Finite Group Databasebeta

← #10690151

Subgroups — #10690151

Order 768.

398 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2987C211yesmaximal, characteristic, complemented
2#83335C211yesmaximal, characteristic
3#71829C211yesmaximal, characteristic, complemented
4#70801C212yesmaximal
5#67751C212yesmaximal, complemented
6#9008—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5477C2211yescharacteristic
9C3×#446—24no—
10C3×#433—22no—
11C3×#432C2212yes—
12#5089—22no—
13#5069C412yes—
14#5067C412yes—
15#5002—22no—
16#4925C2212yes—
17C2×#803C2211yescharacteristic
18#4547—22no—
19#4483—24no—
20#3135—31no—
21#2987D311yescharacteristic, complemented
22#1544—31no—
23#1496—32no—
24#1302—32no—
25C2×#235—21no—
26C3×C4⋊Q8D412yes—
27C3×#125C2×C412yes—
28C3×C22⋊Q8—22no—
29C3×C4×Q8—22no—
30C3×#117D412yes—
31C2×C3×C42C2311yescharacteristic
32#930—22no—
33C22×C3⋊C8—21no—
34#875—21no—
35#874C2×C411yescharacteristic
36C4×D12—22no—
37#869C2×C411yescharacteristic
38C3×C4⋊C8—24no—
39C3×C4⋊C8D414yes—
40C3×C22⋊C8—22no—
41C3×#97—22no—
42C3×#97—22no—
43C3×C4×C8—22no—
44#830—22no—
45#810—24no—
46#809—24no—
47C4⋊(C3⋊C8)—22no—
48#803—21no—
49#803C2×C412yes—
50C4×C3⋊C8—22no—
51#533C2×D311yescharacteristic
52#532—31no—
53#491—32no—
54#490—32no—
55#489—62no—
56#446—24no—
57#433—22no—
58#432C2×D312yes—
59#431—32no—
60#416—62no—
61C2×#97—31no—
62#332—62no—
63D4⋊C8—64no—
64C2×C3×Q8—22no—
65C22×C3×C4C2×D411yescharacteristic
66C22×C3×C4#4112yes—
67C2×#67—41no—
68#235—44no—
69C2×D12—21no—
70C2×D12—22no—
71C2×C4×D3—41no—
72C2×Dic6—21no—
73C2×C3×C8—24no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4C22⋊C414yes—
77C3×C22⋊C4—22no—
78C3×C42—22no—
79C3×C42C22×C411yescharacteristic
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D412yes—
82(C2×D3)⋊C4—42no—
83C4⋊Dic3—22no—
84C2×C3⋊C8—22no—
85C2×C3⋊C8—42no—
86C2×C3⋊C8—42no—
87C2×#41—61no—
88C22×C8—61no—
89C4⋊Q8—31no—
90C4⋊Q8D1212yes—
91#127—31no—
92#125C4×D312yes—
93#124—61no—
94C22⋊Q8—22no—
95#121—62no—
96C4×Q8—22no—
97C4×D4—62no—
98#117#6712yes—
99C2×C42C22×D311yescharacteristic
100C4⋊C8—24no—
101C4⋊C8—62no—
102C4⋊C8#6714yes—
103#103—64no—
104#102—64no—
105C22⋊C8—22no—
106C22⋊C8—62no—
107#97—22no—
108#97—22no—
109#97—32no—
110#97—61no—
111C4×C8—22no—
112C4×C8—62no—
113C23×C3#12311yescharacteristic
114C22×D3—41no—
115C3×Q8—82no—
116C2×C3×C4—22no—
117C2×C3×C4—24no—
118C2×C3×C4—42no—
119C2×C3×C4C2×C22⋊C411yescharacteristic
120C2×C3×C4C4×D412yes—
121C2×C3×C4#12011yescharacteristic
122C2×C3×C4#12112yes—
123#67—82no—
124C2×Dic3—41no—
125D12—44no—
126D12—42no—
127C4×D3—82no—
128Dic6—42no—
129C3×C8—44no—
130C3⋊C8—42no—
131C3⋊C8—82no—
132#41—124no—
133C2×Q8—22no—
134C2×Q8—61no—
135C2×D4—61no—
136C2×D4—62no—
137C2×D4—121no—
138C22×C4—121no—
139C22×C4#23512yes—
140C22×C4C2×#6711yescharacteristic
141C2×C8—24no—
142C2×C8—62no—
143C2×C8—122no—
144C2×C8—122no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—62no—
148C4⋊C4(C2×D3)⋊C414yes—
149C22⋊C4—22no—
150C22⋊C4—122no—
151C42—22no—
152C42C2×C4×D311yescharacteristic
153C42C2×D1211yescharacteristic
154C42C2×#6712yes—
155C22×C3—22no—
156C22×C3—21no—
157C22×C3#38611yescharacteristic
158C2×D3—82no—
159C2×D3—81no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—21no—
163C3×C4—44no—
164C3×C4#45412yes—
165Dic3—81no—
166C23—121no—
167C23#92911yescharacteristic
168Q8—82no—
169Q8—122no—
170D4—122no—
171D4—124no—
172D4—242no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—121no—
177C2×C4—242no—
178C2×C4C2×(C2×D3)⋊C411yescharacteristic
179C2×C4C4×#6712yes—
180C2×C4#93012yes—
181C2×C4#95311yescharacteristic
182C8—44no—
183C8—122no—
184C8—242no—
185C2×C3—41no—
186C2×C3#188112yes—
187C2×C3#188811yescharacteristic
188D3—161no—
189C22—22no—
190C22—21no—
191C22—242no—
192C22—241no—
193C22#521311yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—44no—
198C4—241no—
199C4#524512yes—
200C3#900811yescharacteristic, complemented, Sylow
201C2—41no—
202C2—481no—
203C2#7166111yescharacteristic
204C2#7202012yes—
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2987C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2987D3
hypocenterC3#9008
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
1. C3×#2987C2semidirect
3. #71829C2semidirect
5. #67751C2semidirect
21. #2987D3semidirect
200. C3#9008semidirect

Maximal subgroups

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

subgroupindex
C3×#29872
#833352
#718292
#708012
#677512
#90083