Finite Group Databasebeta

← #10712651

Subgroups — #10712651

Order 768.

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

normal subgroups only

This group has 470 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#83800C211yesmaximal, characteristic
3#72012C211yesmaximal, characteristic, complemented
4#71450C212yesmaximal, complemented
5#67819C212yesmaximal, complemented
6#13077—31nomaximal, Sylow
7C2×C3×#131C2211yescharacteristic
8#5833—21no—
9C2×#904C2211yescharacteristic
10C3×#504—24no—
11C2×C3×M32—22no—
12C2×C3×M32C2212yes—
13#5240—21no—
14#5137C412yescomplemented
15#5135C412yescomplemented
16#5133—21no—
17#5114—22no—
18#5106C2212yes—
19C2×#817C2211yescharacteristic
20#4557—21no—
21#4518—24no—
22#3407D311yescharacteristic, complemented
23#3060—31no—
24#2145—32no—
25#1812—31no—
26#1386—32no—
27C2×C3×#41D411yescharacteristic
28C2×#235—21no—
29C3×#131—22no—
30C3×#131—22no—
31C2×C3×M16C2×C412yes—
32C2×C3×M16D411yescharacteristic
33C22×C3×C8D412yes—
34C22×C3×C8C2311yescharacteristic
35#946—41no—
36#939—42no—
37#938—41no—
38#930—22no—
39#926—22no—
40C2×C4⋊Dic3—21no—
41#924—22no—
42C2×Dic12C2×C411yescharacteristic
43#904—22no—
44#904—22no—
45C2×D24C2×C411yescharacteristic
46C2×#204—21no—
47C3×M32—24no—
48C3×M32—22no—
49C3×M32D414yescomplemented
50C2×C3×C16—24no—
51C2×C3×C16—22no—
52#834—42no—
53#821—22no—
54#818—22no—
55#817—21no—
56#817—22no—
57#817C2×C412yes—
58#816—22no—
59C2×#135—31no—
60C2×#131C2×D311yescharacteristic
61#537—61no—
62#504—24no—
63C2×M32—22no—
64C2×M32C2×D312yes—
65#470—32no—
66#469—32no—
67#465—62no—
68#427—32no—
69C2×#106—31no—
70#417—61no—
71#410—61no—
72#360—64no—
73H(3,ℤ/4)—61no—
74C3×#41—22no—
75C3×#41—22no—
76C2×C3×Q8SD1611yescharacteristic
77C2×C3×D4D812yes—
78C2×C3×D4SD1611yescharacteristic
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C2×D411yescharacteristic
81C2×#67—41no—
82C2×#67—42no—
83C22×Dic3—41no—
84#235—42no—
85#235—42no—
86C2×D12—21no—
87C2×C4×D3—41no—
88C2×Dic6—21no—
89C3×M16—22no—
90C3×M16—24no—
91C2×C3×C8—22no—
92C2×C3×C8—22no—
93C2×C3×C8C22⋊C412yes—
94C2×C3×C8C22×C411yescharacteristic
95C2×C3×C8C2×D412yes—
96C2×C3×C8C2×D411yescharacteristic
97C2×C3×C8#4112yes—
98#217—42no—
99#217—42no—
100(C2×D3)⋊C4—42no—
101C4⋊Dic3—22no—
102C4⋊Dic3—41no—
103C4⋊Dic3—42no—
104Dic3⋊C4—42no—
105C4×Dic3—42no—
106Dic12—22no—
107D24—22no—
108#204—41no—
109#204—41no—
110C3×C16—24no—
111C2×#41—61no—
112C2×#41D1211yescharacteristic
113#135—62no—
114#135—62no—
115C2×Q16—31no—
116C2×SD16—61no—
117C2×D8—31no—
118#131—22no—
119#131—22no—
120C2×M16C4×D312yes—
121C2×M16D1211yescharacteristic
122C22×C8#6712yes—
123C22×C8C22×D311yescharacteristic
124C22⋊Q8—62no—
125C22⋊Q8—121no—
126#121—62no—
127#121—121no—
128#121—122no—
129#117—62no—
130C2×C4⋊C4—61no—
131M32—24no—
132M32—22no—
133M32#6714yescomplemented
134C2×C16—24no—
135C2×C16—22no—
136#107—62no—
137#106—32no—
138#106—61no—
139#106—62no—
140#103—62no—
141#102—62no—
142#99—122no—
143C23×C3#10212yes—
144C23×C3C2×D811yescharacteristic
145C22×D3—41no—
146C3×Q8—22no—
147C3×D4—24no—
148C3×D4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4#10212yes—
152C2×C3×C4C2×C22⋊C411yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4C2×SD1611yescharacteristic
155C2×C3×C4#13512yes—
156#67—82no—
157#67—162no—
158C2×Dic3—41no—
159C2×Dic3—42no—
160C2×Dic3—82no—
161C2×Dic3—161no—
162D12—41no—
163D12—41no—
164C4×D3—82no—
165Dic6—41no—
166Dic6—41no—
167C3×C8—22no—
168C3×C8C4×D412yes—
169C3×C8#12011yescharacteristic
170C3×C8#12311yescharacteristic
171#41—22no—
172#41—22no—
173#41—122no—
174#41—122no—
175C2×Q8—61no—
176C2×Q8#20411yescharacteristic
177C2×D4—61no—
178C2×D4—121no—
179C2×D4—122no—
180C2×D4#20411yescharacteristic
181C2×D4D2412yes—
182C22×C4—121no—
183C22×C4—121no—
184C22×C4(C2×D3)⋊C412yes—
185C22×C4C2×D1211yescharacteristic
186Q16—62no—
187SD16—121no—
188SD16—121no—
189D8—62no—
190M16—22no—
191M16—24no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8(C2×D3)⋊C412yes—
195C2×C8C2×C4×D311yescharacteristic
196C2×C8#23512yes—
197C2×C8C2×#6712yes—
198C2×C8C2×#6711yescharacteristic
199C4⋊C4—62no—
200C4⋊C4—122no—
201C4⋊C4—121no—
202C4⋊C4—122no—
203C22⋊C4—122no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C42—122no—
207C16—24no—
208C22×C3—22no—
209C22×C3—41no—
210C22×C3—42no—
211C22×C3C2×#10211yescharacteristic
212C22×C3#46612yes—
213C2×D3—81no—
214C2×D3—81no—
215C2×D3—81no—
216C3×C4—22no—
217C3×C4#38611yescharacteristic
218C3×C4#41611yescharacteristic
219C3×C4#46512yes—
220Dic3—82no—
221Dic3—81no—
222Dic3—162no—
223C23—121no—
224C23#82112yes—
225C23C2×D2411yescharacteristic
226Q8—22no—
227Q8—121no—
228Q8—121no—
229D4—24no—
230D4—22no—
231D4—121no—
232D4—121no—
233D4—242no—
234D4—482no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—122no—
238C2×C4—121no—
239C2×C4—242no—
240C2×C4—242no—
241C2×C4—481no—
242C2×C4#82112yes—
243C2×C4C2×#20411yescharacteristic
244C2×C4#90412yes—
245C2×C4C2×(C2×D3)⋊C411yescharacteristic
246C2×C4#93012yes—
247C8—22no—
248C8C4×#6712yes—
249C8#92911yescharacteristic
250C8#95311yescharacteristic
251C2×C3—21no—
252C2×C3—22no—
253C2×C3—42no—
254C2×C3#189311yescharacteristic
255D3—161no—
256C22—22no—
257C22—41no—
258C22—42no—
259C22—241no—
260C22—241no—
261C22—241no—
262C22C2×#82111yescharacteristic
263C22#511512yes—
264C4—22no—
265C4—242no—
266C4—241no—
267C4—482no—
268C4#511311yescharacteristic
269C4#511412yes—
270C4#521311yescharacteristic
271C3#1307711yescharacteristic, complemented, Sylow
272C2—21no—
273C2—22no—
274C2—42no—
275C2—481no—
276C2#7197311yescharacteristic
subgroupG/H
centerC2#71973
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#3407C2
socleC2×C3#1893
prime radical#83800C2
cosocleC22×C3×C8C23
hypercenter#3407D3
hypocenterC3#13077
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. #71450C2semidirect
5. #67819C2semidirect
14. #5137C4semidirect
15. #5135C4semidirect
22. #3407D3semidirect
49. C3×M32D4semidirect
133. M32#67semidirect
271. C3#13077semidirect

Maximal subgroups

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

subgroupindex
C3×#34072
#838002
#720122
#714502
#678192
#130773