Finite Group Databasebeta

← #11714232

Subgroups — #11714232

Order 768.

277 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 21 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#85654C211yesmaximal, characteristic
2#8441—31nomaximal, Sylow
3Q8⋊SL(2,𝔽3)C411yescharacteristic, complemented
4C2×GU(2,𝔽3)—41nomaximal
5SL(2,𝔽3)⋊C8—42nomaximal
6#3483D311yescharacteristic
7#1962—31no—
8C4×#104—32no—
9#1525—31no—
10#1489—32no—
11C2×#41⋊C3—41no—
12C2×#41⋊C3—81no—
13C4×SL(2,𝔽3)—42no—
14GU(2,𝔽3)—42no—
15Q82Dic311yescharacteristic
16#555—31no—
17#555—61no—
18#550—32no—
19#532—31no—
20#532—61no—
21C4×#41—32no—
22#423—31no—
23C2×#104—62no—
24C2×#104—61no—
25C2×#104—61no—
26C4×M16—32no—
27C43—31no—
28#339—62no—
29#329—61no—
30#329—61no—
31Q8⋊C8—62no—
32D4⋊C8—62no—
33#41⋊C3—41no—
34#41⋊C3—41no—
35#41⋊C3—81no—
36#41⋊C3—81no—
37C2×SL(2,𝔽3)—42no—
38C2×SL(2,𝔽3)—41no—
39C2×SL(2,𝔽3)—81no—
40C2×C3⋊C8—161no—
41C2×#41—21no—
42C2×#41—31no—
43C2×#41—32no—
44C2×#41S411yescharacteristic
45C2×M16—62no—
46C2×M16—61no—
47C4⋊Q8—31no—
48C4⋊Q8—61no—
49#127—31no—
50#127—61no—
51#124—31no—
52#124—31no—
53#124—61no—
54#124—61no—
55#121—62no—
56#121—62no—
57#121—62no—
58C4×Q8—32no—
59C4×Q8S412yes—
60C4×D4—32no—
61C4×D4—32no—
62C4×D4—62no—
63C4×D4—62no—
64#117—32no—
65#117—62no—
66C2×C42—31no—
67C2×C42—61no—
68C2×C42—62no—
69C2×C42—61no—
70C4⋊C8—62no—
71#104—64no—
72#104—64no—
73#104—122no—
74#104—122no—
75#97—62no—
76C4×C8—62no—
77C2×C3×C4—161no—
78SL(2,𝔽3)—41no—
79SL(2,𝔽3)—42no—
80SL(2,𝔽3)—81no—
81C3⋊C8—162no—
82#41—41no—
83#41—41no—
84#41—62no—
85#41—62no—
86#41—81no—
87#41—81no—
88#41—122no—
89#41—121no—
90#41—121no—
91C2×Q8—21no—
92C2×Q8—31no—
93C2×Q8SL(2,ℤ/4)12yes—
94C2×Q8SL(2,ℤ/4)11yescharacteristic
95C2×D4—32no—
96C2×D4—31no—
97C2×D4—31no—
98C2×D4—62no—
99C2×D4—61no—
100C2×D4—61no—
101C22×C4—32no—
102C22×C4—31no—
103C22×C4—62no—
104C22×C4—61no—
105C22×C4—121no—
106M16—64no—
107M16—122no—
108M16—121no—
109M16—121no—
110C2×C8—122no—
111C2×C8—121no—
112C4⋊C4—32no—
113C4⋊C4—32no—
114C4⋊C4—62no—
115C22⋊C4—62no—
116C22⋊C4—62no—
117C22⋊C4—62no—
118C42—31no—
119C42—31no—
120C42—32no—
121C42—61no—
122C42—61no—
123C42—64no—
124C42—62no—
125C42—122no—
126C42—121no—
127C22×C3—161no—
128C3×C4—161no—
129C3×C4—161no—
130C23—31no—
131C23—61no—
132Q8—32no—
133Q8—41no—
134Q8—81no—
135Q8—121no—
136Q8GU(2,𝔽3)12yescomplemented
137D4—62no—
138D4—62no—
139D4—122no—
140D4—122no—
141D4—121no—
142D4—121no—
143D4—241no—
144D4—241no—
145C2×C4—32no—
146C2×C4—32no—
147C2×C4—31no—
148C2×C4—62no—
149C2×C4—62no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—61no—
153C2×C4—122no—
154C2×C4—122no—
155C2×C4—124no—
156C2×C4—122no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4#102011yescharacteristic
160C8—124no—
161C8—241no—
162C2×C3—162no—
163C2×C3—161no—
164C22—62no—
165C22—61no—
166C22—122no—
167C22—121no—
168C22#593611yescharacteristic
169C4—32no—
170C4—32no—
171C4—62no—
172C4—61no—
173C4—124no—
174C4—121no—
175C4#593311yescharacteristic
176C4#593511yescharacteristic
177C3—161noSylow
178C2—121no—
179C2—241no—
180C2#8556312yes—
181C2#8556511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4#1020
commutator, hypocenterQ8⋊SL(2,𝔽3)C4
Fitting#3483D3
socleC22#5936
prime radical, cosocle#85654C2
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. Q8⋊SL(2,𝔽3)C4semidirect
136. Q8GU(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#856542
#84413
C2×GU(2,𝔽3)4
SL(2,𝔽3)⋊C84