Finite Group Databasebeta

← #83144

Subgroups — #83144

Order 384.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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×#875C211yesmaximal, characteristic, complemented
3#5068C212yesmaximal, complemented
4#5059C212yesmaximal, complemented
5#5050C211yesmaximal, characteristic, complemented
6#5046C211yesmaximal, characteristic, complemented
7#5040C211yesmaximal, characteristic, complemented
8#5036C211yesmaximal, characteristic
9#5028C212yesmaximal, complemented
10#5016C212yesmaximal, complemented
11#4925C211yesmaximal, characteristic, complemented
12#3211—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×#126—21no—
16C3×#125C2211yescharacteristic, complemented
17C3×#124C2211yescharacteristic, complemented
18C3×#123—21no—
19C3×C22⋊Q8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×C4×Q8C2211yescharacteristic, complemented
22C3×C4×D4—21no—
23C3×C4×D4C2211yescharacteristic, complemented
24C3×#117—21no—
25C2×C3×C42C2211yescharacteristic
26C2×#216C2212yescomplemented
27C2×#215C2212yes—
28C2×#214C2212yescomplemented
29C2×#213C2212yes—
30C2×(C2×D3)⋊C4—21no—
31C2×(C2×D3)⋊C4—21no—
32#875—21no—
33#875—21no—
34#875C2212yes—
35#875C2212yes—
36#830C2211yescharacteristic
37#830C2211yescharacteristic
38#810C2212yescomplemented
39#810C2212yescomplemented
40#809C2212yes—
41#809C2212yes—
42C4⋊(C3⋊C8)C2212yescomplemented
43C4×C3⋊C8C2212yescomplemented
44#529D311yescharacteristic, complemented
45C2×#124—31no—
46#495—32no—
47#488—32no—
48#464—32no—
49#463—32no—
50#451—31no—
51#450—31no—
52#448—31no—
53#447—31no—
54#431—31no—
55C23×D3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4D411yescharacteristic
61C22×C3×C4D411yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—21no—
64C2×D12—21no—
65C2×D12—42no—
66C2×D12C2312yes—
67C2×Dic6—21no—
68C2×Dic6—42no—
69C2×Dic6C2312yes—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4C2311yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C22⋊C4—21no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42—21no—
82C3×C42D411yescharacteristic
83C3×C42D411yescharacteristic
84C3×C42C2311yescharacteristic
85C3×C42C2311yescharacteristic
86#216—42no—
87#215—42no—
88#214—42no—
89#213—42no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—22no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C22×Q8—31no—
97C22×D4—31no—
98C2×Q16—32no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×D8—32no—
102#126—21no—
103#125C2×D311yescharacteristic, complemented
104#124—32no—
105#124—32no—
106#124—61no—
107#124—61no—
108#124C2×D311yescharacteristic, complemented
109#123—21no—
110C22⋊Q8C2×D311yescharacteristic
111#121C2×D311yescharacteristic
112C4×Q8C2×D311yescharacteristic, complemented
113C4×D4—21no—
114C4×D4C2×D311yescharacteristic, complemented
115#117—21no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118C2×C42C2×D311yescharacteristic
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—31no—
125C22⋊C8—31no—
126C4×C8—32no—
127C23×C3—21no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C22×D3—42no—
131C22×D3—42no—
132C22×D3—42no—
133C3×Q8—22no—
134C3×D4—22no—
135C3×D4—42no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—41no—
142C2×C3×C4—42no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2411yescharacteristic
149C2×Dic3—22no—
150C2×Dic3—42no—
151D12—22no—
152D12—22no—
153D12—42no—
154Dic6—22no—
155Dic6—22no—
156Dic6—42no—
157C3⋊C8—22no—
158C3⋊C8—22no—
159C24—61no—
160C2×Q8—32no—
161C2×Q8—61no—
162C2×Q8—122no—
163C2×Q8C22×D311yescharacteristic
164C2×D4—21no—
165C2×D4—32no—
166C2×D4—61no—
167C2×D4—122no—
168C2×D4C22×D311yescharacteristic
169C22×C4—21no—
170C22×C4—61no—
171C22×C4#6711yescharacteristic
172C22×C4#6711yescharacteristic
173C22×C4C22×D311yescharacteristic
174Q16—122no—
175SD16—122no—
176SD16—122no—
177D8—122no—
178C2×C8—32no—
179C2×C8—32no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4C22×D311yescharacteristic
185C4⋊C4C22×D311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C42—21no—
196C42#6711yescharacteristic
197C42#6711yescharacteristic
198C42C22×D311yescharacteristic
199C42C22×D311yescharacteristic
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3#13512yes—
206C22×C3C22×D411yescharacteristic
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—84no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—21no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4H(5,𝔽2)12yes—
220Dic3—42no—
221C23—21no—
222C23—62no—
223C23—122no—
224C23—122no—
225C23—122no—
226C23C2×#6711yescharacteristic
227Q8—22no—
228Q8—62no—
229Q8—62no—
230Q8—122no—
231D4—22no—
232D4—42no—
233D4—62no—
234D4—62no—
235D4—122no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—62no—
244C2×C4—122no—
245C2×C4C2×#6711yescharacteristic
246C2×C4C2×#6711yescharacteristic
247C2×C4C2×#6711yescharacteristic
248C2×C4C2×#6711yescharacteristic
249C2×C4C2×#6711yescharacteristic
250C2×C4C23×D311yescharacteristic
251C8—62no—
252C8—62no—
253C2×C3—22no—
254C2×C3—81no—
255C2×C3#53511yescharacteristic
256C2×C3C2×#13511yescharacteristic
257C2×C3#57511yescharacteristic
258D3—82no—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—244no—
268C22#95012yes—
269C22C22×#6711yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—122no—
279C4#100912yes—
280C3#321111yescharacteristic, complemented, Sylow
281C2—22no—
282C2—81no—
283C2—242no—
284C2C2×#95011yescharacteristic
285C2#582211yescharacteristic
286C2#583011yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#529C2
socleC22×C3C22×D4
prime radical#5036C2
hypercenter#529D3
hypocenterC3#3211
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×#875C2semidirect
3. #5068C2semidirect
4. #5059C2semidirect
5. #5050C2semidirect
6. #5046C2semidirect
7. #5040C2semidirect
9. #5028C2semidirect
10. #5016C2semidirect
11. #4925C2semidirect
16. C3×#125C22semidirect
17. C3×#124C22semidirect
21. C3×C4×Q8C22semidirect
23. C3×C4×D4C22semidirect
26. C2×#216C22semidirect
28. C2×#214C22semidirect
38. #810C22semidirect
39. #810C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
43. C4×C3⋊C8C22semidirect
44. #529D3semidirect
103. #125C2×D3semidirect
108. #124C2×D3semidirect
112. C4×Q8C2×D3semidirect
114. C4×D4C2×D3semidirect
280. C3#3211semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
C2×#8752
#50682
#50592
#50502
#50462
#50402
#50362
#50282
#50162
#49252
#32113