Finite Group Databasebeta

← #72267913

Subgroups — #72267913

Order 1536.

884 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 29 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#355895C311yesmaximal, characteristic, complemented, Sylow
2C2×#5948—41nomaximal
3C2×#338⋊C3—44nomaximal
4C2×#2879—31no—
5C22⋊#338—34no—
6#5948—42no—
7C2×C22⋊SL(2,𝔽3)—41no—
8#338⋊C3—44no—
9C23×#41A411yescharacteristic
10C23×M16—32no—
11#2879—68no—
12C2×#408—38no—
13C2×#407—34no—
14C2×#338—34no—
15C2×#338A414yes—
16C22⋊SL(2,𝔽3)—41no—
17C2×#41⋊C3—164no—
18C2×C4×A4—161no—
19C22×#41—34no—
20C22×#41—34no—
21C22×#41—42no—
22C22×#41—122no—
23C23×Q8SL(2,𝔽3)11yescharacteristic
24C23×D4—31no—
25C24×C4—31no—
26C22×M16—34no—
27C22×M16—34no—
28C22×M16—68no—
29C22×M16—124no—
30C22×M16—122no—
31C22×M16—122no—
32C23×C8—34no—
33#408—616no—
34#408—68no—
35#408—68no—
36#407—68no—
37#407—68no—
38C2×C22⋊C8—316no—
39#338—44no—
40#338—124no—
41C22×A4—161no—
42#41⋊C3—164no—
43#41⋊C3—164no—
44C2×SL(2,𝔽3)—164no—
45C4×A4—162no—
46C25—31no—
47C2×#41—34no—
48C2×#41—38no—
49C2×#41—34no—
50C2×#41—34no—
51C2×#41—124no—
52C2×#41—124no—
53C2×#41—124no—
54C2×#41—124no—
55C2×#41—124no—
56C2×#41—124no—
57C2×#41—128no—
58C2×#41C42⋊C314yes—
59C22×Q8—34no—
60C22×Q8—41no—
61C22×Q8—121no—
62C22×D4—34no—
63C22×D4—38no—
64C22×D4—121no—
65C22×D4—122no—
66C22×D4—121no—
67C23×C4—32no—
68C23×C4—34no—
69C23×C4—34no—
70C23×C4—32no—
71C23×C4—31no—
72C23×C4—64no—
73C23×C4—62no—
74C23×C4—62no—
75C23×C4C22⋊A411yescharacteristic
76C2×M16—68no—
77C2×M16—68no—
78C2×M16—68no—
79C2×M16—64no—
80C2×M16—64no—
81C2×M16—124no—
82C2×M16—124no—
83C2×M16—1216no—
84C2×M16—1216no—
85C22×C8—38no—
86C22×C8—68no—
87C22×C8—124no—
88C22×C8—124no—
89C22⋊C8—632no—
90C2×A4—162no—
91C2×A4—161no—
92C2×C3×C4—641no—
93SL(2,𝔽3)—164no—
94C24—32no—
95C24—34no—
96C24—62no—
97C24—61no—
98C24—61no—
99C24—242no—
100C24C22⋊SL(2,𝔽3)11yescharacteristic
101#41—44no—
102#41—44no—
103#41—124no—
104#41—124no—
105#41—128no—
106#41—124no—
107#41—124no—
108#41—128no—
109#41—124no—
110#41—124no—
111C2×Q8—34no—
112C2×Q8—124no—
113C2×Q8—124no—
114C2×Q8#95⋊C314yes—
115C2×D4—34no—
116C2×D4—38no—
117C2×D4—34no—
118C2×D4—128no—
119C2×D4—128no—
120C2×D4—124no—
121C2×D4—124no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—32no—
125C22×C4—38no—
126C22×C4—34no—
127C22×C4—32no—
128C22×C4—34no—
129C22×C4—34no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4—68no—
133C22×C4—62no—
134C22×C4—62no—
135C22×C4—68no—
136C22×C4—64no—
137C22×C4—68no—
138C22×C4—68no—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4—61no—
143C22×C4—61no—
144C22×C4#99712yes—
145M16—128no—
146M16—128no—
147M16—1216no—
148C2×C8—68no—
149C2×C8—68no—
150C2×C8—68no—
151C2×C8—1216no—
152C22×C3—641no—
153A4—161no—
154C3×C4—642no—
155C23—31no—
156C23—31no—
157C23—34no—
158C23—42no—
159C23—62no—
160C23—62no—
161C23—64no—
162C23—64no—
163C23—124no—
164C23—128no—
165C23—242no—
166C23—242no—
167C23—244no—
168C23#543⋊C311yescharacteristic
169Q8—44no—
170Q8—124no—
171D4—124no—
172D4—128no—
173D4—124no—
174C2×C4—32no—
175C2×C4—31no—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—34no—
180C2×C4—64no—
181C2×C4—64no—
182C2×C4—68no—
183C2×C4—68no—
184C2×C4—62no—
185C2×C4—64no—
186C2×C4—62no—
187C2×C4—64no—
188C2×C4—64no—
189C2×C4—64no—
190C2×C4C22⋊(C42⋊C3)11yescharacteristic
191C8—128no—
192C2×C3—641no—
193C2×C3—642no—
194C22—31no—
195C22—31no—
196C22—41no—
197C22—62no—
198C22—64no—
199C22—68no—
200C22—121no—
201C22—124no—
202C22—124no—
203C22—244no—
204C22—244no—
205C22C22⋊(#95⋊C3)11yescharacteristic
206C4—32no—
207C4—64no—
208C4#2273⋊C312yes—
209C3—641noSylow
210C2—61no—
211C2—61no—
212C2—124no—
213C2#13794⋊C311yescharacteristic
214C2#1170942712yes—
subgroupG/H
center, hypercenterC2×C4C22⋊(C42⋊C3)
commutator, Fitting, cosocle, hypocenter#355895C3
FrattiniC23×C4C22⋊A4
socleC22C22⋊(#95⋊C3)
LCSG ▷ #355895
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #355895C3semidirect

Maximal subgroups

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

subgroupindex
#3558953
C2×#59484
C2×#338⋊C34