Finite Group Databasebeta

← 2−1+4⋊H(3,𝔽3) #11722580

Subgroups — 2−1+4⋊H(3,𝔽3) #11722580

Order 864.

263 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 53 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#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×Q8⋊H(3,𝔽3)C212yesmaximal, complemented
3C32×2−1+4C311yesmaximal, characteristic, complemented
4C3×2−1+4⋊C3C313yesmaximal, complemented
5D4×H(3,𝔽3)—41nomaximal
6Q8⋊H(3,𝔽3)C2211yescharacteristic
7C32×#41—31no—
8C32×#41—32no—
9C32×#41C2×C311yescharacteristic, complemented
10C2×C32×Q8—31no—
11C2×C32×Q8C2×C312yescomplemented
12C3×#41⋊C3C2×C313yescomplemented
13C2×C3×SL(2,𝔽3)C2×C316yescomplemented
14C22×H(3,𝔽3)—42no—
15C4×H(3,𝔽3)—41no—
16C3×2−1+4—31no—
17C3×2−1+4C3211yescharacteristic
182−1+4⋊C3—33no—
19C32×Q8—32no—
20C32×Q8—31no—
21C32×Q8C22×C311yescharacteristic
22C32×D4—31no—
23C32×D4—32no—
24C32×D4—43no—
25C32×D4A411yescharacteristic
26C2×C32×C4—32no—
27C2×C32×C4—32no—
28C2×C32×C4—31no—
29C3×SL(2,𝔽3)C22×C313yes—
30C2×H(3,𝔽3)—41no—
31C2×H(3,𝔽3)—82no—
32C3×#41—33no—
33C3×#41—36no—
34C3×#41—31no—
35C3×#41—32no—
36C3×#41—31no—
37C3×#41C2×C3211yescharacteristic
38C2×C3×Q8—31no—
39C2×C3×Q8—33no—
40C2×C3×Q8—32no—
41C2×C3×Q8C2×C3212yes—
42#41⋊C3—33no—
43C2×SL(2,𝔽3)—36no—
44C22×C32—31no—
45C22×C32—46no—
46C22×C32C2×A412yes—
47C32×C4—32no—
48C32×C4—31no—
49C32×C4—43no—
50C32×C4C2×A411yescharacteristic
512−1+4H(3,𝔽3)11yescharacteristic, complemented, Sylow
52H(3,𝔽3)—41noSylow
53C3×Q8—32no—
54C3×Q8—36no—
55C3×Q8—33no—
56C3×Q8—31no—
57C3×Q8—31no—
58C3×Q8C22×C3211yescharacteristic
59C3×D4—31no—
60C3×D4—33no—
61C3×D4—36no—
62C3×D4—32no—
63C3×D4—31no—
64C3×D4—123no—
65C3×D4C3×A411yescharacteristic
66C2×C3×C4—36no—
67C2×C3×C4—32no—
68C2×C3×C4—36no—
69C2×C3×C4—33no—
70C2×C3×C4—32no—
71C2×C3×C4—31no—
72SL(2,𝔽3)—33no—
73C2×C32—22no—
74C2×C32—43no—
75C2×C32—61no—
76C2×C32—86no—
77C2×C32C22×A411yescharacteristic
78#41—31no—
79#41—32no—
80#41C2×H(3,𝔽3)11yescharacteristic, complemented
81C2×Q8—31no—
82C2×Q8C2×H(3,𝔽3)12yescomplemented
83C22×C3—33no—
84C22×C3—31no—
85C22×C3—32no—
86C22×C3—126no—
87C22×C3C2×C3×A412yes—
88C3×C4—33no—
89C3×C4—36no—
90C3×C4—31no—
91C3×C4—32no—
92C3×C4—31no—
93C3×C4—123no—
94C3×C4C2×C3×A411yescharacteristic
95C32—43no—
96C322−1+4⋊C311yescharacteristic, complemented
97Q8—32no—
98Q8—31no—
99Q8C22×H(3,𝔽3)11yescharacteristic
100D4—31no—
101D4—32no—
102D4C22⋊H(3,𝔽3)11yescharacteristic
103C2×C4—32no—
104C2×C4—32no—
105C2×C4—31no—
106C2×C3—22no—
107C2×C3—31no—
108C2×C3—62no—
109C2×C3—63no—
110C2×C3—61no—
111C2×C3—123no—
112C2×C3—246no—
113C2×C3C22×C3×A411yescharacteristic
114C22—31no—
115C22C2×C22⋊H(3,𝔽3)12yes—
116C4—31no—
117C4—32no—
118C4C2×C22⋊H(3,𝔽3)11yescharacteristic
119C3—31no—
120C3—123no—
121C3C3×2−1+4⋊C311yescharacteristic
122C2—22no—
123C2—61no—
124C2C22×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C22×C3×A4
commutator, cosocleC3×Q8C22×C32
FittingC32×2−1+4C3
hypercenterC32×D4A4
hypocenterQ8C22×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C2×C3 ◁ C32×D4

Complements

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

NH ≅ G/Ntype
1. #41⋊H(3,𝔽3)C2semidirect
2. C2×Q8⋊H(3,𝔽3)C2semidirect
3. C32×2−1+4C3semidirect
4. C3×2−1+4⋊C3C3semidirect
9. C32×#41C2×C3semidirect
11. C2×C32×Q8C2×C3semidirect
12. C3×#41⋊C3C2×C3semidirect
13. C2×C3×SL(2,𝔽3)C2×C3semidirect
51. 2−1+4H(3,𝔽3)semidirect
80. #41C2×H(3,𝔽3)semidirect
82. C2×Q8C2×H(3,𝔽3)semidirect
96. C322−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
#41⋊H(3,𝔽3)2
C2×Q8⋊H(3,𝔽3)2
C32×2−1+43
C3×2−1+4⋊C33
D4×H(3,𝔽3)4