Finite Group Databasebeta

← #11714231

Subgroups — #11714231

Order 768.

244 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 23 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#41⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#8439—31nomaximal, Sylow
3Q8⋊SL(2,𝔽3)C411yescharacteristic, complemented
4C2×GU(2,𝔽3)—41nomaximal
5SL(2,𝔽3)⋊C8—41nomaximal
6SL(2,𝔽3)⋊C8—41nomaximal
7Q8×#41D311yescharacteristic
8#1806—31no—
9C4×#103—31no—
10#1503—31no—
11#1493—31no—
12#1487—31no—
13#1476—31no—
14C2×#41⋊C3—41no—
15C4×SL(2,𝔽3)—41no—
16C4×SL(2,𝔽3)—41no—
17C4×SL(2,𝔽3)—81no—
18GU(2,𝔽3)—42no—
19Q82Dic311yescharacteristic
20#557—31no—
21#554—31no—
22#554—61no—
23D4×Q8—31no—
24#533—31no—
25#533—61no—
26C4×#41—31no—
27C2×C4×Q8—31no—
28#431—31no—
29#423—31no—
30C2×#104—61no—
31C2×#103—61no—
32C2×C4×C8—31no—
33C4×C4⋊C4—31no—
34#340—61no—
35#339—61no—
36#328—61no—
37#328—61no—
38Q8⋊C8—61no—
39Q8⋊C8—61no—
40Q8⋊C8—61no—
41Q8⋊C8—61no—
42Q8⋊C8—61no—
43D4⋊C8—61no—
44#41⋊C3—41no—
45#41⋊C3—41no—
46C2×SL(2,𝔽3)—41no—
47C2×SL(2,𝔽3)—41no—
48C2×SL(2,𝔽3)—41no—
49C2×SL(2,𝔽3)—81no—
50C2×C3⋊C8—161no—
51C2×#41S411yescharacteristic
52C22×Q8—31no—
53C2×M16—61no—
54C22×C8—61no—
55C4⋊Q8—31no—
56C4⋊Q8—31no—
57C4⋊Q8—61no—
58C4⋊Q8—61no—
59#125—61no—
60#125—61no—
61#125—61no—
62C22⋊Q8—31no—
63C22⋊Q8—31no—
64C22⋊Q8—61no—
65C22⋊Q8—61no—
66C22⋊Q8—61no—
67C22⋊Q8—61no—
68C22⋊Q8—61no—
69C4×Q8—21no—
70C4×Q8—31no—
71C4×Q8—31no—
72C4×Q8—31no—
73C4×Q8—31no—
74C4×Q8—31no—
75C4×Q8—31no—
76C4×Q8—61no—
77C4×Q8—61no—
78C4×Q8S411yescharacteristic
79C4×Q8S411yescharacteristic
80C4×D4—31no—
81C4×D4—31no—
82C4×D4—61no—
83C4×D4—61no—
84#117—31no—
85#117—61no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—61no—
88C2×C4⋊C4—61no—
89C2×C42—31no—
90C2×C42—61no—
91C2×C42—61no—
92C4⋊C8—61no—
93C4⋊C8—61no—
94C4⋊C8—61no—
95#104—121no—
96#104—121no—
97#103—62no—
98#103—62no—
99C22⋊C8—61no—
100C4×C8—61no—
101C4×C8—61no—
102C4×C8—61no—
103#95—61no—
104C2×C3×C4—161no—
105SL(2,𝔽3)—41no—
106SL(2,𝔽3)—41no—
107SL(2,𝔽3)—41no—
108SL(2,𝔽3)—81no—
109C3⋊C8—162no—
110#41—31no—
111#41—31no—
112#41#22611yescharacteristic
113#41GL(2,𝔽3)11yescharacteristic
114C2×Q8—21no—
115C2×Q8—31no—
116C2×Q8—31no—
117C2×Q8—61no—
118C2×Q8—61no—
119C2×Q8—61no—
120C2×Q8—121no—
121C2×Q8SL(2,ℤ/4)11yescharacteristic
122C2×Q8SL(2,ℤ/4)11yescharacteristic
123C2×Q8SL(2,ℤ/4)11yescharacteristic
124C2×D4—31no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—61no—
129C22×C4—61no—
130C22×C4—121no—
131M16—121no—
132M16—121no—
133C2×C8—62no—
134C2×C8—121no—
135C2×C8—121no—
136C2×C8—121no—
137C2×C8—121no—
138C4⋊C4—31no—
139C4⋊C4—31no—
140C4⋊C4—31no—
141C4⋊C4—31no—
142C4⋊C4—31no—
143C4⋊C4—31no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C4⋊C4—62no—
153C4⋊C4—121no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C42—31no—
158C42—31no—
159C42—31no—
160C42—31no—
161C42—61no—
162C42—61no—
163C42—61no—
164C42—121no—
165C42—121no—
166C22×C3—161no—
167C3×C4—161no—
168C3×C4—161no—
169C23—31no—
170Q8—31no—
171Q8—31no—
172Q8—41no—
173Q8—61no—
174Q8—61no—
175Q8—81no—
176Q8—121no—
177Q8—121no—
178Q8—121no—
179Q8#85911yescharacteristic
180Q8GU(2,𝔽3)11yescharacteristic, complemented
181D4—31no—
182D4—31no—
183D4—61no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—122no—
201C2×C4—241no—
202C2×C4—241no—
203C2×C4—241no—
204C2×C4#102011yescharacteristic
205C8—122no—
206C8—241no—
207C8—241no—
208C2×C3—161no—
209C2×C3—161no—
210C2×C3—161no—
211C22—31no—
212C22—31no—
213C22—61no—
214C22—61no—
215C22#593611yescharacteristic
216C4—31no—
217C4—31no—
218C4—31no—
219C4—31no—
220C4—61no—
221C4—61no—
222C4—61no—
223C4—121no—
224C4—121no—
225C4—242no—
226C4#593011yescharacteristic
227C4#593111yescharacteristic
228C3—161noSylow
229C2—61no—
230C2—61no—
231C2#8556211yescharacteristic
232C2#8556311yescharacteristic
233C2#8556711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4#1020
commutator, hypocenterQ8⋊SL(2,𝔽3)C4
FittingQ8×#41D3
socleC22#5936
prime radical, cosocle#41⋊SL(2,𝔽3)C2
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. Q8⋊SL(2,𝔽3)C4semidirect
180. Q8GU(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#41⋊SL(2,𝔽3)2
#84393
C2×GU(2,𝔽3)4
SL(2,𝔽3)⋊C84