Finite Group Databasebeta

← #81582

Subgroups — #81582

Order 384.

430 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 115 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#5726C211yesmaximal, characteristic, complemented
2#5718C211yesmaximal, characteristic, complemented
3C3×#486C211yesmaximal, characteristic, complemented
4#5094C211yesmaximal, characteristic, complemented
5#5072C211yesmaximal, characteristic, complemented
6#4782C212yesmaximal, complemented
7#4781C212yesmaximal, complemented
8#4777C212yesmaximal, complemented
9#4771C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3332—31nomaximal, Sylow
13C2×#239—21no—
14C2×#237C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×#127C2211yescharacteristic
17#946—21no—
18#940—21no—
19#939—22no—
20#938C2212yes—
21D4×Dic3—21no—
22C2×#213C2212yes—
23C2×#204C2211yescharacteristic
24C2×ΓL(1,𝔽25)C2212yes—
25#895C2212yes—
26#894—22no—
27D3×C4⋊C4C2212yes—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30C4×Dic6C2211yescharacteristic
31C3×#102C2212yes—
32C3×#102C2212yes—
33C3×C4×C8C2211yescharacteristic
34#832C2212yes—
35#821C2211yescharacteristic
36(C2×D3)⋊C8C2212yes—
37#817C2211yescharacteristic
38#816C2211yescharacteristic
39#810C2212yes—
40#808C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#550—31no—
43#547—31no—
44#486D311yescharacteristic, complemented
45#481—32no—
46#475—31no—
47#469—32no—
48#459—31no—
49#449—32no—
50C4×SD16—31no—
51#436—31no—
52#417—32no—
53C2×C3×Q8—21no—
54C2×C3×D4—21no—
55C2×C3×D4—21no—
56C2×C3×D4C2312yes—
57C2×#67—22no—
58C2×#67—22no—
59C22×Dic3—22no—
60#239—44no—
61#237—24no—
62#237—42no—
63C2×D12—22no—
64C2×D12D411yescharacteristic
65C2×D12C2311yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3—21no—
68C2×C4×D3C2312yes—
69C2×Dic6C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C2×C3×C8C2311yescharacteristic
72C3×C4⋊C4—21no—
73C3×C4⋊C4C2312yes—
74C3×C42C2311yescharacteristic
75#217—22no—
76#213—42no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C2×C3⋊C8C2312yes—
87#204—22no—
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—31no—
92C2×D8—32no—
93C2×M16—32no—
94C4⋊Q8C2×D311yescharacteristic
95#127—61no—
96#127C2×D311yescharacteristic
97#123—62no—
98C22⋊Q8—61no—
99#121—32no—
100#121—32no—
101#121—62no—
102C4×Q8—31no—
103C4×D4—31no—
104C4×D4—31no—
105C4×D4—61no—
106C2×C4⋊C4—32no—
107#106—31no—
108#106—32no—
109C4⋊C8—31no—
110#103—31no—
111#103—32no—
112#102—32no—
113#102—31no—
114#102C2×D312yes—
115#102C2×D312yes—
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic
118C23×C3—22no—
119C22×D3—21no—
120C22×D3C2×D412yes—
121C3×Q8—42no—
122C3×D4—22no—
123C3×D4—22no—
124C3×D4—42no—
125C3×D4—81no—
126C2×C3×C4—22no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2411yescharacteristic
130#67—44no—
131#67—82no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—44no—
136C2×Dic3C2×D412yes—
137D12—22no—
138D12—21no—
139D12—21no—
140D12—44no—
141C4×D3—22no—
142C4×D3—22no—
143C4×D3—44no—
144C4×D3—42no—
145Dic6—21no—
146Dic6#4112yescomplemented
147C3×C8—21no—
148C3×C8—21no—
149C3⋊C8—22no—
150#41—64no—
151#41—122no—
152#41—124no—
153C2×Q8—21no—
154C2×Q8—31no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—31no—
158C2×D4—31no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4C22×D312yes—
163C22×C4—32no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—61no—
167SD16—62no—
168D8—122no—
169M16—122no—
170C2×C8—32no—
171C2×C8C22×D311yescharacteristic
172C2×C8C22×D311yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—62no—
178C4⋊C4—61no—
179C4⋊C4C22×D312yes—
180C22⋊C4—32no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C22⋊C4—61no—
184C42—61no—
185C42C22×D311yescharacteristic
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C3×C4—21no—
197C3×C4—42no—
198C3×C4AGL(1,ℤ/8)12yes—
199C3×C4C2×#4111yescharacteristic
200C3×C4H(5,𝔽2)11yescharacteristic
201Dic3—22no—
202Dic3—22no—
203Dic3—41no—
204Dic3—41no—
205C23—22no—
206C23—32no—
207C23—61no—
208Q8—32no—
209Q8—42no—
210Q8—61no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—62no—
215D4—61no—
216D4—61no—
217D4—81no—
218D4—124no—
219D4—124no—
220D4—242no—
221C2×C4—22no—
222C2×C4—32no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—124no—
229C2×C4—124no—
230C2×C4—122no—
231C2×C4D3×D411yescharacteristic
232C2×C4D3×D411yescharacteristic
233C2×C4C23×D311yescharacteristic
234C8—21no—
235C8—21no—
236C8—62no—
237C2×C3—82no—
238C2×C3#54611yescharacteristic
239C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
240C2×C3#57711yescharacteristic
241D3—42no—
242D3—81no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—62no—
247C22—62no—
248C22—62no—
249C22—121no—
250C22—121no—
251C22—121no—
252C22C2×D3×D411yescharacteristic
253C4—21no—
254C4—42no—
255C4—62no—
256C4—62no—
257C4—121no—
258C4—121no—
259C4#91112yes—
260C4D3×#4111yescharacteristic
261C4#100911yescharacteristic
262C3#333211yescharacteristic, complemented, Sylow
263C2—82no—
264C2—122no—
265C2—241no—
266C2#567611yescharacteristic
267C2C2×#91111yescharacteristic
268C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
hypercenter#486D3
hypocenterC3#3332
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5726C2semidirect
2. #5718C2semidirect
3. C3×#486C2semidirect
4. #5094C2semidirect
5. #5072C2semidirect
6. #4782C2semidirect
7. #4781C2semidirect
8. #4777C2semidirect
9. #4771C2semidirect
10. C4×#204C2semidirect
11. #4688C2semidirect
44. #486D3semidirect
146. Dic6#41semidirect
262. C3#3332semidirect

Maximal subgroups

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

subgroupindex
#57262
#57182
C3×#4862
#50942
#50722
#47822
#47812
#47772
#47712
C4×#2042
#46882
#33323