Finite Group Databasebeta

← #11722388

Subgroups — #11722388

Order 864.

199 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C2×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#88611C211yesmaximal, characteristic, complemented
3#88610C211yesmaximal, characteristic, complemented
4C3×#983—33nomaximal
5C3×#942—31nomaximal
6#6235—41nomaximal
7Q8⋊H(3,𝔽3)C2211yescharacteristic
8C2×C32×Q8D311yescharacteristic, complemented
9C3×#235—31no—
10C2×C3×SL(2,𝔽3)D313yescomplemented
11C3×GL(2,𝔽3)—33no—
12C3×#226—33no—
13C3×#216—31no—
14C3×#216—31no—
15C3×#215—31no—
16C3×#215—31no—
17C3×#208—31no—
18C22×H(3,𝔽3)—41no—
19C2×#272—41no—
20#1084—41no—
21#983—33no—
22C3×#137—91no—
23#942—31no—
24C32×Q8—31no—
25C32×Q8C2×D311yescharacteristic
26C2×C32×C4—31no—
27C3×#67—61no—
28C3×#67—123no—
29C3×D12—31no—
30C3×C4×D3—61no—
31C3×Dic6—31no—
32C3×SL(2,𝔽3)C2×D313yes—
33C3×C3⋊C8—31no—
34C3×C3⋊C8—31no—
35C2×H(3,𝔽3)—41no—
36C2×H(3,𝔽3)—81no—
37#272—81no—
38C3×#41—91no—
39C2×C3×Q8—31no—
40C2×C3×Q8C3⋊D311yescharacteristic
41#235—31no—
42C2×SL(2,𝔽3)—33no—
43GL(2,𝔽3)—33no—
44#226—33no—
45C3×Q16—91no—
46C3×Q16—91no—
47C3×SD16—91no—
48C3×SD16—91no—
49C3×M16—91no—
50#216—31no—
51#216—31no—
52#215—31no—
53#215—31no—
54#208—31no—
55C22×C32—43no—
56C22×C32S411yescharacteristic
57C2×C3×D3—61no—
58C2×C3×D3—123no—
59C32×C4—31no—
60C32×C4—31no—
61C3×Dic3—61no—
62C3×Dic3—123no—
63#137—91noSylow
64H(3,𝔽3)—41noSylow
65C3×Q8—31no—
66C3×Q8—31no—
67C3×Q8—31no—
68C3×Q8—61no—
69C3×Q8—91no—
70C3×Q8C2×C3⋊D311yescharacteristic
71C3×D4—91no—
72C3×D4—181no—
73C2×C3×C4—31no—
74C2×C3×C4—31no—
75C2×C3×C4—61no—
76C2×C3×C4—181no—
77#67—61no—
78#67—123no—
79D12—31no—
80C4×D3—61no—
81Dic6—31no—
82SL(2,𝔽3)—33no—
83C3×C8—91no—
84C3×C8—91no—
85C3⋊C8—31no—
86C3⋊C8—31no—
87C2×C32—21no—
88C2×C32—43no—
89C2×C32—83no—
90C2×C32C2×S411yescharacteristic
91C3×D3—121no—
92C3×D3—243no—
93#41—91no—
94C2×Q8#27211yescharacteristic, complemented
95Q16—91no—
96Q16—91no—
97SD16—91no—
98SD16—91no—
99M16—91no—
100C22×C3—31no—
101C22×C3—123no—
102C22×C3—181no—
103C22×C3C3⋊S411yescharacteristic
104C2×D3—61no—
105C2×D3—123no—
106C3×C4—31no—
107C3×C4—31no—
108C3×C4—31no—
109C3×C4—31no—
110C3×C4—61no—
111C3×C4—61no—
112C3×C4—181no—
113Dic3—61no—
114Dic3—123no—
115C32—43no—
116C32#98311yescharacteristic, complemented
117Q8—31no—
118Q8—91no—
119Q8C2×#27211yescharacteristic
120D4—91no—
121D4—181no—
122C2×C4—31no—
123C2×C4—181no—
124C8—91no—
125C8—91no—
126C2×C3—21no—
127C2×C3—31no—
128C2×C3—61no—
129C2×C3—123no—
130C2×C3—243no—
131C2×C3—361no—
132C2×C3C2×C3⋊S411yescharacteristic
133D3—121no—
134D3—243no—
135C22—181no—
136C22#625811yescharacteristic
137C4—31no—
138C4—31no—
139C4—181no—
140C3—31no—
141C3—123no—
142C3#6426411yescharacteristic
143C2—21no—
144C2—361no—
145C2C2×#625811yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C2×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C22
FittingC2×C32×Q8D3
hypercenterC22×C3C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×Q8⋊H(3,𝔽3)C2semidirect
2. #88611C2semidirect
3. #88610C2semidirect
8. C2×C32×Q8D3semidirect
10. C2×C3×SL(2,𝔽3)D3semidirect
94. C2×Q8#272semidirect
116. C32#983semidirect

Maximal subgroups

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

subgroupindex
C2×Q8⋊H(3,𝔽3)2
#886112
#886102
C3×#9833
C3×#9423
#62354