Finite Group Databasebeta

← #11713675

Subgroups — #11713675

Order 768.

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

normal subgroups only

This group has 467 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
1C2×#557⋊C3C211yesmaximal, characteristic
2C2×#5391C211yesmaximal, characteristic
3#73059C214yesmaximal
4C4×#859C211yesmaximal, characteristic
5#25836—31nomaximal, Sylow
6#557⋊C3—22no—
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C2×#859C2212yes—
11#5391—22no—
12#4975—41nomaximal
13C2×#557D311yescharacteristic
14C2×#498—31no—
15C2×#477—31no—
16#2567—31no—
17#2059—34no—
18#1988—31no—
19#1984—31no—
20#1865—34no—
21C4×#103—31no—
22C22×SL(2,𝔽3)C2311yescharacteristic
23C2×C3×C4⋊C4—41no—
24C2×C4⋊Dic3—41no—
25C2×Dic3⋊C4—44no—
26C2×C4×Dic3—41no—
27C4×SL(2,𝔽3)—22no—
28C4×SL(2,𝔽3)D412yes—
29#859—22no—
30#859Q814yescomplemented
31#557—22no—
32#557—62no—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8—31no—
35C2×#125—32no—
36C2×C4×Q8—31no—
37C2×C4×Q8C2×D312yes—
38C2×C4×Q8C2×D311yescharacteristic
39#498—64no—
40#477—62no—
41#477—62no—
42C2×#107—32no—
43C2×#106—34no—
44C2×C4⋊C8—31no—
45C2×#103—32no—
46C2×#103—32no—
47C2×C4×C8—31no—
48#395—62no—
49#391—61no—
50#389—34no—
51#384—61no—
52C4×C4⋊C4—31no—
53#340—32no—
54#340—34no—
55#336—34no—
56C22×C3×C4—41no—
57C22×C3×C4—42no—
58C22×Dic3—42no—
59C22×Dic3—42no—
60C2×SL(2,𝔽3)C2×D411yescharacteristic
61C2×SL(2,𝔽3)C2×Q812yes—
62C3×C4⋊C4—82no—
63C4⋊Dic3—82no—
64Dic3⋊C4—88no—
65C4×Dic3—44no—
66C22×Q8—31no—
67C22×Q8—61no—
68C22×Q8C22×D311yescharacteristic
69C22×C8—32no—
70C22×C8—32no—
71C4⋊Q8—34no—
72C4⋊Q8—62no—
73C4⋊Q8—62no—
74C4⋊Q8—62no—
75#125—64no—
76#125—64no—
77C4×Q8—22no—
78C4×Q8—32no—
79C4×Q8—62no—
80C4×Q8—62no—
81C4×Q8—62no—
82C4×Q8#6712yes—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—61no—
89C2×C4⋊C4—61no—
90C2×C4⋊C4—64no—
91C2×C4⋊C4S411yescharacteristic
92C2×C42—31no—
93C2×C42—32no—
94C2×C42—61no—
95#107—64no—
96#106—68no—
97C4⋊C8—62no—
98#103—34no—
99#103—34no—
100#103—62no—
101#103—62no—
102C4×C8—62no—
103#95—61no—
104#95—64no—
105C23×C3—41no—
106C2×C3×C4—42no—
107C2×C3×C4—42no—
108C2×C3×C4—42no—
109C2×C3×C4—82no—
110C2×C3×C4—84no—
111C2×Dic3—44no—
112C2×Dic3—44no—
113C2×Dic3—44no—
114C2×Dic3—82no—
115C2×Dic3—84no—
116SL(2,𝔽3)C4⋊Q811yescharacteristic
117C2×Q8—32no—
118C2×Q8—31no—
119C2×Q8—62no—
120C2×Q8—62no—
121C2×Q8—62no—
122C2×Q8—122no—
123C2×Q8—242no—
124C2×Q8D3×Q812yes—
125C2×Q8C2×#6711yescharacteristic
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—32no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4C2×S412yes—
132C22×C4C2×S411yescharacteristic
133C2×C8—34no—
134C2×C8—62no—
135C2×C8—62no—
136C2×C8—62no—
137C2×C8—64no—
138C4⋊C4—22no—
139C4⋊C4—32no—
140C4⋊C4—34no—
141C4⋊C4—34no—
142C4⋊C4—64no—
143C4⋊C4—61no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—64no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—122no—
150C4⋊C4—244no—
151C42—32no—
152C42—32no—
153C42—62no—
154C42—62no—
155C42—122no—
156C22×C3—42no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—42no—
160C22×C3—41no—
161C3×C4—44no—
162C3×C4—162no—
163Dic3—48no—
164Dic3—162no—
165C23C22×S411yescharacteristic
166Q8—32no—
167Q8—31no—
168Q8—61no—
169Q8—124no—
170Q8—244no—
171Q8#94411yescharacteristic
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—32no—
179C2×C4—62no—
180C2×C4—64no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—122no—
185C2×C4—122no—
186C2×C4—124no—
187C2×C4—122no—
188C2×C4#98412yes—
189C2×C4#98512yes—
190C2×C4GL(2,ℤ/4)12yes—
191C8—64no—
192C8—122no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—42no—
196C2×C3—42no—
197C2×C3—41no—
198C22Q8×S412yes—
199C22C2×#98411yescharacteristic
200C22C2×#98511yescharacteristic
201C22#592612yes—
202C22C2×GL(2,ℤ/4)11yescharacteristic
203C4—32no—
204C4—31no—
205C4—31no—
206C4—42no—
207C4—62no—
208C4—122no—
209C4—124no—
210C4—242no—
211C4#542514yes—
212C3—41noSylow
213C2#8541512yes—
214C2#8541812yes—
215C2#8550211yescharacteristic
216C2C2×#542511yescharacteristic
217C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
30. #859Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53912
#730592
C4×#8592
#258363
#49754