Finite Group Databasebeta

← #11712604

Subgroups — #11712604

Order 768.

1341 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1341 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C24×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5420C212yesmaximal, complemented
3#73148C214yesmaximal, complemented
4#20599—31nomaximal, Sylow
5C23×SL(2,𝔽3)—24no—
6C23×SL(2,𝔽3)C414yescomplemented
7C23×SL(2,𝔽3)C2212yes—
8C23×SL(2,𝔽3)C2211yescharacteristic
9#5420—24no—
10C2×#859C2214yescomplemented
11#5247—41nomaximal
12C24×Q8D311yescharacteristic
13C2×#419—32no—
14C2×#419—32no—
15C2×#407—31no—
16#2280—31no—
17#1787—34no—
18#1787—34no—
19C25×C3—41no—
20C22×SL(2,𝔽3)—28no—
21C22×SL(2,𝔽3)—24no—
22C22×SL(2,𝔽3)C2×C414yes—
23C22×SL(2,𝔽3)C2×C412yes—
24C22×SL(2,𝔽3)D414yes—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×#217—44no—
27C2×#217—42no—
28#859—24no—
29C23×Q8—24no—
30C23×Q8—31no—
31C23×Q8—34no—
32C23×Q8—32no—
33C23×Q8—64no—
34C23×Q8Dic314yescomplemented
35C23×Q8C2×D312yes—
36C23×Q8C2×D311yescharacteristic
37C24×C4—31no—
38C22×M16—32no—
39C2×#117—32no—
40#419—64no—
41#419—68no—
42#419—64no—
43C2×#103—38no—
44C2×#103—34no—
45C2×#103—34no—
46#407—64no—
47C2×C22⋊C8—34no—
48#381—61no—
49#380—34no—
50#379—61no—
51C4×C22⋊C4—62no—
52C24×C3—44no—
53C24×C3—42no—
54C24×C3—41no—
55C24×C3—88no—
56C24×C3—84no—
57C22×Dic3—44no—
58C2×SL(2,𝔽3)—24no—
59C2×SL(2,𝔽3)C22⋊C414yescomplemented
60C2×SL(2,𝔽3)C22×C411yescharacteristic
61C2×SL(2,𝔽3)C2×D412yes—
62#217—84no—
63#217—88no—
64C25S411yescharacteristic
65C22×Q8—28no—
66C22×Q8—24no—
67C22×Q8—34no—
68C22×Q8—32no—
69C22×Q8—31no—
70C22×Q8—32no—
71C22×Q8—34no—
72C22×Q8—38no—
73C22×Q8—34no—
74C22×Q8—38no—
75C22×Q8—62no—
76C22×Q8—61no—
77C22×Q8—64no—
78C22×Q8—64no—
79C22×Q8—68no—
80C22×Q8—616no—
81C22×Q8—68no—
82C22×Q8—64no—
83C22×Q8—68no—
84C22×Q8—616no—
85C22×Q8C2×Dic314yes—
86C22×Q8C2×Dic312yes—
87C22×Q8#6714yes—
88C22×Q8C22×D311yescharacteristic
89C23×C4—31no—
90C23×C4—34no—
91C23×C4—34no—
92C23×C4—32no—
93C23×C4—32no—
94C23×C4—31no—
95C23×C4—64no—
96C23×C4—64no—
97C2×M16—64no—
98C2×M16—64no—
99C2×M16—64no—
100C22×C8—34no—
101#117—64no—
102#117—64no—
103C2×C4⋊C4—34no—
104C2×C22⋊C4—62no—
105C2×C22⋊C4—64no—
106C2×C42—62no—
107#103—68no—
108#103—64no—
109#103—616no—
110#103—64no—
111C22⋊C8—68no—
112#95—64no—
113C23×C3—48no—
114C23×C3—44no—
115C23×C3—42no—
116C23×C3—44no—
117C23×C3—41no—
118C23×C3—84no—
119C23×C3—88no—
120C23×C3—84no—
121C23×C3—84no—
122C23×C3—816no—
123C23×C3—832no—
124C2×Dic3—84no—
125C2×Dic3—88no—
126SL(2,𝔽3)C2×C22⋊C411yescharacteristic
127C24—28no—
128C24—24no—
129C24SL(2,ℤ/4)14yes—
130C24C2×S412yes—
131C24C2×S411yescharacteristic
132C2×Q8—24no—
133C2×Q8—34no—
134C2×Q8—32no—
135C2×Q8—32no—
136C2×Q8—34no—
137C2×Q8—31no—
138C2×Q8—34no—
139C2×Q8—38no—
140C2×Q8—38no—
141C2×Q8—316no—
142C2×Q8—61no—
143C2×Q8—62no—
144C2×Q8—64no—
145C2×Q8—62no—
146C2×Q8—64no—
147C2×Q8—68no—
148C2×Q8—616no—
149C2×Q8—64no—
150C2×Q8—68no—
151C2×Q8—616no—
152C2×Q8—68no—
153C2×Q8—632no—
154C2×Q8—64no—
155C2×Q8—616no—
156C2×Q8—68no—
157C2×Q8#21714yescomplemented
158C2×Q8C22×Dic311yescharacteristic
159C2×Q8C2×#6712yes—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—34no—
163C22×C4—34no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—34no—
167C22×C4—34no—
168C22×C4—38no—
169C22×C4—34no—
170C22×C4—32no—
171C22×C4—64no—
172C22×C4—68no—
173C22×C4—68no—
174C22×C4—64no—
175C22×C4—616no—
176C22×C4—68no—
177C22×C4—64no—
178C22×C4—64no—
179M16—128no—
180C2×C8—64no—
181C2×C8—64no—
182C2×C8—64no—
183C4⋊C4—64no—
184C4⋊C4—64no—
185C22⋊C4—128no—
186C22⋊C4—124no—
187C42—124no—
188C22×C3—44no—
189C22×C3—41no—
190C22×C3—42no—
191C22×C3—48no—
192C22×C3—44no—
193C22×C3—816no—
194C22×C3—88no—
195C22×C3—84no—
196C22×C3—84no—
197C22×C3—84no—
198C22×C3—832no—
199Dic3—164no—
200C23—232no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—216no—
205C23—28no—
206C23#98318yes—
207C23C2×SL(2,ℤ/4)14yes—
208C23C2×SL(2,ℤ/4)12yes—
209C23GL(2,ℤ/4)14yes—
210C23C22×S411yescharacteristic
211Q8—34no—
212Q8—31no—
213Q8—32no—
214Q8—38no—
215Q8—61no—
216Q8—64no—
217Q8—62no—
218Q8—64no—
219Q8—68no—
220Q8—616no—
221Q8C2×#21711yescharacteristic
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—32no—
225C2×C4—34no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—38no—
229C2×C4—34no—
230C2×C4—34no—
231C2×C4—34no—
232C2×C4—64no—
233C2×C4—68no—
234C2×C4—64no—
235C2×C4—616no—
236C2×C4—64no—
237C2×C4—68no—
238C2×C4—128no—
239C2×C4—124no—
240C8—124no—
241C2×C3—44no—
242C2×C3—41no—
243C2×C3—42no—
244C2×C3—88no—
245C2×C3—84no—
246C22—232no—
247C22—216no—
248C22—28no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22#542018yescomplemented
253C22#543214yes—
254C22C2×#98314yes—
255C22C22×SL(2,ℤ/4)11yescharacteristic
256C22C2×GL(2,ℤ/4)12yes—
257C4—32no—
258C4—31no—
259C4—31no—
260C4—34no—
261C4—64no—
262C4—244no—
263C3—41noSylow
264C2—28no—
265C2—24no—
266C2C2×#542012yes—
267C2#8550714yes—
268C2C2×#543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC24×Q8D3
prime radicalC24×SL(2,𝔽3)C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC25S4
hypocenterSL(2,𝔽3)C2×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
2. C2×#5420C2semidirect
3. #73148C2semidirect
6. C23×SL(2,𝔽3)C4semidirect
10. C2×#859C22semidirect
34. C23×Q8Dic3semidirect
59. C2×SL(2,𝔽3)C22⋊C4semidirect
157. C2×Q8#217semidirect
252. C22#5420semidirect

Maximal subgroups

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

subgroupindex
C24×SL(2,𝔽3)2
C2×#54202
#731482
#205993
#52474