Finite Group Databasebeta

← #87509

Subgroups — #87509

Order 384.

272 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 17 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#5948C211yesmaximal, characteristic, complemented
2#5931C211yesmaximal, characteristic, complemented
3#5930C211yesmaximal, characteristic, complemented
4#3002—31nomaximal, Sylow
5C22⋊SL(2,𝔽3)C2211yescharacteristic
6#985—42nomaximal
7C4×S4—41nomaximal
8C22×#41D311yescharacteristic, complemented
9C2×#135—32no—
10#525—31no—
11#452—32no—
12#451—31no—
13#450—31no—
14#449—32no—
15#448—31no—
16#447—31no—
17#416—32no—
18C2×C22⋊C8—31no—
19C2×S4—41no—
20#41⋊C3—42no—
21#41⋊C3—81no—
22C4×A4—41no—
23SL(2,ℤ/4)—41no—
24GL(2,𝔽3)—42no—
25#226—42no—
26C2×#41—32no—
27C2×#41—32no—
28C2×#41—31no—
29C2×#41—61no—
30C2×#41—61no—
31C22×Q8C2×D311yescharacteristic
32C22×D4—31no—
33C23×C4—31no—
34#135—62no—
35#135—62no—
36#135—62no—
37#135—62no—
38C2×Q16—32no—
39C2×SD16—32no—
40C2×SD16—32no—
41C2×D8—32no—
42C22×C8—32no—
43#123—61no—
44C22⋊Q8—31no—
45C22⋊Q8—31no—
46#121—31no—
47#121—31no—
48#120—61no—
49C4×D4—61no—
50C4×D4—61no—
51#117—31no—
52#103—32no—
53#103—32no—
54#102—32no—
55#102—32no—
56C22⋊C8—31no—
57C22⋊C8—31no—
58C22⋊C8—32no—
59C2×A4—41no—
60S4—42no—
61C4×D3—161no—
62SL(2,𝔽3)—42no—
63SL(2,𝔽3)—81no—
64C24—31no—
65#41—21no—
66#41—32no—
67#41—32no—
68#41—32no—
69#41—61no—
70#41—62no—
71#41—62no—
72#41—62no—
73#41—61no—
74#41—61no—
75#41—62no—
76#41S412yescomplemented
77C2×Q8—32no—
78C2×Q8—31no—
79C2×Q8—61no—
80C2×D4—32no—
81C2×D4—31no—
82C2×D4—61no—
83C2×D4—62no—
84C2×D4—62no—
85C2×D4—61no—
86C2×D4—61no—
87C2×D4—61no—
88C22×C4—31no—
89C22×C4—31no—
90C22×C4—31no—
91C22×C4—31no—
92C22×C4—31no—
93C22×C4—62no—
94C22×C4—62no—
95C22×C4—61no—
96C22×C4S411yescharacteristic
97Q16—62no—
98Q16—62no—
99SD16—62no—
100SD16—62no—
101SD16—62no—
102SD16—62no—
103D8—62no—
104D8—62no—
105C2×C8—32no—
106C2×C8—32no—
107C2×C8—62no—
108C2×C8—62no—
109C4⋊C4—31no—
110C4⋊C4—31no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C22⋊C4—61no—
114C22⋊C4—61no—
115C22⋊C4—61no—
116C22⋊C4—61no—
117C22⋊C4—61no—
118C42—61no—
119C2×D3—161no—
120A4—41no—
121C3×C4—161no—
122Dic3—161no—
123C23—31no—
124C23—31no—
125C23—62no—
126C23—61no—
127C23—61no—
128C23—61no—
129C23—121no—
130C23C2×S411yescharacteristic
131Q8—21no—
132Q8—32no—
133Q8—61no—
134Q8—62no—
135Q8C2×S412yes—
136D4—32no—
137D4—32no—
138D4—62no—
139D4—62no—
140D4—61no—
141D4—62no—
142D4—61no—
143D4—121no—
144D4—122no—
145D4—122no—
146D4—121no—
147C2×C4—31no—
148C2×C4—32no—
149C2×C4—31no—
150C2×C4—32no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—121no—
162C2×C4—121no—
163C8—62no—
164C8—62no—
165C2×C3—161no—
166D3—162no—
167C22—31no—
168C22—31no—
169C22—32no—
170C22—61no—
171C22—61no—
172C22—61no—
173C22—61no—
174C22—61no—
175C22—122no—
176C22—122no—
177C22—121no—
178C22—121no—
179C22—121no—
180C22#98511yescharacteristic, complemented
181C4—32no—
182C4—31no—
183C4—61no—
184C4—121no—
185C4—121no—
186C4—121no—
187C4#102011yescharacteristic
188C3—161noSylow
189C2—31no—
190C2—31no—
191C2—62no—
192C2—121no—
193C2—241no—
194C2C2×#102011yescharacteristic
subgroupG/H
center, hypercenterC4#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
FrattiniC2C2×#1020
FittingC22×#41D3
socleC23C2×S4
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5948C2semidirect
2. #5931C2semidirect
3. #5930C2semidirect
8. C22×#41D3semidirect
76. #41S4semidirect
180. C22#985semidirect

Maximal subgroups

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

subgroupindex
#59482
#59312
#59302
#30023
#9854
C4×S44