Finite Group Databasebeta

← #85454

Subgroups — #85454

Order 384.

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

normal subgroups only

This group has 257 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, complemented
2C2×#986C211yesmaximal, characteristic, complemented
3#568⋊C3C211yesmaximal, characteristic, complemented
4#5407C212yesmaximal, complemented
5#5406C212yesmaximal, complemented
6#3068—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#986—21no—
9#986C2212yes—
10#986C2211yescharacteristic, complemented
11#985—21no—
12#984C2211yescharacteristic, complemented
13#983—21no—
14C2×GL(2,𝔽3)—21no—
15#908—41nomaximal
16#131⋊C3C2212yescomplemented
17#575—31no—
18C2×AGL(1,ℤ/8)—31no—
19C2×AGL(1,ℤ/8)—31no—
20#568D311yescharacteristic, complemented
21#472—32no—
22#472—32no—
23#471—32no—
24#471—32no—
25#430—31no—
26#413—31no—
27C2×#100—31no—
28#235—41no—
29C2×D12—41no—
30#41⋊C3D411yescharacteristic
31#41⋊C3C2311yescharacteristic
32C2×SL(2,𝔽3)D411yescharacteristic
33GL(2,𝔽3)—22no—
34GL(2,𝔽3)—21no—
35#226—21no—
36C3×M16—41no—
37D24—42no—
38#204—42no—
39C2×#41—31no—
40C2×#41C2×D311yescharacteristic
41C22×D4—31no—
42#137—31no—
43#137—31no—
44#137—61no—
45AGL(1,ℤ/8)—32no—
46AGL(1,ℤ/8)—31no—
47AGL(1,ℤ/8)—32no—
48AGL(1,ℤ/8)—31no—
49AGL(1,ℤ/8)—61no—
50AGL(1,ℤ/8)—62no—
51AGL(1,ℤ/8)—61no—
52AGL(1,ℤ/8)—61no—
53AGL(1,ℤ/8)—61no—
54AGL(1,ℤ/8)—61no—
55#135—61no—
56#135—61no—
57C2×SD16—32no—
58C2×SD16—61no—
59C2×D8—32no—
60C2×D8—61no—
61#131—32no—
62#131C2×D312yescomplemented
63C2×M16—31no—
64C2×M16—31no—
65C2×M16—31no—
66C2×M16—31no—
67#108—32no—
68#108—32no—
69#101—32no—
70#100—32no—
71#100—32no—
72#100—32no—
73C22×D3—81no—
74C2×C3×C4—41no—
75#67—81no—
76D12—42no—
77D12—41no—
78D12—81no—
79C4×D3—81no—
80Dic6—41no—
81SL(2,𝔽3)C2×D411yescharacteristic
82C3×C8—42no—
83C24—61no—
84#41—31no—
85#41—31no—
86#41—61no—
87#41—61no—
88#41—61no—
89#41—61no—
90#41D1211yescharacteristic
91#41C22×D311yescharacteristic
92C2×Q8—31no—
93C2×Q8D1211yescharacteristic
94C2×D4—31no—
95C2×D4—31no—
96C2×D4—32no—
97C2×D4—61no—
98C2×D4—61no—
99C2×D4—61no—
100C2×D4—61no—
101C2×D4—61no—
102C2×D4—61no—
103C22×C4—31no—
104C22×C4—61no—
105Q16—61no—
106Q16—61no—
107SD16—62no—
108SD16—61no—
109SD16—61no—
110SD16—62no—
111SD16—61no—
112SD16—62no—
113SD16—61no—
114SD16—62no—
115D8—62no—
116D8—62no—
117D8—62no—
118D8—61no—
119D8—61no—
120D8—62no—
121M16—31no—
122M16—32no—
123M16—32no—
124M16—31no—
125M16—31no—
126M16—32no—
127M16—61no—
128M16—61no—
129M16S411yescharacteristic
130C2×C8—32no—
131C2×C8—61no—
132C2×C8—61no—
133C2×C8—61no—
134C22×C3—41no—
135C2×D3—81no—
136C2×D3—82no—
137C2×D3—162no—
138C3×C4—41no—
139C3×C4—41no—
140Dic3—81no—
141C23—31no—
142C23—61no—
143C23—62no—
144C23—121no—
145C23—121no—
146C23—242no—
147Q8—31no—
148Q8—61no—
149Q8—61no—
150Q8C2×D1211yescharacteristic
151D4—31no—
152D4—31no—
153D4—61no—
154D4—62no—
155D4—61no—
156D4—61no—
157D4—62no—
158D4—121no—
159D4—121no—
160D4—121no—
161D4—121no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—61no—
168C2×C4—121no—
169C2×C4—121no—
170C2×C4C2×S411yescharacteristic
171C8—32no—
172C8—62no—
173C8—61no—
174C8—61no—
175C8C2×S412yes—
176C2×C3—41no—
177C2×C3—81no—
178D3—161no—
179D3—162no—
180C22—31no—
181C22—31no—
182C22—121no—
183C22—121no—
184C22—122no—
185C22—242no—
186C22—241no—
187C22—242no—
188C22—242no—
189C22C4⋊S411yescharacteristic
190C4—31no—
191C4—31no—
192C4—121no—
193C4C4⋊S411yescharacteristic
194C4C22×S411yescharacteristic
195C3—41noSylow
196C2—21no—
197C2—61no—
198C2—61no—
199C2—242no—
200C2—241no—
201C2C2×C4⋊S411yescharacteristic
subgroupG/H
center, socleC2C2×C4⋊S4
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#568D3
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
1. #5923C2semidirect
2. C2×#986C2semidirect
3. #568⋊C3C2semidirect
4. #5407C2semidirect
5. #5406C2semidirect
10. #986C22semidirect
12. #984C22semidirect
16. #131⋊C3C22semidirect
20. #568D3semidirect
62. #131C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59232
C2×#9862
#568⋊C32
#54072
#54062
#30683
#9084