Finite Group Databasebeta

← #11713719

Subgroups — #11713719

Order 768.

567 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 63 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, complemented
2C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3#73065C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25935—31nomaximal, Sylow
6C22×GL(2,𝔽3)C2212yes—
7#557⋊C3—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C4⋊GL(2,𝔽3)—22no—
12#4988—41nomaximal
13C2×#557D311yescharacteristic, complemented
14C2×#486—31no—
15C2×#460—31no—
16#2548—31no—
17#2025—34no—
18#1988—31no—
19#1984—31no—
20#1880—34no—
21C4×#103—31no—
22C22×D12—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×GL(2,𝔽3)—22no—
25C2×GL(2,𝔽3)—24no—
26C2×C3×C4⋊C4—41no—
27C2×(C2×D3)⋊C4—44no—
28C2×C4×Dic3—41no—
29C4×SL(2,𝔽3)—22no—
30C4×SL(2,𝔽3)D412yes—
31#859—22no—
32C22×SD16—32no—
33#557—22no—
34#557—62no—
35C2×C4⋊Q8—31no—
36C2×#127—31no—
37C2×#125—32no—
38C2×#124—61no—
39C2×C4×Q8—31no—
40C2×C4×Q8C2×D312yes—
41C2×C4×Q8C2×D311yescharacteristic
42#486—64no—
43#460—62no—
44#460—62no—
45C2×C4⋊C8—31no—
46C2×#103—32no—
47C2×#102—34no—
48C2×#102—32no—
49C2×C4×C8—31no—
50#394—62no—
51#390—34no—
52#390—61no—
53C4×C4⋊C4—31no—
54#340—34no—
55#340—32no—
56#336—34no—
57C23×D3—42no—
58C22×C3×C4—41no—
59C22×C3×C4—42no—
60C22×Dic3—42no—
61C2×D12—44no—
62C2×D12—82no—
63C2×D12—82no—
64C2×SL(2,𝔽3)C2×D411yescharacteristic
65C2×SL(2,𝔽3)#4112yes—
66GL(2,𝔽3)—42no—
67C3×C4⋊C4—82no—
68(C2×D3)⋊C4—88no—
69C4×Dic3—82no—
70C22×Q8—31no—
71C22×Q8C22×D311yescharacteristic
72C22×D4—32no—
73C22×D4—61no—
74C22×D4—61no—
75C2×SD16—64no—
76C2×SD16—62no—
77C2×SD16—64no—
78C2×SD16—62no—
79C22×C8—32no—
80C22×C8—32no—
81C4⋊Q8—34no—
82C4⋊Q8—62no—
83#127—62no—
84#127—62no—
85#125—64no—
86#125—64no—
87#124—124no—
88C4×Q8—22no—
89C4×Q8—32no—
90C4×Q8—62no—
91C4×Q8—62no—
92C4×Q8—62no—
93C4×Q8#6712yes—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—61no—
99C2×C4⋊C4S411yescharacteristic
100C2×C22⋊C4—64no—
101C2×C22⋊C4—64no—
102C2×C42—31no—
103C2×C42—32no—
104C2×C42—61no—
105C4⋊C8—62no—
106#103—62no—
107#103—62no—
108#102—64no—
109#102—68no—
110C4×C8—62no—
111#95—61no—
112C23×C3—41no—
113C22×D3—44no—
114C22×D3—84no—
115C22×D3—82no—
116C22×D3—84no—
117C22×D3—82no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—42no—
121C2×C3×C4—82no—
122C2×C3×C4—84no—
123C2×Dic3—44no—
124C2×Dic3—82no—
125C2×Dic3—82no—
126D12—88no—
127SL(2,𝔽3)#12411yescharacteristic, complemented
128C24—62no—
129C2×Q8—32no—
130C2×Q8—31no—
131C2×Q8—62no—
132C2×Q8—62no—
133C2×Q8—62no—
134C2×Q8#23912yes—
135C2×Q8C2×#6711yescharacteristic
136C2×D4—62no—
137C2×D4—62no—
138C2×D4—64no—
139C2×D4—64no—
140C2×D4—122no—
141C2×D4—122no—
142C2×D4—122no—
143C2×D4—122no—
144C2×D4—242no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4—62no—
149C22×C4C2×S412yes—
150C22×C4C2×S411yescharacteristic
151SD16—124no—
152SD16—122no—
153SD16—122no—
154C2×C8—62no—
155C2×C8—62no—
156C2×C8—62no—
157C2×C8—62no—
158C2×C8—64no—
159C4⋊C4—22no—
160C4⋊C4—32no—
161C4⋊C4—34no—
162C4⋊C4—64no—
163C4⋊C4—61no—
164C4⋊C4—62no—
165C4⋊C4—64no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C22⋊C4—128no—
169C22⋊C4—244no—
170C42—32no—
171C42—32no—
172C42—62no—
173C42—62no—
174C42—122no—
175C22×C3—42no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—42no—
179C22×C3—41no—
180C2×D3—84no—
181C2×D3—88no—
182C2×D3—162no—
183C2×D3—164no—
184C2×D3—162no—
185C3×C4—44no—
186C3×C4—162no—
187Dic3—84no—
188C23—124no—
189C23—122no—
190C23—122no—
191C23—124no—
192C23—122no—
193C23C22×S411yescharacteristic
194Q8—32no—
195Q8—31no—
196Q8—61no—
197Q8—124no—
198Q8#94711yescharacteristic
199D4—122no—
200D4—124no—
201D4—122no—
202D4—244no—
203D4—244no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—32no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4#98512yes—
219C2×C4#98612yes—
220C2×C4GL(2,ℤ/4)12yes—
221C8—122no—
222C8—122no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226C2×C3—42no—
227C2×C3—42no—
228D3—164no—
229C22—242no—
230C22—242no—
231C22—242no—
232C22—244no—
233C22—244no—
234C22#591912yes—
235C22C2×#98511yescharacteristic
236C22C2×#98611yescharacteristic
237C22#592612yes—
238C22C2×GL(2,ℤ/4)11yescharacteristic
239C4—31no—
240C4—32no—
241C4—31no—
242C4—42no—
243C4—62no—
244C4—122no—
245C4—242no—
246C4#542714yes—
247C3—41noSylow
248C2—482no—
249C2#8542012yes—
250C2#8542212yes—
251C2#8550811yescharacteristic
252C2C2×#542711yescharacteristic
253C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
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
1. C2×#557⋊C3C2semidirect
3. #73065C2semidirect
4. C4×#859C2semidirect
13. C2×#557D3semidirect
127. SL(2,𝔽3)#124semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×C4⋊GL(2,𝔽3)2
#730652
C4×#8592
#259353
#49884