Finite Group Databasebeta

← #10712636

Subgroups — #10712636

Order 768.

490 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3407C211yesmaximal, characteristic, complemented
2#83799C211yesmaximal, characteristic
3#72012C211yesmaximal, characteristic, complemented
4#71440C212yesmaximal, complemented
5#67818C212yesmaximal, complemented
6#13074—31nomaximal, Sylow
7C2×C3×#131C2211yescharacteristic
8#5830—21no—
9C22×D24C2211yescharacteristic
10C3×#504—24no—
11C2×C3×M32—22no—
12C2×C3×M32C2212yes—
13#5240—21no—
14#5135C414yescomplemented
15#5131—21no—
16#5115—22no—
17#5106C2212yes—
18#5106C2211yescharacteristic
19#4557—21no—
20#4509—24no—
21#3407D311yescharacteristic, complemented
22#3058—31no—
23#2140—32no—
24#1812—31no—
25#1384—32no—
26C2×C3×#41D411yescharacteristic
27C22×D12—21no—
28C3×#131—22no—
29C3×#131—22no—
30C2×C3×M16C2×C412yes—
31C2×C3×M16D411yescharacteristic
32C22×C3×C8D412yes—
33C22×C3×C8C2311yescharacteristic
34#947—41no—
35#940—41no—
36#937—42no—
37#930—24no—
38#926—21no—
39#926—22no—
40C2×D24—22no—
41C2×D24—22no—
42C2×D24—21no—
43C2×D24C2×C412yes—
44C3×M32—24no—
45C3×M32—22no—
46C3×M32—22no—
47C2×C3×C16—24no—
48C2×C3×C16D414yescomplemented
49#834—42no—
50#821—24no—
51#818—21no—
52#818—22no—
53#817—22no—
54#817C2×C412yes—
55C22×D8—31no—
56C2×#131C2×D311yescharacteristic
57#535—61no—
58#504—24no—
59C2×M32—22no—
60C2×M32C2×D312yes—
61#469—34no—
62#466—62no—
63#427—31no—
64#427—32no—
65#418—61no—
66#410—61no—
67#357—64no—
68H(3,ℤ/4)—61no—
69C23×D3—41no—
70C3×#41—22no—
71C3×#41—22no—
72C2×C3×Q8SD1611yescharacteristic
73C2×C3×D4D812yes—
74C2×C3×D4SD1611yescharacteristic
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C2×D411yescharacteristic
77C2×#67—44no—
78C2×D12—22no—
79C2×D12—42no—
80C2×D12—42no—
81C2×D12—41no—
82C3×M16—22no—
83C3×M16—24no—
84C2×C3×C8—22no—
85C2×C3×C8—22no—
86C2×C3×C8C22⋊C412yes—
87C2×C3×C8C22×C411yescharacteristic
88C2×C3×C8C2×D412yes—
89C2×C3×C8C2×D411yescharacteristic
90C2×C3×C8#4112yes—
91#217—41no—
92#217—42no—
93(C2×D3)⋊C4—44no—
94C4⋊Dic3—22no—
95C4⋊Dic3—42no—
96C4×Dic3—41no—
97C4×Dic3—42no—
98D24—24no—
99D24—41no—
100D24—41no—
101C3×C16—24no—
102C2×#41D1211yescharacteristic
103C22×D4—61no—
104C2×D8—32no—
105C2×D8—62no—
106C2×D8—62no—
107C2×D8—61no—
108#131—22no—
109#131—22no—
110C2×M16C4×D312yes—
111C2×M16D1211yescharacteristic
112C22×C8#6712yes—
113C22×C8C22×D311yescharacteristic
114#127—121no—
115#124—121no—
116#121—64no—
117#120—122no—
118#117—61no—
119#117—62no—
120M32—24no—
121M32—22no—
122M32—22no—
123C2×C16—24no—
124C2×C16#6714yescomplemented
125#107—61no—
126#107—62no—
127#106—32no—
128#106—62no—
129#102—64no—
130#99—122no—
131C23×C3#10212yes—
132C23×C3C2×D811yescharacteristic
133C22×D3—42no—
134C22×D3—82no—
135C22×D3—162no—
136C3×Q8—22no—
137C3×D4—24no—
138C3×D4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4#10212yes—
142C2×C3×C4C2×C22⋊C411yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4C2×SD1611yescharacteristic
145C2×C3×C4#13512yes—
146#67—164no—
147C2×Dic3—42no—
148C2×Dic3—82no—
149D12—42no—
150D12—42no—
151D12—82no—
152C3×C8—22no—
153C3×C8C4×D412yes—
154C3×C8#12011yescharacteristic
155C3×C8#12311yescharacteristic
156C24—121no—
157#41—22no—
158#41—22no—
159C2×Q8#20411yescharacteristic
160C2×D4—62no—
161C2×D4—124no—
162C2×D4—121no—
163C2×D4—122no—
164C2×D4—122no—
165C2×D4#20411yescharacteristic
166C2×D4D2412yes—
167C22×C4(C2×D3)⋊C412yes—
168C22×C4C2×D1211yescharacteristic
169D8—64no—
170D8—121no—
171D8—121no—
172M16—22no—
173M16—24no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8(C2×D3)⋊C412yes—
177C2×C8C2×C4×D311yescharacteristic
178C2×C8#23512yes—
179C2×C8C2×#6712yes—
180C2×C8C2×#6711yescharacteristic
181C4⋊C4—62no—
182C4⋊C4—122no—
183C22⋊C4—121no—
184C22⋊C4—124no—
185C22⋊C4—122no—
186C42—121no—
187C42—122no—
188C16—24no—
189C22×C3—22no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3C2×#10211yescharacteristic
193C22×C3#46612yes—
194C2×D3—84no—
195C2×D3—82no—
196C2×D3—164no—
197C3×C4—22no—
198C3×C4#38611yescharacteristic
199C3×C4#41611yescharacteristic
200C3×C4#46512yes—
201Dic3—82no—
202Dic3—162no—
203C23—122no—
204C23—242no—
205C23—482no—
206C23#82112yes—
207C23C2×D2411yescharacteristic
208Q8—22no—
209D4—24no—
210D4—22no—
211D4—122no—
212D4—122no—
213D4—242no—
214D4—484no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4#82112yes—
220C2×C4C2×#20411yescharacteristic
221C2×C4#90412yes—
222C2×C4C2×(C2×D3)⋊C411yescharacteristic
223C2×C4#93012yes—
224C8—22no—
225C8C4×#6712yes—
226C8#92911yescharacteristic
227C8#95311yescharacteristic
228C2×C3—21no—
229C2×C3—22no—
230C2×C3—42no—
231C2×C3#189311yescharacteristic
232D3—162no—
233C22—22no—
234C22—41no—
235C22—42no—
236C22—244no—
237C22—242no—
238C22—484no—
239C22C2×#82111yescharacteristic
240C22#511512yes—
241C4—22no—
242C4—242no—
243C4—482no—
244C4#511311yescharacteristic
245C4#511412yes—
246C4#521311yescharacteristic
247C3#1307411yescharacteristic, complemented, Sylow
248C2—21no—
249C2—22no—
250C2—42no—
251C2—482no—
252C2#7197311yescharacteristic
subgroupG/H
centerC2#71973
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#3407C2
socleC2×C3#1893
prime radical#83799C2
cosocleC22×C3×C8C23
hypercenter#3407D3
hypocenterC3#13074
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
1. C3×#3407C2semidirect
3. #72012C2semidirect
4. #71440C2semidirect
5. #67818C2semidirect
14. #5135C4semidirect
21. #3407D3semidirect
48. C2×C3×C16D4semidirect
124. C2×C16#67semidirect
247. C3#13074semidirect

Maximal subgroups

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

subgroupindex
C3×#34072
#837992
#720122
#714402
#678182
#130743