Finite Group Databasebeta

← #11713941

Subgroups — #11713941

Order 768.

1320 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C24×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5419C213yesmaximal, complemented
3#73148C213yesmaximal, complemented
4#32882—31nomaximal, Sylow
5C23×SL(2,𝔽3)—26no—
6C23×SL(2,𝔽3)C2213yes—
7C22×#226C2211yescharacteristic, complemented
8#5419—26no—
9C2×#859C2213yescomplemented
10#5247—41nomaximal
11C24×Q8D311yescharacteristic
12C2×#451—33no—
13C2×#451—33no—
14#2848—31no—
15#1787—33no—
16#1787—33no—
17#1779—31no—
18C25×C3—41no—
19C22×SL(2,𝔽3)—28no—
20C22×SL(2,𝔽3)—26no—
21C22×SL(2,𝔽3)D416yes—
22C22×SL(2,𝔽3)C2311yescharacteristic
23C2×#226—23no—
24C2×#217—43no—
25C2×#217—43no—
26#859—23no—
27C23×Q8—26no—
28C23×Q8—33no—
29C23×Q8—66no—
30C23×Q8C2×D313yes—
31C24×C4—31no—
32C22×Q16—33no—
33C22×Q16—31no—
34C2×C22⋊Q8—33no—
35#451—66no—
36#451—612no—
37#451—66no—
38C2×#103—36no—
39C2×#103—33no—
40C2×#103—33no—
41C2×C22⋊C8—33no—
42C2×C22⋊C8—33no—
43#385—63no—
44#380—33no—
45#379—61no—
46C24×C3—44no—
47C24×C3—43no—
48C24×C3—86no—
49C24×C3—86no—
50C22×Dic3—41no—
51C22×Dic3—43no—
52C2×SL(2,𝔽3)—26no—
53C2×SL(2,𝔽3)C2×D413yes—
54#226—41no—
55#217—86no—
56#217—86no—
57C25S411yescharacteristic
58C22×Q8—28no—
59C22×Q8—26no—
60C22×Q8—33no—
61C22×Q8—36no—
62C22×Q8—312no—
63C22×Q8—31no—
64C22×Q8—61no—
65C22×Q8—66no—
66C22×Q8—612no—
67C22×Q8—616no—
68C22×Q8—66no—
69C22×Q8—612no—
70C22×Q8—624no—
71C22×Q8#6716yes—
72C22×Q8C22×D311yescharacteristic
73C23×C4—33no—
74C23×C4—33no—
75C23×C4—66no—
76C23×C4—66no—
77C2×Q16—66no—
78C2×Q16—63no—
79C2×Q16—612no—
80C2×Q16—63no—
81C22×C8—33no—
82C22×C8—31no—
83C22⋊Q8—66no—
84C22⋊Q8—66no—
85C2×C4⋊C4—33no—
86C2×C4⋊C4—63no—
87C2×C22⋊C4—63no—
88C2×C22⋊C4—63no—
89#103—66no—
90#103—63no—
91#103—612no—
92#103—63no—
93C22⋊C8—66no—
94C22⋊C8—66no—
95#95—63no—
96C23×C3—46no—
97C23×C3—46no—
98C23×C3—46no—
99C23×C3—41no—
100C23×C3—86no—
101C23×C3—86no—
102C23×C3—824no—
103C23×C3—88no—
104C23×C3—824no—
105C2×Dic3—83no—
106C2×Dic3—83no—
107C2×Dic3—86no—
108SL(2,𝔽3)#12011yescharacteristic
109C24—26no—
110C24—26no—
111C24#22614yescomplemented
112C24C2×S413yes—
113C2×Q8—26no—
114C2×Q8—36no—
115C2×Q8—33no—
116C2×Q8—312no—
117C2×Q8—63no—
118C2×Q8—612no—
119C2×Q8—66no—
120C2×Q8—63no—
121C2×Q8—63no—
122C2×Q8—612no—
123C2×Q8—648no—
124C2×Q8—624no—
125C2×Q8—66no—
126C2×Q8—612no—
127C2×Q8—624no—
128C2×Q8C2×#6713yes—
129C22×C4—31no—
130C22×C4—36no—
131C22×C4—36no—
132C22×C4—33no—
133C22×C4—66no—
134C22×C4—66no—
135C22×C4—612no—
136C22×C4—68no—
137C22×C4—624no—
138C22×C4—66no—
139C22×C4—63no—
140C22×C4—61no—
141Q16—123no—
142Q16—1212no—
143Q16—121no—
144C2×C8—66no—
145C2×C8—63no—
146C2×C8—63no—
147C4⋊C4—63no—
148C4⋊C4—63no—
149C4⋊C4—126no—
150C22⋊C4—126no—
151C22⋊C4—126no—
152C22×C3—41no—
153C22×C3—43no—
154C22×C3—43no—
155C22×C3—412no—
156C22×C3—812no—
157C22×C3—86no—
158C22×C3—812no—
159C22×C3—86no—
160C22×C3—832no—
161Dic3—163no—
162Dic3—161no—
163C23—28no—
164C23—224no—
165C23—26no—
166C23—26no—
167C23—224no—
168C23C2×#22616yes—
169C23#98316yes—
170C23GL(2,ℤ/4)16yes—
171C23C22×S411yescharacteristic
172Q8—33no—
173Q8—312no—
174Q8—61no—
175Q8—66no—
176Q8—612no—
177Q8—616no—
178Q8—123no—
179Q8—121no—
180Q8#95311yescharacteristic
181C2×C4—33no—
182C2×C4—33no—
183C2×C4—36no—
184C2×C4—66no—
185C2×C4—612no—
186C2×C4—66no—
187C2×C4—624no—
188C2×C4—66no—
189C2×C4—126no—
190C2×C4—123no—
191C2×C4—123no—
192C8—123no—
193C8—121no—
194C2×C3—43no—
195C2×C3—41no—
196C2×C3—43no—
197C2×C3—812no—
198C22—232no—
199C22—212no—
200C22—26no—
201C22—26no—
202C22—212no—
203C22#5419112yescomplemented
204C22C22×#22611yescharacteristic
205C22C2×#98313yes—
206C22C2×GL(2,ℤ/4)13yes—
207C4—33no—
208C4—31no—
209C4—66no—
210C4—243no—
211C4—241no—
212C3—41noSylow
213C2—212no—
214C2C2×#541913yes—
215C2#8550713yes—
216C2#8554611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC24×Q8D3
prime radicalC24×SL(2,𝔽3)C2
hypercenterC25S4
hypocenterSL(2,𝔽3)#120
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
2. C2×#5419C2semidirect
3. #73148C2semidirect
7. C22×#226C22semidirect
9. C2×#859C22semidirect
111. C24#226semidirect
203. C22#5419semidirect

Maximal subgroups

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

subgroupindex
C24×SL(2,𝔽3)2
C2×#54192
#731482
#328823
#52474