Finite Group Databasebeta

← #10692741

Subgroups — #10692741

Order 768.

620 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2380C211yesmaximal, characteristic, complemented
2#83163C211yesmaximal, characteristic
3#71760C211yesmaximal, characteristic, complemented
4#69564C212yesmaximal, complemented
5#67633C212yesmaximal, complemented
6#9428—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×#932—22no—
9#5477C2211yescharacteristic
10C3×#386—22no—
11C3×#386—24no—
12C3×#383C2212yes—
13#5076C412yescomplemented
14#5074—22no—
15#5073C412yescomplemented
16#4926C2212yes—
17C4×#208C2211yescharacteristic
18#4544—22no—
19#4451—24no—
20#3151—31no—
21#2380D311yescharacteristic, complemented
22#1555—31no—
23#1514—32no—
24#1288—32no—
25C22×C3×D4—22no—
26C22×C3×D4D412yes—
27C23×C3×C4—22no—
28C2×#235—21no—
29C3×#127—22no—
30C3×#127C2×C412yes—
31C3×#127D412yes—
32C2×C3×C22⋊C4—24no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C42C2311yescharacteristic
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×#95—22no—
46C3×#95—22no—
47C3×#95D414yes—
48#830—22no—
49C4⋊(C3⋊C8)—22no—
50#803—21no—
51#803—22no—
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#432—32no—
60C4×M16—31no—
61#386—22no—
62#386—24no—
63#383C2×D312yes—
64#331—62no—
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×C4—24no—
76C22×C3×C4—84no—
77C22×C3×C4—82no—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D412yes—
80C2×#67—41no—
81#235—42no—
82#235—42no—
83C2×D12—21no—
84C2×D12—22no—
85C2×C4×D3—41no—
86C2×Dic6—21no—
87C3×C22⋊C4—84no—
88C3×C22⋊C4—82no—
89C3×C42C22×C411yescharacteristic
90C3×C42C2×D411yescharacteristic
91C3×C42#4112yes—
92#214—48no—
93#213—48no—
94(C2×D3)⋊C4—42no—
95C4⋊Dic3—22no—
96#208—24no—
97C2×C3⋊C8—22no—
98C2×C3⋊C8—42no—
99C2×#41—61no—
100C22×D4—22no—
101C22×D4#6712yes—
102C23×C4—22no—
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#127D1212yes—
113#124—61no—
114#121—62no—
115C4×D4—62no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—22no—
118C2×C42C22×D311yescharacteristic
119C4⋊C8—62no—
120C22⋊C8—62no—
121#97—61no—
122#97—62no—
123C4×C8—32no—
124#95—22no—
125#95—22no—
126#95#6714yes—
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×C4—48no—
141C2×C3×C4—44no—
142C2×C3×C4—88no—
143C2×C3×C4C2×C22⋊C411yescharacteristic
144C2×C3×C4C4×D412yes—
145C2×C3×C4#12112yes—
146C2×C3×C4#12311yescharacteristic
147#67—82no—
148C2×Dic3—41no—
149D12—42no—
150D12—82no—
151C4×D3—81no—
152C4×D3—81no—
153Dic6—42no—
154C3⋊C8—24no—
155C3⋊C8—82no—
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—61no—
166C2×D4—62no—
167C2×D4—82no—
168C2×D4—82no—
169C2×D4—121no—
170C2×D4(C2×D3)⋊C414yes—
171C22×C4—24no—
172C22×C4—84no—
173C22×C4—82no—
174C22×C4—121no—
175C22×C4C2×#6711yescharacteristic
176C22×C4C2×#6712yes—
177SD16—128no—
178D8—128no—
179M16—64no—
180C2×C8—62no—
181C2×C8—122no—
182C4⋊C4—62no—
183C22⋊C4—84no—
184C22⋊C4—82no—
185C22⋊C4—122no—
186C42C2×C4×D311yescharacteristic
187C42C2×D1211yescharacteristic
188C42#23512yes—
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—
203C3×C4—84no—
204Dic3—81no—
205C23—24no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—42no—
210C23—121no—
211C23#95311yescharacteristic
212Q8—122no—
213D4—44no—
214D4—82no—
215D4—84no—
216D4—84no—
217D4—122no—
218D4—242no—
219D4—242no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—88no—
225C2×C4—121no—
226C2×C4—241no—
227C2×C4—241no—
228C2×C4C2×(C2×D3)⋊C411yescharacteristic
229C2×C4C4×#6712yes—
230C2×C4#92911yescharacteristic
231C2×C4#93012yes—
232C8—64no—
233C8—242no—
234C2×C3—22no—
235C2×C3—84no—
236C2×C3#188711yescharacteristic
237C2×C3#190412yes—
238D3—161no—
239C22—22no—
240C22—44no—
241C22—48no—
242C22—82no—
243C22—82no—
244C22—84no—
245C22—241no—
246C22—242no—
247C22#521311yescharacteristic
248C22#524112yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—84no—
254C4—241no—
255C3#942811yescharacteristic, complemented, Sylow
256C2—22no—
257C2—84no—
258C2—481no—
259C2#7165011yescharacteristic
260C2#7165112yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2380C2
socleC22×C3#386
prime radical#83163C2
cosocleC2×C3×C42C23
hypercenter#2380D3
hypocenterC3#9428
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
1. C3×#2380C2semidirect
3. #71760C2semidirect
4. #69564C2semidirect
5. #67633C2semidirect
13. #5076C4semidirect
15. #5073C4semidirect
21. #2380D3semidirect
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
C3×#23802
#831632
#717602
#695642
#676332
#94283