Finite Group Databasebeta

← #10697952

Subgroups — #10697952

Order 768.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#83374C212yesmaximal
2#79069C211yesmaximal, characteristic
3C3×#1994C211yesmaximal, characteristic
4#69396C212yesmaximal
5C3⋊#1309C211yesmaximal, characteristic
6#10303—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5470C2212yes—
9C3×#423C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5137—22no—
12#5094—22no—
13#5090—22no—
14#5052—24no—
15#5031—22no—
16#5020—22no—
17#4924C2212yes—
18#4851—22no—
19#4849—22no—
20C3×#339—22no—
21#4523—24no—
22#3255—31no—
23#3255—32no—
24#1994—32no—
25#1994D311yescharacteristic
26#1309—31no—
27C2×C3×M16D412yes—
28C3×C4⋊Q8D412yes—
29C3×#125—21no—
30C3×C22⋊Q8—22no—
31C3×C4×Q8—22no—
32C3×#117—21no—
33C2×C3×C42—22no—
34C2×C3×C42—21no—
35C2×C3×C42C2311yescharacteristic
36C2×#216—24no—
37#926—22no—
38#924—22no—
39#924—22no—
40C2×#208D414yes—
41C2×Dic12—22no—
42#870—22no—
43#869D412yes—
44#869D412yes—
45C4×Dic6—22no—
46C4×Dic6—22no—
47C3×C4⋊C8—22no—
48#835—24no—
49#817—22no—
50#816—24no—
51#810—24no—
52#808—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#533—32no—
56#533C2×D311yescharacteristic
57#475—62no—
58#475—62no—
59#475—62no—
60#470—64no—
61#470—62no—
62#462—62no—
63#462—62no—
64#462—62no—
65#423—32no—
66#423C2×D311yescharacteristic
67C43C2×D311yescharacteristic
68#339—22no—
69#339—64no—
70C2×C3×Q8—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C2×Dic6—22no—
75C2×Dic6—22no—
76C3×M16—42no—
77C2×C3×C8—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—41no—
82C3×C42—24no—
83C3×C42—22no—
84C3×C42—42no—
85C3×C42—41no—
86C3×C42—41no—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90#217—42no—
91#216—84no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96Dic3⋊C4—42no—
97Dic3⋊C4—42no—
98C4×Dic3—42no—
99#208—44no—
100C2×C3⋊C8—24no—
101Dic12—82no—
102C2×Q16—62no—
103C2×Q16—64no—
104C2×M16—34no—
105C2×M16#6712yes—
106C4⋊Q8—32no—
107C4⋊Q8—32no—
108C4⋊Q8D1212yes—
109#125—21no—
110#125—62no—
111C22⋊Q8—22no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114C4×Q8—22no—
115C4×Q8—62no—
116C4×Q8—62no—
117#117—21no—
118#117—62no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42C22×D311yescharacteristic
122#106—64no—
123#106—62no—
124C4⋊C8—22no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127#103—64no—
128#103—64no—
129#103—64no—
130C23×C3#12711yescharacteristic
131C3×Q8—42no—
132C3×Q8—42no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—42no—
138C2×C3×C4—44no—
139C2×C3×C4—42no—
140C2×C3×C4#12012yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12112yes—
143C2×C3×C4#12111yescharacteristic
144C2×Dic3—42no—
145C2×Dic3—42no—
146Dic6—42no—
147Dic6—42no—
148Dic6—42no—
149Dic6—42no—
150C3×C8—42no—
151C3⋊C8—44no—
152C2×Q8—22no—
153C2×Q8—62no—
154C2×Q8—62no—
155C22×C4—41no—
156C22×C4D3×D412yes—
157C22×C4C2×#6711yescharacteristic
158Q16—242no—
159Q16—244no—
160M16—42no—
161M16—124no—
162C2×C8—22no—
163C2×C8—64no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C22⋊C4—41no—
174C22⋊C4—122no—
175C42—24no—
176C42—22no—
177C42—41no—
178C42—42no—
179C42—41no—
180C42—122no—
181C42C2×D1211yescharacteristic
182C42#23511yescharacteristic
183C42D3×D412yes—
184C22×C3—21no—
185C22×C3—22no—
186C22×C3#39211yescharacteristic
187C3×C4—21no—
188C3×C4—22no—
189C3×C4—44no—
190C3×C4—82no—
191C3×C4#45511yescharacteristic
192C3×C4#45512yes—
193C3×C4#45612yes—
194C3×C4#45611yescharacteristic
195Dic3—82no—
196Dic3—82no—
197C23#94011yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—122no—
201Q8—122no—
202Q8—122no—
203Q8—122no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4#88212yes—
214C2×C4#88412yes—
215C2×C4#93011yescharacteristic
216C2×C4#93711yescharacteristic
217C8—42no—
218C8—124no—
219C2×C3—41no—
220C2×C3#200711yescharacteristic
221C2×C3#200712yes—
222C22—21no—
223C22—22no—
224C22#496011yescharacteristic
225C4—21no—
226C4—22no—
227C4—44no—
228C4—82no—
229C4—242no—
230C4—242no—
231C4#482512yes—
232C4#482612yes—
233C4#506111yescharacteristic
234C4#509511yescharacteristic
235C3#1030311yescharacteristic, complemented, Sylow
236C2—41no—
237C2#6969411yescharacteristic
238C2#7039312yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1994D3
hypocenterC3#10303
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
235. C3#10303semidirect

Maximal subgroups

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

subgroupindex
#833742
#790692
C3×#19942
#693962
C3⋊#13092
#103033