Finite Group Databasebeta

← #69672

Subgroups — #69672

Order 384.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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#5812C211yesmaximal, characteristic
2C3×#391C211yesmaximal, characteristic
3C2×#836C211yesmaximal, characteristic
4#4964C212yesmaximal
5#4523C212yesmaximal
6#1857—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C42C2211yescharacteristic
10Q8×Dic3—22no—
11Q8×Dic3—22no—
12#926—22no—
13#926C2212yes—
14C2×#208C2212yes—
15C3×#95—22no—
16#836C414yescomplemented
17#808—22no—
18#807—22no—
19#519—31no—
20#428—32no—
21C2×#101—31no—
22#391D311yescharacteristic
23#339—32no—
24C2×C3×Q8—22no—
25C2×C3×Q8—22no—
26C2×C3×Q8C2×C412yes—
27C2×C3×Q8Q812yes—
28C22×C3×C4—21no—
29C22×C3×C4—21no—
30C22×C3×C4D412yes—
31C22×C3×C4C2311yescharacteristic
32C3×C4⋊C4—22no—
33C3×C42—22no—
34#217—22no—
35C4⋊Dic3—22no—
36C4⋊Dic3—22no—
37C4⋊Dic3D412yes—
38C4⋊Dic3D412yes—
39C4×Dic3—24no—
40C4×Dic3—22no—
41#208—22no—
42#208C2×C414yes—
43C2×C3⋊C8—22no—
44C22×Q8C2×D311yescharacteristic
45C2×M16—32no—
46C4×Q8—62no—
47C4×Q8—62no—
48#117—32no—
49#117—62no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C42C2×D311yescharacteristic
52#107—62no—
53#106—62no—
54#101—34no—
55#95—22no—
56C23×C3C2×D411yescharacteristic
57C3×Q8—24no—
58C3×Q8—24no—
59C2×C3×C4—22no—
60C2×C3×C4—22no—
61C2×C3×C4—22no—
62C2×C3×C4—41no—
63C2×C3×C4—41no—
64C2×C3×C4—42no—
65C2×C3×C4—41no—
66C2×C3×C4C4⋊C414yes—
67C2×C3×C4C22×C411yescharacteristic
68C2×C3×C4C2×D412yes—
69C2×C3×C4C2×Q811yescharacteristic
70C2×C3×C4#4112yes—
71C2×Dic3—22no—
72C2×Dic3—22no—
73C3⋊C8—24no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8Dic612yes—
77C2×Q8C4×D312yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4#6712yes—
81C22×C4C22×D311yescharacteristic
82M16—34no—
83M16—62no—
84C2×C8—62no—
85C4⋊C4—22no—
86C4⋊C4—32no—
87C4⋊C4—32no—
88C4⋊C4—62no—
89C4⋊C4—62no—
90C22⋊C4—62no—
91C42—22no—
92C42—64no—
93C42—62no—
94C22×C3—21no—
95C22×C3—21no—
96C22×C3C2×C4⋊C411yescharacteristic
97C22×C3#12011yescharacteristic
98C22×C3#12311yescharacteristic
99C3×C4—24no—
100C3×C4—42no—
101C3×C4—81no—
102C3×C4C4×D412yes—
103C3×C4C22⋊Q812yes—
104Dic3—42no—
105Dic3—42no—
106C23C2×#6711yescharacteristic
107Q8—24no—
108Q8—24no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—62no—
117C2×C4—62no—
118C2×C4Dic3⋊C414yes—
119C2×C4C2×Dic611yescharacteristic
120C2×C4C2×C4×D311yescharacteristic
121C2×C4D3×D412yes—
122C2×C4#23712yes—
123C8—64no—
124C2×C3—21no—
125C2×C3—21no—
126C2×C3#38511yescharacteristic
127C2×C3#45411yescharacteristic
128C2×C3#45611yescharacteristic
129C22—21no—
130C22—21no—
131C22C2×Dic3⋊C411yescharacteristic
132C22#93511yescharacteristic
133C22#93711yescharacteristic
134C4—24no—
135C4—42no—
136C4—81no—
137C4—122no—
138C4—122no—
139C4#87812yes—
140C4#67⋊C412yes—
141C3#185711yescharacteristic, complemented, Sylow
142C2—21no—
143C2—21no—
144C2#494611yescharacteristic
145C2#506211yescharacteristic
146C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#391C2
socleC22×C3#120
prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenter#391D3
hypocenterC3#1857
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #836C4semidirect
141. C3#1857semidirect

Maximal subgroups

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

subgroupindex
#58122
C3×#3912
C2×#8362
#49642
#45232
#18573