Finite Group Databasebeta

← #11714080

Subgroups — #11714080

Order 768.

397 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C42×SL(2,𝔽3)C211yesmaximal, characteristic
2#73062C212yesmaximal
3C4⋊#859C211yesmaximal, characteristic
4C4⋊#859C211yesmaximal, characteristic
5#73059C212yesmaximal
6#33073—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2212yes—
12#4928—41nomaximal
13#2859—31no—
14C42×Q8D311yescharacteristic
15#1948—31no—
16#1947—31no—
17#1920—32no—
18#1918—32no—
19#1865—32no—
20#1863—32no—
21#1831—31no—
22#1803—31no—
23#1803—31no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25C2×C3×C42—41no—
26C2×C4⋊Dic3—41no—
27C2×C4⋊Dic3—41no—
28C2×Dic3⋊C4—42no—
29C4×SL(2,𝔽3)—21no—
30C4×SL(2,𝔽3)—21no—
31C4×SL(2,𝔽3)Q812yes—
32#859—22no—
33#859—22no—
34C3⋊#95—42no—
35C2×#125—61no—
36C2×C4×Q8C2×D311yescharacteristic
37C2×C4×Q8C2×D311yescharacteristic
38C2×C4×Q8C2×D311yescharacteristic
39C2×#107—32no—
40C2×#106—32no—
41C2×C4⋊C8—31no—
42C2×C4⋊C8—31no—
43C2×#103—32no—
44C2×#103—32no—
45#389—32no—
46#389—32no—
47#389—61no—
48#389—61no—
49#384—31no—
50#384—31no—
51#382—61no—
52C4×C4⋊C4—31no—
53C4×C4⋊C4—31no—
54C4×C4⋊C4—31no—
55C43—31no—
56#340—32no—
57#340—32no—
58#340—32no—
59#340—32no—
60#336—32no—
61#336—32no—
62C22×C3×C4—41no—
63C22×C3×C4—41no—
64C22×C3×C4—41no—
65C22×Dic3—42no—
66C22×Dic3—42no—
67C2×SL(2,𝔽3)C2×Q811yescharacteristic
68C2×SL(2,𝔽3)#4111yescharacteristic
69C2×SL(2,𝔽3)#4111yescharacteristic
70C3×C42—82no—
71C4⋊Dic3—44no—
72C4⋊Dic3—82no—
73Dic3⋊C4—84no—
74C22×Q8C22×D311yescharacteristic
75C22×C8—32no—
76C22×C8—32no—
77#125—124no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—32no—
81C4×Q8—61no—
82C4×Q8—61no—
83C4×Q8Dic612yes—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—61no—
90C2×C4⋊C4—61no—
91C2×C4⋊C4—62no—
92C2×C4⋊C4—62no—
93C2×C4⋊C4—61no—
94C2×C4⋊C4—61no—
95C2×C42—31no—
96C2×C42—31no—
97C2×C42—31no—
98C2×C42—31no—
99C2×C42—31no—
100C2×C42—31no—
101C2×C42S411yescharacteristic
102#107—64no—
103#106—64no—
104C4⋊C8—62no—
105C4⋊C8—62no—
106#103—62no—
107#103—62no—
108#103—62no—
109#103—62no—
110#95—61no—
111#95—62no—
112#95—62no—
113C23×C3—41no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4—81no—
120C2×C3×C4—81no—
121C2×C3×C4—81no—
122C2×C3×C4—81no—
123C2×Dic3—44no—
124C2×Dic3—82no—
125C2×Dic3—82no—
126C2×Dic3—82no—
127C2×Dic3—82no—
128C2×Dic3—82no—
129SL(2,𝔽3)#12511yescharacteristic, complemented
130C2×Q8—31no—
131C2×Q8—31no—
132C2×Q8—31no—
133C2×Q8C2×Dic611yescharacteristic
134C2×Q8#23511yescharacteristic
135C2×Q8#23511yescharacteristic
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4C2×S411yescharacteristic
143C22×C4C2×S411yescharacteristic
144C22×C4C2×S411yescharacteristic
145C2×C8—62no—
146C2×C8—62no—
147C2×C8—62no—
148C2×C8—62no—
149C2×C8—62no—
150C2×C8—62no—
151C4⋊C4—32no—
152C4⋊C4—32no—
153C4⋊C4—32no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C4⋊C4—124no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C4⋊C4—124no—
167C42—22no—
168C42—32no—
169C42—32no—
170C42—61no—
171C42—61no—
172C42—61no—
173C42—61no—
174C42—62no—
175C42—62no—
176C42—62no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C3×C4—44no—
185C3×C4—82no—
186C3×C4—82no—
187Dic3—84no—
188Dic3—162no—
189C23C22×S411yescharacteristic
190Q8—31no—
191Q8—31no—
192Q8—31no—
193Q8—61no—
194Q8#87011yescharacteristic
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—32no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4A4⋊Q812yes—
221C2×C4#98312yes—
222C2×C4#98412yes—
223C2×C4#98512yes—
224C2×C4#98612yes—
225C8—122no—
226C8—122no—
227C2×C3—41no—
228C2×C3—41no—
229C2×C3—41no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C22C2×A4⋊Q811yescharacteristic
235C22#591211yescharacteristic
236C22#591211yescharacteristic
237C22C2×#98311yescharacteristic
238C22C2×#98411yescharacteristic
239C22C2×#98511yescharacteristic
240C22C2×#98611yescharacteristic
241C4—22no—
242C4—22no—
243C4—31no—
244C4—31no—
245C4—31no—
246C4—31no—
247C4—62no—
248C4—121no—
249C4—121no—
250C4—242no—
251C4—242no—
252C4#538914yes—
253C3—41noSylow
254C2#8528411yescharacteristic
255C2C2×#538911yescharacteristic
256C2#8531911yescharacteristic
257C2#8533211yescharacteristic
258C2#8533511yescharacteristic
259C2#8533711yescharacteristic
260C2#8534511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#125
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
129. SL(2,𝔽3)#125semidirect

Maximal subgroups

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

subgroupindex
C42×SL(2,𝔽3)2
#730622
C4⋊#8592
#730592
#330733
#49284