Finite Group Databasebeta

← #83146

Subgroups — #83146

Order 384.

490 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 125 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×#529C211yesmaximal, characteristic, complemented
2C2×#874C211yesmaximal, characteristic, complemented
3#5067C212yesmaximal, complemented
4#5058C212yesmaximal, complemented
5#5046C212yesmaximal, complemented
6#5036C212yesmaximal, complemented
7#5027C212yesmaximal, complemented
8#5015C212yesmaximal, complemented
9#4926C211yesmaximal, characteristic, complemented
10#3217—31nomaximal, Sylow
11C22×D12—21no—
12C22×D12—21no—
13C22×D12C2212yes—
14C3×#126—21no—
15C3×#125C2211yescharacteristic, complemented
16C3×#124C2211yescharacteristic, complemented
17C3×#123—21no—
18C3×C22⋊Q8C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic, complemented
21C3×C4×D4—21no—
22C3×C4×D4C2211yescharacteristic, complemented
23C3×#117—21no—
24C2×C3×C42C2211yescharacteristic
25C2×#215C2214yescomplemented
26C2×#213C2214yescomplemented
27#874—21no—
28#874—21no—
29#874C2212yes—
30#874C2212yes—
31#830C2212yes—
32#809C2214yescomplemented
33#809C2214yescomplemented
34C4⋊(C3⋊C8)C2212yescomplemented
35#803C2212yescomplemented
36C2×#127—31no—
37#529D311yescharacteristic, complemented
38#496—32no—
39#490—32no—
40#460—32no—
41#459—32no—
42#450—32no—
43#447—32no—
44#432—31no—
45C23×D3—22no—
46C2×C3×Q8C2311yescharacteristic
47C2×C3×D4—21no—
48C2×C3×D4C2311yescharacteristic
49C22×C3×C4—21no—
50C22×C3×C4D411yescharacteristic
51C22×C3×C4D411yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C2×D12—22no—
54C2×D12—22no—
55C2×D12—22no—
56C2×D12—22no—
57C2×D12—22no—
58C2×D12—44no—
59C2×D12—44no—
60C2×D12—44no—
61C2×D12C2314yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2311yescharacteristic
67C3×C4⋊C4C2311yescharacteristic
68C3×C22⋊C4—21no—
69C3×C22⋊C4—21no—
70C3×C22⋊C4—21no—
71C3×C22⋊C4—21no—
72C3×C22⋊C4—21no—
73C3×C42—21no—
74C3×C42D411yescharacteristic
75C3×C42D411yescharacteristic
76C3×C42C2311yescharacteristic
77C3×C42C2311yescharacteristic
78#215—44no—
79#213—44no—
80C2×C3⋊C8C2314yes—
81C22×D4—32no—
82C22×D4—61no—
83C22×D4—61no—
84C2×SD16—34no—
85C2×D8—34no—
86#127—32no—
87#127—32no—
88#127—61no—
89#127—61no—
90#126—21no—
91#125C2×D311yescharacteristic, complemented
92#124C2×D311yescharacteristic, complemented
93#123—21no—
94C22⋊Q8C2×D311yescharacteristic
95#121C2×D311yescharacteristic
96C4×Q8C2×D311yescharacteristic, complemented
97C4×D4—21no—
98C4×D4C2×D311yescharacteristic, complemented
99#117—21no—
100C2×C42C2×D311yescharacteristic
101C4⋊C8—32no—
102#102—34no—
103#102—34no—
104C22⋊C8—32no—
105#97—32no—
106C23×C3—21no—
107C23×C3C2×D411yescharacteristic
108C22×D3—24no—
109C22×D3—44no—
110C22×D3—44no—
111C22×D3—44no—
112C3×Q8—22no—
113C3×D4—22no—
114C3×D4—42no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—42no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128D12—24no—
129D12—24no—
130D12—44no—
131D12—44no—
132D12—44no—
133D12—44no—
134D12—44no—
135C3⋊C8—24no—
136C24—62no—
137C2×Q8C22×D311yescharacteristic
138C2×D4—21no—
139C2×D4—34no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4—62no—
143C2×D4—62no—
144C2×D4—62no—
145C2×D4—124no—
146C2×D4—124no—
147C2×D4—124no—
148C2×D4C22×D311yescharacteristic
149C22×C4—21no—
150C22×C4#6711yescharacteristic
151C22×C4#6711yescharacteristic
152C22×C4C22×D311yescharacteristic
153SD16—124no—
154D8—124no—
155C2×C8—34no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C42#6711yescharacteristic
169C42#6711yescharacteristic
170C42C22×D311yescharacteristic
171C42C22×D311yescharacteristic
172C22×C3—21no—
173C22×C3—21no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3AGL(1,ℤ/8)12yes—
178C22×C3C22×D411yescharacteristic
179C2×D3—44no—
180C2×D3—44no—
181C2×D3—44no—
182C2×D3—88no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—21no—
186C3×C4—21no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4—41no—
190C3×C4AGL(1,ℤ/8)12yes—
191C3×C4H(5,𝔽2)12yes—
192C23—21no—
193C23—64no—
194C23—124no—
195C23—124no—
196C23—124no—
197C23C2×#6711yescharacteristic
198Q8—22no—
199D4—22no—
200D4—42no—
201D4—64no—
202D4—64no—
203D4—124no—
204D4—124no—
205D4—124no—
206D4—124no—
207D4—124no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4C2×#6711yescharacteristic
216C2×C4C2×#6711yescharacteristic
217C2×C4C2×#6711yescharacteristic
218C2×C4C2×#6711yescharacteristic
219C2×C4C2×#6711yescharacteristic
220C2×C4C23×D311yescharacteristic
221C8—64no—
222C2×C3—22no—
223C2×C3—81no—
224C2×C3#53511yescharacteristic
225C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
226C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
227D3—84no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—124no—
234C22—124no—
235C22—124no—
236C22—248no—
237C22#94912yes—
238C22C22×#6711yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4#94912yes—
247C4#100912yes—
248C3#321711yescharacteristic, complemented, Sylow
249C2—22no—
250C2—81no—
251C2—244no—
252C2C2×#94911yescharacteristic
253C2C2×#94911yescharacteristic
254C2#583011yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#529C2
socleC22×C3C22×D4
hypercenter#529D3
hypocenterC3#3217
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. C2×#874C2semidirect
3. #5067C2semidirect
4. #5058C2semidirect
5. #5046C2semidirect
6. #5036C2semidirect
7. #5027C2semidirect
8. #5015C2semidirect
9. #4926C2semidirect
15. C3×#125C22semidirect
16. C3×#124C22semidirect
20. C3×C4×Q8C22semidirect
22. C3×C4×D4C22semidirect
25. C2×#215C22semidirect
26. C2×#213C22semidirect
32. #809C22semidirect
33. #809C22semidirect
34. C4⋊(C3⋊C8)C22semidirect
35. #803C22semidirect
37. #529D3semidirect
91. #125C2×D3semidirect
92. #124C2×D3semidirect
96. C4×Q8C2×D3semidirect
98. C4×D4C2×D3semidirect
248. C3#3217semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
C2×#8742
#50672
#50582
#50462
#50362
#50272
#50152
#49262
#32173