Finite Group Databasebeta

← #10697923

Subgroups — #10697923

Order 768.

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

normal subgroups only

This group has 528 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#83222C212yesmaximal
3#71774C212yesmaximal
4#69396C211yesmaximal, characteristic
5#67611C211yesmaximal, characteristic
6#10297—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8#5470C2211yescharacteristic
9C3×#391—24no—
10C3×#382—22no—
11C3×C43C2211yescharacteristic
12#5228—24no—
13#5092—22no—
14#5090—24no—
15#5079—22no—
16#4925C2212yes—
17#4924C2211yescharacteristic
18#4546—24no—
19#4523—22no—
20#3209—32no—
21#2862D311yescharacteristic
22#1994—31no—
23#1548—32no—
24#1276—31no—
25C22×C3×Q8D414yes—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8—22no—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C42—22no—
33C2×C3×C42D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×#216—28no—
36#926—21no—
37#924—22no—
38C2×#208—21no—
39#870—21no—
40#869D412yes—
41C4×Dic6—22no—
42C3×#95—28no—
43#835—28no—
44#830—22no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)D414yes—
47C4×C3⋊C8—22no—
48#533—31no—
49C2×C4⋊Q8C2×D312yes—
50#494—62no—
51#487—62no—
52#462—64no—
53#451—64no—
54#431—32no—
55#423—31no—
56#391—24no—
57#382—22no—
58C43C2×D311yescharacteristic
59#339—62no—
60#333—64no—
61C2×C3×Q8—24no—
62C2×C3×Q8—24no—
63C2×C3×Q8—48no—
64C22×C3×C4—22no—
65C22×C3×C4—24no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×D411yescharacteristic
68C2×Dic6—22no—
69C3×C4⋊C4—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—42no—
73C3×C42—22no—
74C3×C42—44no—
75C3×C42Q1614yescomplemented
76C3×C42C2×D411yescharacteristic
77C3×C42C2×D412yes—
78C3×C42#4111yescharacteristic
79#217—41no—
80#216—88no—
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83Dic3⋊C4—42no—
84C4×Dic3—41no—
85#208—42no—
86C2×C3⋊C8—24no—
87C2×C3⋊C8—41no—
88C22×Q8#6714yes—
89C2×Q16—68no—
90C2×M16—61no—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—32no—
96#125—61no—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—61no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C42—22no—
103C2×C42#6712yes—
104C2×C42C22×D311yescharacteristic
105C4⋊C8—34no—
106C4⋊C8—62no—
107#103—68no—
108C22⋊C8—62no—
109C4×C8—62no—
110#95—28no—
111C23×C3#12011yescharacteristic
112C3×Q8—48no—
113C3×Q8—84no—
114C2×C3×C4—24no—
115C2×C3×C4—24no—
116C2×C3×C4—44no—
117C2×C3×C4—48no—
118C2×C3×C4—44no—
119C2×C3×C4#12012yes—
120C2×C3×C4#12111yescharacteristic
121C2×C3×C4#12112yes—
122C2×C3×C4#12711yescharacteristic
123C2×C3×C4C2×Q1612yes—
124C2×C3×C4#13712yes—
125C2×Dic3—42no—
126Dic6—44no—
127C3⋊C8—44no—
128C3⋊C8—81no—
129C2×Q8—24no—
130C2×Q8—24no—
131C2×Q8—48no—
132C2×Q8—62no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4C2×#6712yes—
136C22×C4C2×#6711yescharacteristic
137Q16—248no—
138M16—122no—
139C2×C8—64no—
140C2×C8—121no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—122no—
148C22⋊C4—121no—
149C42—22no—
150C42—44no—
151C42—121no—
152C42#21614yescomplemented
153C42D3×D411yescharacteristic
154C42D3×D412yes—
155C42#23711yescharacteristic
156C22×C3—22no—
157C22×C3#39211yescharacteristic
158C22×C3#45112yes—
159C3×C4—22no—
160C3×C4—44no—
161C3×C4—84no—
162C3×C4#45514yes—
163C3×C4#46212yes—
164C3×C4#49411yescharacteristic
165C3×C4#49711yescharacteristic
166Dic3—82no—
167C23#95311yescharacteristic
168Q8—48no—
169Q8—84no—
170Q8—124no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—44no—
174C2×C4—48no—
175C2×C4—44no—
176C2×C4—122no—
177C2×C4#93712yes—
178C2×C4#93811yescharacteristic
179C2×C4#93912yes—
180C2×C4#94011yescharacteristic
181C2×C4#94212yes—
182C2×C4C2×#21612yes—
183C8—124no—
184C8—241no—
185C2×C3—22no—
186C2×C3#200412yes—
187C2×C3#201611yescharacteristic
188C22—22no—
189C22#522311yescharacteristic
190C22#522812yes—
191C4—22no—
192C4—44no—
193C4—84no—
194C4—242no—
195C4#506114yes—
196C4#508611yescharacteristic
197C4#509012yes—
198C4#509211yescharacteristic
199C3#1029711yescharacteristic, complemented, Sylow
200C2—22no—
201C2#7173112yes—
202C2#7179311yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2862D3
hypocenterC3#10297
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
75. 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
C3×#28622
#832222
#717742
#693962
#676112
#102973