Finite Group Databasebeta

← #11713718

Subgroups — #11713718

Order 768.

619 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 71 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3#73065C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25932—31nomaximal, Sylow
6C22×GL(2,𝔽3)C2212yes—
7C2×#947—41nomaximal
8Q8×SL(2,𝔽3)—22no—
9C2×C4×SL(2,𝔽3)C2212yes—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C4⋊GL(2,𝔽3)—22no—
13C2×Q82D311yescharacteristic, complemented
14C2×#486—31no—
15C2×#460—31no—
16#2548—31no—
17#2023—34no—
18#1987—31no—
19#1984—31no—
20#1880—34no—
21C4×#103—31no—
22C22×C3×Q8—41no—
23C22×D12—41no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25C2×GL(2,𝔽3)—22no—
26C2×GL(2,𝔽3)—24no—
27#947—48no—
28C2×(C2×D3)⋊C4—44no—
29C2×C4×Dic3—41no—
30C4×SL(2,𝔽3)—22no—
31C4×SL(2,𝔽3)D412yes—
32#859—22no—
33C22×SD16—32no—
34Q82—22no—
35Q82—62no—
36C2×C4⋊Q8—31no—
37C2×C4⋊Q8—32no—
38C2×#127—31no—
39C2×#124—61no—
40C2×C4×Q8—31no—
41C2×C4×Q8C2×D312yes—
42C2×C4×Q8C2×D311yescharacteristic
43#486—64no—
44#460—62no—
45#460—62no—
46C2×C4⋊C8—31no—
47C2×#103—32no—
48C2×#102—34no—
49C2×#102—32no—
50C2×C4×C8—31no—
51#394—62no—
52#390—34no—
53#390—61no—
54C4×C4⋊C4—31no—
55#340—34no—
56#340—32no—
57#336—34no—
58C23×D3—42no—
59C2×C3×Q8—44no—
60C2×C3×Q8—82no—
61C2×C3×Q8—82no—
62C22×C3×C4—41no—
63C22×C3×C4—42no—
64C22×Dic3—42no—
65C2×D12—44no—
66C2×D12—82no—
67C2×D12—82no—
68C2×SL(2,𝔽3)C2×D411yescharacteristic
69C2×SL(2,𝔽3)#4112yes—
70GL(2,𝔽3)—42no—
71(C2×D3)⋊C4—416no—
72C4×Dic3—44no—
73C22×Q8—61no—
74C22×Q8S411yescharacteristic
75C22×Q8C22×D311yescharacteristic
76C22×D4—32no—
77C22×D4—61no—
78C22×D4—61no—
79C2×SD16—64no—
80C2×SD16—62no—
81C2×SD16—64no—
82C2×SD16—62no—
83C22×C8—32no—
84C22×C8—32no—
85C4⋊Q8—34no—
86C4⋊Q8—62no—
87C4⋊Q8—64no—
88C4⋊Q8—64no—
89#127—62no—
90#127—62no—
91#124—124no—
92C4×Q8—22no—
93C4×Q8—32no—
94C4×Q8—62no—
95C4×Q8—62no—
96C4×Q8—62no—
97C4×Q8#6712yes—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C22⋊C4—64no—
104C2×C22⋊C4—64no—
105C2×C42—31no—
106C2×C42—32no—
107C2×C42—61no—
108C4⋊C8—62no—
109#103—62no—
110#103—62no—
111#102—64no—
112#102—68no—
113C4×C8—62no—
114#95—61no—
115C23×C3—41no—
116C22×D3—44no—
117C22×D3—82no—
118C22×D3—84no—
119C22×D3—82no—
120C22×D3—84no—
121C3×Q8—88no—
122C2×C3×C4—42no—
123C2×C3×C4—44no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—82no—
127C2×C3×C4—82no—
128C2×Dic3—44no—
129C2×Dic3—82no—
130C2×Dic3—82no—
131D12—88no—
132SL(2,𝔽3)#12411yescharacteristic, complemented
133C24—62no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—32no—
137C2×Q8—31no—
138C2×Q8—64no—
139C2×Q8—242no—
140C2×Q8GL(2,𝔽3)14yescomplemented
141C2×Q8#23912yes—
142C2×Q8C2×#6711yescharacteristic
143C2×D4—62no—
144C2×D4—62no—
145C2×D4—64no—
146C2×D4—64no—
147C2×D4—122no—
148C2×D4—122no—
149C2×D4—122no—
150C2×D4—122no—
151C2×D4—242no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—62no—
156C22×C4C2×S412yes—
157C22×C4C2×S411yescharacteristic
158SD16—124no—
159SD16—122no—
160SD16—122no—
161C2×C8—62no—
162C2×C8—62no—
163C2×C8—62no—
164C2×C8—62no—
165C2×C8—64no—
166C4⋊C4—32no—
167C4⋊C4—34no—
168C4⋊C4—64no—
169C4⋊C4—61no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C22⋊C4—128no—
175C22⋊C4—244no—
176C42—32no—
177C42—32no—
178C42—62no—
179C42—62no—
180C42—122no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—41no—
186C2×D3—84no—
187C2×D3—88no—
188C2×D3—162no—
189C2×D3—164no—
190C2×D3—162no—
191C3×C4—44no—
192C3×C4—84no—
193Dic3—84no—
194C23—122no—
195C23—122no—
196C23—124no—
197C23—124no—
198C23—122no—
199C23C22×S411yescharacteristic
200Q8—28no—
201Q8—32no—
202Q8—31no—
203Q8—61no—
204Q8—244no—
205Q8#94711yescharacteristic, complemented
206D4—122no—
207D4—124no—
208D4—122no—
209D4—244no—
210D4—244no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—32no—
218C2×C4—62no—
219C2×C4—64no—
220C2×C4—62no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4C2×GL(2,𝔽3)14yes—
225C2×C4C2×GL(2,𝔽3)12yes—
226C2×C4#98312yes—
227C2×C4GL(2,ℤ/4)12yes—
228C8—122no—
229C8—122no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—42no—
234C2×C3—42no—
235D3—164no—
236C22—242no—
237C22—242no—
238C22—242no—
239C22—244no—
240C22—244no—
241C22C22×GL(2,𝔽3)11yescharacteristic
242C22C2×#98311yescharacteristic
243C22#591912yes—
244C22#592512yes—
245C22C2×GL(2,ℤ/4)11yescharacteristic
246C4—24no—
247C4—31no—
248C4—32no—
249C4—31no—
250C4—62no—
251C4—122no—
252C4—242no—
253C4#542114yes—
254C3—41noSylow
255C2—482no—
256C2#8543012yes—
257C2#8543312yes—
258C2C2×#542111yescharacteristic
259C2#8550811yescharacteristic
260C2#8553011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×Q82D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
hypercenterC22×Q8S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×SL(2,𝔽3)C2semidirect
3. #73065C2semidirect
4. C4×#859C2semidirect
13. C2×Q82D3semidirect
132. SL(2,𝔽3)#124semidirect
140. C2×Q8GL(2,𝔽3)semidirect
205. Q8#947semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SL(2,𝔽3)2
C2×C4⋊GL(2,𝔽3)2
#730652
C4×#8592
#259323
C2×#9474