Finite Group Databasebeta

← #10692715

Subgroups — #10692715

Order 768.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3151C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic, complemented
3#71799C211yesmaximal, characteristic, complemented
4#71760C212yesmaximal, complemented
5#67670C212yesmaximal, complemented
6#9423—31nomaximal, Sylow
7C2×C3×AGL(1,ℤ/8)—22no—
8C3×#532C2211yescharacteristic
9C2×C3×#127C2211yescharacteristic
10C3×#496—22no—
11C3×#493C412yes—
12C3×#492C412yes—
13C3×C4×M16C2211yescharacteristic
14#5234—22no—
15#5011—24no—
16#4926C2212yes—
17#4926C2212yes—
18#4544—24no—
19#4540—22no—
20#3151D311yescharacteristic
21#2980—31no—
22#1555—32no—
23#1536—31no—
24#1291—32no—
25C2×C3×#41D412yes—
26C22×C3×D4—21no—
27C22×C3×D4—22no—
28C3×AGL(1,ℤ/8)—44no—
29C3×AGL(1,ℤ/8)—44no—
30C2×C3×SD16—24no—
31C2×C3×D8—24no—
32C2×C3×M16—21no—
33C3×C4⋊Q8C2×C411yescharacteristic
34C3×#127—22no—
35C3×#127—21no—
36C3×#127C2×C412yes—
37C3×#127C2×C411yescharacteristic
38C3×#124D412yes—
39C3×#121—22no—
40C3×C4×D4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×#208—22no—
43C22×C3⋊C8—22no—
44C3×#97—21no—
45C3×C4×C8C2×C412yes—
46#830—24no—
47#830—22no—
48#830—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51#803—22no—
52#803—22no—
53C2×AGL(1,ℤ/8)—22no—
54#532C2×D311yescharacteristic
55C2×#127C2×D311yescharacteristic
56#496—22no—
57#493Dic312yes—
58#492Dic312yes—
59#435—64no—
60#432—32no—
61#432—32no—
62#408—62no—
63C4×M16C2×D311yescharacteristic
64#331—64no—
65#329—62no—
66C24×C3—41no—
67C3×#41—42no—
68C3×#41—42no—
69C2×C3×Q8C22⋊C412yes—
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—24no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—84no—
79C2×C3×D4C22⋊C412yes—
80C22×C3×C4—22no—
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4#4112yes—
83C3×SD16—48no—
84C3×D8—48no—
85C3×M16—24no—
86C2×C3×C8—22no—
87C3×C4⋊C4—22no—
88C3×C22⋊C4—24no—
89C3×C42C22×C411yescharacteristic
90C3×C42C2×D412yes—
91C3×C42C2×D411yescharacteristic
92#208—82no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—42no—
95C2×C3⋊C8—82no—
96C2×#41#6712yes—
97C22×D4—22no—
98C22×D4—21no—
99AGL(1,ℤ/8)—44no—
100AGL(1,ℤ/8)—44no—
101C2×SD16—24no—
102C2×D8—24no—
103C2×M16—21no—
104C2×M16—62no—
105C22×C8—62no—
106C4⋊Q8C2×Dic311yescharacteristic
107#127—22no—
108#127—21no—
109#127C2×Dic312yes—
110#127C2×Dic311yescharacteristic
111#124#6712yes—
112#121—22no—
113C4×D4—22no—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—34no—
116C4⋊C8—62no—
117C22⋊C8—64no—
118C22⋊C8—62no—
119C22⋊C8—62no—
120#97—21no—
121#97—62no—
122#97—62no—
123C4×C8C2×Dic312yes—
124C23×C3—22no—
125C23×C3—42no—
126C23×C3—81no—
127C23×C3—81no—
128C23×C3—84no—
129C23×C3#12311yescharacteristic
130C3×Q8—42no—
131C3×D4—42no—
132C3×D4—44no—
133C3×D4—82no—
134C3×D4—84no—
135C3×D4—84no—
136C3×D4—82no—
137C3×D4—82no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4C2×C22⋊C411yescharacteristic
144C2×C3×C4C4×D412yes—
145C2×C3×C4#12011yescharacteristic
146C2×C3×C4#12112yes—
147C3×C8—24no—
148C3⋊C8—44no—
149C3⋊C8—82no—
150C24—41no—
151#41—42no—
152#41—42no—
153C2×Q8#21712yes—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—84no—
163C2×D4#21712yes—
164C22×C4—22no—
165C22×C4#23712yes—
166C22×C4C2×#6711yescharacteristic
167SD16—48no—
168D8—48no—
169M16—24no—
170M16—242no—
171C2×C8—22no—
172C2×C8—64no—
173C2×C8—122no—
174C2×C8—242no—
175C4⋊C4—22no—
176C22⋊C4—24no—
177C42D3×D412yes—
178C42C22×Dic311yescharacteristic
179C42C2×#6711yescharacteristic
180C22×C3—21no—
181C22×C3—22no—
182C22×C3—42no—
183C22×C3—44no—
184C22×C3—82no—
185C22×C3—84no—
186C22×C3—162no—
187C22×C3—162no—
188C22×C3#38611yescharacteristic
189C3×C4—21no—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—42no—
193C3×C4#45312yes—
194C23—22no—
195C23—42no—
196C23—81no—
197C23—81no—
198C23—84no—
199C23#93511yescharacteristic
200Q8—42no—
201D4—42no—
202D4—44no—
203D4—82no—
204D4—84no—
205D4—84no—
206D4—82no—
207D4—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4D4×Dic312yes—
214C2×C4#93711yescharacteristic
215C2×C4#93912yes—
216C2×C4C2×#21711yescharacteristic
217C8—24no—
218C8—124no—
219C8—242no—
220C2×C3—41no—
221C2×C3—82no—
222C2×C3—162no—
223C2×C3#188711yescharacteristic
224C2×C3#190312yes—
225C22—21no—
226C22—22no—
227C22—42no—
228C22—44no—
229C22—82no—
230C22—84no—
231C22—162no—
232C22—162no—
233C22#522011yescharacteristic
234C4—21no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4#507712yes—
239C3#942311yescharacteristic, complemented, Sylow
240C2—41no—
241C2—82no—
242C2—162no—
243C2#7176311yescharacteristic
244C2#7209012yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3151C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3151D3
hypocenterC3#9423
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
2. #83279C2semidirect
3. #71799C2semidirect
4. #71760C2semidirect
5. #67670C2semidirect
239. C3#9423semidirect

Maximal subgroups

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

subgroupindex
C3×#31512
#832792
#717992
#717602
#676702
#94233