Finite Group Databasebeta

← #71949

Subgroups — #71949

Order 384.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C3×H(3,𝔽4)C211yesmaximal, characteristic
2#5060C211yesmaximal, characteristic, complemented
3#5055C212yesmaximal, complemented
4#4540C211yesmaximal, characteristic, complemented
5#4540C212yesmaximal, complemented
6#1659—31nomaximal, Sylow
7C3×#124—22no—
8C3×#124—21no—
9C3×#124C2212yes—
10C3×#124C2211yescharacteristic
11C3×#120—22no—
12C3×#120—21no—
13C2×#216—21no—
14C2×#214—21no—
15#869C2211yescharacteristic, complemented
16#835—22no—
17#832—22no—
18#803C2212yescomplemented
19#803C2211yescharacteristic, complemented
20H(3,𝔽4)D311yescharacteristic
21#497—31no—
22#489—32no—
23#329—31no—
24#329—32no—
25C24×C3—21no—
26C2×C3×Q8D412yes—
27C2×C3×Q8D411yescharacteristic
28C2×C3×D4—21no—
29C2×C3×D4—22no—
30C2×C3×D4D412yes—
31C2×C3×D4D411yescharacteristic
32C2×Dic6—21no—
33C3×C22⋊C4—22no—
34C3×C22⋊C4—21no—
35C3×C22⋊C4—22no—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—22no—
38C3×C42—21no—
39C3×C42C2311yescharacteristic
40#216—42no—
41#214—42no—
42C4⋊Dic3—22no—
43C2×C3⋊C8—21no—
44C2×C3⋊C8—22no—
45C2×Q16—61no—
46C2×SD16—61no—
47C4⋊Q8—31no—
48#124—22no—
49#124—21no—
50#124C2×D312yes—
51#124C2×D311yescharacteristic
52#120—22no—
53#120—21no—
54#103—62no—
55#102—62no—
56#97—32no—
57#97—31no—
58C23×C3—21no—
59C23×C3—22no—
60C23×C3—81no—
61C23×C3—82no—
62C3×Q8—41no—
63C3×Q8—42no—
64C3×D4—41no—
65C3×D4—42no—
66C3×D4—81no—
67C3×D4—82no—
68C2×C3×C4—21no—
69C2×C3×C4—22no—
70C2×C3×C4C2×D412yes—
71C2×C3×C4C2×D411yescharacteristic
72C2×Dic3—41no—
73Dic6—41no—
74Dic6—41no—
75C3⋊C8—22no—
76C3⋊C8—42no—
77C24—21no—
78C2×Q8—61no—
79C2×Q8#6712yes—
80C2×Q8#6711yescharacteristic
81C2×D4—21no—
82C2×D4—22no—
83C2×D4#6712yes—
84C2×D4#6711yescharacteristic
85Q16—122no—
86SD16—122no—
87C2×C8—61no—
88C2×C8—62no—
89C4⋊C4—62no—
90C22⋊C4—22no—
91C22⋊C4—21no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C42—21no—
96C42C22×D311yescharacteristic
97C22×C3—41no—
98C22×C3—42no—
99C22×C3—42no—
100C22×C3—42no—
101C22×C3—42no—
102C22×C3—84no—
103C22×C3#12011yescharacteristic
104C3×C4—21no—
105C3×C4—22no—
106C3×C4—41no—
107C3×C4—42no—
108Dic3—81no—
109C23—21no—
110C23—22no—
111C23—81no—
112C23—82no—
113Q8—41no—
114Q8—42no—
115Q8—121no—
116Q8—121no—
117D4—41no—
118D4—42no—
119D4—81no—
120D4—82no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—121no—
124C2×C4C2×#6712yes—
125C2×C4C2×#6711yescharacteristic
126C8—62no—
127C8—122no—
128C2×C3—81no—
129C2×C3—82no—
130C2×C3#45311yescharacteristic
131C2×C3#45512yes—
132C22—41no—
133C22—42no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—84no—
138C22#95311yescharacteristic
139C4—21no—
140C4—22no—
141C4—41no—
142C4—42no—
143C4—241no—
144C3#165911yescharacteristic, complemented, Sylow
145C2—81no—
146C2—82no—
147C2#524111yescharacteristic
148C2#524212yes—
subgroupG/H
centerC22#953
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
Fitting, prime radicalC3×H(3,𝔽4)C2
socleC22×C3#120
hypercenterH(3,𝔽4)D3
hypocenterC3#1659
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
2. #5060C2semidirect
3. #5055C2semidirect
4. #4540C2semidirect
5. #4540C2semidirect
15. #869C22semidirect
18. #803C22semidirect
19. #803C22semidirect
144. C3#1659semidirect

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽4)2
#50602
#50552
#45402
#16593