20C2×#226—24no—
21C2×Dic12—44no—
22#869—42no—
23#131⋊C3D414yes—
24C4×SL(2,𝔽3)D412yes—
25C3×C4×C8—41no—
26C2×#137—34no—
27C22×Q16—34no—
28C2×#131C2×D312yes—
29C2×C4⋊Q8—32no—
30C4×#41C2×D311yescharacteristic
31#494—62no—
32#494—61no—
33#494—61no—
34#473—68no—
35#473—68no—
36#462—64no—
37#461—64no—
38C2×#108—34no—
39#423—32no—
40#419—34no—
41C2×#101—38no—
42#405—32no—
43C4×M16—31no—
44C2×C4×C8—31no—
45C2×Dic6—44no—
46#41⋊C3C2×D412yes—
47C2×SL(2,𝔽3)C2×D411yescharacteristic
48#226—44no—
49C2×C3×C8—42no—
50C3×C42—41no—
51C4⋊Dic3—84no—
52Dic12—88no—
53C2×#41C22×D311yescharacteristic
54C22×Q8—34no—
55#137—68no—
56#137—68no—
57C2×Q16—68no—
58C2×Q16—68no—
59C2×Q16—64no—
60C2×Q16—64no—
61C2×Q16—64no—
62C2×SD16—64no—
63#131—34no—
64#131D1214yes—
65C2×M16—32no—
66C2×M16—34no—
67C22×C8—32no—
68C4⋊Q8—64no—
69C4⋊Q8—62no—
70C4⋊Q8—62no—
71C4×Q8D1212yes—
72C4×D4—32no—
73#117—31no—
74C2×C4⋊C4—64no—
75C2×C42—31no—
76#108—68no—
77C4⋊C8—64no—
78#103—68no—
79#101—616no—
80#97—61no—
81C4×C8—32no—
82C4×C8—31no—
83C4×C8S411yescharacteristic
84C2×C3×C4—42no—
85C2×C3×C4—41no—
86C2×Dic3—84no—
87Dic6—88no—
88SL(2,𝔽3)#12711yescharacteristic
89C3×C8—44no—
90#41—32no—
91#41C2×D1212yes—
92C2×Q8—68no—
93C2×Q8—64no—
94C2×Q8—68no—
95C2×Q8—64no—
96C2×Q8C2×D1211yescharacteristic
97C2×D4—31no—
98C22×C4—31no—
99C22×C4—32no—
100C22×C4—64no—
101Q16—128no—
102Q16—124no—
103Q16—124no—
104Q16—128no—
105SD16—128no—
106M16—34no—
107M16—62no—
108M16—68no—
109C2×C8—32no—
110C2×C8—34no—
111C2×C8—34no—
112C2×C8—64no—
113C2×C8C2×S412yes—
114C4⋊C4—32no—
115C4⋊C4—124no—
116C4⋊C4—124no—
117C22⋊C4—61no—
118C42—32no—
119C42—31no—
120C42C2×S411yescharacteristic
121C22×C3—41no—
122C3×C4—42no—
123C3×C4—44no—
124Dic3—164no—
125C23—31no—
126Q8—31no—
127Q8—124no—
128Q8—124no—
129Q8—128no—
130Q8#87411yescharacteristic
131D4—32no—
132D4—61no—
133C2×C4—31no—
134C2×C4—32no—
135C2×C4—32no—
136C2×C4—32no—
137C2×C4—64no—
138C2×C4—128no—
139C2×C4—124no—
140C2×C4C4⋊S412yes—
141C2×C4C22×S411yescharacteristic
142C8—34no—
143C8—124no—
144C8C4⋊S414yes—
145C2×C3—41no—
146C2×C3—42no—
147C22—32no—
148C22—62no—
149C22C2×C4⋊S411yescharacteristic
150C4—31no—
151C4—31no—
152C4—62no—
153C4—244no—
154C4#540314yescomplemented
155C4C2×C4⋊S412yes—
156C3—41noSylow
157C2—62no—
158C2#8528911yescharacteristic
159C2C2×#540312yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, prime radical, 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
154. C4#5403semidirect

Maximal subgroups

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

subgroupindex
C2×#54032
#853362
C4×#131⋊C32
#331913
#46994