Finite Group Databasebeta

← #68353

Subgroups — #68353

Order 384.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#5683C211yesmaximal, characteristic
2#5060C211yesmaximal, characteristic, complemented
3#4714C212yesmaximal, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5#4464C212yesmaximal, complemented
6#1659—31nomaximal, Sylow
7C3×#124C2211yescharacteristic
8#947—21no—
9C2×#216—21no—
10#937—21no—
11C2×#214—21no—
12#885—22no—
13#882—22no—
14#875C2212yes—
15#869C2211yescharacteristic, complemented
16C3×#97C2212yescomplemented
17#821—22no—
18#816—22no—
19#803C2211yescharacteristic, complemented
20H(3,𝔽4)—31no—
21#497—31no—
22#489—32no—
23#329—32no—
24#329D311yescharacteristic, complemented
25C23×D3—21no—
26C2×C3×Q8D411yescharacteristic
27C2×C3×D4D411yescharacteristic
28C2×#67—22no—
29C2×D12—21no—
30C2×D12D412yes—
31C2×Dic6—21no—
32C2×Dic6D412yes—
33C2×C3×C8—22no—
34C3×C22⋊C4—22no—
35C3×C42C2311yescharacteristic
36#217—21no—
37#216—42no—
38#214—42no—
39(C2×D3)⋊C4—22no—
40(C2×D3)⋊C4—22no—
41(C2×D3)⋊C4—22no—
42C4⋊Dic3—22no—
43C4×Dic3—21no—
44C2×C3⋊C8—21no—
45C2×Q16—61no—
46C2×SD16—61no—
47C4⋊Q8—31no—
48#124—32no—
49#124—62no—
50#124—61no—
51#124C2×D311yescharacteristic
52#120—62no—
53#120—61no—
54#103—62no—
55#102—62no—
56#97—31no—
57#97C2×D312yescomplemented
58C23×C3—21no—
59C22×D3—22no—
60C22×D3—81no—
61C22×D3—82no—
62C3×Q8—41no—
63C3×D4—41no—
64C2×C3×C4—21no—
65C2×C3×C4C2×D411yescharacteristic
66C2×C3×C4C2×D412yes—
67#67—82no—
68C2×Dic3—22no—
69C2×Dic3—41no—
70D12—42no—
71D12—81no—
72Dic6—42no—
73Dic6—41no—
74Dic6—41no—
75C3×C8—42no—
76C3⋊C8—22no—
77C24—61no—
78C2×Q8—32no—
79C2×Q8—61no—
80C2×Q8D1211yescharacteristic
81C2×D4—32no—
82C2×D4—61no—
83C2×D4—62no—
84C2×D4D1211yescharacteristic
85Q16—122no—
86SD16—122no—
87C2×C8—22no—
88C2×C8—61no—
89C4⋊C4—62no—
90C22⋊C4—22no—
91C22⋊C4—62no—
92C22⋊C4—61no—
93C22⋊C4—62no—
94C22⋊C4—62no—
95C42—61no—
96C42C22×D311yescharacteristic
97C22×C3—41no—
98C22×C3—42no—
99C22×C3#12011yescharacteristic
100C2×D3—42no—
101C2×D3—42no—
102C2×D3—42no—
103C2×D3—84no—
104C3×C4—21no—
105C3×C4—22no—
106C3×C4—41no—
107Dic3—42no—
108Dic3—81no—
109C23—21no—
110C23—62no—
111C23—241no—
112C23—242no—
113Q8—41no—
114Q8—122no—
115Q8—121no—
116Q8—121no—
117D4—41no—
118D4—122no—
119D4—241no—
120D4—242no—
121C2×C4—21no—
122C2×C4—62no—
123C2×C4—121no—
124C2×C4C2×D1211yescharacteristic
125C2×C4D3×D412yes—
126C8—42no—
127C8—62no—
128C2×C3—81no—
129C2×C3#45311yescharacteristic
130C2×C3#45512yes—
131D3—82no—
132C22—41no—
133C22—42no—
134C22—122no—
135C22—122no—
136C22—122no—
137C22—244no—
138C22#88211yescharacteristic
139C4—21no—
140C4—22no—
141C4—41no—
142C4—122no—
143C4—241no—
144C3#165911yescharacteristic, complemented, Sylow
145C2—81no—
146C2—242no—
147C2#474711yescharacteristic
148C2#482512yes—
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radical#5683C2
hypercenter#329D3
hypocenterC3#1659
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #5060C2semidirect
3. #4714C2semidirect
4. C3×#329C2semidirect
5. #4464C2semidirect
15. #869C22semidirect
16. C3×#97C22semidirect
19. #803C22semidirect
24. #329D3semidirect
57. #97C2×D3semidirect
144. C3#1659semidirect

Maximal subgroups

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

subgroupindex
#56832
#50602
#47142
C3×#3292
#44642
#16593