Finite Group Databasebeta

← #10698019

Subgroups — #10698019

Order 768.

734 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×C3×#498C211yesmaximal, characteristic, complemented
2C2×#5085C212yesmaximal
3#71787C212yesmaximal, complemented
4#69772C211yesmaximal, characteristic
5C3⋊#1273C211yesmaximal, characteristic, complemented
6#10315—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8C22×#215—24no—
9C2×#874C2211yescharacteristic
10C3×#498—24no—
11C2×C3×#106—22no—
12C2×C3×C4×C8C2211yescharacteristic
13C2×#821—22no—
14#5085—24no—
15#5085—24no—
16#4935C2211yescharacteristic
17C2×C4×C3⋊C8C2212yes—
18#4529—24no—
19C2×#498D311yescharacteristic, complemented
20C2×#492—32no—
21#1990—32no—
22#1957—31no—
23#1273—31no—
24C22×C3×Q8—22no—
25C22×D12—22no—
26C22×D12—21no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4⋊Q8D414yes—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C42C2311yescharacteristic
35C2×#215—216no—
36C2×#215—48no—
37C2×#215—48no—
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—21no—
40C22×C3⋊C8D414yes—
41#874—22no—
42#874—21no—
43#874D412yes—
44C3×#106—28no—
45C3×C4×C8—22no—
46#821—44no—
47C4×C3⋊C8—22no—
48C4×C3⋊C8—22no—
49C22×SD16—64no—
50C2×C4⋊Q8C2×D312yes—
51C2×#127—31no—
52#498—24no—
53#492—64no—
54#492—64no—
55C2×#106—22no—
56C2×#102—62no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384—31no—
60#340—64no—
61C23×D3—41no—
62C2×C3×Q8—24no—
63C2×C3×Q8—42no—
64C2×C3×Q8—44no—
65C2×C3×Q8—44no—
66C22×C3×C4—42no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×D411yescharacteristic
69C22×Dic3—41no—
70C2×D12—24no—
71C2×D12—22no—
72C2×D12—42no—
73C2×D12—44no—
74C2×D12—44no—
75C2×D12—42no—
76C2×D12—41no—
77C2×D12—42no—
78C2×C3×C8—22no—
79C2×C3×C8—22no—
80C2×C3×C8SD1614yescomplemented
81C3×C4⋊C4—24no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C42C2×D411yescharacteristic
86C3×C42C2×D412yes—
87C3×C42#4111yescharacteristic
88#215—432no—
89C4⋊Dic3—42no—
90C4⋊Dic3—44no—
91C2×C3⋊C8—24no—
92C2×C3⋊C8—24no—
93C2×C3⋊C8SD1618yescomplemented
94C22×Q8—22no—
95C22×D4—62no—
96C22×D4—61no—
97C2×SD16—616no—
98C2×SD16—128no—
99C2×SD16—128no—
100C22×C8—34no—
101C22×C8#6712yes—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8#6714yes—
106#127—32no—
107#127—61no—
108#127—62no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—61no—
113C2×C42C22×D311yescharacteristic
114#106—28no—
115#102—124no—
116C4×C8—22no—
117C4×C8—62no—
118C4×C8—62no—
119C23×C3#12711yescharacteristic
120C22×D3—42no—
121C22×D3—82no—
122C22×D3—81no—
123C22×D3—82no—
124C22×D3—81no—
125C3×Q8—48no—
126C3×Q8—84no—
127C2×C3×C4—44no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12111yescharacteristic
133C2×C3×C4#12112yes—
134C2×C3×C4C2×SD1614yes—
135C2×C3×C4C2×SD1612yes—
136C2×C3×C4AGL(1,ℤ/8)14yes—
137C2×C3×C4#13712yes—
138C2×Dic3—81no—
139C2×Dic3—82no—
140D12—48no—
141D12—44no—
142D12—82no—
143D12—84no—
144C3×C8—24no—
145C3⋊C8—28no—
146C24—121no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—124no—
154C2×D4—124no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—122no—
158C2×D4—121no—
159C22×C4—42no—
160C22×C4—121no—
161C22×C4D3×D412yes—
162C22×C4C2×#6711yescharacteristic
163SD16—1232no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—38no—
167C2×C8—64no—
168C2×C8—64no—
169C2×C8#21514yescomplemented
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—124no—
175C4⋊C4—122no—
176C42D3×D411yescharacteristic
177C42#23711yescharacteristic
178C42C2×#6712yes—
179C22×C3#39211yescharacteristic
180C22×C3#49212yes—
181C22×C3#49211yescharacteristic
182C22×C3#49612yes—
183C22×C3#49711yescharacteristic
184C2×D3—82no—
185C2×D3—84no—
186C2×D3—161no—
187C2×D3—162no—
188C2×D3—161no—
189C3×C4—84no—
190C3×C4#44812yes—
191C3×C4#45014yes—
192C3×C4#46014yes—
193C3×C4#46112yes—
194Dic3—161no—
195C23—122no—
196C23—242no—
197C23—242no—
198C23—241no—
199C23—241no—
200C23#94011yescharacteristic
201Q8—48no—
202Q8—84no—
203D4—128no—
204D4—124no—
205D4—244no—
206D4—242no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—242no—
211C2×C4—241no—
212C2×C4D3×SD1614yes—
213C2×C4#91414yes—
214C2×C4#93712yes—
215C2×C4#93811yescharacteristic
216C2×C4#93912yes—
217C2×C4C2×#21512yes—
218C2×C4#94212yes—
219C2×C4#95311yescharacteristic
220C8—24no—
221C8—68no—
222C2×C3#170511yescharacteristic
223C2×C3#171412yes—
224C2×C3#171511yescharacteristic
225C2×C3#201311yescharacteristic
226C2×C3#201412yes—
227D3—162no—
228C22—242no—
229C22—244no—
230C22—481no—
231C22—482no—
232C22—481no—
233C22#508511yescharacteristic
234C22#508611yescharacteristic
235C22#517512yes—
236C22#517612yes—
237C22#522311yescharacteristic
238C4—84no—
239C4—481no—
240C4#508312yes—
241C4#516314yes—
242C4#516914yes—
243C4#522712yes—
244C3#1031511yescharacteristic, complemented, Sylow
245C2—482no—
246C2#7133211yescharacteristic
247C2#7133512yes—
248C2#7133611yescharacteristic
249C2#7179111yescharacteristic
250C2#7205612yes—
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC23×C3#127
prime radicalC2×#874C22
hypercenterC2×#498D3
hypocenterC3#10315
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×C3×#498C2semidirect
3. #71787C2semidirect
5. C3⋊#1273C2semidirect
19. C2×#498D3semidirect
80. C2×C3×C8SD16semidirect
93. C2×C3⋊C8SD16semidirect
169. C2×C8#215semidirect
244. C3#10315semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×#50852
#717872
#697722
C3⋊#12732
#103153