Finite Group Databasebeta

← #10690511

Subgroups — #10690511

Order 768.

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

normal subgroups only

This group has 426 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
1#83334C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic, complemented
3C3×#1526C211yesmaximal, characteristic, complemented
4#69853C212yesmaximal
5#67667C212yesmaximal, complemented
6#9070—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5477C2211yescharacteristic
9C3×C4×M16C2212yes—
10#5093C412yes—
11#5088C412yes—
12#5068—22no—
13#5024—22no—
14Q8×C3⋊C8—22no—
15#5002—22no—
16#5000—22no—
17#4925C2212yes—
18C2×C4×C3⋊C8C2211yescharacteristic
19C3×#330—22no—
20#4459—22no—
21#4459—22no—
22#3131—31no—
23#2986—31no—
24#1526D311yescharacteristic, complemented
25#1484—32no—
26#1278—32no—
27C2×#235—21no—
28C2×C3×M16—22no—
29C3×C4⋊Q8C2×C412yes—
30C3×#125D412yes—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117D412yes—
34C2×C3×C42C2311yescharacteristic
35#930—22no—
36C22×C3⋊C8—21no—
37#875—21no—
38#874C2×C411yescharacteristic
39C4×D12—22no—
40#869C2×C411yescharacteristic
41C3×#97—22no—
42C3×C4×C8—22no—
43#830—22no—
44#810—24no—
45#809—24no—
46C4⋊(C3⋊C8)—24no—
47C4⋊(C3⋊C8)D412yes—
48C4⋊(C3⋊C8)D412yes—
49#803—22no—
50C4×C3⋊C8—22no—
51C4×C3⋊C8—21no—
52C4×C3⋊C8—22no—
53C4×C3⋊C8C2×C412yes—
54#533C2×D311yescharacteristic
55#532—31no—
56#488—62no—
57#487—32no—
58#486—32no—
59#446—62no—
60C8×Q8—62no—
61#433—62no—
62#431—32no—
63#416—62no—
64C4×M16C2×D312yes—
65C2×C4×C8—31no—
66#330—22no—
67D4⋊C8—62no—
68D4⋊C8—62no—
69C2×C3×Q8—22no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4#4112yes—
72C2×#67—41no—
73#235—44no—
74C2×D12—21no—
75C2×D12—22no—
76C2×C4×D3—41no—
77C2×Dic6—21no—
78C3×M16—24no—
79C3×M16—42no—
80C2×C3×C8—42no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4C22⋊C414yes—
84C3×C22⋊C4—22no—
85C3×C42—22no—
86C3×C42C22×C411yescharacteristic
87C3×C42C2×D412yes—
88C3×C42C2×D411yescharacteristic
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91C2×C3⋊C8—22no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—22no—
94C2×C3⋊C8—42no—
95C2×#41—61no—
96C2×M16—22no—
97C22×C8—61no—
98C4⋊Q8—31no—
99C4⋊Q8C4×D312yes—
100#127—31no—
101#125#6712yes—
102#124—61no—
103C22⋊Q8—22no—
104#121—62no—
105C4×Q8—22no—
106C4×D4—62no—
107#117D1212yes—
108C2×C42C22×D311yescharacteristic
109C4⋊C8—32no—
110C4⋊C8—32no—
111C4⋊C8—64no—
112#103—64no—
113#102—64no—
114C22⋊C8—62no—
115#97—22no—
116#97—62no—
117C4×C8—22no—
118C4×C8—32no—
119C4×C8—62no—
120C4×C8—62no—
121C4×C8—61no—
122C23×C3#12311yescharacteristic
123C22×D3—41no—
124C3×Q8—42no—
125C3×Q8—42no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—24no—
130C2×C3×C4C2×C22⋊C411yescharacteristic
131C2×C3×C4C4×D412yes—
132C2×C3×C4#12011yescharacteristic
133C2×C3×C4#12112yes—
134#67—82no—
135C2×Dic3—41no—
136D12—42no—
137D12—42no—
138D12—42no—
139C4×D3—82no—
140Dic6—42no—
141C3×C8—44no—
142C3⋊C8—42no—
143C3⋊C8—42no—
144C3⋊C8—42no—
145#41—124no—
146C2×Q8—22no—
147C2×Q8—61no—
148C2×D4—62no—
149C2×D4—61no—
150C2×D4—121no—
151C22×C4—121no—
152C22×C4C2×D1211yescharacteristic
153C22×C4#23712yes—
154M16—24no—
155M16—42no—
156C2×C8—42no—
157C2×C8—62no—
158C2×C8—62no—
159C2×C8—62no—
160C2×C8—122no—
161C4⋊C4—22no—
162C4⋊C4—24no—
163C4⋊C4—62no—
164C4⋊C4(C2×D3)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—122no—
167C42—22no—
168C42C2×C4×D311yescharacteristic
169C42D3×D412yes—
170C42C2×#6711yescharacteristic
171C22×C3—22no—
172C22×C3—21no—
173C22×C3#38611yescharacteristic
174C2×D3—82no—
175C2×D3—81no—
176C3×C4—22no—
177C3×C4—21no—
178C3×C4—44no—
179C3×C4#44312yes—
180C3×C4#44412yes—
181C3×C4#45412yes—
182Dic3—81no—
183C23—121no—
184C23#88611yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—122no—
188D4—122no—
189D4—122no—
190D4—122no—
191D4—242no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—121no—
197C2×C4—242no—
198C2×C4#67⋊C412yes—
199C2×C4#88211yescharacteristic
200C2×C4C2×(C2×D3)⋊C411yescharacteristic
201C2×C4#93912yes—
202C8—44no—
203C8—122no—
204C8—122no—
205C8—122no—
206C2×C3—41no—
207C2×C3#187612yes—
208C2×C3#188811yescharacteristic
209D3—161no—
210C22—21no—
211C22—22no—
212C22—241no—
213C22—242no—
214C22#495811yescharacteristic
215C4—21no—
216C4—22no—
217C4—44no—
218C4—241no—
219C4#475312yes—
220C4#482312yes—
221C4#482412yes—
222C3#907011yescharacteristic, complemented, Sylow
223C2—41no—
224C2—481no—
225C2#7010411yescharacteristic
226C2#7037712yes—
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1526C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1526D3
hypocenterC3#9070
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #83332C2semidirect
3. C3×#1526C2semidirect
5. #67667C2semidirect
24. #1526D3semidirect
222. C3#9070semidirect

Maximal subgroups

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

subgroupindex
#833342
#833322
C3×#15262
#698532
#676672
#90703