Finite Group Databasebeta

← #10692714

Subgroups — #10692714

Order 768.

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

normal subgroups only

This group has 540 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×#2980C211yesmaximal, characteristic, complemented
2#83287C211yesmaximal, characteristic
3#71799C211yesmaximal, characteristic, complemented
4#70818C212yesmaximal, complemented
5#67753C212yesmaximal, complemented
6#9423—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#949—22no—
9C2×#874C2211yescharacteristic
10C3×#435—24no—
11C3×#432C2212yes—
12C3×#408—22no—
13#5085C412yes—
14#5073C412yes—
15#5058—22no—
16#4926C2212yes—
17C4×#208C2211yescharacteristic
18#4540—22no—
19#4485—24no—
20#3151—31no—
21#2980D311yescharacteristic, complemented
22#1555—32no—
23#1536—31no—
24#1291—32no—
25C2×C3×#41D412yes—
26C22×D12—21no—
27C22×D12—22no—
28C2×C3×M16—22no—
29C22×C3×C8—22no—
30C3×C4⋊Q8C2×C411yescharacteristic
31C3×#127C2×C411yescharacteristic
32C3×#124D412yes—
33C3×#121—22no—
34C3×C4×D4—22no—
35C2×C3×C42C2311yescharacteristic
36#949—44no—
37#949—44no—
38C2×#215—24no—
39C2×#213—24no—
40C2×#208—21no—
41#874—22no—
42#874—21no—
43#874C2×C412yes—
44C3×C4⋊C8D414yes—
45C3×C22⋊C8—24no—
46C3×C22⋊C8—22no—
47C3×#97—22no—
48#830—22no—
49C4⋊(C3⋊C8)—22no—
50#803—21no—
51#803—22no—
52C4×C3⋊C8C2×C412yes—
53C2×AGL(1,ℤ/8)—62no—
54#532C2×D311yescharacteristic
55C2×#127—31no—
56#496—62no—
57#493—32no—
58#492—32no—
59#435—24no—
60#432—32no—
61#432C2×D312yes—
62#408—22no—
63C4×M16—31no—
64#331—64no—
65#329—62no—
66C23×D3—41no—
67C3×#41—42no—
68C3×#41—42no—
69C2×C3×Q8C22⋊C412yes—
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4C22⋊C412yes—
73C22×C3×C4—22no—
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4112yes—
76C2×D12—22no—
77C2×D12—24no—
78C2×D12—42no—
79C2×D12—42no—
80C2×D12—41no—
81C2×D12—42no—
82C2×D12—84no—
83C3×M16—82no—
84C2×C3×C8—24no—
85C2×C3×C8—82no—
86C3×C4⋊C4—22no—
87C3×C22⋊C4—24no—
88C3×C42C22×C411yescharacteristic
89C3×C42C2×D411yescharacteristic
90C3×C42C2×D412yes—
91#215—48no—
92#213—48no—
93#208—24no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—42no—
96C2×#41#6712yes—
97C22×D4—62no—
98C22×D4—61no—
99AGL(1,ℤ/8)—124no—
100AGL(1,ℤ/8)—124no—
101C2×SD16—64no—
102C2×D8—64no—
103C2×M16—22no—
104C2×M16—61no—
105C22×C8—22no—
106C4⋊Q8C4×D311yescharacteristic
107#127—32no—
108#127—61no—
109#127—62no—
110#127C4×D311yescharacteristic
111#124D1212yes—
112#121—22no—
113C4×D4—22no—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—62no—
116C4⋊C8#6714yes—
117C22⋊C8—24no—
118C22⋊C8—22no—
119C22⋊C8—62no—
120#97—22no—
121#97—61no—
122#97—62no—
123C4×C8—32no—
124C23×C3—22no—
125C23×C3#12311yescharacteristic
126C22×D3—42no—
127C22×D3—81no—
128C22×D3—81no—
129C22×D3—84no—
130C3×Q8—42no—
131C3×D4—42no—
132C3×D4—82no—
133C3×D4—82no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4C2×C22⋊C411yescharacteristic
140C2×C3×C4C4×D412yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12112yes—
143D12—44no—
144D12—84no—
145D12—84no—
146D12—82no—
147C3×C8—44no—
148C3⋊C8—24no—
149C3⋊C8—82no—
150C24—121no—
151#41—42no—
152#41—42no—
153C2×Q8(C2×D3)⋊C412yes—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—62no—
157C2×D4—64no—
158C2×D4—122no—
159C2×D4—122no—
160C2×D4—122no—
161C2×D4—121no—
162C2×D4—244no—
163C2×D4(C2×D3)⋊C412yes—
164C22×C4—22no—
165C22×C4#23512yes—
166C22×C4C2×#6711yescharacteristic
167SD16—128no—
168D8—128no—
169M16—64no—
170M16—82no—
171C2×C8—24no—
172C2×C8—62no—
173C2×C8—82no—
174C2×C8—122no—
175C4⋊C4—22no—
176C22⋊C4—24no—
177C42C2×C4×D311yescharacteristic
178C42C2×D1211yescharacteristic
179C42C2×#6712yes—
180C22×C3—21no—
181C22×C3—22no—
182C22×C3—42no—
183C22×C3—44no—
184C22×C3#38611yescharacteristic
185C2×D3—82no—
186C2×D3—84no—
187C2×D3—162no—
188C2×D3—162no—
189C3×C4—21no—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—42no—
193C3×C4#45312yes—
194C23—22no—
195C23—122no—
196C23—241no—
197C23—241no—
198C23—244no—
199C23#92911yescharacteristic
200Q8—42no—
201D4—42no—
202D4—82no—
203D4—82no—
204D4—124no—
205D4—244no—
206D4—244no—
207D4—242no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C2×(C2×D3)⋊C411yescharacteristic
214C2×C4C4×#6712yes—
215C2×C4#93012yes—
216C2×C4#95311yescharacteristic
217C8—44no—
218C8—64no—
219C8—242no—
220C2×C3—41no—
221C2×C3—82no—
222C2×C3#188711yescharacteristic
223C2×C3#190312yes—
224D3—162no—
225C22—22no—
226C22—21no—
227C22—44no—
228C22—42no—
229C22—244no—
230C22—242no—
231C22—482no—
232C22—482no—
233C22#521311yescharacteristic
234C4—22no—
235C4—22no—
236C4—21no—
237C4—42no—
238C4#524112yes—
239C3#942311yescharacteristic, complemented, Sylow
240C2—41no—
241C2—82no—
242C2—482no—
243C2#7165011yescharacteristic
244C2#7202312yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2980C2
socleC22×C3#386
prime radical#83287C2
cosocleC2×C3×C42C23
hypercenter#2980D3
hypocenterC3#9423
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
1. C3×#2980C2semidirect
3. #71799C2semidirect
4. #70818C2semidirect
5. #67753C2semidirect
21. #2980D3semidirect
239. C3#9423semidirect

Maximal subgroups

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

subgroupindex
C3×#29802
#832872
#717992
#708182
#677532
#94233