Finite Group Databasebeta

← #11713690

Subgroups — #11713690

Order 768.

531 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C4⋊#994C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic, complemented
3#73153C214yesmaximal, complemented
4C4×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5#25875—31nomaximal, Sylow
6C2×#994C2212yes—
7C4⋊(#41⋊C3)—21no—
8C4⋊(#41⋊C3)—21no—
9C4×#41⋊C3C2211yescharacteristic
10#5426C2212yes—
11C2×GU(2,𝔽3)C2212yes—
12#5391—21no—
13#5386—21no—
14#5079—41nomaximal
15#3498D311yescharacteristic
16#3141—31no—
17#3079—31no—
18#2858—31no—
19#2037—34no—
20#1965—31no—
21#1962—31no—
22#1792—34no—
23C4×#104—31no—
24#994—24no—
25C2×#41⋊C3—22no—
26C2×#41⋊C3C2311yescharacteristic
27C3×C4⋊Q8—41no—
28#869—41no—
29C4×SL(2,𝔽3)D411yescharacteristic
30C4×SL(2,𝔽3)D411yescharacteristic
31GU(2,𝔽3)—22no—
32#859—22no—
33#835—44no—
34C4×C3⋊C8—41no—
35C2×H(5,𝔽2)C2×D312yes—
36#550—21no—
37#550—21no—
38D42—61no—
39D42—61no—
40#535—32no—
41#532—31no—
42C2×C4⋊Q8—31no—
43C2×#127—31no—
44C2×#124—62no—
45C4×#41C2×D311yescharacteristic
46#489—62no—
47#487—61no—
48#486—61no—
49#477—61no—
50#476—61no—
51#475—61no—
52#474—61no—
53#428—32no—
54#423—31no—
55C2×#104—32no—
56C2×#104—34no—
57#417—34no—
58#417—32no—
59C2×#100—34no—
60C4×M16—31no—
61#390—34no—
62#390—62no—
63C43—31no—
64#339—32no—
65C2×C3×Q8—42no—
66C2×Dic6—81no—
67#41⋊C3—42no—
68#41⋊C3#4112yes—
69C2×SL(2,𝔽3)C2×D411yescharacteristic
70C3×C4⋊C4—81no—
71C3×C4⋊C4—81no—
72C3×C42—41no—
73C4⋊Dic3—42no—
74C4⋊Dic3—81no—
75C2×C3⋊C8—42no—
76H(5,𝔽2)—24no—
77H(5,𝔽2)—64no—
78C2×#41—22no—
79C2×#41—32no—
80C2×#41C22×D311yescharacteristic
81C22×Q8—61no—
82C22×D4—32no—
83C22×D4—32no—
84C22×D4—32no—
85C22×D4—62no—
86C2×M16—32no—
87C2×M16—32no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8S411yescharacteristic
93#127—31no—
94#127—61no—
95#127—61no—
96#127—61no—
97#124—32no—
98#124—128no—
99#121—61no—
100#121—61no—
101#121—62no—
102#121—62no—
103#120—64no—
104C4×Q8#6711yescharacteristic
105C4×Q8#6711yescharacteristic
106C4×D4—31no—
107C4×D4—31no—
108C4×D4—61no—
109C4×D4—61no—
110#117—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—61no—
113C2×C22⋊C4—68no—
114C2×C42—31no—
115C2×C42—32no—
116C2×C42—61no—
117C2×C42—61no—
118#107—62no—
119#106—62no—
120C4⋊C8—61no—
121C4⋊C8—61no—
122#104—62no—
123#104—62no—
124#104—124no—
125#103—62no—
126#103—64no—
127#102—62no—
128#102—64no—
129#100—124no—
130#97—61no—
131C4×C8—61no—
132C3×Q8—84no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—82no—
137C2×Dic3—82no—
138Dic6—162no—
139SL(2,𝔽3)#12411yescharacteristic, complemented
140C3⋊C8—82no—
141C24—34no—
142#41—32no—
143#41—42no—
144#41—64no—
145#41—64no—
146#41—122no—
147#41#23712yes—
148C2×Q8—121no—
149C2×Q8—124no—
150C2×Q8—121no—
151C2×Q8C2×#6711yescharacteristic
152C2×Q8C2×S412yes—
153C2×D4—31no—
154C2×D4—32no—
155C2×D4—62no—
156C2×D4—64no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—61no—
162C2×D4—61no—
163C2×D4—124no—
164C2×D4—124no—
165C2×D4—124no—
166C2×D4—124no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—62no—
173M16—62no—
174M16—64no—
175M16—62no—
176C2×C8—62no—
177C2×C8—62no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—31no—
181C4⋊C4—31no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—121no—
185C4⋊C4—121no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—61no—
189C22⋊C4—1216no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42—62no—
194C42—62no—
195C42—64no—
196C42—122no—
197C42C2×S411yescharacteristic
198C22×C3—41no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—42no—
202C3×C4—162no—
203Dic3—162no—
204C23—31no—
205C23—68no—
206C23—62no—
207C23—124no—
208C23—124no—
209Q8—24no—
210Q8—31no—
211Q8—242no—
212Q8—242no—
213Q8#93611yescharacteristic
214D4—32no—
215D4—61no—
216D4—64no—
217D4—122no—
218D4—122no—
219D4—124no—
220D4—122no—
221D4—122no—
222D4—242no—
223D4—242no—
224C2×C4—22no—
225C2×C4—31no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—64no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—124no—
238C2×C4—122no—
239C2×C4GL(2,ℤ/4)11yescharacteristic
240C2×C4GL(2,ℤ/4)11yescharacteristic
241C2×C4C22×S411yescharacteristic
242C8—122no—
243C8—122no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3—41no—
247C22—32no—
248C22—61no—
249C22—61no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22—248no—
254C22C2×GL(2,ℤ/4)11yescharacteristic
255C4—31no—
256C4—31no—
257C4—42no—
258C4—61no—
259C4—61no—
260C4—124no—
261C4—242no—
262C4#542812yes—
263C4#542912yes—
264C4#591412yes—
265C3—41noSylow
266C2—62no—
267C2—242no—
268C2#8549211yescharacteristic
269C2C2×#542811yescharacteristic
270C2C2×#542911yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3498D3
prime radicalC4⋊#994C2
hypercenterC4⋊Q8S4
hypocenterSL(2,𝔽3)#124
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
2. #85337C2semidirect
3. #73153C2semidirect
4. C4×GU(2,𝔽3)C2semidirect
139. SL(2,𝔽3)#124semidirect

Maximal subgroups

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

subgroupindex
C4⋊#9942
#853372
#731532
C4×GU(2,𝔽3)2
#258753
#50794