Finite Group Databasebeta

← #11712605

Subgroups — #11712605

Order 768.

1133 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1133 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C23×#41⋊C3C211yesmaximal, characteristic
2C2×#5420C212yesmaximal, complemented
3#73151C214yesmaximal, complemented
4#20600—31nomaximal, Sylow
5C22×#41⋊C3—24no—
6C22×#41⋊C3C414yescomplemented
7C22×#41⋊C3C2212yes—
8C23×SL(2,𝔽3)C2211yescharacteristic
9#5420—24no—
10C2×#859C2214yescomplemented
11#5211—41nomaximal
12C23×#41D311yescharacteristic
13C2×#419—32no—
14C2×#418—32no—
15C2×#407—31no—
16#2280—31no—
17#1788—34no—
18#1788—34no—
19C23×C3×C4—41no—
20C2×#41⋊C3—28no—
21C2×#41⋊C3—24no—
22C2×#41⋊C3C2×C414yes—
23C22×SL(2,𝔽3)C2×C412yes—
24C22×SL(2,𝔽3)D414yes—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×#217—42no—
27#859—24no—
28C3⋊#95—44no—
29C22×#41—24no—
30C22×#41—34no—
31C22×#41—32no—
32C22×#41—64no—
33C22×#41Dic314yescomplemented
34C22×#41C2×D312yes—
35C23×Q8C2×D311yescharacteristic
36C23×D4—31no—
37C24×C4—31no—
38C22×M16—32no—
39C2×#117—32no—
40#419—64no—
41#419—64no—
42#418—68no—
43C2×#103—34no—
44C2×#103—34no—
45C2×#102—38no—
46#407—64no—
47C2×C22⋊C8—34no—
48#381—61no—
49#380—34no—
50#379—61no—
51C4×C22⋊C4—62no—
52C24×C3—41no—
53C22×C3×C4—44no—
54C22×C3×C4—42no—
55C22×C3×C4—84no—
56C22×Dic3—44no—
57#41⋊C3—24no—
58C2×SL(2,𝔽3)C22⋊C414yescomplemented
59C2×SL(2,𝔽3)C22×C411yescharacteristic
60C2×SL(2,𝔽3)C2×D412yes—
61#217—84no—
62C25—31no—
63C2×#41—28no—
64C2×#41—24no—
65C2×#41—38no—
66C2×#41—34no—
67C2×#41—64no—
68C2×#41—68no—
69C2×#41—616no—
70C2×#41—68no—
71C2×#41—64no—
72C2×#41—68no—
73C2×#41—616no—
74C2×#41C2×Dic314yes—
75C22×Q8—32no—
76C22×Q8—31no—
77C22×Q8—34no—
78C22×Q8C2×Dic312yes—
79C22×Q8#6714yes—
80C22×Q8C22×D311yescharacteristic
81C22×D4—34no—
82C22×D4—32no—
83C22×D4—38no—
84C22×D4—62no—
85C22×D4—61no—
86C22×D4—64no—
87C23×C4—31no—
88C23×C4—34no—
89C23×C4—34no—
90C23×C4—32no—
91C23×C4—32no—
92C23×C4—64no—
93C23×C4—64no—
94C23×C4S411yescharacteristic
95C2×M16—64no—
96C2×M16—64no—
97C2×M16—64no—
98C22×C8—34no—
99#117—64no—
100#117—64no—
101C2×C4⋊C4—34no—
102C2×C22⋊C4—64no—
103C2×C22⋊C4—62no—
104C2×C42—62no—
105#103—68no—
106#103—64no—
107#103—64no—
108#102—616no—
109C22⋊C8—68no—
110#95—64no—
111C23×C3—42no—
112C23×C3—44no—
113C23×C3—41no—
114C23×C3—84no—
115C2×C3×C4—44no—
116C2×C3×C4—84no—
117C2×C3×C4—88no—
118C2×Dic3—84no—
119C2×Dic3—88no—
120SL(2,𝔽3)C2×C22⋊C411yescharacteristic
121C24—32no—
122C24—34no—
123C24—64no—
124C24—124no—
125C24C2×S411yescharacteristic
126#41—24no—
127#41—616no—
128#41—68no—
129#41—632no—
130#41—64no—
131#41—616no—
132#41—68no—
133C2×Q8—34no—
134C2×Q8—32no—
135C2×Q8—32no—
136C2×Q8—34no—
137C2×Q8—38no—
138C2×Q8—31no—
139C2×Q8—61no—
140C2×Q8—62no—
141C2×Q8—64no—
142C2×Q8#21714yes—
143C2×Q8C22×Dic311yescharacteristic
144C2×Q8C2×#6712yes—
145C2×D4—34no—
146C2×D4—38no—
147C2×D4—316no—
148C2×D4—62no—
149C2×D4—64no—
150C2×D4—64no—
151C2×D4—68no—
152C2×D4—68no—
153C2×D4—616no—
154C22×C4—24no—
155C22×C4—32no—
156C22×C4—32no—
157C22×C4—34no—
158C22×C4—34no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—34no—
162C22×C4—34no—
163C22×C4—38no—
164C22×C4—64no—
165C22×C4—68no—
166C22×C4—68no—
167C22×C4—64no—
168C22×C4—616no—
169C22×C4—68no—
170C22×C4—64no—
171C22×C4SL(2,ℤ/4)14yes—
172C22×C4C2×S412yes—
173M16—128no—
174C2×C8—64no—
175C2×C8—64no—
176C2×C8—64no—
177C4⋊C4—64no—
178C4⋊C4—64no—
179C22⋊C4—128no—
180C22⋊C4—124no—
181C42—124no—
182C22×C3—44no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3—44no—
186C22×C3—84no—
187C22×C3—88no—
188C3×C4—84no—
189Dic3—164no—
190C23—24no—
191C23—34no—
192C23—64no—
193C23—64no—
194C23—64no—
195C23—68no—
196C23—128no—
197C23—1216no—
198C23C2×SL(2,ℤ/4)12yes—
199C23GL(2,ℤ/4)14yes—
200C23C22×S411yescharacteristic
201Q8—34no—
202Q8—31no—
203Q8—32no—
204Q8—61no—
205Q8—64no—
206Q8—62no—
207Q8C2×#21711yescharacteristic
208D4—38no—
209D4—64no—
210D4—68no—
211D4—616no—
212C2×C4—24no—
213C2×C4—28no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—32no—
217C2×C4—34no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—38no—
221C2×C4—34no—
222C2×C4—34no—
223C2×C4—64no—
224C2×C4—68no—
225C2×C4—64no—
226C2×C4—616no—
227C2×C4—128no—
228C2×C4—124no—
229C2×C4C2×SL(2,ℤ/4)14yes—
230C8—124no—
231C2×C3—42no—
232C2×C3—44no—
233C2×C3—41no—
234C2×C3—84no—
235C22—24no—
236C22—28no—
237C22—64no—
238C22—64no—
239C22—64no—
240C22—1216no—
241C22—128no—
242C22#543214yes—
243C22#592314yes—
244C22C22×SL(2,ℤ/4)11yescharacteristic
245C22C2×GL(2,ℤ/4)12yes—
246C4—24no—
247C4—32no—
248C4—31no—
249C4—31no—
250C4—34no—
251C4—244no—
252C3—41noSylow
253C2—24no—
254C2—124no—
255C2#8551912yes—
256C2#8552214yes—
257C2C2×#543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC23×#41D3
prime radicalC23×#41⋊C3C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)C2×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#5420C2semidirect
3. #73151C2semidirect
6. C22×#41⋊C3C4semidirect
10. C2×#859C22semidirect
33. C22×#41Dic3semidirect
58. C2×SL(2,𝔽3)C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#41⋊C32
C2×#54202
#731512
#206003
#52114