Finite Group Databasebeta

← #85455

Subgroups — #85455

Order 384.

243 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 36 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#5923C211yesmaximal, characteristic
2C2×#984C211yesmaximal, characteristic, complemented
3#568⋊C3C211yesmaximal, characteristic, complemented
4#5406C212yesmaximal, complemented
5#5403C212yesmaximal, complemented
6#3069—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#986C2211yescharacteristic
9#985—21no—
10#984—21no—
11#984C2211yescharacteristic
12#984C2212yes—
13#983—21no—
14C2×#226—21no—
15#909—41nomaximal
16#131⋊C3C2212yes—
17#575—31no—
18C2×#137—31no—
19C2×#137—31no—
20#568D311yescharacteristic, complemented
21#473—32no—
22#473—32no—
23#471—32no—
24#471—32no—
25#430—31no—
26#413—31no—
27C2×#101—31no—
28#235—41no—
29C2×Dic6—41no—
30#41⋊C3D411yescharacteristic
31#41⋊C3C2311yescharacteristic
32C2×SL(2,𝔽3)D411yescharacteristic
33GL(2,𝔽3)—21no—
34#226—22no—
35#226—21no—
36C3×M16—41no—
37Dic12—42no—
38#204—42no—
39C2×#41—31no—
40C2×#41C2×D311yescharacteristic
41C22×Q8—31no—
42#137—32no—
43#137—31no—
44#137—32no—
45#137—31no—
46#137—61no—
47#137—62no—
48#137—61no—
49#137—61no—
50#137—61no—
51#137—61no—
52AGL(1,ℤ/8)—31no—
53AGL(1,ℤ/8)—31no—
54AGL(1,ℤ/8)—61no—
55#135—61no—
56#135—61no—
57C2×Q16—32no—
58C2×Q16—61no—
59C2×SD16—32no—
60C2×SD16—61no—
61#131—32no—
62#131C2×D312yes—
63C2×M16—31no—
64C2×M16—31no—
65C2×M16—31no—
66C2×M16—31no—
67#108—32no—
68#108—32no—
69#101—32no—
70#101—32no—
71#101—32no—
72#100—32no—
73C2×C3×C4—41no—
74#67—81no—
75C2×Dic3—81no—
76D12—41no—
77C4×D3—81no—
78Dic6—42no—
79Dic6—41no—
80Dic6—81no—
81SL(2,𝔽3)C2×D411yescharacteristic
82C3×C8—42no—
83#41—31no—
84#41—31no—
85#41—61no—
86#41—61no—
87#41—61no—
88#41—61no—
89#41D1211yescharacteristic
90#41C22×D311yescharacteristic
91C2×Q8—32no—
92C2×Q8—31no—
93C2×Q8—61no—
94C2×Q8—61no—
95C2×Q8—61no—
96C2×Q8—61no—
97C2×Q8—61no—
98C2×Q8D1211yescharacteristic
99C2×D4—31no—
100C2×D4—31no—
101C2×D4—61no—
102C22×C4—31no—
103C22×C4—61no—
104C22×C4—61no—
105Q16—62no—
106Q16—62no—
107Q16—61no—
108Q16—62no—
109Q16—61no—
110Q16—62no—
111SD16—62no—
112SD16—62no—
113SD16—62no—
114SD16—61no—
115SD16—61no—
116SD16—61no—
117SD16—61no—
118SD16—62no—
119D8—61no—
120D8—61no—
121M16—31no—
122M16—32no—
123M16—32no—
124M16—32no—
125M16—31no—
126M16—31no—
127M16—61no—
128M16—61no—
129M16S411yescharacteristic
130C2×C8—32no—
131C2×C8—61no—
132C2×C8—61no—
133C2×C8—61no—
134C22×C3—41no—
135C2×D3—81no—
136C3×C4—41no—
137C3×C4—41no—
138Dic3—82no—
139Dic3—81no—
140C23—31no—
141C23—61no—
142Q8—31no—
143Q8—62no—
144Q8—61no—
145Q8—62no—
146Q8—61no—
147Q8—121no—
148Q8—121no—
149Q8C2×D1211yescharacteristic
150D4—31no—
151D4—31no—
152D4—61no—
153D4—61no—
154D4—61no—
155D4—121no—
156D4—121no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—61no—
163C2×C4—62no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4—121no—
167C2×C4—121no—
168C2×C4C2×S411yescharacteristic
169C8—32no—
170C8—61no—
171C8—61no—
172C8—62no—
173C8C2×S412yes—
174C2×C3—41no—
175C2×C3—81no—
176D3—161no—
177C22—31no—
178C22—31no—
179C22—121no—
180C22—121no—
181C22—241no—
182C22C4⋊S411yescharacteristic
183C4—31no—
184C4—31no—
185C4—121no—
186C4—122no—
187C4C4⋊S411yescharacteristic
188C4C22×S411yescharacteristic
189C3—41noSylow
190C2—21no—
191C2—61no—
192C2—61no—
193C2—241no—
194C2C2×C4⋊S411yescharacteristic
subgroupG/H
center, socleC2C2×C4⋊S4
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#568D3
prime radical#5923C2
hypercenterM16S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×#984C2semidirect
3. #568⋊C3C2semidirect
4. #5406C2semidirect
5. #5403C2semidirect
20. #568D3semidirect

Maximal subgroups

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

subgroupindex
#59232
C2×#9842
#568⋊C32
#54062
#54032
#30693
#9094