Finite Group Databasebeta

← #10698027

Subgroups — #10698027

Order 768.

554 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#5079C212yesmaximal
3#71787C212yesmaximal
4#69771C211yesmaximal, characteristic
5C3⋊#1273C211yesmaximal, characteristic
6#10317—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8C2×#869C2211yescharacteristic
9C3×#498—24no—
10C2×C3×#106—22no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#835—24no—
13C2×#816—22no—
14#5079—24no—
15#5079—24no—
16#4935C2211yescharacteristic
17C2×C4×C3⋊C8C2212yes—
18#4529—24no—
19C2×#498D311yescharacteristic
20C2×#487—32no—
21#1990—32no—
22#1956—31no—
23#1273—31no—
24C22×C3×Q8—22no—
25C22×Dic6—21no—
26C22×C3×C8D412yes—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8D414yes—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3—21no—
37C22×C3⋊C8D414yes—
38#869—22no—
39#869—21no—
40#869D412yes—
41C3×#106—28no—
42C3×C4×C8—22no—
43#835—216no—
44#816—44no—
45C4×C3⋊C8—22no—
46C4×C3⋊C8—22no—
47C2×C4⋊Q8—31no—
48C2×C4⋊Q8C2×D312yes—
49#498—24no—
50#487—64no—
51#487—64no—
52C2×#106—22no—
53C2×#103—64no—
54C2×#103—62no—
55C2×C4×C8—32no—
56C2×C4×C8C2×D311yescharacteristic
57#384—31no—
58#340—64no—
59C2×C3×Q8—24no—
60C2×C3×Q8—42no—
61C2×C3×Q8—44no—
62C2×C3×Q8—44no—
63C22×C3×C4—42no—
64C22×C3×C4C2×D412yes—
65C22×C3×C4C2×D411yescharacteristic
66C22×Dic3—41no—
67C22×Dic3—41no—
68C2×Dic6—22no—
69C2×Dic6—42no—
70C2×Dic6—42no—
71C2×Dic6—41no—
72C2×C3×C8—22no—
73C2×C3×C8—22no—
74C2×C3×C8SD1614yes—
75C3×C4⋊C4—24no—
76C3×C4⋊C4—24no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C42C2×D411yescharacteristic
80C3×C42C2×D412yes—
81C3×C42#4111yescharacteristic
82C4⋊Dic3—24no—
83C4⋊Dic3—42no—
84C4⋊Dic3—42no—
85C4⋊Dic3—44no—
86C2×C3⋊C8—24no—
87C2×C3⋊C8—24no—
88C2×C3⋊C8—24no—
89C22×Q8—22no—
90C22×Q8—61no—
91C22×C8—34no—
92C22×C8#6712yes—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—32no—
97C4⋊Q8—61no—
98C4⋊Q8—62no—
99C4⋊Q8#6714yes—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—61no—
105C2×C42C22×D311yescharacteristic
106#106—28no—
107#103—616no—
108#103—124no—
109C4×C8—22no—
110C4×C8—62no—
111C4×C8—62no—
112C23×C3#12711yescharacteristic
113C3×Q8—48no—
114C3×Q8—84no—
115C2×C3×C4—44no—
116C2×C3×C4—82no—
117C2×C3×C4—82no—
118C2×C3×C4#12012yes—
119C2×C3×C4#12011yescharacteristic
120C2×C3×C4#12111yescharacteristic
121C2×C3×C4#12112yes—
122C2×C3×C4C2×SD1612yes—
123C2×C3×C4#13514yes—
124C2×C3×C4#13714yes—
125C2×C3×C4#13712yes—
126C2×Dic3—42no—
127C2×Dic3—81no—
128C2×Dic3—81no—
129C2×Dic3—81no—
130C2×Dic3—82no—
131Dic6—44no—
132Dic6—82no—
133C3×C8—24no—
134C3⋊C8—44no—
135C2×Q8—24no—
136C2×Q8—42no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—62no—
140C2×Q8—121no—
141C2×Q8—122no—
142C2×Q8—122no—
143C22×C4—42no—
144C22×C4—121no—
145C22×C4—121no—
146C22×C4D3×D412yes—
147C22×C4C2×#6711yescharacteristic
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—64no—
151C2×C8—64no—
152C2×C8—64no—
153C2×C8#21514yes—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—64no—
159C4⋊C4—122no—
160C4⋊C4—124no—
161C4⋊C4—122no—
162C42D3×D411yescharacteristic
163C42#23711yescharacteristic
164C42C2×#6712yes—
165C22×C3#39211yescharacteristic
166C22×C3#49211yescharacteristic
167C22×C3#49512yes—
168C22×C3#49712yes—
169C22×C3#49711yescharacteristic
170C3×C4—84no—
171C3×C4#44812yes—
172C3×C4#45214yes—
173C3×C4#46112yes—
174C3×C4#46414yes—
175Dic3—82no—
176Dic3—161no—
177C23#94011yescharacteristic
178Q8—48no—
179Q8—84no—
180Q8—124no—
181Q8—242no—
182C2×C4—44no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—122no—
186C2×C4—241no—
187C2×C4—241no—
188C2×C4—242no—
189C2×C4—241no—
190C2×C4#91514yes—
191C2×C4#91614yes—
192C2×C4#93712yes—
193C2×C4#93811yescharacteristic
194C2×C4#93912yes—
195C2×C4C2×#21512yes—
196C2×C4#94212yes—
197C2×C4#95311yescharacteristic
198C8—24no—
199C8—124no—
200C2×C3#170811yescharacteristic
201C2×C3#171611yescharacteristic
202C2×C3#171812yes—
203C2×C3#201311yescharacteristic
204C2×C3#201512yes—
205C22#508511yescharacteristic
206C22#508611yescharacteristic
207C22#516712yes—
208C22#516812yes—
209C22#522311yescharacteristic
210C4—84no—
211C4—242no—
212C4—481no—
213C4#508312yes—
214C4#516614yes—
215C4#517314yes—
216C4#522712yes—
217C3#1031711yescharacteristic, complemented, Sylow
218C2#7133411yescharacteristic
219C2#7133811yescharacteristic
220C2#7133912yes—
221C2#7179111yescharacteristic
222C2#7205812yes—
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socle, prime radicalC23×C3#127
hypercenterC2×#498D3
hypocenterC3#10317
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
217. C3#10317semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×#50792
#717872
#697712
C3⋊#12732
#103173