Finite Group Databasebeta

← #10698025

Subgroups — #10698025

Order 768.

586 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 159 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×#5080C212yesmaximal
2C2×#4680C211yesmaximal, characteristic
3C3×#1956C211yesmaximal, characteristic
4#69579C212yesmaximal
5C3⋊#1273C211yesmaximal, characteristic
6#10316—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×#869C2212yes—
9C2×C3×#103—22no—
10C2×C3×C4×C8C2211yescharacteristic
11C3×#384C2211yescharacteristic
12C2×#817—22no—
13#5080—24no—
14#5080—24no—
15C2×#810—24no—
16C2×#808—24no—
17C2×C4×C3⋊C8C2212yes—
18#4680—24no—
19C2×#498—31no—
20C2×#498—32no—
21#1956—32no—
22#1956D311yescharacteristic
23#1273—31no—
24C22×C3×Q8—21no—
25C22×Dic6—22no—
26C22×C3×C8D412yes—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—21no—
29C3×C4⋊Q8D412yes—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3—22no—
36C22×C3⋊C8D414yes—
37#869—22no—
38#869—22no—
39#869—22no—
40#869D414yes—
41C3×#103—44no—
42C3×C4×C8—22no—
43#817—28no—
44#810—48no—
45#808—216no—
46C4×C3⋊C8—22no—
47C4×C3⋊C8—22no—
48C2×C4⋊Q8—32no—
49C2×C4⋊Q8C2×D311yescharacteristic
50#498—64no—
51#498—64no—
52#498—64no—
53C2×#106—64no—
54C2×#106—62no—
55C2×#103—22no—
56C2×#103—64no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384C2×D311yescharacteristic
60C2×C3×Q8—22no—
61C2×C3×Q8—41no—
62C2×C3×Q8—42no—
63C2×C3×Q8—42no—
64C22×C3×C4—41no—
65C22×C3×C4—41no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×D411yescharacteristic
68C22×Dic3—42no—
69C2×Dic6—24no—
70C2×Dic6—44no—
71C2×Dic6—44no—
72C2×Dic6—42no—
73C2×C3×C8—22no—
74C2×C3×C8—22no—
75C2×C3×C8SD1614yes—
76C3×C4⋊C4—24no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—44no—
79C3×C4⋊C4—42no—
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D412yes—
82C3×C42#4111yescharacteristic
83C4⋊Dic3—24no—
84C4⋊Dic3—24no—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87C2×C3⋊C8—24no—
88C2×C3⋊C8—24no—
89C2×C3⋊C8SD1618yes—
90C22×Q8—21no—
91C22×Q8—62no—
92C22×C8—34no—
93C22×C8#6712yes—
94C4⋊Q8—21no—
95C4⋊Q8—22no—
96C4⋊Q8—34no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4⋊Q8D1212yes—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—62no—
106C2×C42C22×D311yescharacteristic
107#106—616no—
108#106—68no—
109#103—44no—
110#103—128no—
111C4×C8—22no—
112C4×C8—62no—
113C4×C8—62no—
114C23×C3#12711yescharacteristic
115C3×Q8—44no—
116C3×Q8—82no—
117C2×C3×C4—42no—
118C2×C3×C4—81no—
119C2×C3×C4—81no—
120C2×C3×C4—82no—
121C2×C3×C4—81no—
122C2×C3×C4#12012yes—
123C2×C3×C4#12011yescharacteristic
124C2×C3×C4#12112yes—
125C2×C3×C4#12111yescharacteristic
126C2×C3×C4C2×SD1614yes—
127C2×C3×C4C2×SD1612yes—
128C2×C3×C4#13714yes—
129C2×C3×C4#13712yes—
130C2×Dic3—44no—
131C2×Dic3—82no—
132C2×Dic3—82no—
133Dic6—48no—
134Dic6—84no—
135C3×C8—24no—
136C3⋊C8—28no—
137C2×Q8—22no—
138C2×Q8—41no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—64no—
142C2×Q8—122no—
143C2×Q8—124no—
144C2×Q8—124no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—122no—
148C22×C4D3×D412yes—
149C22×C4C2×#6711yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—38no—
153C2×C8—64no—
154C2×C8—64no—
155C2×C8#21414yes—
156C4⋊C4—24no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C42C2×D1211yescharacteristic
165C42#23511yescharacteristic
166C42D3×D412yes—
167C22×C3#39211yescharacteristic
168C22×C3#49212yes—
169C22×C3#49211yescharacteristic
170C22×C3#49712yes—
171C22×C3#49711yescharacteristic
172C3×C4—82no—
173C3×C4—161no—
174C3×C4#44812yes—
175C3×C4#44814yes—
176C3×C4#46112yes—
177C3×C4#46114yes—
178Dic3—84no—
179C23#94011yescharacteristic
180Q8—44no—
181Q8—82no—
182Q8—128no—
183Q8—244no—
184C2×C4—42no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—81no—
189C2×C4—124no—
190C2×C4—242no—
191C2×C4—242no—
192C2×C4#88212yes—
193C2×C4#88412yes—
194C2×C4D3×SD1614yes—
195C2×C4#91814yes—
196C2×C4#93011yescharacteristic
197C2×C4C2×#21412yes—
198C2×C4#93711yescharacteristic
199C2×C4#95112yes—
200C8—24no—
201C8—68no—
202C2×C3#170511yescharacteristic
203C2×C3#171612yes—
204C2×C3#171611yescharacteristic
205C2×C3#201311yescharacteristic
206C2×C3#201312yes—
207C22#496011yescharacteristic
208C22#506011yescharacteristic
209C22#507611yescharacteristic
210C22#517512yes—
211C22#519012yes—
212C4—82no—
213C4—161no—
214C4—244no—
215C4#480614yes—
216C4#481814yes—
217C4#501912yes—
218C4#504012yes—
219C3#1031611yescharacteristic, complemented, Sylow
220C2#6966111yescharacteristic
221C2#7028212yes—
222C2#7121712yes—
223C2#7125411yescharacteristic
224C2#7129911yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1956C2
socle, prime radicalC23×C3#127
hypercenter#1956D3
hypocenterC3#10316
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
219. C3#10316semidirect

Maximal subgroups

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

subgroupindex
C2×#50802
C2×#46802
C3×#19562
#695792
C3⋊#12732
#103163