Finite Group Databasebeta

← #10697934

Subgroups — #10697934

Order 768.

576 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 111 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#83323C212yesmaximal
2#73311C211yesmaximal, characteristic
3C3×#1994C211yesmaximal, characteristic
4#70506C212yesmaximal
5#67611C211yesmaximal, characteristic
6#10299—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869—22no—
9C2×#869C2212yes—
10C3×#423C2211yescharacteristic
11C3×C43C2211yescharacteristic
12#5080—22no—
13#5079—22no—
14#5043—24no—
15#5020—24no—
16#4935—24no—
17#4925C2212yes—
18C3×#339—22no—
19#4482—24no—
20#3229—32no—
21#2868—31no—
22#1994D311yescharacteristic
23#1546—32no—
24#1276—31no—
25C22×Dic6—22no—
26C2×C3×M16—21no—
27C3×C4⋊Q8D412yes—
28C3×#125—21no—
29C3×C22⋊Q8—22no—
30C3×C4×Q8—22no—
31C3×#117—21no—
32C2×C3×C42—22no—
33C2×C3×C42D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—28no—
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3D414yes—
38#869—22no—
39#869—22no—
40#869—22no—
41#869—22no—
42#869—44no—
43#869—44no—
44C3×C4⋊C8—22no—
45#835—28no—
46#830—22no—
47#808—28no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#533C2×D311yescharacteristic
51C2×C4⋊Q8—32no—
52C2×C4⋊Q8—62no—
53#498—62no—
54#487—62no—
55#483—64no—
56#475—64no—
57#431—32no—
58#423C2×D311yescharacteristic
59#384—64no—
60C43C2×D311yescharacteristic
61#339—22no—
62#332—64no—
63C2×C3×Q8—22no—
64C22×C3×C4—22no—
65C22×C3×C4C2×D412yes—
66C22×C3×C4C2×D411yescharacteristic
67C22×Dic3—24no—
68C2×Dic6—42no—
69C2×Dic6—42no—
70C2×Dic6—44no—
71C2×Dic6—44no—
72C3×M16—42no—
73C2×C3×C8—41no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—42no—
77C3×C22⋊C4—41no—
78C3×C42—28no—
79C3×C42—22no—
80C3×C42—41no—
81C3×C42SD1614yes—
82C3×C42C2×D411yescharacteristic
83C3×C42C2×D412yes—
84C3×C42#4111yescharacteristic
85C4⋊Dic3—24no—
86C4⋊Dic3—24no—
87C4⋊Dic3—416no—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90C2×C3⋊C8—24no—
91C22×Q8—62no—
92C2×M16—21no—
93C4⋊Q8—62no—
94C4⋊Q8—62no—
95C4⋊Q8—62no—
96C4⋊Q8—62no—
97C4⋊Q8—124no—
98C4⋊Q8—124no—
99C4⋊Q8D1212yes—
100#125—21no—
101C22⋊Q8—22no—
102C4×Q8—22no—
103#117—21no—
104C2×C4⋊C4—34no—
105C2×C4⋊C4—68no—
106C2×C4⋊C4—62no—
107C2×C42—22no—
108C2×C42#6712yes—
109C2×C42C22×D311yescharacteristic
110#106—68no—
111C4⋊C8—22no—
112C4⋊C8—34no—
113#103—68no—
114C22⋊C8—62no—
115C4×C8—62no—
116C23×C3#12011yescharacteristic
117C3×Q8—44no—
118C2×C3×C4—28no—
119C2×C3×C4—24no—
120C2×C3×C4—24no—
121C2×C3×C4—42no—
122C2×C3×C4#12012yes—
123C2×C3×C4#12112yes—
124C2×C3×C4#12111yescharacteristic
125C2×C3×C4#12711yescharacteristic
126C2×C3×C4C2×SD1612yes—
127C2×C3×C4AGL(1,ℤ/8)12yes—
128C2×Dic3—44no—
129C2×Dic3—48no—
130Dic6—84no—
131Dic6—82no—
132Dic6—82no—
133C3×C8—81no—
134C3⋊C8—44no—
135C2×Q8—22no—
136C2×Q8—122no—
137C2×Q8—122no—
138C2×Q8—124no—
139C2×Q8—124no—
140C22×C4—22no—
141C22×C4—64no—
142C22×C4D3×D412yes—
143C22×C4C2×#6711yescharacteristic
144M16—42no—
145C2×C8—41no—
146C2×C8—64no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—64no—
151C4⋊C4—64no—
152C4⋊C4—1216no—
153C4⋊C4—122no—
154C4⋊C4—122no—
155C22⋊C4—41no—
156C42—28no—
157C42—22no—
158C42—41no—
159C42#21514yes—
160C42C2×D1211yescharacteristic
161C42#23511yescharacteristic
162C42D3×D412yes—
163C22×C3—22no—
164C22×C3#39211yescharacteristic
165C22×C3#45012yes—
166C3×C4—28no—
167C3×C4—22no—
168C3×C4—82no—
169C3×C4#45614yes—
170C3×C4#46012yes—
171C3×C4#49211yescharacteristic
172C3×C4#49611yescharacteristic
173Dic3—84no—
174C23#93711yescharacteristic
175Q8—44no—
176Q8—244no—
177Q8—242no—
178Q8—242no—
179C2×C4—28no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—124no—
184C2×C4—128no—
185C2×C4#88212yes—
186C2×C4#88412yes—
187C2×C4#93011yescharacteristic
188C2×C4#94011yescharacteristic
189C2×C4C2×#21512yes—
190C2×C4#94912yes—
191C8—81no—
192C8—124no—
193C2×C3—22no—
194C2×C3#200512yes—
195C2×C3#201411yescharacteristic
196C22—22no—
197C22#496011yescharacteristic
198C22#504612yes—
199C4—28no—
200C4—22no—
201C4—82no—
202C4—244no—
203C4#482614yes—
204C4#502712yes—
205C4#505811yescharacteristic
206C4#508511yescharacteristic
207C3#1029911yescharacteristic, complemented, Sylow
208C2—22no—
209C2#6968811yescharacteristic
210C2#7038912yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1994D3
hypocenterC3#10299
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

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
#833232
#733112
C3×#19942
#705062
#676112
#102993