Finite Group Databasebeta

← #11713678

Subgroups — #11713678

Order 768.

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

normal subgroups only

This group has 483 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×#5391C211yesmaximal, characteristic
3#73062C214yesmaximal
4C4×#859C211yesmaximal, characteristic
5#25841—31nomaximal, Sylow
6Q8×SL(2,𝔽3)—22no—
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C2×#859C2212yes—
11#5391—22no—
12#5229—41nomaximal
13C2×Q82D311yescharacteristic
14C2×#500—31no—
15C2×#477—31no—
16#2567—31no—
17#2057—34no—
18#1987—31no—
19#1984—31no—
20#1863—34no—
21C4×#103—31no—
22C22×C3×Q8—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×C4⋊Dic3—41no—
25C2×C4×Dic3—41no—
26C4×SL(2,𝔽3)—22no—
27C4×SL(2,𝔽3)D412yes—
28#859—22no—
29#859Q814yescomplemented
30C3⋊#95—44no—
31Q82—22no—
32Q82—62no—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8—32no—
35C2×C4⋊Q8—31no—
36C2×C4×Q8—31no—
37C2×C4×Q8C2×D312yes—
38C2×C4×Q8C2×D311yescharacteristic
39#500—64no—
40#477—62no—
41#477—62no—
42C2×#107—32no—
43C2×#107—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—
56C2×C3×Q8—44no—
57C2×C3×Q8—82no—
58C2×C3×Q8—82no—
59C22×C3×C4—41no—
60C22×C3×C4—42no—
61C22×Dic3—42no—
62C22×Dic3—42no—
63C2×SL(2,𝔽3)C2×D411yescharacteristic
64C2×SL(2,𝔽3)C2×Q812yes—
65C4⋊Dic3—82no—
66C4×Dic3—82no—
67C22×Q8—61no—
68C22×Q8—61no—
69C22×Q8S411yescharacteristic
70C22×Q8C22×D311yescharacteristic
71C22×C8—32no—
72C22×C8—32no—
73C4⋊Q8—34no—
74C4⋊Q8—62no—
75C4⋊Q8—62no—
76C4⋊Q8—62no—
77C4⋊Q8—64no—
78C4⋊Q8—64no—
79C4×Q8—22no—
80C4×Q8—32no—
81C4×Q8—62no—
82C4×Q8—62no—
83C4×Q8—62no—
84C4×Q8#6712yes—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—61no—
92C2×C4⋊C4—64no—
93C2×C42—31no—
94C2×C42—32no—
95C2×C42—61no—
96#107—64no—
97#107—68no—
98C4⋊C8—62no—
99#103—34no—
100#103—34no—
101#103—62no—
102#103—62no—
103C4×C8—62no—
104#95—61no—
105#95—64no—
106C23×C3—41no—
107C3×Q8—88no—
108C2×C3×C4—42no—
109C2×C3×C4—44no—
110C2×C3×C4—42no—
111C2×C3×C4—42no—
112C2×C3×C4—82no—
113C2×C3×C4—82no—
114C2×Dic3—44no—
115C2×Dic3—82no—
116C2×Dic3—82no—
117C2×Dic3—82no—
118C2×Dic3—84no—
119SL(2,𝔽3)C4⋊Q811yescharacteristic
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—32no—
123C2×Q8—31no—
124C2×Q8—64no—
125C2×Q8—122no—
126C2×Q8—242no—
127C2×Q8—242no—
128C2×Q8#22614yes—
129C2×Q8D3×Q812yes—
130C2×Q8C2×#6711yescharacteristic
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4C2×S412yes—
137C22×C4C2×S411yescharacteristic
138C2×C8—34no—
139C2×C8—62no—
140C2×C8—62no—
141C2×C8—62no—
142C2×C8—64no—
143C4⋊C4—32no—
144C4⋊C4—34no—
145C4⋊C4—34no—
146C4⋊C4—64no—
147C4⋊C4—61no—
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—122no—
154C4⋊C4—244no—
155C42—32no—
156C42—32no—
157C42—62no—
158C42—62no—
159C42—122no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3—41no—
165C3×C4—44no—
166C3×C4—84no—
167Dic3—84no—
168Dic3—162no—
169C23C22×S411yescharacteristic
170Q8—28no—
171Q8—32no—
172Q8—31no—
173Q8—61no—
174Q8—244no—
175Q8—244no—
176Q8#94411yescharacteristic
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—32no—
180C2×C4—32no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—62no—
185C2×C4—64no—
186C2×C4—64no—
187C2×C4—62no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4—124no—
191C2×C4—122no—
192C2×C4C2×#22614yes—
193C2×C4C2×#22612yes—
194C2×C4#98312yes—
195C2×C4GL(2,ℤ/4)12yes—
196C8—64no—
197C8—122no—
198C2×C3—41no—
199C2×C3—42no—
200C2×C3—42no—
201C2×C3—41no—
202C2×C3—41no—
203C22C22×#22611yescharacteristic
204C22C2×#98311yescharacteristic
205C22Q8×S412yes—
206C22#592512yes—
207C22C2×GL(2,ℤ/4)11yescharacteristic
208C4—24no—
209C4—32no—
210C4—31no—
211C4—31no—
212C4—62no—
213C4—122no—
214C4—124no—
215C4—242no—
216C4#541914yes—
217C3—41noSylow
218C2Q8×#22612yes—
219C2#8543112yes—
220C2C2×#541911yescharacteristic
221C2#8550211yescharacteristic
222C2#8553011yescharacteristic
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
29. #859Q8semidirect

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×#53912
#730622
C4×#8592
#258413
#52294