Finite Group Databasebeta

← #10697394

Subgroups — #10697394

Order 768.

498 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 79 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#83358C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic, complemented
4#69432C212yesmaximal, complemented
5#67716C211yesmaximal, characteristic, complemented
6#10213—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5477C2212yes—
9C3×#423C2211yescharacteristic
10C3×#376C2211yescharacteristic
11#5135—22no—
12#5089—22no—
13#5086—22no—
14#5047—24no—
15#5031—22no—
16#5027—22no—
17#4926C2212yes—
18#4843—22no—
19#4842—22no—
20C3×#339—22no—
21#4451—24no—
22#3253—31no—
23#3216—32no—
24#1995D311yescharacteristic, complemented
25#1505—32no—
26#1307—31no—
27C2×#235—22no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D412yes—
30C3×#125—21no—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117—21no—
34C2×C3×C42—21no—
35C2×C3×C42C2311yescharacteristic
36C2×#216—24no—
37C2×#215—24no—
38#930—22no—
39#930—22no—
40C2×D24—22no—
41#875—22no—
42#874D412yes—
43C4×D12—22no—
44C4×D12—22no—
45#869D412yes—
46C3×C4⋊C8—22no—
47C3×#95—22no—
48#830D414yes—
49#821—24no—
50#817—22no—
51#810—24no—
52#809—24no—
53C4⋊(C3⋊C8)—22no—
54#803—22no—
55#533C2×D311yescharacteristic
56#532—32no—
57#497—62no—
58#489—62no—
59#476—62no—
60#469—62no—
61#462—62no—
62#460—62no—
63#459—62no—
64#452—64no—
65#432—32no—
66#423C2×D311yescharacteristic
67#376C2×D311yescharacteristic
68#339—22no—
69#327—64no—
70C2×C3×Q8—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C2×#67—42no—
75#235—84no—
76C2×D12—22no—
77C2×D12—22no—
78C2×D12—22no—
79C2×C4×D3—42no—
80C2×Dic6—22no—
81C3×M16—42no—
82C2×C3×C8—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—42no—
86C3×C22⋊C4—41no—
87C3×C42—22no—
88C3×C42—41no—
89C3×C42—41no—
90C3×C42C2×D411yescharacteristic
91C3×C42C2×D412yes—
92C3×C42#4111yescharacteristic
93#216—84no—
94#215—84no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C2×C3⋊C8—24no—
100D24—82no—
101C2×#41—62no—
102C2×Q16—64no—
103C2×SD16—64no—
104C2×D8—62no—
105C2×M16#6712yes—
106C4⋊Q8—32no—
107C4⋊Q8D1212yes—
108#127—32no—
109#125—21no—
110#124—62no—
111C22⋊Q8—22no—
112#121—62no—
113#121—62no—
114C4×Q8—22no—
115C4×D4—62no—
116C4×D4—62no—
117#117—21no—
118C2×C42—21no—
119C2×C42C22×D311yescharacteristic
120#106—62no—
121C4⋊C8—22no—
122C4⋊C8—62no—
123#103—64no—
124#102—64no—
125#102—64no—
126C22⋊C8—34no—
127#97—62no—
128#95—22no—
129C23×C3#12711yescharacteristic
130C22×D3—42no—
131C3×Q8—42no—
132C3×Q8—42no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—82no—
139C2×C3×C4#12012yes—
140C2×C3×C4#12011yescharacteristic
141C2×C3×C4#12112yes—
142C2×C3×C4#12111yescharacteristic
143#67—84no—
144C2×Dic3—42no—
145D12—44no—
146D12—42no—
147D12—42no—
148D12—82no—
149C4×D3—84no—
150Dic6—44no—
151C3×C8—42no—
152C3⋊C8—44no—
153#41—244no—
154C2×Q8—22no—
155C2×Q8—62no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—122no—
160C22×C4—41no—
161C22×C4—122no—
162C22×C4D3×D412yes—
163C22×C4C2×#6711yescharacteristic
164Q16—244no—
165SD16—244no—
166D8—242no—
167M16—42no—
168C2×C8—22no—
169C2×C8—64no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C22⋊C4—41no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C42—22no—
179C42—41no—
180C42—41no—
181C42C2×D1211yescharacteristic
182C42#23511yescharacteristic
183C42D3×D412yes—
184C22×C3—21no—
185C22×C3—22no—
186C22×C3#39211yescharacteristic
187C2×D3—82no—
188C2×D3—82no—
189C2×D3—82no—
190C3×C4—21no—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4—82no—
194C3×C4—82no—
195C3×C4#45511yescharacteristic
196C3×C4#45611yescharacteristic
197Dic3—82no—
198C23—122no—
199C23#94011yescharacteristic
200Q8—42no—
201Q8—42no—
202Q8—124no—
203D4—124no—
204D4—122no—
205D4—122no—
206D4—242no—
207D4—244no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—122no—
215C2×C4—244no—
216C2×C4#88212yes—
217C2×C4#88412yes—
218C2×C4#93011yescharacteristic
219C2×C4#93711yescharacteristic
220C8—42no—
221C8—124no—
222C2×C3—41no—
223C2×C3#200711yescharacteristic
224C2×C3#200812yes—
225D3—162no—
226C22—22no—
227C22—21no—
228C22—242no—
229C22—242no—
230C22—242no—
231C22#496011yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—82no—
236C4—82no—
237C4—242no—
238C4#506111yescharacteristic
239C4#509511yescharacteristic
240C3#1021311yescharacteristic, complemented, Sylow
241C2—41no—
242C2—482no—
243C2#6969411yescharacteristic
244C2#7040912yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1995D3
hypocenterC3#10213
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
1. #83358C2semidirect
3. C3×#1995C2semidirect
4. #69432C2semidirect
5. #67716C2semidirect
24. #1995D3semidirect
240. C3#10213semidirect

Maximal subgroups

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

subgroupindex
#833582
#790642
C3×#19952
#694322
#677162
#102133