Finite Group Databasebeta

← #10698003

Subgroups — #10698003

Order 768.

638 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 151 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×#5070C212yesmaximal
2C2×#4680C211yesmaximal, characteristic, complemented
3C3×#1957C211yesmaximal, characteristic
4#69579C212yesmaximal, complemented
5C3⋊#1273C211yesmaximal, characteristic, complemented
6#10312—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×#869C2212yes—
9C2×C3×#102—22no—
10C2×C3×C4×C8C2211yescharacteristic
11C3×#384C2211yescharacteristic
12C2×#832—24no—
13C2×#817—22no—
14#5070—24no—
15#5070—24no—
16C2×#810—24no—
17C2×C4×C3⋊C8C2212yes—
18#4680—24no—
19C2×#498—31no—
20C2×#486—32no—
21#1957D311yescharacteristic
22#1956—32no—
23#1273—31no—
24C22×C3×D4—22no—
25C22×C3×D4—21no—
26C22×Dic6—22no—
27C22×C3×C8D412yes—
28C3×#127—22no—
29C3×#127—21no—
30C3×#127D412yes—
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×#102—44no—
42C3×C4×C8—22no—
43#832—216no—
44#817—28no—
45#810—48no—
46C4×C3⋊C8—22no—
47C4×C3⋊C8—22no—
48C2×C4⋊Q8—32no—
49C2×#127C2×D311yescharacteristic
50#498—64no—
51#486—64no—
52#486—64no—
53C2×#106—62no—
54C2×#103—64no—
55C2×#102—22no—
56C2×#102—64no—
57C2×C4×C8—32no—
58C2×C4×C8C2×D311yescharacteristic
59#384C2×D311yescharacteristic
60C24×C3—41no—
61C2×C3×D4—24no—
62C2×C3×D4—22no—
63C2×C3×D4—41no—
64C2×C3×D4—42no—
65C2×C3×D4—42no—
66C2×C3×D4—44no—
67C2×C3×D4—44no—
68C2×C3×D4—42no—
69C22×C3×C4—41no—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C22×Dic3—42no—
73C2×Dic6—24no—
74C2×Dic6—44no—
75C2×Dic6—44no—
76C2×Dic6—42no—
77C2×C3×C8—22no—
78C2×C3×C8—22no—
79C2×C3×C8SD1614yes—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—44no—
82C3×C42C2×D411yescharacteristic
83C3×C42C2×D412yes—
84C3×C42#4111yescharacteristic
85C4⋊Dic3—24no—
86C4⋊Dic3—24no—
87C4⋊Dic3—42no—
88C4⋊Dic3—42no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8—24no—
92C22×Q8—62no—
93C22×D4—22no—
94C22×D4—21no—
95C22×C8—34no—
96C22×C8#6712yes—
97C4⋊Q8—34no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4⋊Q8—62no—
101#127—21no—
102#127—22no—
103#127D1212yes—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C42C22×D311yescharacteristic
109#106—68no—
110#103—128no—
111#102—44no—
112#102—616no—
113C4×C8—22no—
114C4×C8—62no—
115C4×C8—62no—
116C23×C3—42no—
117C23×C3—81no—
118C23×C3—82no—
119C23×C3—82no—
120C23×C3—81no—
121C23×C3#12711yescharacteristic
122C3×D4—48no—
123C3×D4—44no—
124C3×D4—82no—
125C3×D4—84no—
126C2×C3×C4—81no—
127C2×C3×C4—82no—
128C2×C3×C4#12012yes—
129C2×C3×C4#12011yescharacteristic
130C2×C3×C4#12112yes—
131C2×C3×C4#12111yescharacteristic
132C2×C3×C4C2×SD1612yes—
133C2×C3×C4#13514yes—
134C2×C3×C4AGL(1,ℤ/8)14yes—
135C2×C3×C4#13712yes—
136C2×Dic3—44no—
137C2×Dic3—82no—
138C2×Dic3—82no—
139Dic6—48no—
140Dic6—84no—
141C3×C8—24no—
142C3⋊C8—44no—
143C24—41no—
144C2×Q8—64no—
145C2×Q8—122no—
146C2×Q8—124no—
147C2×Q8—124no—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4—42no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—41no—
157C22×C4—122no—
158C22×C4D3×D412yes—
159C22×C4C2×#6711yescharacteristic
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—64no—
163C2×C8—64no—
164C2×C8—64no—
165C2×C8#21414yes—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—64no—
169C4⋊C4—64no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C42C2×D1211yescharacteristic
173C42#23511yescharacteristic
174C42D3×D412yes—
175C22×C3—84no—
176C22×C3—82no—
177C22×C3—161no—
178C22×C3—161no—
179C22×C3—162no—
180C22×C3#39211yescharacteristic
181C22×C3#49211yescharacteristic
182C22×C3#49512yes—
183C22×C3#49612yes—
184C22×C3#49711yescharacteristic
185C3×C4—161no—
186C3×C4#44812yes—
187C3×C4#44914yes—
188C3×C4#46112yes—
189C3×C4#46314yes—
190Dic3—84no—
191C23—42no—
192C23—82no—
193C23—82no—
194C23—81no—
195C23—81no—
196C23#94011yescharacteristic
197Q8—128no—
198Q8—244no—
199D4—44no—
200D4—48no—
201D4—82no—
202D4—84no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—124no—
206C2×C4—242no—
207C2×C4—242no—
208C2×C4#88212yes—
209C2×C4#88412yes—
210C2×C4#91114yes—
211C2×C4#91614yes—
212C2×C4#93011yescharacteristic
213C2×C4C2×#21412yes—
214C2×C4#93711yescharacteristic
215C2×C4#95112yes—
216C8—24no—
217C8—124no—
218C2×C3—162no—
219C2×C3#170811yescharacteristic
220C2×C3#171511yescharacteristic
221C2×C3#172012yes—
222C2×C3#201212yes—
223C2×C3#201311yescharacteristic
224C22—82no—
225C22—84no—
226C22—162no—
227C22—161no—
228C22—161no—
229C22#496011yescharacteristic
230C22#506011yescharacteristic
231C22#507611yescharacteristic
232C22#515412yes—
233C22#516812yes—
234C4—161no—
235C4—244no—
236C4#478114yes—
237C4#478714yes—
238C4#501912yes—
239C4#504012yes—
240C3#1031211yescharacteristic, complemented, Sylow
241C2—162no—
242C2#6966111yescharacteristic
243C2#7021412yes—
244C2#7119112yes—
245C2#7125611yescharacteristic
246C2#7127711yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1957C2
socleC23×C3#127
prime radicalC2×C3×#127C22
hypercenter#1957D3
hypocenterC3#10312
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×#4680C2semidirect
4. #69579C2semidirect
5. C3⋊#1273C2semidirect
240. C3#10312semidirect

Maximal subgroups

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

subgroupindex
C2×#50702
C2×#46802
C3×#19572
#695792
C3⋊#12732
#103123