Finite Group Databasebeta

← #11713712

Subgroups — #11713712

Order 768.

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

normal subgroups only

This group has 507 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
1#3503⋊C3C211yesmaximal, characteristic, complemented
2#85346C211yesmaximal, characteristic
3#73078C214yesmaximal, complemented
4C4×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5#25924—31nomaximal, Sylow
6C2×#986C2212yes—
7Q8×SL(2,𝔽3)—21no—
8#557⋊C3—21no—
9#520⋊C3C2212yes—
10C4×#41⋊C3C2211yescharacteristic
11C2×GU(2,𝔽3)C2212yes—
12C4⋊GL(2,𝔽3)—21no—
13C4⋊GL(2,𝔽3)—21no—
14#5088—41nomaximal
15#3503D311yescharacteristic, complemented
16#3138—31no—
17#3038—31no—
18#2858—31no—
19#2027—34no—
20#1997—31no—
21#1994—31no—
22#1883—34no—
23C4×#104—31no—
24C2×#41⋊C3C2311yescharacteristic
25#986—24no—
26C2×GL(2,𝔽3)—22no—
27C3×C4⋊Q8—41no—
28#874—41no—
29C4×SL(2,𝔽3)—22no—
30C4×SL(2,𝔽3)D411yescharacteristic
31C4×SL(2,𝔽3)D411yescharacteristic
32GU(2,𝔽3)—22no—
33#809—44no—
34C4×C3⋊C8—41no—
35C2×AGL(1,ℤ/8)—32no—
36Q82—21no—
37#557—21no—
38#554—61no—
39D4×Q8—61no—
40#544—32no—
41#533—31no—
42C2×C4⋊Q8—31no—
43C2×#127—31no—
44C2×#124—62no—
45#520C2×D312yes—
46C4×#41C2×D311yescharacteristic
47#490—62no—
48#486—61no—
49#486—61no—
50#460—61no—
51#460—61no—
52#459—61no—
53#459—61no—
54#423—31no—
55C2×#104—32no—
56#418—34no—
57#418—32no—
58C4×M16—31no—
59#390—34no—
60#390—62no—
61C43—31no—
62#343—34no—
63#339—34no—
64#339—32no—
65C2×C3×Q8—81no—
66C2×D12—42no—
67C2×D12—81no—
68C2×D12—81no—
69#41⋊C3#4112yes—
70C2×SL(2,𝔽3)C2×D411yescharacteristic
71GL(2,𝔽3)—42no—
72C3×C4⋊C4—42no—
73C3×C4⋊C4—81no—
74C3×C42—41no—
75C2×C3⋊C8—42no—
76C2×#41C22×D311yescharacteristic
77C22×Q8—32no—
78C22×D4—32no—
79C22×D4—61no—
80C22×D4—61no—
81AGL(1,ℤ/8)—64no—
82AGL(1,ℤ/8)—64no—
83C2×SD16—62no—
84C2×D8—62no—
85C2×M16—32no—
86C2×M16—32no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—62no—
92C4⋊Q8S411yescharacteristic
93#127—61no—
94#127—61no—
95#127—61no—
96#127—61no—
97#125—32no—
98#125—62no—
99#124—128no—
100C22⋊Q8—61no—
101C22⋊Q8—61no—
102C22⋊Q8—64no—
103C4×Q8—22no—
104C4×Q8—61no—
105C4×Q8—61no—
106C4×Q8#6711yescharacteristic
107C4×Q8#6711yescharacteristic
108C4×D4—31no—
109C4×D4—31no—
110C4×D4—62no—
111#117—31no—
112#117—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C22⋊C4—68no—
116C2×C42—31no—
117C2×C42—32no—
118C2×C42—61no—
119C2×C42—61no—
120C4⋊C8—61no—
121C4⋊C8—61no—
122#104—62no—
123#104—62no—
124#102—62no—
125#102—62no—
126#102—64no—
127#102—64no—
128#97—61no—
129C4×C8—61no—
130C22×D3—82no—
131C3×Q8—162no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—82no—
136D12—84no—
137D12—162no—
138D12—162no—
139SL(2,𝔽3)#12411yescharacteristic, complemented
140C3⋊C8—82no—
141C24—62no—
142#41—32no—
143#41#23912yes—
144C2×Q8—21no—
145C2×Q8—62no—
146C2×Q8—61no—
147C2×Q8—124no—
148C2×Q8C2×#6711yescharacteristic
149C2×D4—31no—
150C2×D4—62no—
151C2×D4—64no—
152C2×D4—64no—
153C2×D4—62no—
154C2×D4—121no—
155C2×D4—121no—
156C2×D4—124no—
157C2×D4—124no—
158C2×D4—121no—
159C2×D4—121no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—34no—
164C22×C4—62no—
165SD16—122no—
166SD16—122no—
167D8—124no—
168M16—62no—
169M16—62no—
170M16—64no—
171C2×C8—62no—
172C2×C8—62no—
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—32no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—61no—
182C4⋊C4C2×S412yes—
183C22⋊C4—62no—
184C22⋊C4—61no—
185C22⋊C4—1216no—
186C42—31no—
187C42—31no—
188C42—31no—
189C42—62no—
190C42—62no—
191C42—62no—
192C42—64no—
193C42—122no—
194C42C2×S411yescharacteristic
195C22×C3—41no—
196C2×D3—162no—
197C2×D3—162no—
198C2×D3—162no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—42no—
202C3×C4—162no—
203C23—31no—
204C23—124no—
205C23—124no—
206C23—124no—
207C23—122no—
208Q8—31no—
209Q8—42no—
210Q8—122no—
211Q8—242no—
212Q8#94711yescharacteristic
213D4—32no—
214D4—61no—
215D4—122no—
216D4—122no—
217D4—124no—
218D4—242no—
219D4—242no—
220D4—242no—
221D4—242no—
222C2×C4—22no—
223C2×C4—31no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—64no—
231C2×C4—62no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—124no—
235C2×C4GL(2,ℤ/4)11yescharacteristic
236C2×C4GL(2,ℤ/4)11yescharacteristic
237C2×C4C22×S411yescharacteristic
238C8—122no—
239C8—122no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—41no—
243D3—322no—
244C22—32no—
245C22—61no—
246C22—61no—
247C22—248no—
248C22—242no—
249C22—242no—
250C22—242no—
251C22C2×GL(2,ℤ/4)11yescharacteristic
252C4—31no—
253C4—31no—
254C4—42no—
255C4—61no—
256C4—61no—
257C4—124no—
258C4—122no—
259C4#542912yes—
260C4#543112yes—
261C4#591912yes—
262C3—41noSylow
263C2—62no—
264C2—482no—
265C2#8550811yescharacteristic
266C2C2×#542911yescharacteristic
267C2C2×#543111yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3503D3
prime radical#85346C2
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
1. #3503⋊C3C2semidirect
3. #73078C2semidirect
4. C4×GU(2,𝔽3)C2semidirect
15. #3503D3semidirect
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
#3503⋊C32
#853462
#730782
C4×GU(2,𝔽3)2
#259243
#50884