Finite Group Databasebeta

← #11713676

Subgroups — #11713676

Order 768.

545 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 75 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
2C2×#5386C211yesmaximal, characteristic
3#73059C214yesmaximal
4C4×#859C211yesmaximal, characteristic
5#25837—31nomaximal, Sylow
6C2×#944—41nomaximal
7Q8×SL(2,𝔽3)—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2212yes—
12#5386—22no—
13C2×Q82D311yescharacteristic
14C2×#498—31no—
15C2×#475—31no—
16#2546—31no—
17#2058—34no—
18#1987—31no—
19#1984—31no—
20#1865—34no—
21C4×#103—31no—
22C22×C3×Q8—41no—
23C22×Dic6—41no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25#944—48no—
26C2×Dic3⋊C4—44no—
27C2×C4×Dic3—41no—
28C4×SL(2,𝔽3)—22no—
29C4×SL(2,𝔽3)D412yes—
30#859—22no—
31#859Q814yescomplemented
32Q82—22no—
33Q82—62no—
34C2×C4⋊Q8—31no—
35C2×C4⋊Q8—32no—
36C2×C4⋊Q8—31no—
37C2×C4⋊Q8—61no—
38C2×C4×Q8—31no—
39C2×C4×Q8C2×D312yes—
40C2×C4×Q8C2×D311yescharacteristic
41#498—64no—
42#475—62no—
43#475—62no—
44C2×#106—32no—
45C2×#106—34no—
46C2×C4⋊C8—31no—
47C2×#103—32no—
48C2×#103—32no—
49C2×C4×C8—31no—
50#398—62no—
51#391—61no—
52#389—34no—
53C4×C4⋊C4—31no—
54#340—32no—
55#340—34no—
56#336—34no—
57C2×C3×Q8—44no—
58C2×C3×Q8—82no—
59C2×C3×Q8—82no—
60C22×C3×C4—41no—
61C22×C3×C4—42no—
62C22×Dic3—42no—
63C22×Dic3—42no—
64C2×Dic6—44no—
65C2×Dic6—82no—
66C2×Dic6—82no—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68C2×SL(2,𝔽3)C2×Q812yes—
69Dic3⋊C4—416no—
70C4×Dic3—44no—
71C22×Q8—61no—
72C22×Q8—61no—
73C22×Q8S411yescharacteristic
74C22×Q8C22×D311yescharacteristic
75C22×C8—32no—
76C22×C8—32no—
77C4⋊Q8—34no—
78C4⋊Q8—62no—
79C4⋊Q8—62no—
80C4⋊Q8—62no—
81C4⋊Q8—64no—
82C4⋊Q8—64no—
83C4⋊Q8—124no—
84C4×Q8—22no—
85C4×Q8—32no—
86C4×Q8—62no—
87C4×Q8—62no—
88C4×Q8—62no—
89C4×Q8#6712yes—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—61no—
97C2×C4⋊C4—64no—
98C2×C42—31no—
99C2×C42—32no—
100C2×C42—61no—
101#106—64no—
102#106—68no—
103C4⋊C8—62no—
104#103—34no—
105#103—34no—
106#103—62no—
107#103—62no—
108C4×C8—62no—
109#95—61no—
110#95—64no—
111C23×C3—41no—
112C3×Q8—88no—
113C2×C3×C4—42no—
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—82no—
118C2×C3×C4—82no—
119C2×Dic3—44no—
120C2×Dic3—44no—
121C2×Dic3—44no—
122C2×Dic3—44no—
123C2×Dic3—82no—
124C2×Dic3—82no—
125Dic6—88no—
126SL(2,𝔽3)C4⋊Q811yescharacteristic, complemented
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—32no—
130C2×Q8—31no—
131C2×Q8—64no—
132C2×Q8—122no—
133C2×Q8—122no—
134C2×Q8—122no—
135C2×Q8—242no—
136C2×Q8GL(2,𝔽3)14yes—
137C2×Q8D3×Q812yes—
138C2×Q8C2×#6711yescharacteristic
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—62no—
143C22×C4—62no—
144C22×C4C2×S412yes—
145C22×C4C2×S411yescharacteristic
146C2×C8—34no—
147C2×C8—62no—
148C2×C8—62no—
149C2×C8—62no—
150C2×C8—64no—
151C4⋊C4—32no—
152C4⋊C4—34no—
153C4⋊C4—34no—
154C4⋊C4—64no—
155C4⋊C4—61no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—122no—
162C4⋊C4—128no—
163C42—32no—
164C42—32no—
165C42—62no—
166C42—62no—
167C42—122no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—41no—
173C3×C4—44no—
174C3×C4—84no—
175Dic3—48no—
176Dic3—84no—
177C23C22×S411yescharacteristic
178Q8—28no—
179Q8—32no—
180Q8—31no—
181Q8—61no—
182Q8—244no—
183Q8—244no—
184Q8#94411yescharacteristic, complemented
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—32no—
192C2×C4—62no—
193C2×C4—64no—
194C2×C4—64no—
195C2×C4—62no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4C2×GL(2,𝔽3)14yes—
202C2×C4C2×GL(2,𝔽3)12yes—
203C2×C4#98312yes—
204C2×C4GL(2,ℤ/4)12yes—
205C8—64no—
206C8—122no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3—41no—
212C22C22×GL(2,𝔽3)11yescharacteristic
213C22C2×#98311yescharacteristic
214C22Q8×S412yes—
215C22#592412yes—
216C22C2×GL(2,ℤ/4)11yescharacteristic
217C4—24no—
218C4—32no—
219C4—31no—
220C4—31no—
221C4—62no—
222C4—122no—
223C4—124no—
224C4—242no—
225C4#542114yes—
226C3—41noSylow
227C2#8542612yes—
228C2Q8×GL(2,𝔽3)12yes—
229C2#8550211yescharacteristic
230C2C2×#542111yescharacteristic
231C2#8552711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×Q82D3
hypercenterC22×Q8S4
hypocenterSL(2,𝔽3)C4⋊Q8
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
31. #859Q8semidirect
126. SL(2,𝔽3)C4⋊Q8semidirect
184. Q8#944semidirect

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×#53862
#730592
C4×#8592
#258373
C2×#9444