Finite Group Databasebeta

← #11713720

Subgroups — #11713720

Order 768.

495 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 71 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×C4⋊#226C211yesmaximal, characteristic
3#73061C214yesmaximal
4C4×#859C211yesmaximal, characteristic
5#25936—31nomaximal, Sylow
6C22×#226C2212yes—
7Q8×SL(2,𝔽3)—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C4⋊#226—22no—
12#5229—41nomaximal
13C2×Q82D311yescharacteristic
14C2×#487—31no—
15C2×#462—31no—
16#2568—31no—
17#2024—34no—
18#1987—31no—
19#1984—31no—
20#1878—34no—
21C4×#103—31no—
22C22×C3×Q8—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×#226—22no—
25C2×#226—24no—
26C2×C4⋊Dic3—41no—
27C2×C4×Dic3—41no—
28C4×SL(2,𝔽3)—22no—
29C4×SL(2,𝔽3)D412yes—
30#859—22no—
31C3⋊#95—44no—
32C22×Q16—32no—
33Q82—22no—
34Q82—62no—
35C2×C4⋊Q8—31no—
36C2×C4⋊Q8—32no—
37C2×C4⋊Q8—31no—
38C2×C4×Q8—31no—
39C2×C4×Q8C2×D312yes—
40C2×C4×Q8C2×D311yescharacteristic
41#487—64no—
42#462—62no—
43#462—62no—
44C2×C4⋊C8—31no—
45C2×#103—34no—
46C2×#103—32no—
47C2×#103—32no—
48C2×C4×C8—31no—
49#400—62no—
50#391—34no—
51#391—61no—
52#382—61no—
53C4×C4⋊C4—31no—
54#340—34no—
55#340—32no—
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×SL(2,𝔽3)C2×D411yescharacteristic
65C2×SL(2,𝔽3)#4112yes—
66#226—42no—
67C4⋊Dic3—82no—
68C4×Dic3—82no—
69C22×Q8—32no—
70C22×Q8—61no—
71C22×Q8S411yescharacteristic
72C22×Q8C22×D311yescharacteristic
73C2×Q16—64no—
74C2×Q16—62no—
75C2×Q16—64no—
76C2×Q16—62no—
77C22×C8—32no—
78C22×C8—32no—
79C4⋊Q8—34no—
80C4⋊Q8—62no—
81C4⋊Q8—62no—
82C4⋊Q8—62no—
83C4⋊Q8—64no—
84C4⋊Q8—64no—
85C4×Q8—22no—
86C4×Q8—32no—
87C4×Q8—62no—
88C4×Q8—62no—
89C4×Q8—62no—
90C4×Q8#6712yes—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—61no—
97C2×C4⋊C4—61no—
98C2×C42—31no—
99C2×C42—32no—
100C2×C42—61no—
101C4⋊C8—62no—
102#103—64no—
103#103—68no—
104#103—62no—
105#103—62no—
106C4×C8—62no—
107#95—61no—
108#95—64no—
109#95—64no—
110C23×C3—41no—
111C3×Q8—88no—
112C2×C3×C4—42no—
113C2×C3×C4—44no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—82no—
117C2×C3×C4—82no—
118C2×Dic3—44no—
119C2×Dic3—82no—
120C2×Dic3—82no—
121C2×Dic3—82no—
122C2×Dic3—84no—
123SL(2,𝔽3)#12411yescharacteristic
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—32no—
127C2×Q8—31no—
128C2×Q8—64no—
129C2×Q8—62no—
130C2×Q8—62no—
131C2×Q8—64no—
132C2×Q8—64no—
133C2×Q8—242no—
134C2×Q8#22614yescomplemented
135C2×Q8#23912yes—
136C2×Q8C2×#6711yescharacteristic
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4C2×S412yes—
143C22×C4C2×S411yescharacteristic
144Q16—124no—
145Q16—122no—
146Q16—122no—
147C2×C8—62no—
148C2×C8—62no—
149C2×C8—62no—
150C2×C8—62no—
151C2×C8—64no—
152C4⋊C4—32no—
153C4⋊C4—34no—
154C4⋊C4—64no—
155C4⋊C4—61no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—122no—
161C4⋊C4—122no—
162C42—32no—
163C42—32no—
164C42—62no—
165C42—62no—
166C42—122no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—41no—
172C3×C4—44no—
173C3×C4—84no—
174Dic3—84no—
175Dic3—162no—
176C23C22×S411yescharacteristic
177Q8—28no—
178Q8—32no—
179Q8—31no—
180Q8—61no—
181Q8—122no—
182Q8—124no—
183Q8—122no—
184Q8—244no—
185Q8#94711yescharacteristic
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—32no—
189C2×C4—32no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—62no—
194C2×C4—64no—
195C2×C4—62no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—124no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4C2×#22614yes—
202C2×C4C2×#22612yes—
203C2×C4#98312yes—
204C2×C4GL(2,ℤ/4)12yes—
205C8—122no—
206C8—122no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3—41no—
212C22C22×#22611yescharacteristic
213C22C2×#98311yescharacteristic
214C22#591912yes—
215C22#592412yes—
216C22C2×GL(2,ℤ/4)11yescharacteristic
217C4—24no—
218C4—31no—
219C4—32no—
220C4—31no—
221C4—62no—
222C4—122no—
223C4—242no—
224C4—242no—
225C4#541914yes—
226C3—41noSylow
227C2#8542812yes—
228C2#8543212yes—
229C2C2×#541911yescharacteristic
230C2#8550811yescharacteristic
231C2#8552711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×Q82D3
hypercenterC22×Q8S4
hypocenterSL(2,𝔽3)#124
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
134. C2×Q8#226semidirect

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×C4⋊#2262
#730612
C4×#8592
#259363
#52294