Finite Group Databasebeta

← #11714118

Subgroups — #11714118

Order 768.

369 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 63 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#85337C212yesmaximal
2C4×#131⋊C3C211yesmaximal, characteristic
3#73125C214yesmaximal
4#33220—31nomaximal, Sylow
5C2×#131⋊C3C2212yes—
6C4×#41⋊C3C2211yescharacteristic
7#5426C2214yes—
8#5391—22no—
9#5386—22no—
10#4680—41nomaximal
11C2×#498—31no—
12#3079—32no—
13C4×#131D311yescharacteristic
14#1997—32no—
15#1951—34no—
16#1848—31no—
17#1815—34no—
18C2×#41⋊C3C2311yescharacteristic
19#869—42no—
20#131⋊C3Q814yescomplemented
21C4×SL(2,𝔽3)D412yes—
22#859—24no—
23C3×C4×C8—41no—
24#817—44no—
25C2×#131C2×D312yes—
26C2×C4⋊Q8—32no—
27C4×#41C2×D311yescharacteristic
28#498—62no—
29#498—61no—
30#498—61no—
31#477—62no—
32#476—62no—
33#475—62no—
34#474—62no—
35C2×#108—34no—
36#428—34no—
37C2×#106—34no—
38#423—32no—
39#417—34no—
40#405—32no—
41C4×M16—31no—
42C2×C4×C8—31no—
43#343—38no—
44C2×Dic6—82no—
45#41⋊C3C2×Q812yes—
46C2×SL(2,𝔽3)C2×D411yescharacteristic
47C2×C3×C8—42no—
48C3×C42—41no—
49C4⋊Dic3—44no—
50C4⋊Dic3—82no—
51C2×#41C22×D311yescharacteristic
52C22×Q8—62no—
53#131—34no—
54#131Dic614yescomplemented
55C2×M16—32no—
56C2×M16—34no—
57C22×C8—32no—
58C4⋊Q8—62no—
59C4⋊Q8—62no—
60C4⋊Q8—62no—
61C4⋊Q8—62no—
62C4×Q8D1212yes—
63C4×D4—32no—
64#117—31no—
65C2×C4⋊C4—34no—
66C2×C4⋊C4—62no—
67C2×C42—31no—
68#108—68no—
69#107—64no—
70#106—64no—
71#106—64no—
72#106—64no—
73C4⋊C8—62no—
74C4⋊C8—62no—
75#103—64no—
76#102—64no—
77#97—61no—
78C4×C8—32no—
79C4×C8—31no—
80C4×C8S411yescharacteristic
81C2×C3×C4—42no—
82C2×C3×C4—41no—
83C2×Dic3—84no—
84Dic6—164no—
85SL(2,𝔽3)C4⋊Q811yescharacteristic
86C3×C8—44no—
87#41—32no—
88#41C2×Dic612yes—
89C2×Q8—122no—
90C2×Q8—122no—
91C2×Q8—244no—
92C2×Q8C2×D1211yescharacteristic
93C2×D4—31no—
94C22×C4—31no—
95C22×C4—32no—
96C22×C4—64no—
97M16—34no—
98M16—62no—
99M16—68no—
100C2×C8—32no—
101C2×C8—34no—
102C2×C8—34no—
103C2×C8—64no—
104C2×C8C2×S412yes—
105C4⋊C4—32no—
106C4⋊C4—64no—
107C4⋊C4—64no—
108C4⋊C4—122no—
109C4⋊C4—122no—
110C22⋊C4—61no—
111C42—32no—
112C42—31no—
113C42C2×S411yescharacteristic
114C22×C3—41no—
115C3×C4—42no—
116C3×C4—44no—
117Dic3—164no—
118C23—31no—
119Q8—31no—
120Q8—244no—
121Q8—244no—
122Q8#86911yescharacteristic
123D4—32no—
124D4—61no—
125C2×C4—31no—
126C2×C4—32no—
127C2×C4—32no—
128C2×C4—32no—
129C2×C4—64no—
130C2×C4—124no—
131C2×C4—244no—
132C2×C4C4⋊S412yes—
133C2×C4C22×S411yescharacteristic
134C8—34no—
135C8—124no—
136C8A4⋊Q814yes—
137C2×C3—41no—
138C2×C3—42no—
139C22—32no—
140C22—62no—
141C22C2×C4⋊S411yescharacteristic
142C4—31no—
143C4—31no—
144C4—62no—
145C4—244no—
146C4#540614yes—
147C4C2×A4⋊Q812yes—
148C3—41noSylow
149C2—62no—
150C2#8528311yescharacteristic
151C2C2×#540612yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×#131D3
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
20. #131⋊C3Q8semidirect
54. #131Dic6semidirect

Maximal subgroups

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

subgroupindex
#853372
C4×#131⋊C32
#731252
#332203
#46804