Finite Group Databasebeta

← #11714078

Subgroups — #11714078

Order 768.

389 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C42×SL(2,𝔽3)C211yesmaximal, characteristic
2#73062C214yesmaximal
3C4⋊#859C212yesmaximal
4#33069—31nomaximal, Sylow
5C2×C4×SL(2,𝔽3)C2212yes—
6C2×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×#859C2214yes—
8#4936—41nomaximal
9#2859—31no—
10C42×Q8D311yescharacteristic
11#1947—32no—
12#1921—34no—
13#1863—34no—
14#1831—31no—
15#1803—32no—
16C22×SL(2,𝔽3)C2311yescharacteristic
17C2×C3×C42—41no—
18C2×C4⋊Dic3—42no—
19C4×SL(2,𝔽3)—22no—
20C4×SL(2,𝔽3)Q812yes—
21#859—24no—
22C3⋊#95—44no—
23C2×#125—61no—
24C2×C4×Q8C2×D312yes—
25C2×C4×Q8C2×D311yescharacteristic
26C2×#107—34no—
27C2×C4⋊C8—32no—
28C2×#103—34no—
29#389—34no—
30#389—62no—
31#384—32no—
32#382—61no—
33C4×C4⋊C4—32no—
34C4×C4⋊C4—31no—
35C43—31no—
36#340—34no—
37#340—34no—
38#336—34no—
39C22×C3×C4—41no—
40C22×C3×C4—42no—
41C22×Dic3—44no—
42C2×SL(2,𝔽3)C2×Q811yescharacteristic
43C2×SL(2,𝔽3)#4112yes—
44C3×C42—44no—
45C4⋊Dic3—84no—
46C22×Q8C22×D311yescharacteristic
47C22×C8—34no—
48#125—124no—
49C4×Q8—22no—
50C4×Q8—32no—
51C4×Q8—62no—
52C4×Q8Dic612yes—
53C2×C4⋊C4—31no—
54C2×C4⋊C4—32no—
55C2×C4⋊C4—34no—
56C2×C4⋊C4—64no—
57C2×C4⋊C4—62no—
58C2×C4⋊C4—62no—
59C2×C42—31no—
60C2×C42—32no—
61C2×C42—31no—
62C2×C42—32no—
63C2×C42S411yescharacteristic
64#107—68no—
65C4⋊C8—64no—
66#103—64no—
67#103—64no—
68#95—61no—
69#95—64no—
70C23×C3—41no—
71C2×C3×C4—42no—
72C2×C3×C4—42no—
73C2×C3×C4—44no—
74C2×C3×C4—42no—
75C2×C3×C4—82no—
76C2×C3×C4—82no—
77C2×Dic3—84no—
78C2×Dic3—84no—
79C2×Dic3—84no—
80SL(2,𝔽3)#12511yescharacteristic, complemented
81C2×Q8—32no—
82C2×Q8—31no—
83C2×Q8C2×Dic611yescharacteristic
84C2×Q8#23512yes—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—32no—
88C22×C4—64no—
89C22×C4C2×S412yes—
90C22×C4C2×S411yescharacteristic
91C2×C8—64no—
92C2×C8—64no—
93C2×C8—64no—
94C4⋊C4—32no—
95C4⋊C4—34no—
96C4⋊C4—61no—
97C4⋊C4—62no—
98C4⋊C4—64no—
99C4⋊C4—64no—
100C4⋊C4—124no—
101C4⋊C4—128no—
102C4⋊C4—124no—
103C42—32no—
104C42—32no—
105C42—62no—
106C42—62no—
107C42—62no—
108C42—122no—
109C42#22614yes—
110C22×C3—41no—
111C22×C3—42no—
112C22×C3—41no—
113C22×C3—41no—
114C22×C3—42no—
115C3×C4—44no—
116C3×C4—84no—
117Dic3—164no—
118C23C22×S411yescharacteristic
119Q8—31no—
120Q8—32no—
121Q8—61no—
122Q8#87011yescharacteristic
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—31no—
126C2×C4—31no—
127C2×C4—32no—
128C2×C4—32no—
129C2×C4—32no—
130C2×C4—62no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4—62no—
134C2×C4—124no—
135C2×C4—124no—
136C2×C4—124no—
137C2×C4A4⋊Q812yes—
138C2×C4C2×#22614yes—
139C2×C4C2×#22612yes—
140C2×C4#98612yes—
141C8—124no—
142C2×C3—41no—
143C2×C3—41no—
144C2×C3—42no—
145C2×C3—41no—
146C2×C3—42no—
147C22C2×A4⋊Q811yescharacteristic
148C22#591212yes—
149C22C22×#22611yescharacteristic
150C22C2×#98611yescharacteristic
151C22#592312yes—
152C4—24no—
153C4—32no—
154C4—31no—
155C4—31no—
156C4—62no—
157C4—122no—
158C4—244no—
159C4#539114yes—
160C3—41noSylow
161C2#8528411yescharacteristic
162C2C2×#539111yescharacteristic
163C2#8532212yes—
164C2#8533911yescharacteristic
165C2#8534812yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#125
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
80. SL(2,𝔽3)#125semidirect

Maximal subgroups

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

subgroupindex
C42×SL(2,𝔽3)2
#730622
C4⋊#8592
#330693
#49364