Finite Group Databasebeta

← #11713699

Subgroups — #11713699

Order 768.

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

normal subgroups only

This group has 595 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×#5391C211yesmaximal, characteristic, complemented
3#73148C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25888—31nomaximal, Sylow
6C23×SL(2,𝔽3)C2212yes—
7D4×SL(2,𝔽3)—22no—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C2×#859C2212yes—
11#5391—22no—
12#5221—41nomaximal
13C2×D4×Q8D311yescharacteristic
14C2×#487—31no—
15C2×#477—31no—
16#2563—31no—
17#2036—34no—
18#1981—31no—
19#1954—31no—
20#1787—34no—
21C4×#103—31no—
22C22×C3×D4—41no—
23C22×SL(2,𝔽3)—22no—
24C22×SL(2,𝔽3)—24no—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×#217—44no—
27C2×C4⋊Dic3—41no—
28C2×C4×Dic3—41no—
29C4×SL(2,𝔽3)D412yes—
30#859—22no—
31#859—22no—
32C23×Q8C2×D312yes—
33D4×Q8—22no—
34D4×Q8—62no—
35C2×C4⋊Q8—31no—
36C2×C4⋊Q8—31no—
37C2×C22⋊Q8—32no—
38C2×C4×Q8C2×D311yescharacteristic
39C2×C4×D4—31no—
40#487—64no—
41#477—62no—
42#477—62no—
43C2×#107—32no—
44C2×C4⋊C8—31no—
45C2×#103—34no—
46C2×#103—32no—
47C2×#103—32no—
48C2×#103—34no—
49C2×C22⋊C8—34no—
50C2×C4×C8—31no—
51#393—62no—
52#390—61no—
53#382—61no—
54#380—34no—
55C4×C4⋊C4—31no—
56#340—32no—
57C24×C3—42no—
58C2×C3×D4—44no—
59C2×C3×D4—82no—
60C2×C3×D4—82no—
61C22×C3×C4—41no—
62C22×Dic3—42no—
63C22×Dic3—42no—
64C2×SL(2,𝔽3)—42no—
65C2×SL(2,𝔽3)C2×D411yescharacteristic
66C2×SL(2,𝔽3)#4112yes—
67#217—88no—
68C4⋊Dic3—82no—
69C4×Dic3—82no—
70C22×Q8—24no—
71C22×Q8—22no—
72C22×Q8—32no—
73C22×Q8—61no—
74C22×Q8—62no—
75C22×Q8—64no—
76C22×Q8C22×D311yescharacteristic
77C22×D4S411yescharacteristic
78C23×C4—32no—
79C22×C8—32no—
80C22×C8—32no—
81C4⋊Q8—34no—
82C4⋊Q8—62no—
83C4⋊Q8—62no—
84C4⋊Q8—62no—
85C22⋊Q8—64no—
86C22⋊Q8—64no—
87C4×Q8—32no—
88C4×Q8#6712yes—
89C4×D4—62no—
90C4×D4—62no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4—61no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—64no—
99C2×C42—31no—
100C2×C42—61no—
101#107—64no—
102C4⋊C8—62no—
103#103—62no—
104#103—62no—
105#103—68no—
106#103—62no—
107#103—62no—
108#103—124no—
109C22⋊C8—124no—
110C4×C8—62no—
111#95—61no—
112#95—64no—
113C23×C3—44no—
114C23×C3—41no—
115C23×C3—84no—
116C23×C3—82no—
117C23×C3—84no—
118C23×C3—82no—
119C3×D4—88no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×Dic3—44no—
124C2×Dic3—82no—
125C2×Dic3—82no—
126C2×Dic3—82no—
127C2×Dic3—84no—
128SL(2,𝔽3)#12411yescharacteristic
129C24C2×S412yes—
130C2×Q8—31no—
131C2×Q8—32no—
132C2×Q8—34no—
133C2×Q8—42no—
134C2×Q8—64no—
135C2×Q8—64no—
136C2×Q8—62no—
137C2×Q8—122no—
138C2×Q8—124no—
139C2×Q8—124no—
140C2×Q8—122no—
141C2×Q8—242no—
142C2×Q8#23712yes—
143C2×Q8C2×#6711yescharacteristic
144C2×D4—22no—
145C2×D4—22no—
146C2×D4#22614yes—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—64no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—64no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4C2×S411yescharacteristic
157C2×C8—62no—
158C2×C8—62no—
159C2×C8—64no—
160C2×C8—62no—
161C2×C8—62no—
162C4⋊C4—32no—
163C4⋊C4—61no—
164C4⋊C4—64no—
165C4⋊C4—62no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C22⋊C4—64no—
172C22⋊C4—244no—
173C42—32no—
174C42—32no—
175C42—122no—
176C22×C3—42no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3—41no—
181C22×C3—84no—
182C22×C3—88no—
183C22×C3—162no—
184C22×C3—164no—
185C22×C3—162no—
186C3×C4—44no—
187Dic3—84no—
188Dic3—162no—
189C23—24no—
190C23—22no—
191C23—24no—
192C23—22no—
193C23C2×#22614yes—
194C23C22×S411yescharacteristic
195Q8—32no—
196Q8—31no—
197Q8—61no—
198Q8—68no—
199Q8—122no—
200Q8—122no—
201Q8—244no—
202Q8#93611yescharacteristic
203D4—28no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—32no—
209C2×C4—62no—
210C2×C4—64no—
211C2×C4—62no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—124no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—124no—
219C2×C4—122no—
220C2×C4C2×#22612yes—
221C2×C4#98312yes—
222C2×C4GL(2,ℤ/4)12yes—
223C8—122no—
224C8—122no—
225C2×C3—41no—
226C2×C3—42no—
227C2×C3—42no—
228C2×C3—41no—
229C2×C3—41no—
230C2×C3—164no—
231C22—28no—
232C22—24no—
233C22—42no—
234C22—44no—
235C22—42no—
236C22#591412yes—
237C22C22×#22611yescharacteristic
238C22C2×#98311yescharacteristic
239C22#592612yes—
240C22C2×GL(2,ℤ/4)11yescharacteristic
241C4—31no—
242C4—32no—
243C4—31no—
244C4—62no—
245C4—122no—
246C4—242no—
247C4—242no—
248C4#541914yes—
249C3—41noSylow
250C2—44no—
251C2#8539312yes—
252C2#8540012yes—
253C2#8549211yescharacteristic
254C2C2×#541911yescharacteristic
255C2#8554011yescharacteristic
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×#5391C2semidirect
3. #73148C2semidirect
4. C4×#859C2semidirect

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×#53912
#731482
C4×#8592
#258883
#52214