Finite Group Databasebeta

← #11714090

Subgroups — #11714090

Order 768.

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

normal subgroups only

This group has 349 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
2#73062C213yesmaximal
3#73061C213yesmaximal
4#33093—31nomaximal, Sylow
5C22×#226C2211yescharacteristic
6C2×C4×SL(2,𝔽3)C2213yes—
7C2×#859C2213yes—
8C3⋊#401—41nomaximal
9#2645—31no—
10#2276D311yescharacteristic
11#1928—33no—
12#1918—33no—
13#1878—33no—
14#1863—33no—
15#1840—31no—
16C22×SL(2,𝔽3)C2311yescharacteristic
17C2×#226—23no—
18C4×SL(2,𝔽3)—23no—
19#859—23no—
20C3×#95—41no—
21C3⋊#95—43no—
22C3⋊#95—43no—
23C22×Q16—31no—
24C2×C4×Q8C2×D313yes—
25C2×#107—33no—
26C2×#103—33no—
27C2×#103—33no—
28#401—61no—
29#400—63no—
30#391—33no—
31#389—33no—
32C4×C4⋊C4—33no—
33#376—31no—
34#340—33no—
35#340—33no—
36#336—33no—
37#336—33no—
38C22×C3×C4—43no—
39C22×Dic3—41no—
40C22×Dic3—43no—
41C2×SL(2,𝔽3)#4113yes—
42#226—41no—
43C22×Q8—31no—
44C22×Q8C22×D311yescharacteristic
45C2×Q16—63no—
46C2×Q16—63no—
47C22×C8—31no—
48C22×C8—33no—
49C4×Q8—23no—
50C4×Q8—63no—
51C2×C4⋊C4—33no—
52C2×C4⋊C4—33no—
53C2×C4⋊C4—63no—
54C2×C42—33no—
55C2×C42—61no—
56#107—66no—
57#103—66no—
58#103—63no—
59#103—63no—
60#95—33no—
61#95—63no—
62#95—63no—
63#95—63no—
64#95S411yescharacteristic
65C23×C3—41no—
66C2×C3×C4—83no—
67C2×C3×C4—86no—
68C2×Dic3—83no—
69C2×Dic3—83no—
70C2×Dic3—83no—
71C2×Dic3—83no—
72SL(2,𝔽3)#12611yescharacteristic
73C2×Q8—33no—
74C2×Q8—63no—
75C2×Q8—63no—
76C2×Q8#23513yes—
77C22×C4—31no—
78C22×C4—33no—
79C22×C4—61no—
80C22×C4—63no—
81C22×C4C2×S413yes—
82Q16—123no—
83Q16—121no—
84C2×C8—63no—
85C2×C8—63no—
86C2×C8—63no—
87C2×C8—63no—
88C4⋊C4—36no—
89C4⋊C4—63no—
90C4⋊C4—63no—
91C4⋊C4—63no—
92C4⋊C4—243no—
93C42—63no—
94C42—63no—
95C42—241no—
96C22×C3—43no—
97C22×C3—41no—
98C22×C3—43no—
99C3×C4—163no—
100Dic3—163no—
101Dic3—161no—
102C23C22×S411yescharacteristic
103Q8—33no—
104Q8—61no—
105Q8—121no—
106Q8—123no—
107Q8#87611yescharacteristic
108C2×C4—23no—
109C2×C4—26no—
110C2×C4—33no—
111C2×C4—33no—
112C2×C4—63no—
113C2×C4—63no—
114C2×C4—63no—
115C2×C4—123no—
116C2×C4—123no—
117C2×C4—123no—
118C2×C4—123no—
119C8—123no—
120C8—121no—
121C2×C3—41no—
122C2×C3—43no—
123C2×C3—43no—
124C22#591213yes—
125C22#592511yescharacteristic
126C22#592713yes—
127C4—31no—
128C4—33no—
129C4—43no—
130C4—123no—
131C4—241no—
132C4—243no—
133C3—41noSylow
134C2#8529111yescharacteristic
135C2#8535213yes—
136C2#8537113yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
Fitting#2276D3
hypercenter#95S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Maximal subgroups

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

subgroupindex
#731672
#730622
#730612
#330933
C3⋊#4014