Finite Group Databasebeta

← #10697911

Subgroups — #10697911

Order 768.

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

normal subgroups only

This group has 504 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
1C3×#2862C211yesmaximal, characteristic
2#83224C212yesmaximal
3#71775C212yesmaximal
4#69396C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic
6#10295—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8#5470C2211yescharacteristic
9C3×#391—24no—
10C3×#382—22no—
11C3×C43C2211yescharacteristic
12#5093—22no—
13#5091—24no—
14#5078—22no—
15#4970—24no—
16#4925C2212yes—
17#4924C2211yescharacteristic
18#4547—24no—
19#4523—22no—
20#3230—32no—
21#2862D311yescharacteristic
22#1994—31no—
23#1549—32no—
24#1276—31no—
25C22×C3×Q8—22no—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4D414yes—
32C2×C3×C42—22no—
33C2×C3×C42D412yes—
34C2×C3×C42C2311yescharacteristic
35#926—21no—
36#924—22no—
37C2×#208—21no—
38#870—21no—
39#869D412yes—
40C4×Dic6—22no—
41C3×#95—28no—
42#830—22no—
43#810—28no—
44#807—28no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)D414yes—
47C4×C3⋊C8—22no—
48#533—31no—
49C2×C4⋊Q8C2×D312yes—
50#500—62no—
51#487—62no—
52#484—64no—
53#477—64no—
54#431—32no—
55#423—31no—
56#391—24no—
57#382—22no—
58C43C2×D311yescharacteristic
59#339—62no—
60#332—64no—
61C2×C3×Q8—48no—
62C2×C3×Q8—42no—
63C2×C3×Q8—42no—
64C22×C3×C4—22no—
65C22×C3×C4—24no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×D411yescharacteristic
68C2×Dic6—22no—
69C3×C4⋊C4—24no—
70C3×C4⋊C4—24no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—42no—
73C3×C42—22no—
74C3×C42—44no—
75C3×C42Q1614yes—
76C3×C42C2×D411yescharacteristic
77C3×C42C2×D412yes—
78C3×C42#4111yescharacteristic
79#217—41no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82Dic3⋊C4—42no—
83C4×Dic3—41no—
84#208—42no—
85C2×C3⋊C8—24no—
86C2×C3⋊C8—41no—
87C22×Q8—22no—
88C2×M16—61no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—32no—
94#125—61no—
95C22⋊Q8—62no—
96C4×Q8—62no—
97#117—61no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4#6714yes—
100C2×C42—22no—
101C2×C42#6712yes—
102C2×C42C22×D311yescharacteristic
103#107—68no—
104C4⋊C8—34no—
105C4⋊C8—62no—
106#103—68no—
107C22⋊C8—62no—
108C4×C8—62no—
109#95—28no—
110C23×C3#12011yescharacteristic
111C3×Q8—84no—
112C3×Q8—84no—
113C2×C3×C4—24no—
114C2×C3×C4—24no—
115C2×C3×C4—44no—
116C2×C3×C4—48no—
117C2×C3×C4—44no—
118C2×C3×C4#12012yes—
119C2×C3×C4#12111yescharacteristic
120C2×C3×C4#12112yes—
121C2×C3×C4#12711yescharacteristic
122C2×C3×C4C2×Q1612yes—
123C2×C3×C4#13712yes—
124C2×Dic3—42no—
125Dic6—44no—
126C3⋊C8—44no—
127C3⋊C8—81no—
128C2×Q8—48no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—62no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4C2×#6712yes—
135C22×C4C2×#6711yescharacteristic
136M16—122no—
137C2×C8—64no—
138C2×C8—121no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—122no—
146C22⋊C4—121no—
147C42—22no—
148C42—44no—
149C42—121no—
150C42#21614yes—
151C42D3×D411yescharacteristic
152C42D3×D412yes—
153C42#23711yescharacteristic
154C22×C3—22no—
155C22×C3#39211yescharacteristic
156C22×C3#45112yes—
157C3×C4—22no—
158C3×C4—44no—
159C3×C4—84no—
160C3×C4#45414yes—
161C3×C4#46212yes—
162C3×C4#49411yescharacteristic
163C3×C4#49711yescharacteristic
164Dic3—82no—
165C23#95311yescharacteristic
166Q8—84no—
167Q8—84no—
168Q8—124no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—48no—
173C2×C4—44no—
174C2×C4—122no—
175C2×C4#93712yes—
176C2×C4#93811yescharacteristic
177C2×C4#93912yes—
178C2×C4#94011yescharacteristic
179C2×C4#94212yes—
180C2×C4C2×#21612yes—
181C8—124no—
182C8—241no—
183C2×C3—22no—
184C2×C3#200312yes—
185C2×C3#201611yescharacteristic
186C22—22no—
187C22#522311yescharacteristic
188C22#522812yes—
189C4—22no—
190C4—44no—
191C4—84no—
192C4—242no—
193C4#506214yes—
194C4#508611yescharacteristic
195C4#509012yes—
196C4#509211yescharacteristic
197C3#1029511yescharacteristic, complemented, Sylow
198C2—22no—
199C2#7173012yes—
200C2#7179311yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2862D3
hypocenterC3#10295
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

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
C3×#28622
#832242
#717752
#693962
#676112
#102953