Finite Group Databasebeta

← #10697951

Subgroups — #10697951

Order 768.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3255C211yesmaximal, characteristic
2#83374C212yesmaximal
3#71839C212yesmaximal
4#70514C211yesmaximal, characteristic
5C3⋊#1309C211yesmaximal, characteristic
6#10303—31nomaximal, Sylow
7C3×#533C2212yes—
8#5470C2211yescharacteristic
9C3×#475—22no—
10C3×#470—22no—
11C3×#462—22no—
12C3×#423C2211yescharacteristic
13#5094—22no—
14#5090—22no—
15#5052—24no—
16#5031—22no—
17#5020—22no—
18C42×Dic3C2211yescharacteristic
19#4924C2212yes—
20#4555—22no—
21#4532—24no—
22#3255—32no—
23#3255D311yescharacteristic
24#1994—31no—
25#1994—32no—
26#1309—31no—
27C2×C3×Q16—22no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D412yes—
30C3×C4⋊Q8D412yes—
31C3×#125—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C3×#117—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#216—24no—
39#926—21no—
40#924—22no—
41C2×C4×Dic3—22no—
42C2×C4×Dic3—21no—
43C2×#208D414yes—
44#870—21no—
45#869D412yes—
46C4×Dic6—22no—
47C3×#106—22no—
48C3×C4⋊C8—22no—
49C3×#103—24no—
50#835—24no—
51#810—24no—
52#808—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#533—31no—
56#533C2×D312yes—
57#475—22no—
58#475—62no—
59#475—62no—
60#470—22no—
61#470—64no—
62#462—22no—
63#462—62no—
64#462—62no—
65#423—32no—
66#423C2×D311yescharacteristic
67C43—31no—
68#339—64no—
69#339—62no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C22×Dic3—41no—
75C2×Dic6—22no—
76C3×Q16—82no—
77C3×M16—42no—
78C2×C3×C8—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C22⋊C4—42no—
86C3×C42—42no—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90#217—41no—
91#216—84no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94Dic3⋊C4—42no—
95C4×Dic3—24no—
96C4×Dic3—22no—
97C4×Dic3—42no—
98C4×Dic3—41no—
99C4×Dic3—41no—
100#208—44no—
101C2×C3⋊C8—24no—
102C2×Q16—22no—
103C2×Q16—64no—
104C2×M16—34no—
105C2×M16#6712yes—
106C4⋊Q8—32no—
107C4⋊Q8#6712yes—
108C4⋊Q8#6712yes—
109#125—22no—
110#125—61no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—62no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—62no—
117#117—22no—
118#117—61no—
119C2×C42—62no—
120C2×C42—61no—
121C2×C42C22×D311yescharacteristic
122#106—22no—
123#106—64no—
124C4⋊C8—22no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127#103—24no—
128#103—64no—
129#103—64no—
130C23×C3#12711yescharacteristic
131C3×Q8—42no—
132C3×Q8—42no—
133C3×Q8—42no—
134C3×Q8—42no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4#12012yes—
142C2×C3×C4#12011yescharacteristic
143C2×C3×C4#12111yescharacteristic
144C2×C3×C4#12112yes—
145C2×Dic3—44no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148Dic6—42no—
149Dic6—42no—
150C3×C8—42no—
151C3⋊C8—44no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—62no—
155C22×C4—121no—
156C22×C4D3×D412yes—
157C22×C4C2×#6711yescharacteristic
158Q16—82no—
159Q16—244no—
160M16—42no—
161M16—124no—
162C2×C8—22no—
163C2×C8—64no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C22⋊C4—42no—
174C22⋊C4—121no—
175C42—42no—
176C42—64no—
177C42—62no—
178C42—122no—
179C42—121no—
180C42—121no—
181C42D3×D411yescharacteristic
182C42#23711yescharacteristic
183C42C2×#6712yes—
184C22×C3—22no—
185C22×C3—21no—
186C22×C3#39211yescharacteristic
187C3×C4—22no—
188C3×C4—21no—
189C3×C4—82no—
190C3×C4—82no—
191C3×C4#45512yes—
192C3×C4#45511yescharacteristic
193C3×C4#45612yes—
194C3×C4#45611yescharacteristic
195Dic3—44no—
196Dic3—82no—
197C23#94011yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—122no—
203Q8—122no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—124no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4#93712yes—
214C2×C4#93811yescharacteristic
215C2×C4#93912yes—
216C2×C4#95311yescharacteristic
217C8—42no—
218C8—124no—
219C2×C3—41no—
220C2×C3#200712yes—
221C2×C3#200711yescharacteristic
222C22—22no—
223C22—21no—
224C22#522311yescharacteristic
225C4—22no—
226C4—21no—
227C4—82no—
228C4—82no—
229C4—124no—
230C4—242no—
231C4#520212yes—
232C4#520512yes—
233C4#524211yescharacteristic
234C4#524611yescharacteristic
235C3#1030311yescharacteristic, complemented, Sylow
236C2—41no—
237C2#7188111yescharacteristic
238C2#7211012yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3255D3
hypocenterC3#10303
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

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
C3×#32552
#833742
#718392
#705142
C3⋊#13092
#103033