Finite Group Databasebeta

← #11713697

Subgroups — #11713697

Order 768.

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

normal subgroups only

This group has 641 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3#73148C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25884—31nomaximal, Sylow
6C23×SL(2,𝔽3)C2212yes—
7C2×#936—41nomaximal
8D4×SL(2,𝔽3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2212yes—
12#5386—22no—
13C2×D4×Q8D311yescharacteristic
14C2×#487—31no—
15C2×#475—31no—
16#2547—31no—
17#2035—34no—
18#1981—31no—
19#1954—31no—
20#1787—34no—
21C4×#103—31no—
22C22×C3×D4—41no—
23C22×Dic6—41no—
24C22×SL(2,𝔽3)—22no—
25C22×SL(2,𝔽3)—24no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×#217—44no—
28#936—48no—
29C2×C4×Dic3—41no—
30C4×SL(2,𝔽3)D412yes—
31#859—22no—
32#859—22no—
33C23×Q8C2×D312yes—
34D4×Q8—22no—
35D4×Q8—62no—
36C2×C4⋊Q8—31no—
37C2×C4⋊Q8—31no—
38C2×#124—61no—
39C2×C22⋊Q8—32no—
40C2×C4×Q8C2×D311yescharacteristic
41C2×C4×D4—31no—
42#487—64no—
43#475—62no—
44#475—62no—
45C2×#106—32no—
46C2×C4⋊C8—31no—
47C2×#103—34no—
48C2×#103—32no—
49C2×#103—32no—
50C2×#103—34no—
51C2×C22⋊C8—34no—
52C2×C4×C8—31no—
53#397—62no—
54#391—61no—
55#380—34no—
56C4×C4⋊C4—31no—
57#340—32no—
58C24×C3—42no—
59C2×C3×D4—44no—
60C2×C3×D4—82no—
61C2×C3×D4—82no—
62C22×C3×C4—41no—
63C22×Dic3—42no—
64C22×Dic3—42no—
65C2×Dic6—44no—
66C2×Dic6—82no—
67C2×Dic6—82no—
68C2×SL(2,𝔽3)—42no—
69C2×SL(2,𝔽3)C2×D411yescharacteristic
70C2×SL(2,𝔽3)#4112yes—
71#217—416no—
72C4×Dic3—44no—
73C22×Q8—24no—
74C22×Q8—22no—
75C22×Q8—32no—
76C22×Q8—61no—
77C22×Q8—62no—
78C22×Q8—64no—
79C22×Q8C22×D311yescharacteristic
80C22×D4S411yescharacteristic
81C23×C4—32no—
82C22×C8—32no—
83C22×C8—32no—
84C4⋊Q8—34no—
85C4⋊Q8—62no—
86C4⋊Q8—62no—
87C4⋊Q8—62no—
88#124—124no—
89C22⋊Q8—64no—
90C22⋊Q8—64no—
91C4×Q8—32no—
92C4×Q8#6712yes—
93C4×D4—62no—
94C4×D4—62no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—61no—
100C2×C4⋊C4—61no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—64no—
103C2×C42—31no—
104C2×C42—61no—
105#106—64no—
106C4⋊C8—62no—
107#103—62no—
108#103—62no—
109#103—68no—
110#103—62no—
111#103—62no—
112#103—124no—
113C22⋊C8—124no—
114C4×C8—62no—
115#95—61no—
116#95—64no—
117C23×C3—44no—
118C23×C3—41no—
119C23×C3—82no—
120C23×C3—84no—
121C23×C3—82no—
122C23×C3—84no—
123C3×D4—88no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×Dic3—44no—
128C2×Dic3—44no—
129C2×Dic3—82no—
130C2×Dic3—82no—
131C2×Dic3—82no—
132C2×Dic3—82no—
133Dic6—88no—
134SL(2,𝔽3)#12411yescharacteristic, complemented
135C24C2×S412yes—
136C2×Q8—31no—
137C2×Q8—32no—
138C2×Q8—34no—
139C2×Q8—42no—
140C2×Q8—64no—
141C2×Q8—64no—
142C2×Q8—62no—
143C2×Q8—124no—
144C2×Q8—122no—
145C2×Q8—122no—
146C2×Q8—122no—
147C2×Q8—124no—
148C2×Q8—122no—
149C2×Q8#23712yes—
150C2×Q8C2×#6711yescharacteristic
151C2×D4—22no—
152C2×D4—22no—
153C2×D4GL(2,𝔽3)14yes—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—64no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4—64no—
161C22×C4—62no—
162C22×C4—62no—
163C22×C4C2×S411yescharacteristic
164C2×C8—62no—
165C2×C8—62no—
166C2×C8—64no—
167C2×C8—62no—
168C2×C8—62no—
169C4⋊C4—32no—
170C4⋊C4—61no—
171C4⋊C4—64no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4—122no—
178C22⋊C4—64no—
179C22⋊C4—128no—
180C42—32no—
181C42—32no—
182C42—122no—
183C22×C3—42no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—84no—
189C22×C3—88no—
190C22×C3—162no—
191C22×C3—164no—
192C22×C3—162no—
193C3×C4—44no—
194Dic3—84no—
195Dic3—84no—
196C23—24no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23C2×GL(2,𝔽3)14yes—
201C23C22×S411yescharacteristic
202Q8—32no—
203Q8—31no—
204Q8—61no—
205Q8—68no—
206Q8—122no—
207Q8—122no—
208Q8—244no—
209Q8#93611yescharacteristic, complemented
210D4—28no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—32no—
216C2×C4—62no—
217C2×C4—64no—
218C2×C4—62no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—124no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4C2×GL(2,𝔽3)12yes—
229C2×C4#98312yes—
230C2×C4GL(2,ℤ/4)12yes—
231C8—122no—
232C8—122no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—42no—
236C2×C3—42no—
237C2×C3—41no—
238C2×C3—164no—
239C22—24no—
240C22—28no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22#591412yes—
245C22C22×GL(2,𝔽3)11yescharacteristic
246C22C2×#98311yescharacteristic
247C22#592712yes—
248C22C2×GL(2,ℤ/4)11yescharacteristic
249C4—31no—
250C4—32no—
251C4—31no—
252C4—62no—
253C4—122no—
254C4—242no—
255C4—242no—
256C4#542114yes—
257C3—41noSylow
258C2—44no—
259C2#8539512yes—
260C2#8539712yes—
261C2#8549211yescharacteristic
262C2C2×#542111yescharacteristic
263C2#8553611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×D4×Q8D3
prime radicalC2×D4×SL(2,𝔽3)C2
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5386C2semidirect
3. #73148C2semidirect
4. C4×#859C2semidirect
134. SL(2,𝔽3)#124semidirect
209. Q8#936semidirect

Maximal subgroups

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

subgroupindex
C2×D4×SL(2,𝔽3)2
C2×#53862
#731482
C4×#8592
#258843
C2×#9364