Finite Group Databasebeta

← #10697910

Subgroups — #10697910

Order 768.

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

normal subgroups only

This group has 504 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#83324C212yesmaximal
2#73308C211yesmaximal, characteristic
3C3×#1994C211yesmaximal, characteristic
4#70508C212yesmaximal
5#67611C211yesmaximal, characteristic
6#10295—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869C2212yes—
9C3×#423C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5079—22no—
12#5078—22no—
13#5044—24no—
14#5021—24no—
15#4936—22no—
16#4929—24no—
17#4925C2212yes—
18C3×#339—22no—
19#4486—24no—
20#3230—32no—
21#2862—31no—
22#1994D311yescharacteristic
23#1549—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—22no—
36C2×C4⋊Dic3D414yes—
37#869—22no—
38#869—22no—
39#869—22no—
40#869—22no—
41C3×C4⋊C8—22no—
42#835—28no—
43C3⋊#95—28no—
44#830—22no—
45#807—28no—
46C4⋊(C3⋊C8)D414yes—
47C4×C3⋊C8—22no—
48#533C2×D311yescharacteristic
49C2×C4⋊Q8—32no—
50#500—62no—
51#487—62no—
52#484—64no—
53#477—64no—
54#431—32no—
55#423C2×D311yescharacteristic
56#391—64no—
57#382—62no—
58C43C2×D311yescharacteristic
59#339—22no—
60#332—64no—
61C2×C3×Q8—22no—
62C22×C3×C4—22no—
63C22×C3×C4C2×D412yes—
64C22×C3×C4C2×D411yescharacteristic
65C22×Dic3—24no—
66C2×Dic6—48no—
67C2×Dic6—42no—
68C2×Dic6—42no—
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×C42Q1614yes—
79C3×C42C2×D411yescharacteristic
80C3×C42C2×D412yes—
81C3×C42#4111yescharacteristic
82C4⋊Dic3—24no—
83C4⋊Dic3—24no—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86C2×C3⋊C8—24no—
87C22×Q8—62no—
88C2×M16—21no—
89C4⋊Q8—62no—
90C4⋊Q8—62no—
91C4⋊Q8—62no—
92C4⋊Q8—62no—
93C4⋊Q8D1212yes—
94#125—21no—
95C22⋊Q8—22no—
96C4×Q8—22no—
97#117—21no—
98C2×C4⋊C4—34no—
99C2×C4⋊C4—62no—
100C2×C42—22no—
101C2×C42#6712yes—
102C2×C42C22×D311yescharacteristic
103#107—68no—
104C4⋊C8—22no—
105C4⋊C8—34no—
106#103—68no—
107C22⋊C8—62no—
108C4×C8—62no—
109#95—68no—
110C23×C3#12011yescharacteristic
111C3×Q8—44no—
112C2×C3×C4—24no—
113C2×C3×C4—24no—
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4#12012yes—
117C2×C3×C4#12112yes—
118C2×C3×C4#12111yescharacteristic
119C2×C3×C4#12711yescharacteristic
120C2×C3×C4C2×Q1612yes—
121C2×C3×C4#13712yes—
122C2×Dic3—48no—
123C2×Dic3—44no—
124Dic6—84no—
125Dic6—84no—
126C3×C8—81no—
127C3⋊C8—44no—
128C2×Q8—22no—
129C2×Q8—128no—
130C2×Q8—122no—
131C2×Q8—122no—
132C22×C4—22no—
133C22×C4—64no—
134C22×C4D3×D412yes—
135C22×C4C2×#6711yescharacteristic
136M16—42no—
137C2×C8—41no—
138C2×C8—64no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C22⋊C4—41no—
147C42—22no—
148C42—44no—
149C42—41no—
150C42#21614yes—
151C42C2×D1211yescharacteristic
152C42#23511yescharacteristic
153C42D3×D412yes—
154C22×C3—22no—
155C22×C3#39211yescharacteristic
156C22×C3#45112yes—
157C3×C4—22no—
158C3×C4—44no—
159C3×C4—82no—
160C3×C4#45414yes—
161C3×C4#46212yes—
162C3×C4#49411yescharacteristic
163C3×C4#49711yescharacteristic
164Dic3—84no—
165C23#93711yescharacteristic
166Q8—44no—
167Q8—244no—
168Q8—244no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—128no—
174C2×C4—124no—
175C2×C4#88212yes—
176C2×C4#88412yes—
177C2×C4#93011yescharacteristic
178C2×C4#94011yescharacteristic
179C2×C4C2×#21612yes—
180C2×C4#95112yes—
181C8—81no—
182C8—124no—
183C2×C3—22no—
184C2×C3#200312yes—
185C2×C3#201611yescharacteristic
186C22—22no—
187C22#496011yescharacteristic
188C22#505012yes—
189C4—22no—
190C4—44no—
191C4—82no—
192C4—244no—
193C4#482714yes—
194C4#503112yes—
195C4#506011yescharacteristic
196C4#509211yescharacteristic
197C3#1029511yescharacteristic, complemented, Sylow
198C2—22no—
199C2#6969111yescharacteristic
200C2#7038812yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1994D3
hypocenterC3#10295
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
197. C3#10295semidirect

Maximal subgroups

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

subgroupindex
#833242
#733082
C3×#19942
#705082
#676112
#102953