Finite Group Databasebeta

← #10713915

Subgroups — #10713915

Order 768.

568 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 95 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#83742C212yesmaximal, complemented
2#82009C211yesmaximal, characteristic, complemented
3C3×#2112C211yesmaximal, characteristic, complemented
4#71450C212yesmaximal, complemented
5#67818C211yesmaximal, characteristic, complemented
6#13309—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8#5791—22no—
9C2×#904C2212yes—
10C2×C3×M32C2211yescharacteristic
11C2×C3×M32C2211yescharacteristic
12#5116—22no—
13#5115—22no—
14#5114—24no—
15#5113—22no—
16#5106C2212yes—
17#4909—22no—
18#4908—22no—
19#4904—22no—
20C3×C22⋊C16—22no—
21#4518—24no—
22#3414—31no—
23#3050—32no—
24#2145—32no—
25#2112D311yescharacteristic, complemented
26#1384—31no—
27C23×C3×C4D411yescharacteristic
28C2×#235—22no—
29C22×C3×C8—21no—
30C22×C3×C8—22no—
31C22×C3×C8—22no—
32C22×C3×C8—21no—
33C22×C3×C8D411yescharacteristic
34C22×C3×C8C2311yescharacteristic
35#953—42no—
36#930—24no—
37#929—42no—
38C4×#67—44no—
39#926—22no—
40#924—24no—
41C2×Dic12D412yescomplemented
42#904—22no—
43#904—22no—
44#904—22no—
45#904—22no—
46C2×D24D412yescomplemented
47C2×#204—22no—
48C3×M32—22no—
49C3×M32—22no—
50C2×C3×C16—21no—
51C2×C3×C16D412yescomplemented
52#821—24no—
53#818—22no—
54#817D414yescomplemented
55#816—24no—
56Dic24—22no—
57#800—24no—
58D48—22no—
59C2×#135—32no—
60C23×C8C2×D311yescharacteristic
61#525—62no—
62#510—62no—
63#509—62no—
64#508—62no—
65C2×M32C2×D311yescharacteristic
66C2×M32C2×D311yescharacteristic
67#467—62no—
68#466—62no—
69#465—64no—
70#427—32no—
71#416—62no—
72#360—64no—
73C22⋊C16—22no—
74C24×C3D811yescharacteristic
75C22×C3×C4—22no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—22no—
79C22×C3×C4D811yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C2×#67—44no—
82C2×#67—42no—
83#235—42no—
84#235—42no—
85#235—42no—
86#235—42no—
87C2×D12—22no—
88C2×C4×D3—42no—
89C2×Dic6—22no—
90C2×C3×C8—22no—
91C2×C3×C8—21no—
92C2×C3×C8—41no—
93C2×C3×C8—42no—
94C2×C3×C8—41no—
95C2×C3×C8C2×D412yes—
96C2×C3×C8C2×D412yes—
97C2×C3×C8C2×D411yescharacteristic
98C2×C3×C8#4111yescharacteristic
99#217—44no—
100#217—42no—
101(C2×D3)⋊C4—44no—
102C4⋊Dic3—24no—
103Dic3⋊C4—44no—
104C4×Dic3—42no—
105Dic12—22no—
106Dic12—22no—
107D24—22no—
108D24—22no—
109#204—42no—
110#204—42no—
111C3×C16—22no—
112C3×C16—41no—
113C2×#41—62no—
114C23×C4D1211yescharacteristic
115#135—62no—
116#135—62no—
117#135—62no—
118#135—62no—
119C2×Q16—32no—
120C2×SD16—62no—
121C2×D8—32no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8D1211yescharacteristic
127C22×C8C22×D311yescharacteristic
128#123—122no—
129C22⋊Q8—64no—
130#121—64no—
131#120—122no—
132C4×D4—124no—
133#117—62no—
134Q32—62no—
135SD32—64no—
136D16—62no—
137M32—22no—
138M32—22no—
139C2×C16—21no—
140C2×C16#6712yescomplemented
141#107—62no—
142#106—34no—
143#103—64no—
144#102—64no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—42no—
148C23×C3C2×D811yescharacteristic
149C22×D3—42no—
150C2×C3×C4—22no—
151C2×C3×C4—21no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4#12012yes—
156C2×C3×C4#12111yescharacteristic
157C2×C3×C4#13511yescharacteristic
158C2×C3×C4AGL(1,ℤ/8)12yes—
159#67—88no—
160#67—82no—
161#67—82no—
162C2×Dic3—44no—
163C2×Dic3—42no—
164D12—42no—
165D12—42no—
166C4×D3—82no—
167C4×D3—82no—
168Dic6—42no—
169Dic6—42no—
170C3×C8—41no—
171C3×C8#12011yescharacteristic
172C3×C8#12112yes—
173C3×C8#12711yescharacteristic
174C24D2411yescharacteristic
175#41—122no—
176#41—122no—
177#41—122no—
178#41—122no—
179C2×Q8—62no—
180C2×D4—62no—
181C2×D4—124no—
182C2×D4—122no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—122no—
188C22×C4D2411yescharacteristic
189C22×C4C2×D1211yescharacteristic
190Q16—62no—
191Q16—62no—
192SD16—122no—
193SD16—122no—
194D8—62no—
195D8—62no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8#23511yescharacteristic
202C2×C8D3×D412yes—
203C2×C8D3×D412yes—
204C2×C8C2×#6711yescharacteristic
205C4⋊C4—64no—
206C4⋊C4—124no—
207C22⋊C4—124no—
208C22⋊C4—124no—
209C22⋊C4—122no—
210C42—122no—
211C16—22no—
212C16—41no—
213C22×C3—24no—
214C22×C3—22no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3#44712yes—
220C22×C3#46611yescharacteristic
221C2×D3—84no—
222C2×D3—82no—
223C3×C4—41no—
224C3×C4#39211yescharacteristic
225C3×C4#44912yes—
226C3×C4#46911yescharacteristic
227Dic3—84no—
228Dic3—82no—
229C23—22no—
230C23—21no—
231C23—42no—
232C23—122no—
233C23C2×D2411yescharacteristic
234Q8—122no—
235Q8—122no—
236D4—122no—
237D4—122no—
238D4—248no—
239D4—242no—
240D4—242no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—124no—
247C2×C4—122no—
248C2×C4—242no—
249C2×C4—242no—
250C2×C4#88212yes—
251C2×C4#90411yescharacteristic
252C2×C4#90812yes—
253C2×C4#93011yescharacteristic
254C8—41no—
255C8#88412yes—
256C8#93711yescharacteristic
257C8#94011yescharacteristic
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—42no—
261C2×C3#199911yescharacteristic
262D3—162no—
263C22—24no—
264C22—22no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—244no—
270C22—242no—
271C22#473212yes—
272C22#511511yescharacteristic
273C4—41no—
274C4—244no—
275C4—242no—
276C4#473412yes—
277C4#496011yescharacteristic
278C4#513511yescharacteristic
279C3#1330911yescharacteristic, complemented, Sylow
280C2—22no—
281C2—21no—
282C2—42no—
283C2—482no—
284C2#6996111yescharacteristic
subgroupG/H
centerC4#5135
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#2112C2
socleC2×C3#1999
hypercenter#2112D3
hypocenterC3#13309
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
1. #83742C2semidirect
2. #82009C2semidirect
3. C3×#2112C2semidirect
4. #71450C2semidirect
5. #67818C2semidirect
25. #2112D3semidirect
41. C2×Dic12D4semidirect
46. C2×D24D4semidirect
51. C2×C3×C16D4semidirect
54. #817D4semidirect
140. C2×C16#67semidirect
279. C3#13309semidirect

Maximal subgroups

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

subgroupindex
#837422
#820092
C3×#21122
#714502
#678182
#133093