Finite Group Databasebeta

← #10697478

Subgroups — #10697478

Order 768.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#83223C212yesmaximal
2#78479C211yesmaximal, characteristic
3C3×#1960C211yesmaximal, characteristic
4#69546C212yesmaximal
5#67661C211yesmaximal, characteristic
6#10227—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8#5470C2212yes—
9C2×C3×#103—22no—
10C2×C3×#97C2211yescharacteristic
11C3×#382C2211yescharacteristic
12#5106—22no—
13#5094—24no—
14#5093—22no—
15#5080—22no—
16#4971—24no—
17#4925C2212yes—
18#4702—22no—
19#4702—22no—
20#4452—24no—
21#3229—32no—
22#3162—31no—
23#1960D311yescharacteristic
24#1518—32no—
25#1281—31no—
26C22×C3×Q8—21no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—21no—
30C3×C4⋊Q8D412yes—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C42C2311yescharacteristic
34#926—22no—
35#924—22no—
36#924—22no—
37#870—22no—
38#869D412yes—
39#869D412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C3×#103—44no—
43C3×#97—22no—
44C3×#95—22no—
45#830D414yes—
46#818—24no—
47#817—22no—
48#817—22no—
49#810—28no—
50#808—28no—
51C4⋊(C3⋊C8)—22no—
52C4×C3⋊C8—22no—
53#533—32no—
54C2×C4⋊Q8C2×D311yescharacteristic
55#501—62no—
56#501—62no—
57#498—62no—
58#487—62no—
59#483—64no—
60#475—64no—
61#431—32no—
62#427—62no—
63C2×#103—22no—
64C2×#97C2×D311yescharacteristic
65#382C2×D311yescharacteristic
66#328—64no—
67C2×C3×Q8—22no—
68C2×C3×Q8—41no—
69C2×C3×Q8—42no—
70C2×C3×Q8—42no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×C3×C8—22no—
78C2×C3×C8—22no—
79C2×C3×C8SD1614yes—
80C3×C4⋊C4—24no—
81C3×C4⋊C4—42no—
82C3×C4⋊C4—42no—
83C3×C42C2×D411yescharacteristic
84C3×C42C2×D412yes—
85C3×C42#4111yescharacteristic
86#217—42no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—22no—
91Dic3⋊C4—42no—
92Dic3⋊C4—42no—
93C4×Dic3—42no—
94C2×C3⋊C8—24no—
95C22×Q8—21no—
96C22×C8#6712yes—
97C4⋊Q8—21no—
98C4⋊Q8—22no—
99C4⋊Q8—32no—
100C4⋊Q8—32no—
101C4⋊Q8D1212yes—
102#125—62no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—62no—
106C4×Q8—62no—
107#117—62no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C42C22×D311yescharacteristic
111#107—64no—
112#106—68no—
113#106—62no—
114#106—62no—
115C4⋊C8—62no—
116#103—44no—
117#103—68no—
118C22⋊C8—34no—
119#97—22no—
120C4×C8—62no—
121#95—22no—
122C23×C3#12711yescharacteristic
123C3×Q8—44no—
124C3×Q8—82no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—81no—
128C2×C3×C4—81no—
129C2×C3×C4—82no—
130C2×C3×C4—81no—
131C2×C3×C4#12012yes—
132C2×C3×C4#12011yescharacteristic
133C2×C3×C4#12112yes—
134C2×C3×C4#12111yescharacteristic
135C2×C3×C4C2×SD1612yes—
136C2×C3×C4#13712yes—
137C2×Dic3—42no—
138C2×Dic3—42no—
139Dic6—44no—
140Dic6—82no—
141C3×C8—24no—
142C3⋊C8—44no—
143C2×Q8—22no—
144C2×Q8—41no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—62no—
148C2×Q8—62no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4D3×D412yes—
152C22×C4C2×#6711yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—64no—
156C2×C8#21414yes—
157C4⋊C4—24no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—122no—
167C42—122no—
168C42C2×D1211yescharacteristic
169C42#23511yescharacteristic
170C42D3×D412yes—
171C22×C3—22no—
172C22×C3#39211yescharacteristic
173C22×C3#49211yescharacteristic
174C22×C3#49711yescharacteristic
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—82no—
178C3×C4—161no—
179C3×C4#44812yes—
180C3×C4#46112yes—
181Dic3—82no—
182Dic3—82no—
183C23#94011yescharacteristic
184Q8—44no—
185Q8—82no—
186Q8—124no—
187Q8—242no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4#88212yes—
197C2×C4#88412yes—
198C2×C4#93011yescharacteristic
199C2×C4C2×#21412yes—
200C2×C4#93711yescharacteristic
201C2×C4#95112yes—
202C8—24no—
203C8—124no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3#200912yes—
207C2×C3#201311yescharacteristic
208C22—22no—
209C22#496011yescharacteristic
210C22#506011yescharacteristic
211C22#507611yescharacteristic
212C4—22no—
213C4—22no—
214C4—82no—
215C4—161no—
216C4—242no—
217C4—242no—
218C4#501912yes—
219C4#504012yes—
220C3#1022711yescharacteristic, complemented, Sylow
221C2—21no—
222C2—21no—
223C2#6966111yescharacteristic
224C2#7035112yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1960D3
hypocenterC3#10227
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
220. C3#10227semidirect

Maximal subgroups

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

subgroupindex
#832232
#784792
C3×#19602
#695462
#676612
#102273