Finite Group Databasebeta

← #11714059

Subgroups — #11714059

Order 768.

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

normal subgroups only

This group has 565 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×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5391C211yesmaximal, characteristic, complemented
3C42×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C214yesmaximal, complemented
5#33031—31nomaximal, Sylow
6C22×GL(2,𝔽3)C2212yes—
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C4⋊GL(2,𝔽3)—22no—
11#5391—22no—
12#4939—41nomaximal
13C2×#477—31no—
14C2×#460—31no—
15#2858—31no—
16C42×Q8D311yescharacteristic, complemented
17#1987—31no—
18#1982—31no—
19#1925—34no—
20#1880—34no—
21#1831—31no—
22C22×D12—41no—
23C22×SL(2,𝔽3)C2311yescharacteristic
24C2×GL(2,𝔽3)—22no—
25C2×GL(2,𝔽3)—24no—
26C2×C3×C42—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—
33C2×C4⋊Q8—31no—
34C2×#127—31no—
35C2×#124—62no—
36C2×C4×Q8C2×D312yes—
37C2×C4×Q8C2×D311yescharacteristic
38#477—62no—
39#477—62no—
40#460—62no—
41#460—62no—
42C2×#107—32no—
43C2×C4⋊C8—31no—
44C2×C4⋊C8—31no—
45C2×#103—32no—
46C2×#102—34no—
47C2×#102—32no—
48#390—34no—
49#390—61no—
50#390—61no—
51C4×C4⋊C4—32no—
52C4×C4⋊C4—31no—
53C43—31no—
54#340—34no—
55#336—34no—
56C23×D3—42no—
57C22×C3×C4—41no—
58C22×C3×C4—42no—
59C22×Dic3—42no—
60C2×D12—44no—
61C2×D12—82no—
62C2×D12—82no—
63C2×SL(2,𝔽3)C2×D411yescharacteristic
64C2×SL(2,𝔽3)#4112yes—
65GL(2,𝔽3)—42no—
66C3×C42—82no—
67(C2×D3)⋊C4—88no—
68C4⋊Dic3—82no—
69C22×Q8—61no—
70C22×Q8C22×D311yescharacteristic
71C22×D4—32no—
72C22×D4—61no—
73C22×D4—61no—
74C2×SD16—64no—
75C2×SD16—62no—
76C2×SD16—62no—
77C2×SD16—64no—
78C22×C8—32no—
79C22×C8—32no—
80C4⋊Q8—62no—
81C4⋊Q8—62no—
82#127—62no—
83#127—62no—
84#124—128no—
85C4×Q8—22no—
86C4×Q8—32no—
87C4×Q8—62no—
88C4×Q8D1212yes—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—61no—
93C2×C22⋊C4—64no—
94C2×C22⋊C4—64no—
95C2×C42—31no—
96C2×C42—32no—
97C2×C42—32no—
98C2×C42—31no—
99C2×C42S411yescharacteristic
100#107—64no—
101C4⋊C8—62no—
102C4⋊C8—62no—
103#103—62no—
104#103—62no—
105#102—68no—
106#102—64no—
107#95—61no—
108C23×C3—41no—
109C22×D3—44no—
110C22×D3—84no—
111C22×D3—82no—
112C22×D3—84no—
113C22×D3—82no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—44no—
118C2×C3×C4—82no—
119C2×C3×C4—82no—
120C2×Dic3—82no—
121C2×Dic3—84no—
122D12—88no—
123SL(2,𝔽3)#12411yescharacteristic, complemented
124C24—62no—
125C2×Q8—31no—
126C2×Q8—32no—
127C2×Q8—122no—
128C2×Q8—124no—
129C2×Q8C2×D1211yescharacteristic
130C2×Q8#23512yes—
131C2×D4—62no—
132C2×D4—64no—
133C2×D4—62no—
134C2×D4—64no—
135C2×D4—122no—
136C2×D4—124no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4—122no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4C2×S412yes—
145C22×C4C2×S411yescharacteristic
146SD16—122no—
147SD16—124no—
148SD16—122no—
149C2×C8—62no—
150C2×C8—64no—
151C2×C8—62no—
152C2×C8—64no—
153C4⋊C4—32no—
154C4⋊C4—34no—
155C4⋊C4—61no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—122no—
160C22⋊C4—128no—
161C22⋊C4—128no—
162C42—22no—
163C42—32no—
164C42—32no—
165C42—62no—
166C42—62no—
167C42—62no—
168C42—64no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—42no—
174C2×D3—84no—
175C2×D3—88no—
176C2×D3—164no—
177C2×D3—162no—
178C2×D3—162no—
179C3×C4—44no—
180C3×C4—84no—
181Dic3—162no—
182C23—122no—
183C23—124no—
184C23—122no—
185C23—122no—
186C23—124no—
187C23C22×S411yescharacteristic
188Q8—31no—
189Q8—32no—
190Q8—61no—
191Q8—244no—
192Q8#87511yescharacteristic
193D4—122no—
194D4—122no—
195D4—124no—
196D4—244no—
197D4—244no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—122no—
210C2×C4—124no—
211C2×C4C4⋊S412yes—
212C2×C4#98312yes—
213C2×C4#98512yes—
214C2×C4#98614yes—
215C8—122no—
216C8—122no—
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—42no—
220C2×C3—42no—
221C2×C3—41no—
222D3—164no—
223C22—242no—
224C22—242no—
225C22—244no—
226C22—244no—
227C22—242no—
228C22C2×C4⋊S411yescharacteristic
229C22#591212yes—
230C22C2×#98311yescharacteristic
231C22C2×#98511yescharacteristic
232C22C2×#98612yes—
233C4—24no—
234C4—31no—
235C4—31no—
236C4—32no—
237C4—62no—
238C4—122no—
239C4—242no—
240C4#539514yes—
241C3—41noSylow
242C2—482no—
243C2#8529011yescharacteristic
244C2C2×#539511yescharacteristic
245C2#8533212yes—
246C2#8534512yes—
247C2#8534611yescharacteristic
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×#5391C2semidirect
3. C42×SL(2,𝔽3)C2semidirect
4. #73065C2semidirect
16. C42×Q8D3semidirect
123. 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×C4⋊GL(2,𝔽3)2
C2×#53912
C42×SL(2,𝔽3)2
#730652
#330313
#49394