Finite Group Databasebeta

← #87507

Subgroups — #87507

Order 384.

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

normal subgroups only

This group has 267 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
1C2×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5931C211yesmaximal, characteristic, complemented
3#5930C211yesmaximal, characteristic, complemented
4#3004—31nomaximal, Sylow
5C22⋊SL(2,𝔽3)C2211yescharacteristic
6GL(2,ℤ/4)—41nomaximal
7#983—42nomaximal
8C23×Q8D311yescharacteristic, complemented
9C2×#137—32no—
10#525—31no—
11#451—31no—
12#451—32no—
13#451—31no—
14#448—31no—
15#448—32no—
16#448—31no—
17#419—32no—
18#407—31no—
19C22×A4—41no—
20C2×S4—41no—
21C2×SL(2,𝔽3)—42no—
22C2×SL(2,𝔽3)—81no—
23SL(2,ℤ/4)—41no—
24GL(2,𝔽3)—42no—
25#226—42no—
26C2×#41—31no—
27C22×Q8—31no—
28C22×Q8—32no—
29C22×Q8—32no—
30C22×Q8—61no—
31C22×Q8—61no—
32C22×Q8C2×D311yescharacteristic
33C23×C4—31no—
34#137—62no—
35#137—62no—
36#137—62no—
37#137—62no—
38C2×Q16—32no—
39C2×Q16—32no—
40C2×SD16—32no—
41C2×SD16—32no—
42C2×M16—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#103—32no—
55#103—32no—
56C22⋊C8—32no—
57C22⋊C8—31no—
58C22⋊C8—31no—
59C2×A4—41no—
60C2×A4—81no—
61S4—81no—
62#67—161no—
63SL(2,𝔽3)—42no—
64SL(2,𝔽3)—81no—
65C24S411yescharacteristic
66#41—62no—
67#41—62no—
68C2×Q8—21no—
69C2×Q8—32no—
70C2×Q8—32no—
71C2×Q8—32no—
72C2×Q8—32no—
73C2×Q8—32no—
74C2×Q8—31no—
75C2×Q8—61no—
76C2×Q8—61no—
77C2×Q8—62no—
78C2×Q8—62no—
79C2×Q8—62no—
80C2×Q8—61no—
81C2×Q8—61no—
82C2×Q8—61no—
83C2×Q8—62no—
84C2×Q8S412yescomplemented
85C2×D4—31no—
86C2×D4—61no—
87C2×D4—61no—
88C2×D4—61no—
89C22×C4—31no—
90C22×C4—31no—
91C22×C4—31no—
92C22×C4—31no—
93C22×C4—31no—
94C22×C4—31no—
95C22×C4—62no—
96C22×C4—62no—
97C22×C4—61no—
98Q16—62no—
99Q16—62no—
100Q16—62no—
101Q16—62no—
102SD16—62no—
103SD16—62no—
104SD16—62no—
105SD16—62no—
106M16—62no—
107M16—62no—
108C2×C8—32no—
109C2×C8—32no—
110C4⋊C4—31no—
111C4⋊C4—31no—
112C4⋊C4—61no—
113C4⋊C4—61no—
114C22⋊C4—61no—
115C22⋊C4—61no—
116C22⋊C4—61no—
117C22⋊C4—61no—
118C22⋊C4—61no—
119C42—61no—
120C22×C3—161no—
121C2×D3—161no—
122A4—41no—
123Dic3—161no—
124C23—21no—
125C23—31no—
126C23—31no—
127C23—61no—
128C23—61no—
129C23C2×S411yescharacteristic
130Q8—21no—
131Q8—32no—
132Q8—32no—
133Q8—32no—
134Q8—61no—
135Q8—62no—
136Q8—61no—
137Q8—62no—
138Q8—62no—
139Q8—61no—
140Q8C2×S412yes—
141D4—62no—
142D4—121no—
143D4—122no—
144D4—121no—
145D4—121no—
146D4—121no—
147C2×C4—31no—
148C2×C4—31no—
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—122no—
163C8—62no—
164C8—62no—
165C2×C3—161no—
166C2×C3—321no—
167D3—321no—
168C22—31no—
169C22—31no—
170C22—32no—
171C22—31no—
172C22—61no—
173C22—61no—
174C22—61no—
175C22—121no—
176C22—121no—
177C22—121no—
178C22#98311yescharacteristic, complemented
179C22#102011yescharacteristic
180C4—32no—
181C4—32no—
182C4—61no—
183C4—61no—
184C4—121no—
185C4—121no—
186C4—121no—
187C3—161noSylow
188C2—21no—
189C2—31no—
190C2—31no—
191C2—61no—
192C2—241no—
193C2C2×#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
FittingC23×Q8D3
socleC23C2×S4
hypercenterC22#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊SL(2,𝔽3)C2semidirect
2. #5931C2semidirect
3. #5930C2semidirect
8. C23×Q8D3semidirect
84. C2×Q8S4semidirect
178. C22#983semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊SL(2,𝔽3)2
#59312
#59302
#30043
GL(2,ℤ/4)4
#9834