Finite Group Databasebeta

← #10692739

Subgroups — #10692739

Order 768.

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

normal subgroups only

This group has 620 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
1#83174C211yesmaximal, characteristic, complemented
2#83163C211yesmaximal, characteristic
3C3×#1555C211yesmaximal, characteristic, complemented
4#70835C212yesmaximal, complemented
5#67752C212yesmaximal, complemented
6#9428—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×#932—22no—
9#5477C2211yescharacteristic
10C3×#432C2212yes—
11#5220—22no—
12#5220—24no—
13#5076C412yescomplemented
14#5074—22no—
15#5073C412yescomplemented
16#4937C2212yes—
17C4×#208C2211yescharacteristic
18C3×#331—22no—
19#4470—24no—
20#3151—31no—
21#2380—31no—
22#1555D311yescharacteristic, complemented
23#1514—32no—
24#1288—32no—
25C22×C3×D4—22no—
26C22×C3×D4D412yes—
27C23×Dic3—22no—
28C2×#235—21no—
29C3×#127—22no—
30C3×#127C2×C412yes—
31C3×#127D412yes—
32C2×C3×C42C2311yescharacteristic
33C2×#217—24no—
34C2×#217—22no—
35C2×#214—24no—
36#932—44no—
37#932—44no—
38C2×#213—24no—
39#930—22no—
40C2×#208—21no—
41#875—21no—
42#874C2×C411yescharacteristic, complemented
43C4×D12—22no—
44#869C2×C411yescharacteristic, complemented
45C3×C4⋊C8—22no—
46C3×C22⋊C8—22no—
47C3×#97—22no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95D414yes—
51#803—21no—
52C4×C3⋊C8C2×C412yescomplemented
53C2×AGL(1,ℤ/8)—62no—
54#532—31no—
55C2×#127C2×D311yescharacteristic
56#496—62no—
57#493—32no—
58#492—32no—
59#432C2×D312yes—
60C4×M16—31no—
61#386—64no—
62#386—62no—
63#383—32no—
64#331—22no—
65#327—64no—
66C24×C3—22no—
67C2×C3×D4—24no—
68C2×C3×D4—24no—
69C2×C3×D4—22no—
70C2×C3×D4—42no—
71C2×C3×D4—42no—
72C2×C3×D4—82no—
73C2×C3×D4—82no—
74C2×C3×D4C22⋊C414yes—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D412yes—
77C2×#67—41no—
78C22×Dic3—24no—
79C22×Dic3—84no—
80C22×Dic3—82no—
81#235—42no—
82#235—42no—
83C2×D12—21no—
84C2×D12—22no—
85C2×C4×D3—41no—
86C2×Dic6—21no—
87C2×C3×C8—42no—
88C3×C42C22×C411yescharacteristic
89C3×C42C2×D411yescharacteristic
90C3×C42#4112yes—
91#217—84no—
92#217—82no—
93#214—48no—
94#213—48no—
95(C2×D3)⋊C4—42no—
96C4⋊Dic3—22no—
97#208—24no—
98C2×C3⋊C8—22no—
99C2×#41—61no—
100C22×D4—22no—
101C22×D4D1212yes—
102C23×C4—62no—
103AGL(1,ℤ/8)—124no—
104AGL(1,ℤ/8)—124no—
105C2×SD16—64no—
106C2×D8—64no—
107C2×M16—61no—
108C4⋊Q8—31no—
109#127—22no—
110#127—31no—
111#127C4×D312yes—
112#127#6712yes—
113#124—61no—
114#121—62no—
115C4×D4—62no—
116C2×C22⋊C4—64no—
117C2×C22⋊C4—62no—
118C2×C42C22×D311yescharacteristic
119C4⋊C8—22no—
120C22⋊C8—22no—
121#97—22no—
122#97—61no—
123C4×C8—32no—
124#95—34no—
125#95—62no—
126#95—62no—
127C23×C3—24no—
128C23×C3—44no—
129C23×C3—44no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3#12011yescharacteristic
133C22×D3—41no—
134C3×D4—44no—
135C3×D4—82no—
136C3×D4—84no—
137C3×D4—84no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4C2×C22⋊C411yescharacteristic
141C2×C3×C4C4×D412yes—
142C2×C3×C4#12112yes—
143C2×C3×C4#12311yescharacteristic
144#67—82no—
145C2×Dic3—48no—
146C2×Dic3—41no—
147C2×Dic3—44no—
148C2×Dic3—88no—
149D12—42no—
150D12—82no—
151C4×D3—81no—
152C4×D3—81no—
153Dic6—42no—
154C3×C8—82no—
155C3⋊C8—24no—
156C24—22no—
157#41—122no—
158#41—122no—
159C2×Q8—61no—
160C2×D4—24no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—62no—
166C2×D4—61no—
167C2×D4—82no—
168C2×D4—82no—
169C2×D4—121no—
170C2×D4(C2×D3)⋊C414yes—
171C22×C4—64no—
172C22×C4—121no—
173C22×C4—244no—
174C22×C4—242no—
175C22×C4C2×D1211yescharacteristic
176C22×C4D3×D412yes—
177SD16—128no—
178D8—128no—
179M16—64no—
180C2×C8—42no—
181C2×C8—62no—
182C4⋊C4—62no—
183C22⋊C4—122no—
184C22⋊C4—244no—
185C22⋊C4—242no—
186C42C2×C4×D311yescharacteristic
187C42#23712yes—
188C42C2×#6711yescharacteristic
189C22×C3—22no—
190C22×C3—44no—
191C22×C3—48no—
192C22×C3—82no—
193C22×C3—82no—
194C22×C3—84no—
195C22×C3#38611yescharacteristic
196C22×C3#45312yes—
197C2×D3—81no—
198C2×D3—82no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—22no—
202C3×C4—22no—
203Dic3—84no—
204Dic3—81no—
205C23—24no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—42no—
210C23—121no—
211C23#88211yescharacteristic
212Q8—122no—
213D4—44no—
214D4—82no—
215D4—84no—
216D4—84no—
217D4—122no—
218D4—242no—
219D4—242no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—128no—
223C2×C4—121no—
224C2×C4—124no—
225C2×C4—241no—
226C2×C4—241no—
227C2×C4—248no—
228C2×C4#67⋊C412yes—
229C2×C4#88611yescharacteristic
230C2×C4C2×(C2×D3)⋊C411yescharacteristic
231C2×C4#93912yes—
232C8—64no—
233C8—82no—
234C2×C3—22no—
235C2×C3—84no—
236C2×C3#188711yescharacteristic
237C2×C3#190412yes—
238D3—161no—
239C22—22no—
240C22—48no—
241C22—44no—
242C22—82no—
243C22—82no—
244C22—84no—
245C22—242no—
246C22—241no—
247C22#474712yes—
248C22#495811yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—244no—
254C4—241no—
255C3#942811yescharacteristic, complemented, Sylow
256C2—22no—
257C2—84no—
258C2—481no—
259C2#7007011yescharacteristic
260C2#7031412yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1555C2
socleC22×C3#386
prime radical#83163C2
cosocleC2×C3×C42C23
hypercenter#1555D3
hypocenterC3#9428
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
1. #83174C2semidirect
3. C3×#1555C2semidirect
4. #70835C2semidirect
5. #67752C2semidirect
13. #5076C4semidirect
15. #5073C4semidirect
22. #1555D3semidirect
42. #874C2×C4semidirect
44. #869C2×C4semidirect
52. C4×C3⋊C8C2×C4semidirect
255. C3#9428semidirect

Maximal subgroups

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

subgroupindex
#831742
#831632
C3×#15552
#708352
#677522
#94283