Finite Group Databasebeta

← #11714088

Subgroups — #11714088

Order 768.

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

normal subgroups only

This group has 400 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
1#73167C211yesmaximal, characteristic, complemented
2#73065C213yesmaximal
3#73059C213yesmaximal, complemented
4#33089—31nomaximal, Sylow
5C22×GL(2,𝔽3)C2211yescharacteristic
6C2×C4×SL(2,𝔽3)C2213yes—
7C2×#859C2213yes—
8#4676—41nomaximal
9#2644—31no—
10#2276D311yescharacteristic, complemented
11#1926—33no—
12#1920—33no—
13#1880—33no—
14#1865—33no—
15#1840—31no—
16C22×SL(2,𝔽3)C2311yescharacteristic
17C2×GL(2,𝔽3)—23no—
18C2×(C2×D3)⋊C4—43no—
19C2×Dic3⋊C4—43no—
20C4×SL(2,𝔽3)—23no—
21#859—23no—
22C3×#95—41no—
23C22×SD16—31no—
24C2×C4×Q8C2×D313yes—
25C2×#106—33no—
26C2×#103—33no—
27C2×#102—33no—
28#397—63no—
29#396—61no—
30#390—33no—
31#389—33no—
32C4×C4⋊C4—33no—
33#376—31no—
34#340—33no—
35#340—33no—
36#336—33no—
37#336—33no—
38C23×D3—41no—
39C22×C3×C4—43no—
40C22×Dic3—43no—
41C2×SL(2,𝔽3)#4113yes—
42GL(2,𝔽3)—41no—
43(C2×D3)⋊C4—86no—
44Dic3⋊C4—86no—
45C22×Q8C22×D311yescharacteristic
46C22×D4—31no—
47C2×SD16—63no—
48C2×SD16—63no—
49C22×C8—31no—
50C22×C8—33no—
51C4×Q8—23no—
52C4×Q8—63no—
53C2×C4⋊C4—33no—
54C2×C4⋊C4—33no—
55C2×C4⋊C4—63no—
56C2×C22⋊C4—63no—
57C2×C22⋊C4—63no—
58C2×C42—33no—
59C2×C42—61no—
60#106—66no—
61#103—63no—
62#103—63no—
63#102—66no—
64#95—33no—
65#95—63no—
66#95S411yescharacteristic
67C23×C3—41no—
68C22×D3—42no—
69C22×D3—83no—
70C22×D3—83no—
71C2×C3×C4—83no—
72C2×C3×C4—83no—
73C2×C3×C4—83no—
74C2×Dic3—46no—
75C2×Dic3—83no—
76C2×Dic3—83no—
77SL(2,𝔽3)#12611yescharacteristic
78C24—61no—
79C2×Q8—33no—
80C2×Q8#23513yes—
81C2×D4—63no—
82C2×D4—63no—
83C22×C4—31no—
84C22×C4—33no—
85C22×C4—63no—
86C22×C4C2×S413yes—
87SD16—123no—
88SD16—121no—
89C2×C8—63no—
90C2×C8—63no—
91C2×C8—63no—
92C2×C8—63no—
93C4⋊C4—36no—
94C4⋊C4—63no—
95C4⋊C4—63no—
96C4⋊C4—63no—
97C4⋊C4—243no—
98C22⋊C4—243no—
99C22⋊C4—243no—
100C42—63no—
101C42—63no—
102C42—241no—
103C22×C3—43no—
104C22×C3—41no—
105C22×C3—43no—
106C2×D3—86no—
107C2×D3—163no—
108C2×D3—161no—
109C3×C4—163no—
110Dic3—86no—
111C23—123no—
112C23—121no—
113C23—123no—
114C23C22×S411yescharacteristic
115Q8—33no—
116Q8—61no—
117Q8#87611yescharacteristic
118D4—121no—
119D4—123no—
120C2×C4—23no—
121C2×C4—23no—
122C2×C4—23no—
123C2×C4—33no—
124C2×C4—33no—
125C2×C4—63no—
126C2×C4—63no—
127C2×C4—63no—
128C2×C4—123no—
129C2×C4—123no—
130C2×C4—123no—
131C8—123no—
132C8—121no—
133C2×C3—41no—
134C2×C3—43no—
135C2×C3—43no—
136D3—162no—
137C22—241no—
138C22—243no—
139C22—243no—
140C22#591213yes—
141C22#592411yescharacteristic
142C22#592613yes—
143C4—31no—
144C4—33no—
145C4—43no—
146C4—123no—
147C4—243no—
148C3—41noSylow
149C2—481no—
150C2#8529111yescharacteristic
151C2#8535813yes—
152C2#8536913yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2276D3
prime radicalC22×GL(2,𝔽3)C22
hypercenter#95S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. #73167C2semidirect
3. #73059C2semidirect
10. #2276D3semidirect

Maximal subgroups

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

subgroupindex
#731672
#730652
#730592
#330893
#46764