Finite Group Databasebeta

← #10690152

Subgroups — #10690152

Order 768.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3135C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic, complemented
3#71829C211yesmaximal, characteristic, complemented
4#71803C212yesmaximal
5#67662C212yesmaximal, complemented
6#9008—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#532C2211yescharacteristic
9C3×#491C412yes—
10C3×#490C412yes—
11C3×#489—22no—
12C3×#416—22no—
13C2×C3×#97C2211yescharacteristic
14#5024—24no—
15#5000—22no—
16#4926C2212yes—
17#4925C2212yes—
18#4547—22no—
19D4⋊(C3⋊C8)—24no—
20#3135D311yescharacteristic
21#2987—31no—
22#1544—31no—
23#1496—32no—
24#1302—32no—
25C2×C3×#41—21no—
26C22×C3×C8—21no—
27C3×C4⋊Q8C2×C411yescharacteristic
28C3×C4⋊Q8D412yes—
29C3×#127C2×C411yescharacteristic
30C3×#125C2×C412yes—
31C3×#124—21no—
32C3×C22⋊Q8—22no—
33C3×#121—22no—
34C3×C4×Q8—22no—
35C3×C4×D4—22no—
36C3×#117D412yes—
37C2×C3×C42C2311yescharacteristic
38C3×#103—24no—
39C3×#102—24no—
40C3×#97—21no—
41C3×#97C2×C412yes—
42#830—22no—
43#830—22no—
44C4⋊(C3⋊C8)—24no—
45C4⋊(C3⋊C8)—22no—
46C4⋊(C3⋊C8)D414yes—
47#803—22no—
48#803—22no—
49C4×C3⋊C8—22no—
50C4×C3⋊C8—22no—
51#533C2×D311yescharacteristic
52#532C2×D311yescharacteristic
53#491Dic312yes—
54#490Dic312yes—
55#489—22no—
56#446—64no—
57#433—62no—
58#432—32no—
59#431—32no—
60#416—22no—
61C2×#97C2×D311yescharacteristic
62#332—62no—
63D4⋊C8—64no—
64C3×#41—44no—
65C2×C3×Q8—22no—
66C2×C3×Q8—21no—
67C2×C3×D4—21no—
68C2×C3×D4—22no—
69C2×C3×D4—41no—
70C22×C3×C4—41no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4#4112yes—
73C2×C3×C8—22no—
74C2×C3×C8—42no—
75C3×C4⋊C4—24no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4C22⋊C414yes—
79C3×C22⋊C4—22no—
80C3×C22⋊C4—42no—
81C3×C42—22no—
82C3×C42C22×C411yescharacteristic
83C3×C42C2×D412yes—
84C3×C42C2×D411yescharacteristic
85C2×C3⋊C8—24no—
86C2×C3⋊C8—42no—
87C2×#41—21no—
88C22×C8—21no—
89C4⋊Q8C2×Dic311yescharacteristic
90C4⋊Q8#6712yes—
91#127C2×Dic311yescharacteristic
92#125C2×Dic312yes—
93#124—21no—
94C22⋊Q8—22no—
95#121—22no—
96C4×Q8—22no—
97C4×D4—22no—
98#117#6712yes—
99C2×C42C22×D311yescharacteristic
100C4⋊C8—34no—
101C4⋊C8—64no—
102C4⋊C8—62no—
103#103—24no—
104#102—24no—
105C22⋊C8—62no—
106C22⋊C8—62no—
107#97—21no—
108#97—62no—
109#97—62no—
110#97C2×Dic312yes—
111C4×C8—62no—
112C4×C8—62no—
113C23×C3—41no—
114C23×C3#12311yescharacteristic
115C3×Q8—42no—
116C3×Q8—82no—
117C3×D4—44no—
118C3×D4—42no—
119C3×D4—82no—
120C2×C3×C4—22no—
121C2×C3×C4—24no—
122C2×C3×C4—42no—
123C2×C3×C4—41no—
124C2×C3×C4—82no—
125C2×C3×C4C2×C22⋊C411yescharacteristic
126C2×C3×C4C4×D412yes—
127C2×C3×C4#12011yescharacteristic
128C2×C3×C4#12112yes—
129C3×C8—42no—
130C3⋊C8—44no—
131C3⋊C8—82no—
132#41—44no—
133C2×Q8—22no—
134C2×Q8—21no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—41no—
138C22×C4—41no—
139C22×C4#23712yes—
140C22×C4C2×#6711yescharacteristic
141C2×C8—22no—
142C2×C8—42no—
143C2×C8—64no—
144C2×C8—122no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4#21714yes—
149C22⋊C4—22no—
150C22⋊C4—42no—
151C42—22no—
152C42D3×D412yes—
153C42C22×Dic311yescharacteristic
154C42C2×#6711yescharacteristic
155C22×C3—22no—
156C22×C3—21no—
157C22×C3—82no—
158C22×C3—81no—
159C22×C3#38611yescharacteristic
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—21no—
163C3×C4—44no—
164C3×C4—81no—
165C3×C4#45412yes—
166C23—41no—
167C23#93511yescharacteristic
168Q8—42no—
169Q8—82no—
170D4—42no—
171D4—44no—
172D4—82no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4D4×Dic312yes—
179C2×C4#93711yescharacteristic
180C2×C4#93912yes—
181C2×C4C2×#21711yescharacteristic
182C8—42no—
183C8—124no—
184C8—242no—
185C2×C3—41no—
186C2×C3—161no—
187C2×C3#188112yes—
188C2×C3#188811yescharacteristic
189C22—21no—
190C22—22no—
191C22—81no—
192C22—82no—
193C22#522011yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—44no—
198C4—81no—
199C4#506212yes—
200C3#900811yescharacteristic, complemented, Sylow
201C2—41no—
202C2—161no—
203C2#7172411yescharacteristic
204C2#7211612yes—
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3135C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3135D3
hypocenterC3#9008
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #83333C2semidirect
3. #71829C2semidirect
5. #67662C2semidirect
200. C3#9008semidirect

Maximal subgroups

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

subgroupindex
C3×#31352
#833332
#718292
#718032
#676622
#90083