Finite Group Databasebeta

← #11714056

Subgroups — #11714056

Order 768.

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

normal subgroups only

This group has 601 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×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3C42×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C214yesmaximal, complemented
5#33025—31nomaximal, Sylow
6C22×GL(2,𝔽3)C2212yes—
7C2×#875—41nomaximal
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C4⋊GL(2,𝔽3)—22no—
12#5386—22no—
13C2×#475—31no—
14C2×#460—31no—
15#2858—31no—
16C42×Q8D311yescharacteristic, complemented
17#1988—31no—
18#1982—31no—
19#1926—34no—
20#1880—34no—
21#1831—31no—
22C22×D12—41no—
23C22×Dic6—41no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25C2×GL(2,𝔽3)—22no—
26C2×GL(2,𝔽3)—24no—
27C2×C3×C42—41no—
28C2×(C2×D3)⋊C4—44no—
29#875—48no—
30C4×SL(2,𝔽3)—22no—
31C4×SL(2,𝔽3)D412yes—
32#859—22no—
33C22×SD16—32no—
34C2×C4⋊Q8—31no—
35C2×#127—31no—
36C2×#124—61no—
37C2×#124—61no—
38C2×C4×Q8C2×D312yes—
39C2×C4×Q8C2×D311yescharacteristic
40#475—62no—
41#475—62no—
42#460—62no—
43#460—62no—
44C2×#106—32no—
45C2×C4⋊C8—31no—
46C2×C4⋊C8—31no—
47C2×#103—32no—
48C2×#102—34no—
49C2×#102—32no—
50#390—34no—
51#390—62no—
52C4×C4⋊C4—32no—
53C4×C4⋊C4—31no—
54C43—31no—
55#340—34no—
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×Dic6—44no—
65C2×Dic6—82no—
66C2×Dic6—82no—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68C2×SL(2,𝔽3)#4112yes—
69GL(2,𝔽3)—42no—
70C3×C42—44no—
71(C2×D3)⋊C4—416no—
72C22×Q8—61no—
73C22×Q8C22×D311yescharacteristic
74C22×D4—32no—
75C22×D4—61no—
76C22×D4—61no—
77C2×SD16—64no—
78C2×SD16—62no—
79C2×SD16—62no—
80C2×SD16—64no—
81C22×C8—32no—
82C22×C8—32no—
83C4⋊Q8—62no—
84C4⋊Q8—62no—
85#127—62no—
86#127—62no—
87#124—124no—
88#124—124no—
89C4×Q8—22no—
90C4×Q8—32no—
91C4×Q8—62no—
92C4×Q8D1212yes—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—61no—
97C2×C22⋊C4—64no—
98C2×C22⋊C4—64no—
99C2×C42—31no—
100C2×C42—32no—
101C2×C42—31no—
102C2×C42—32no—
103C2×C42S411yescharacteristic
104#106—64no—
105C4⋊C8—62no—
106C4⋊C8—62no—
107#103—62no—
108#103—62no—
109#102—68no—
110#102—64no—
111#95—61no—
112C23×C3—41no—
113C22×D3—44no—
114C22×D3—82no—
115C22×D3—84no—
116C22×D3—82no—
117C22×D3—84no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—44no—
121C2×C3×C4—42no—
122C2×C3×C4—82no—
123C2×C3×C4—82no—
124C2×Dic3—44no—
125C2×Dic3—82no—
126C2×Dic3—82no—
127D12—88no—
128Dic6—88no—
129SL(2,𝔽3)#12411yescharacteristic, complemented
130C24—62no—
131C2×Q8—31no—
132C2×Q8—32no—
133C2×Q8—122no—
134C2×Q8—122no—
135C2×Q8—122no—
136C2×Q8C2×D1211yescharacteristic
137C2×Q8#23512yes—
138C2×D4—62no—
139C2×D4—64no—
140C2×D4—62no—
141C2×D4—64no—
142C2×D4—122no—
143C2×D4—124no—
144C2×D4—122no—
145C2×D4—122no—
146C2×D4—122no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—62no—
151C22×C4C2×S412yes—
152C22×C4C2×S411yescharacteristic
153SD16—122no—
154SD16—124no—
155SD16—122no—
156C2×C8—62no—
157C2×C8—64no—
158C2×C8—62no—
159C2×C8—64no—
160C4⋊C4—32no—
161C4⋊C4—34no—
162C4⋊C4—61no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—122no—
167C22⋊C4—128no—
168C22⋊C4—128no—
169C42—32no—
170C42—32no—
171C42—62no—
172C42—62no—
173C42—62no—
174C42—122no—
175C42GL(2,𝔽3)14yescomplemented
176C22×C3—41no—
177C22×C3—42no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—42no—
181C2×D3—84no—
182C2×D3—88no—
183C2×D3—162no—
184C2×D3—164no—
185C2×D3—162no—
186C3×C4—44no—
187C3×C4—84no—
188Dic3—84no—
189C23—122no—
190C23—124no—
191C23—122no—
192C23—122no—
193C23—124no—
194C23C22×S411yescharacteristic
195Q8—31no—
196Q8—32no—
197Q8—61no—
198Q8—244no—
199Q8#87511yescharacteristic, complemented
200D4—122no—
201D4—122no—
202D4—124no—
203D4—244no—
204D4—244no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4C4⋊S412yes—
220C2×C4C2×GL(2,𝔽3)14yes—
221C2×C4C2×GL(2,𝔽3)12yes—
222C2×C4#98612yes—
223C8—122no—
224C8—122no—
225C2×C3—41no—
226C2×C3—41no—
227C2×C3—42no—
228C2×C3—41no—
229C2×C3—42no—
230D3—164no—
231C22—242no—
232C22—242no—
233C22—244no—
234C22—242no—
235C22—244no—
236C22C2×C4⋊S411yescharacteristic
237C22#591212yes—
238C22C22×GL(2,𝔽3)11yescharacteristic
239C22C2×#98611yescharacteristic
240C22#592312yes—
241C4—24no—
242C4—31no—
243C4—31no—
244C4—32no—
245C4—62no—
246C4—122no—
247C4—242no—
248C4C4⋊GL(2,𝔽3)14yes—
249C3—41noSylow
250C2—482no—
251C2#8529011yescharacteristic
252C2C2×C4⋊GL(2,𝔽3)11yescharacteristic
253C2#8532512yes—
254C2#8534711yescharacteristic
255C2#8534812yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#5386C2semidirect
3. C42×SL(2,𝔽3)C2semidirect
4. #73065C2semidirect
16. C42×Q8D3semidirect
129. SL(2,𝔽3)#124semidirect
175. C42GL(2,𝔽3)semidirect
199. Q8#875semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊GL(2,𝔽3)2
C2×#53862
C42×SL(2,𝔽3)2
#730652
#330253
C2×#8754