Finite Group Databasebeta

← #10697922

Subgroups — #10697922

Order 768.

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

normal subgroups only

This group has 528 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#83370C212yesmaximal
2#73308C211yesmaximal, characteristic
3C3×#1994C211yesmaximal, characteristic
4#70507C212yesmaximal
5#67611C211yesmaximal, characteristic
6#10297—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869C2212yes—
9C3×#423C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5093—22no—
12#5092—22no—
13#5050—24no—
14#5031—24no—
15#4936—22no—
16#4929—24no—
17#4925C2212yes—
18C3×#339—22no—
19#4481—24no—
20#3209—32no—
21#2862—31no—
22#1994D311yescharacteristic
23#1548—32no—
24#1276—31no—
25C22×Dic6D414yes—
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×#216—28no—
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3—22no—
38#869—22no—
39#869—22no—
40#869—22no—
41#869—22no—
42C3×C4⋊C8—22no—
43C3⋊#95—28no—
44#830—22no—
45#810—28no—
46C4⋊(C3⋊C8)D414yes—
47C4×C3⋊C8—22no—
48#533C2×D311yescharacteristic
49C2×C4⋊Q8—32no—
50#494—62no—
51#487—62no—
52#462—64no—
53#451—64no—
54#431—32no—
55#423C2×D311yescharacteristic
56#391—64no—
57#382—62no—
58C43C2×D311yescharacteristic
59#339—22no—
60#333—64no—
61C2×C3×Q8—22no—
62C22×C3×C4—22no—
63C22×C3×C4C2×D412yes—
64C22×C3×C4C2×D411yescharacteristic
65C22×Dic3—24no—
66C2×Dic6—24no—
67C2×Dic6—24no—
68C2×Dic6—48no—
69C3×M16—42no—
70C2×C3×C8—41no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—42no—
74C3×C22⋊C4—41no—
75C3×C42—22no—
76C3×C42—44no—
77C3×C42—41no—
78C3×C42Q1614yescomplemented
79C3×C42C2×D411yescharacteristic
80C3×C42C2×D412yes—
81C3×C42#4111yescharacteristic
82#216—88no—
83C4⋊Dic3—42no—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87C2×C3⋊C8—24no—
88C22×Q8—34no—
89C2×Q16—68no—
90C2×M16—21no—
91C4⋊Q8—62no—
92C4⋊Q8—62no—
93C4⋊Q8—62no—
94C4⋊Q8—62no—
95C4⋊Q8D1212yes—
96#125—21no—
97C22⋊Q8—22no—
98C4×Q8—22no—
99#117—21no—
100C2×C4⋊C4—62no—
101C2×C4⋊C4—62no—
102C2×C42—22no—
103C2×C42#6712yes—
104C2×C42C22×D311yescharacteristic
105C4⋊C8—22no—
106C4⋊C8—34no—
107#103—68no—
108C22⋊C8—62no—
109C4×C8—62no—
110#95—68no—
111C23×C3#12011yescharacteristic
112C3×Q8—44no—
113C2×C3×C4—24no—
114C2×C3×C4—24no—
115C2×C3×C4—44no—
116C2×C3×C4—42no—
117C2×C3×C4#12012yes—
118C2×C3×C4#12112yes—
119C2×C3×C4#12111yescharacteristic
120C2×C3×C4#12711yescharacteristic
121C2×C3×C4C2×Q1612yes—
122C2×C3×C4#13712yes—
123C2×Dic3—48no—
124C2×Dic3—44no—
125Dic6—48no—
126Dic6—84no—
127C3×C8—81no—
128C3⋊C8—44no—
129C2×Q8—22no—
130C2×Q8—64no—
131C2×Q8—64no—
132C2×Q8—128no—
133C22×C4—22no—
134C22×C4—64no—
135C22×C4D3×D412yes—
136C22×C4C2×#6711yescharacteristic
137Q16—248no—
138M16—42no—
139C2×C8—41no—
140C2×C8—64no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C4⋊C4—122no—
147C4⋊C4—122no—
148C22⋊C4—41no—
149C42—22no—
150C42—44no—
151C42—41no—
152C42#21614yescomplemented
153C42C2×D1211yescharacteristic
154C42#23511yescharacteristic
155C42D3×D412yes—
156C22×C3—22no—
157C22×C3#39211yescharacteristic
158C22×C3#45112yes—
159C3×C4—22no—
160C3×C4—44no—
161C3×C4—82no—
162C3×C4#45514yes—
163C3×C4#46212yes—
164C3×C4#49411yescharacteristic
165C3×C4#49711yescharacteristic
166Dic3—84no—
167C23#93711yescharacteristic
168Q8—44no—
169Q8—128no—
170Q8—244no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—128no—
176C2×C4—124no—
177C2×C4#88212yes—
178C2×C4#88412yes—
179C2×C4#93011yescharacteristic
180C2×C4#94011yescharacteristic
181C2×C4C2×#21612yes—
182C2×C4#95112yes—
183C8—81no—
184C8—124no—
185C2×C3—22no—
186C2×C3#200412yes—
187C2×C3#201611yescharacteristic
188C22—22no—
189C22#496011yescharacteristic
190C22#505012yes—
191C4—22no—
192C4—44no—
193C4—82no—
194C4—244no—
195C4#482514yes—
196C4#503112yes—
197C4#506011yescharacteristic
198C4#509211yescharacteristic
199C3#1029711yescharacteristic, complemented, Sylow
200C2—22no—
201C2#6969111yescharacteristic
202C2#7038612yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1994D3
hypocenterC3#10297
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
78. C3×C42Q16semidirect
152. C42#216semidirect
199. C3#10297semidirect

Maximal subgroups

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

subgroupindex
#833702
#733082
C3×#19942
#705072
#676112
#102973