Finite Group Databasebeta

← #11713669

Subgroups — #11713669

Order 768.

415 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#3503⋊C3C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic
3#73075C214yesmaximal
4C4×GU(2,𝔽3)C211yesmaximal, characteristic
5#25829—31nomaximal, Sylow
6Q8×SL(2,𝔽3)—21no—
7#557⋊C3—21no—
8#520⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10#5426C2212yes—
11C2×GU(2,𝔽3)C2212yes—
12#5391—21no—
13#5386—21no—
14#5078—41nomaximal
15#3503D311yescharacteristic
16#3165—31no—
17#3079—31no—
18#2862—31no—
19#2061—34no—
20#1997—31no—
21#1994—31no—
22#1860—34no—
23C4×#104—31no—
24C2×#41⋊C3C2311yescharacteristic
25C3×C4⋊Q8—41no—
26#869—41no—
27C4×SL(2,𝔽3)—22no—
28C4×SL(2,𝔽3)D411yescharacteristic
29C4×SL(2,𝔽3)D411yescharacteristic
30GU(2,𝔽3)Q814yescomplemented
31#859—22no—
32#807—44no—
33C4×C3⋊C8—41no—
34Q82—21no—
35#557—21no—
36#554—61no—
37D4×Q8—61no—
38#544—32no—
39#533—31no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—31no—
42#520C2×D312yes—
43C4×#41C2×D311yescharacteristic
44#501—62no—
45#500—61no—
46#498—61no—
47#477—61no—
48#476—61no—
49#475—61no—
50#474—61no—
51#428—32no—
52#428—34no—
53#423—31no—
54C2×#104—32no—
55#417—32no—
56C4×M16—31no—
57#391—62no—
58#391—62no—
59#382—34no—
60C43—31no—
61#343—34no—
62#339—32no—
63#339—34no—
64C2×C3×Q8—81no—
65C2×Dic6—81no—
66#41⋊C3C2×Q812yes—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68C3×C4⋊C4—42no—
69C3×C4⋊C4—81no—
70C3×C42—41no—
71C4⋊Dic3—42no—
72C4⋊Dic3—81no—
73C2×C3⋊C8—42no—
74C2×#41C22×D311yescharacteristic
75C22×Q8—32no—
76C22×Q8—61no—
77C2×M16—32no—
78C2×M16—32no—
79C4⋊Q8—31no—
80C4⋊Q8—61no—
81C4⋊Q8—61no—
82C4⋊Q8—61no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8—61no—
86C4⋊Q8—62no—
87C4⋊Q8—61no—
88C4⋊Q8S411yescharacteristic
89#125—32no—
90#125—62no—
91C22⋊Q8—61no—
92C22⋊Q8—61no—
93C22⋊Q8—64no—
94C4×Q8—22no—
95C4×Q8—61no—
96C4×Q8—61no—
97C4×Q8#6711yescharacteristic
98C4×Q8#6711yescharacteristic
99C4×D4—31no—
100C4×D4—31no—
101C4×D4—62no—
102#117—31no—
103#117—32no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—61no—
108C2×C42—31no—
109C2×C42—32no—
110C2×C42—61no—
111C2×C42—61no—
112#107—62no—
113#107—64no—
114#106—62no—
115#106—64no—
116C4⋊C8—61no—
117C4⋊C8—61no—
118#104—34no—
119#104—34no—
120#103—62no—
121#102—62no—
122#97—61no—
123C4×C8—61no—
124#95—68no—
125C3×Q8—162no—
126C2×C3×C4—41no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—82no—
130C2×Dic3—82no—
131Dic6—162no—
132SL(2,𝔽3)C4⋊Q811yescharacteristic
133C3⋊C8—44no—
134#41—32no—
135#41D3×Q812yes—
136C2×Q8—21no—
137C2×Q8—62no—
138C2×Q8—61no—
139C2×Q8—121no—
140C2×Q8—124no—
141C2×Q8—124no—
142C2×Q8—121no—
143C2×Q8C2×#6711yescharacteristic
144C2×D4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—34no—
149C22×C4—62no—
150C22×C4—62no—
151M16—34no—
152M16—62no—
153M16—64no—
154C2×C8—62no—
155C2×C8—62no—
156C4⋊C4—21no—
157C4⋊C4—31no—
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—61no—
166C4⋊C4—62no—
167C4⋊C4—121no—
168C4⋊C4—121no—
169C4⋊C4C2×S412yes—
170C22⋊C4—62no—
171C22⋊C4—61no—
172C42—31no—
173C42—31no—
174C42—31no—
175C42—34no—
176C42—62no—
177C42—62no—
178C42—122no—
179C42—122no—
180C42C2×S411yescharacteristic
181C22×C3—41no—
182C3×C4—42no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4—162no—
186Dic3—162no—
187C23—31no—
188Q8—31no—
189Q8—42no—
190Q8—122no—
191Q8—242no—
192Q8—242no—
193Q8—242no—
194Q8#94411yescharacteristic
195D4—32no—
196D4—61no—
197C2×C4—22no—
198C2×C4—31no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—64no—
206C2×C4—62no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4—124no—
210C2×C4—124no—
211C2×C4—122no—
212C2×C4GL(2,ℤ/4)11yescharacteristic
213C2×C4GL(2,ℤ/4)11yescharacteristic
214C2×C4C22×S411yescharacteristic
215C8—64no—
216C8—122no—
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—41no—
220C22—32no—
221C22—61no—
222C22—61no—
223C22C2×GL(2,ℤ/4)11yescharacteristic
224C4—31no—
225C4—31no—
226C4—42no—
227C4—61no—
228C4—61no—
229C4—124no—
230C4—122no—
231C4—242no—
232C4#542812yes—
233C4#543112yes—
234C4Q8×S412yes—
235C3—41noSylow
236C2—62no—
237C2#8550211yescharacteristic
238C2C2×#542811yescharacteristic
239C2C2×#543111yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3503D3
hypercenterC4⋊Q8S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
30. GU(2,𝔽3)Q8semidirect

Maximal subgroups

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

subgroupindex
#3503⋊C32
#853372
#730752
C4×GU(2,𝔽3)2
#258293
#50784