Finite Group Databasebeta

← #69573

Subgroups — #69573

Order 384.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#5500C211yesmaximal, characteristic
2C3×#384C211yesmaximal, characteristic
3#4964C212yesmaximal
4#4530C211yesmaximal, characteristic
5#4523C212yesmaximal
6#2060—31nomaximal, Sylow
7C2×C3×C4⋊C4—22no—
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C3×C42C2211yescharacteristic
11#926C2212yes—
12#924—22no—
13C2×#208C2212yes—
14#890—22no—
15#888—22no—
16#808—22no—
17#807—22no—
18#544—31no—
19#428—32no—
20#384D311yescharacteristic
21#343—31no—
22#339—32no—
23C22×C3×C4—21no—
24C22×C3×C4—21no—
25C22×C3×C4D412yes—
26C22×C3×C4C2311yescharacteristic
27C2×Dic6—22no—
28C3×C4⋊C4—22no—
29C3×C4⋊C4—22no—
30C3×C4⋊C4—44no—
31C3×C42—21no—
32C3×C42Q812yescomplemented
33#217—22no—
34C4⋊Dic3—22no—
35C4⋊Dic3D412yes—
36C4⋊Dic3D412yes—
37Dic3⋊C4—24no—
38C4×Dic3—22no—
39#208—22no—
40#208Q814yescomplemented
41C2×C3⋊C8—22no—
42C2×M16—32no—
43C4⋊Q8—62no—
44#125—62no—
45C22⋊Q8—62no—
46#117—32no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4C2×D311yescharacteristic
49C2×C4⋊C4C2×D311yescharacteristic
50C2×C42C2×D311yescharacteristic
51#107—62no—
52#106—62no—
53C23×C3C2×D411yescharacteristic
54C2×C3×C4—22no—
55C2×C3×C4—22no—
56C2×C3×C4—22no—
57C2×C3×C4—22no—
58C2×C3×C4—42no—
59C2×C3×C4—42no—
60C2×C3×C4—41no—
61C2×C3×C4C2×D412yes—
62C2×C3×C4C2×Q811yescharacteristic
63C2×C3×C4C2×Q812yes—
64C2×C3×C4#4111yescharacteristic
65C2×Dic3—22no—
66C2×Dic3—22no—
67Dic6—82no—
68C3⋊C8—24no—
69C2×Q8—62no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4#6712yes—
73C22×C4C22×D311yescharacteristic
74M16—34no—
75M16—62no—
76C2×C8—62no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—32no—
80C4⋊C4—32no—
81C4⋊C4—44no—
82C4⋊C4—62no—
83C4⋊C4—64no—
84C22⋊C4—62no—
85C42—21no—
86C42—62no—
87C42Dic612yescomplemented
88C22×C3—21no—
89C22×C3—21no—
90C22×C3#12011yescharacteristic
91C22×C3C22⋊Q811yescharacteristic
92C22×C3C4⋊Q811yescharacteristic
93C3×C4—24no—
94C3×C4—42no—
95C3×C4—81no—
96C3×C4C22⋊Q812yes—
97C3×C4C4⋊Q812yes—
98Dic3—42no—
99Dic3—42no—
100C23C2×#6711yescharacteristic
101Q8—242no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—62no—
110C2×C4—62no—
111C2×C4C2×Dic611yescharacteristic
112C2×C4#23511yescharacteristic
113C2×C4D3×D412yes—
114C2×C4D3×Q812yes—
115C8—64no—
116C2×C3—21no—
117C2×C3—21no—
118C2×C3#39511yescharacteristic
119C2×C3#45311yescharacteristic
120C2×C3#45611yescharacteristic
121C22—21no—
122C22—21no—
123C22#92411yescharacteristic
124C22#93711yescharacteristic
125C22#94411yescharacteristic
126C4—24no—
127C4—42no—
128C4—81no—
129C4—122no—
130C4—122no—
131C4#88812yes—
132C4#89612yes—
133C3#206011yescharacteristic, complemented, Sylow
134C2—21no—
135C2—21no—
136C2#497911yescharacteristic
137C2#507711yescharacteristic
138C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#384C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#384D3
hypocenterC3#2060
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
32. C3×C42Q8semidirect
40. #208Q8semidirect
87. C42Dic6semidirect
133. C3#2060semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#3842
#49642
#45302
#45232
#20603