Finite Group Databasebeta

← #11714119

Subgroups — #11714119

Order 768.

347 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 59 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#85339C212yesmaximal
2C4×#131⋊C3C211yesmaximal, characteristic
3#73126C212yesmaximal
4#73125C212yesmaximal
5#33221—31nomaximal, Sylow
6C2×#131⋊C3C2211yescharacteristic
7C2×#131⋊C3C2211yescharacteristic
8C4×#41⋊C3C2211yescharacteristic
9#5426C2214yes—
10#5389—24no—
11#4683—41nomaximal
12C2×#499—31no—
13#3080—32no—
14C4×#131D311yescharacteristic
15#1998—32no—
16#1952—32no—
17#1951—32no—
18#1848—31no—
19#1816—32no—
20#1815—32no—
21C2×#41⋊C3C2311yescharacteristic
22#870—42no—
23#131⋊C3Q812yes—
24#131⋊C3Q812yes—
25C4×SL(2,𝔽3)D412yes—
26#859—24no—
27C3×C4×C8—41no—
28#818—42no—
29#817—42no—
30C2×#131C2×D311yescharacteristic
31C2×#131C2×D311yescharacteristic
32C2×#125—32no—
33C4×#41C2×D311yescharacteristic
34#499—61no—
35#499—61no—
36#499—61no—
37#499—61no—
38#479—64no—
39#478—64no—
40C2×#108—32no—
41C2×#108—32no—
42#428—34no—
43C2×#107—32no—
44C2×#106—32no—
45#423—32no—
46#417—34no—
47#405—31no—
48#405—31no—
49C4×M16—31no—
50C2×C4×C8—31no—
51#343—34no—
52#343—34no—
53#41⋊C3C2×Q811yescharacteristic
54#41⋊C3C2×Q811yescharacteristic
55C2×SL(2,𝔽3)C2×D411yescharacteristic
56C2×C3×C8—41no—
57C2×C3×C8—41no—
58C3×C42—41no—
59C4⋊Dic3—44no—
60Dic3⋊C4—84no—
61C2×#41C22×D311yescharacteristic
62#131—32no—
63#131—32no—
64#131Dic612yes—
65#131Dic612yes—
66C2×M16—31no—
67C2×M16—31no—
68C2×M16—34no—
69C22×C8—31no—
70C22×C8—31no—
71#125—64no—
72#125—62no—
73#125—62no—
74C4×Q8D1212yes—
75C4×D4—32no—
76#117—31no—
77C2×C4⋊C4—34no—
78C2×C4⋊C4—64no—
79C2×C42—31no—
80#108—64no—
81#108—64no—
82#107—64no—
83#107—62no—
84#107—62no—
85#106—64no—
86#106—62no—
87#106—62no—
88C4⋊C8—64no—
89#103—64no—
90#102—64no—
91#97—61no—
92C4×C8—31no—
93C4×C8—31no—
94C4×C8—31no—
95C4×C8S411yescharacteristic
96C2×C3×C4—42no—
97C2×C3×C4—41no—
98C2×Dic3—84no—
99SL(2,𝔽3)C4⋊Q811yescharacteristic
100C3×C8—42no—
101C3×C8—42no—
102#41—31no—
103#41—31no—
104#41C2×Dic611yescharacteristic
105#41C2×Dic611yescharacteristic
106C2×Q8C2×D1211yescharacteristic
107C2×D4—31no—
108C22×C4—31no—
109C22×C4—32no—
110C22×C4—64no—
111M16—32no—
112M16—32no—
113M16—61no—
114M16—61no—
115M16—64no—
116M16—64no—
117C2×C8—31no—
118C2×C8—31no—
119C2×C8—32no—
120C2×C8—32no—
121C2×C8—32no—
122C2×C8—32no—
123C2×C8—64no—
124C2×C8C2×S411yescharacteristic
125C2×C8C2×S411yescharacteristic
126C4⋊C4—32no—
127C4⋊C4—64no—
128C4⋊C4—64no—
129C4⋊C4—124no—
130C4⋊C4—124no—
131C22⋊C4—61no—
132C42—32no—
133C42—31no—
134C42C2×S411yescharacteristic
135C22×C3—41no—
136C3×C4—41no—
137C3×C4—41no—
138C3×C4—82no—
139Dic3—164no—
140C23—31no—
141Q8—31no—
142Q8#86911yescharacteristic
143D4—31no—
144D4—31no—
145D4—61no—
146C2×C4—31no—
147C2×C4—31no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—32no—
152C2×C4—122no—
153C2×C4—124no—
154C2×C4—244no—
155C2×C4C4⋊S412yes—
156C2×C4C22×S411yescharacteristic
157C8—32no—
158C8—32no—
159C8—124no—
160C8A4⋊Q812yes—
161C8A4⋊Q812yes—
162C2×C3—41no—
163C2×C3—42no—
164C22—31no—
165C22—31no—
166C22—62no—
167C22C2×C4⋊S411yescharacteristic
168C4—22no—
169C4—31no—
170C4—31no—
171C4—62no—
172C4—244no—
173C4C2×A4⋊Q811yescharacteristic
174C4C2×A4⋊Q811yescharacteristic
175C3—41noSylow
176C2—61no—
177C2—61no—
178C2#8528311yescharacteristic
179C2#8545112yes—
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

Maximal subgroups

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

subgroupindex
#853392
C4×#131⋊C32
#731262
#731252
#332213
#46834