Finite Group Databasebeta

← #84017

Subgroups — #84017

Order 384.

350 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 129 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×#578C211yesmaximal, characteristic, complemented
2#5753C211yesmaximal, characteristic, complemented
3#5195C213yesmaximal, complemented
4#5194C213yesmaximal, complemented
5C2×#829C213yesmaximal, complemented
6#5179C213yesmaximal, complemented
7#5140C211yesmaximal, characteristic, complemented
8#3417—31nomaximal, Sylow
9C3×2−1+4—21no—
10#1010—21no—
11C3×#137—23no—
12C3×#135C2213yes—
13C2×C3×Q16C2213yes—
14C3×#131C2211yescharacteristic
15#909—23no—
16C2×Dic12C2213yes—
17#904C2213yes—
18#829C2216yes—
19#829C2213yes—
20#828C2213yes—
21#827C2213yes—
22#826C2211yescharacteristic
23#812C2213yes—
24C2×C3⋊C16C2213yes—
25#578—31no—
26#578D311yescharacteristic, complemented
27#510—33no—
28#510—33no—
29#508—33no—
30C2×Q32—33no—
31#504—31no—
32C3×#41—23no—
33C3×#41—43no—
34C3×#41D411yescharacteristic
35C2×C3×Q8—23no—
36C2×C3×Q8—41no—
37D3×Q8—41no—
38#237—43no—
39#235—23no—
40C2×Dic6—23no—
41C3×Q16—23no—
42C3×Q16C2313yes—
43C3×SD16—23no—
44C3×D8C2311yescharacteristic
45C3×M16D413yes—
46C2×C3×C8C2313yes—
47Dic12—23no—
48Dic12C2313yes—
49D24C2311yescharacteristic
50#204—23no—
51C3⋊C16C2311yescharacteristic
52C3⋊C16C2313yes—
532−1+4—21no—
542−1+4—61no—
55#137—23no—
56#137—63no—
57#135—33no—
58#135C2×D313yes—
59C2×Q16—33no—
60C2×Q16C2×D313yes—
61#131C2×D311yescharacteristic
62Q32—33no—
63Q32—36no—
64SD32—33no—
65SD32—33no—
66D16—31no—
67M32—33no—
68C2×C16—33no—
69C3×Q8—23no—
70C3×Q8—43no—
71C3×Q8D811yescharacteristic
72C3×D4—21no—
73C3×D4—43no—
74C3×D4D813yes—
75C2×C3×C4—43no—
76C2×C3×C4—43no—
77C2×C3×C4C2×D413yes—
78#67—43no—
79C2×Dic3—43no—
80D12—21no—
81C4×D3—43no—
82Dic6—23no—
83Dic6—43no—
84C3×C8C2×D413yes—
85C3×C8C2411yescharacteristic
86#41—23no—
87#41—43no—
88#41—63no—
89#41—123no—
90#41#6711yescharacteristic
91C2×Q8—23no—
92C2×Q8—41no—
93C2×Q8—63no—
94C2×Q8—121no—
95Q16—23no—
96Q16—33no—
97Q16—63no—
98Q16C22×D313yes—
99SD16—23no—
100SD16—63no—
101D8—31no—
102D8C22×D311yescharacteristic
103M16#6713yes—
104C2×C8C22×D313yes—
105C16—31no—
106C16—33no—
107C22×C3—41no—
108C22×C3C2×D813yes—
109C2×D3—41no—
110C3×C4—43no—
111C3×C4C2×D813yes—
112C3×C4C22×D411yescharacteristic
113Dic3—43no—
114Q8—23no—
115Q8—43no—
116Q8—63no—
117Q8—123no—
118Q8#21311yescharacteristic
119D4—21no—
120D4—43no—
121D4—61no—
122D4—123no—
123D4#21313yes—
124C2×C4—43no—
125C2×C4—43no—
126C2×C4—123no—
127C2×C4—123no—
128C2×C4C2×#6713yes—
129C8C2×#6713yes—
130C8C23×D311yescharacteristic
131C2×C3—23no—
132C2×C3—81no—
133C2×C3C22×D811yescharacteristic
134D3—81no—
135C22—41no—
136C22—121no—
137C22C2×#21313yes—
138C4—43no—
139C4—123no—
140C4C2×#21313yes—
141C4C22×#6711yescharacteristic
142C3#341711yescharacteristic, complemented, Sylow
143C2—23no—
144C2—81no—
145C2—241no—
146C2C22×#21311yescharacteristic
subgroupG/H
centerC2C22×#213
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC3×#578C2
socleC2×C3C22×D8
hypercenter#578D3
hypocenterC3#3417
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #41 ◁ #131 ◁ #578

Complements

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

NH ≅ G/Ntype
1. C3×#578C2semidirect
2. #5753C2semidirect
3. #5195C2semidirect
4. #5194C2semidirect
5. C2×#829C2semidirect
6. #5179C2semidirect
7. #5140C2semidirect
26. #578D3semidirect
142. C3#3417semidirect

Maximal subgroups

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

subgroupindex
C3×#5782
#57532
#51952
#51942
C2×#8292
#51792
#51402
#34173