Finite Group Databasebeta

← #10697937

Subgroups — #10697937

Order 768.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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#79271C212yesmaximal
2#73311C211yesmaximal, characteristic
3#70778C212yesmaximal
4#69396C211yesmaximal, characteristic
5C3×#1276C211yesmaximal, characteristic
6#10299—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C2×#869—22no—
9C2×#869C2212yes—
10C3×#431C2212yes—
11C3×C43C2211yescharacteristic
12#4935—24no—
13#4924C2211yescharacteristic
14#4851—24no—
15#4721—24no—
16#4681—22no—
17#4680—22no—
18#4523—22no—
19#4486—24no—
20#3229—32no—
21#2868—31no—
22#1994—31no—
23#1546—32no—
24#1276D311yescharacteristic
25C22×Dic6—22no—
26C2×C3×C42—22no—
27C2×C3×C42D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—21no—
30C2×C4⋊Dic3—28no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3D414yes—
33#924—22no—
34C2×#208—21no—
35#870—21no—
36#869—22no—
37#869—22no—
38#869—22no—
39#869—22no—
40#869—44no—
41#869—44no—
42#869D412yes—
43C4×Dic6—22no—
44C3×C4⋊C8D414yes—
45C3×C22⋊C8—22no—
46C3×C4×C8—22no—
47#817—28no—
48#816—28no—
49C4⋊(C3⋊C8)—22no—
50#533—31no—
51C2×C4⋊Q8—32no—
52C2×C4⋊Q8—62no—
53#498—62no—
54#487—62no—
55#483—64no—
56#475—64no—
57#431C2×D312yes—
58#423—31no—
59#384—64no—
60C43C2×D311yescharacteristic
61#339—62no—
62#332—64no—
63C22×C3×C4—22no—
64C22×C3×C4C2×D411yescharacteristic
65C22×C3×C4C2×D412yes—
66C22×Dic3—24no—
67C2×Dic6—22no—
68C2×Dic6—42no—
69C2×Dic6—42no—
70C2×Dic6—44no—
71C2×Dic6—44no—
72C2×C3×C8—24no—
73C3×C42—28no—
74C3×C42—22no—
75C3×C42SD1614yes—
76C3×C42C2×D412yes—
77C3×C42C2×D411yescharacteristic
78C3×C42#4111yescharacteristic
79#217—41no—
80C4⋊Dic3—24no—
81C4⋊Dic3—22no—
82C4⋊Dic3—24no—
83C4⋊Dic3—22no—
84C4⋊Dic3—416no—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87Dic3⋊C4—42no—
88C4×Dic3—41no—
89#208—42no—
90C2×C3⋊C8—41no—
91C22×Q8—62no—
92C2×M16—61no—
93C4⋊Q8—32no—
94C4⋊Q8—62no—
95C4⋊Q8—62no—
96C4⋊Q8—62no—
97C4⋊Q8—62no—
98C4⋊Q8—124no—
99C4⋊Q8—124no—
100#125—61no—
101C22⋊Q8—62no—
102C4×Q8—62no—
103#117—61no—
104C2×C4⋊C4—34no—
105C2×C4⋊C4—68no—
106C2×C4⋊C4—62no—
107C2×C42—22no—
108C2×C42D1212yes—
109C2×C42C22×D311yescharacteristic
110#106—68no—
111C4⋊C8—62no—
112C4⋊C8D1214yes—
113#103—68no—
114C22⋊C8—22no—
115C4×C8—22no—
116C23×C3#12011yescharacteristic
117C2×C3×C4—28no—
118C2×C3×C4—24no—
119C2×C3×C4—24no—
120C2×C3×C4#12012yes—
121C2×C3×C4#12112yes—
122C2×C3×C4#12111yescharacteristic
123C2×C3×C4#12711yescharacteristic
124C2×C3×C4C2×SD1612yes—
125C2×C3×C4AGL(1,ℤ/8)12yes—
126C2×Dic3—42no—
127C2×Dic3—44no—
128C2×Dic3—48no—
129Dic6—44no—
130Dic6—84no—
131Dic6—82no—
132Dic6—82no—
133C3×C8—44no—
134C3⋊C8—81no—
135C2×Q8—62no—
136C2×Q8—122no—
137C2×Q8—122no—
138C2×Q8—124no—
139C2×Q8—124no—
140C22×C4—22no—
141C22×C4—64no—
142C22×C4C2×D1211yescharacteristic
143C22×C4D3×D412yes—
144M16—122no—
145C2×C8—24no—
146C2×C8—121no—
147C4⋊C4—64no—
148C4⋊C4—62no—
149C4⋊C4—64no—
150C4⋊C4—62no—
151C4⋊C4—1216no—
152C4⋊C4—122no—
153C4⋊C4—122no—
154C4⋊C4—122no—
155C22⋊C4—121no—
156C42—28no—
157C42—22no—
158C42—121no—
159C42#20414yes—
160C42C2×D1212yes—
161C42D3×D411yescharacteristic
162C42#23911yescharacteristic
163C22×C3—22no—
164C22×C3#39211yescharacteristic
165C22×C3#45012yes—
166C3×C4—28no—
167C3×C4—22no—
168C3×C4#45614yes—
169C3×C4#46012yes—
170C3×C4#49211yescharacteristic
171C3×C4#49611yescharacteristic
172Dic3—82no—
173Dic3—84no—
174C23#88211yescharacteristic
175Q8—124no—
176Q8—242no—
177Q8—242no—
178Q8—244no—
179C2×C4—28no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—122no—
183C2×C4—124no—
184C2×C4—128no—
185C2×C4#87411yescharacteristic
186C2×C4#88212yes—
187C2×C4#89512yes—
188C2×C4#89511yescharacteristic
189C2×C4C2×#20412yes—
190C2×C4#90812yes—
191C8—44no—
192C8—241no—
193C2×C3—22no—
194C2×C3#200512yes—
195C2×C3#201411yescharacteristic
196C22—22no—
197C22#467111yescharacteristic
198C22#473112yes—
199C4—28no—
200C4—22no—
201C4—244no—
202C4—242no—
203C4#469311yescharacteristic
204C4#471211yescharacteristic
205C4#482614yes—
206C4#484812yes—
207C3#1029911yescharacteristic, complemented, Sylow
208C2—22no—
209C2#6867611yescharacteristic
210C2#6876812yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1276D3
hypocenterC3#10299
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
207. C3#10299semidirect

Maximal subgroups

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

subgroupindex
#792712
#733112
#707782
#693962
C3×#12762
#102993