Finite Group Databasebeta

← #10697466

Subgroups — #10697466

Order 768.

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

normal subgroups only

This group has 574 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#83215C212yesmaximal, complemented
2#78499C211yesmaximal, characteristic
3C3×#1960C211yesmaximal, characteristic, complemented
4#69545C212yesmaximal, complemented
5#67661C211yesmaximal, characteristic, complemented
6#10225—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×#904—22no—
9#5477C2212yes—
10C2×C3×#103—22no—
11C2×C3×#97C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5088—22no—
14#5087—24no—
15#5078—22no—
16#4967—24no—
17#4925C2212yes—
18#4715—22no—
19#4712—22no—
20#4451—24no—
21#3226—32no—
22#3150—31no—
23#1960D311yescharacteristic, complemented
24#1517—32no—
25#1281—31no—
26C22×C3×Q8—21no—
27C2×#235—22no—
28C22×C3×C8D412yes—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—21no—
31C3×C4⋊Q8D412yes—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C42C2311yescharacteristic
35#930—22no—
36#930—22no—
37C2×Dic12—22no—
38#904—44no—
39#904—44no—
40C2×D24—22no—
41C2×#204—24no—
42#875—22no—
43#874D412yes—
44C4×D12—22no—
45C4×D12—22no—
46#869D412yes—
47C3×#103—44no—
48C3×#97—22no—
49C3×#95—22no—
50#830D414yes—
51#809—28no—
52#807—28no—
53C4⋊(C3⋊C8)—22no—
54C4×C3⋊C8—22no—
55C2×#135—62no—
56#532—32no—
57C2×C4⋊Q8C2×D311yescharacteristic
58#500—62no—
59#497—62no—
60#496—62no—
61#486—62no—
62#480—64no—
63#474—64no—
64#431—32no—
65C2×#103—22no—
66C2×#97C2×D311yescharacteristic
67#382C2×D311yescharacteristic
68#327—64no—
69C2×C3×Q8—22no—
70C2×C3×Q8—41no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C2×#67—42no—
78#235—44no—
79#235—44no—
80C2×D12—22no—
81C2×D12—22no—
82C2×D12—22no—
83C2×C4×D3—42no—
84C2×Dic6—22no—
85C2×C3×C8—22no—
86C2×C3×C8—22no—
87C2×C3×C8D814yes—
88C3×C4⋊C4—24no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—42no—
91C3×C42C2×D411yescharacteristic
92C3×C42C2×D412yes—
93C3×C42#4111yescharacteristic
94(C2×D3)⋊C4—42no—
95(C2×D3)⋊C4—42no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98C2×C3⋊C8—24no—
99Dic12—44no—
100D24—44no—
101#204—48no—
102C2×#41—62no—
103C22×Q8—21no—
104#135—124no—
105#135—124no—
106C2×Q16—62no—
107C2×SD16—64no—
108C2×D8—62no—
109C22×C8#6712yes—
110C4⋊Q8—21no—
111C4⋊Q8—22no—
112C4⋊Q8—32no—
113C4⋊Q8D1212yes—
114#127—32no—
115#124—62no—
116#121—62no—
117#121—62no—
118C4×D4—62no—
119C4×D4—62no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C42C22×D311yescharacteristic
123#107—68no—
124C4⋊C8—62no—
125#103—44no—
126#102—68no—
127C22⋊C8—34no—
128#97—22no—
129C4×C8—62no—
130#95—22no—
131C23×C3#12711yescharacteristic
132C22×D3—42no—
133C3×Q8—44no—
134C3×Q8—82no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—81no—
138C2×C3×C4—81no—
139C2×C3×C4—82no—
140C2×C3×C4—81no—
141C2×C3×C4#12012yes—
142C2×C3×C4#12011yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4#12111yescharacteristic
145C2×C3×C4C2×D812yes—
146C2×C3×C4AGL(1,ℤ/8)12yes—
147#67—82no—
148#67—82no—
149C2×Dic3—42no—
150D12—44no—
151D12—44no—
152D12—82no—
153C4×D3—84no—
154Dic6—44no—
155C3×C8—24no—
156C3⋊C8—44no—
157#41—124no—
158#41—124no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—62no—
164C2×D4—62no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—122no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—122no—
171C22×C4D3×D412yes—
172C22×C4C2×#6711yescharacteristic
173Q16—124no—
174SD16—128no—
175D8—124no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—64no—
179C2×C8#21314yes—
180C4⋊C4—24no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42C2×D1211yescharacteristic
188C42#23511yescharacteristic
189C42D3×D412yes—
190C22×C3—22no—
191C22×C3#39211yescharacteristic
192C22×C3#49311yescharacteristic
193C22×C3#49611yescharacteristic
194C2×D3—82no—
195C2×D3—82no—
196C2×D3—82no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—82no—
200C3×C4—161no—
201C3×C4#44712yes—
202C3×C4#45912yes—
203Dic3—82no—
204C23—122no—
205C23#94011yescharacteristic
206Q8—44no—
207Q8—82no—
208Q8—124no—
209D4—124no—
210D4—124no—
211D4—242no—
212D4—242no—
213D4—242no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—122no—
221C2×C4—244no—
222C2×C4#88212yes—
223C2×C4#88412yes—
224C2×C4#93011yescharacteristic
225C2×C4C2×#21312yes—
226C2×C4#93711yescharacteristic
227C2×C4#94912yes—
228C8—24no—
229C8—124no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3#200912yes—
233C2×C3#201111yescharacteristic
234D3—162no—
235C22—22no—
236C22—242no—
237C22—242no—
238C22—242no—
239C22#496011yescharacteristic
240C22#505811yescharacteristic
241C22#507311yescharacteristic
242C4—22no—
243C4—22no—
244C4—82no—
245C4—161no—
246C4—242no—
247C4#501512yes—
248C4#503612yes—
249C3#1022511yescharacteristic, complemented, Sylow
250C2—21no—
251C2—21no—
252C2—482no—
253C2#6965811yescharacteristic
254C2#7035112yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC22×C3#392
prime radical#78499C2
hypercenter#1960D3
hypocenterC3#10225
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. #83215C2semidirect
3. C3×#1960C2semidirect
4. #69545C2semidirect
5. #67661C2semidirect
23. #1960D3semidirect
249. C3#10225semidirect

Maximal subgroups

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

subgroupindex
#832152
#784992
C3×#19602
#695452
#676612
#102253