Finite Group Databasebeta

← #81851

Subgroups — #81851

Order 384.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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
1D3×#127C211yesmaximal, characteristic, complemented
2#5680C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5169C212yesmaximal, complemented
5#5163C212yesmaximal, complemented
6#5156C212yesmaximal, complemented
7#5150C212yesmaximal, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5058C211yesmaximal, characteristic, complemented
10#4713C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3217—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×SD16C2212yescomplemented
17C2×C3×D8C2212yescomplemented
18C3×#127C2211yescharacteristic
19C3×#124C2211yescharacteristic, complemented
20#946C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic, complemented
22C2×#215C2212yescomplemented
23#940—21no—
24#940—21no—
25#940C2212yes—
26#938C2211yescharacteristic
27D4×Dic3C2211yescharacteristic, complemented
28C2×#213C2212yescomplemented
29#883—21no—
30#67⋊C4—21no—
31#879—21no—
32#877—21no—
33#874C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#870C2211yescharacteristic, complemented
36C3×#97C2211yescharacteristic, complemented
37#832C2212yescomplemented
38#832C2212yescomplemented
39#821C2212yescomplemented
40#821C2212yescomplemented
41(C2×D3)⋊C8C2212yes—
42Dic3⋊C8C2212yescomplemented
43#803C2211yescharacteristic, complemented
44C2×#127—31no—
45#529—31no—
46#496—31no—
47#496D311yescharacteristic, complemented
48#490—31no—
49#490—31no—
50#460—32no—
51#459—32no—
52#450—32no—
53#447—32no—
54#432—31no—
55C23×D3—22no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2312yes—
60C2×C3×D4C2311yescharacteristic
61C2×#67—21no—
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65C22×Dic3—21no—
66D3×D4—42no—
67D3×D4—42no—
68D3×D4—44no—
69D3×D4—44no—
70C2×D12—21no—
71C2×D12—21no—
72C2×D12C2312yes—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C3×SD16—42no—
77C3×D8—42no—
78C2×C3×C8C2312yes—
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42C2311yescharacteristic
82#217—21no—
83#217—21no—
84#215—42no—
85#213—42no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2311yescharacteristic
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94C4×Dic3D411yescharacteristic
95C4×Dic3D411yescharacteristic
96C4×Dic3C2311yescharacteristic
97C2×C3⋊C8C2312yes—
98C22×D4—32no—
99C22×D4—61no—
100C22×D4—61no—
101C2×SD16—32no—
102C2×SD16C2×D312yescomplemented
103C2×D8—32no—
104C2×D8C2×D312yescomplemented
105#127—32no—
106#127—31no—
107#127—61no—
108#127—61no—
109#127C2×D311yescharacteristic
110#126—61no—
111#125—31no—
112#124C2×D311yescharacteristic, complemented
113#123—61no—
114C22⋊Q8—31no—
115#121—31no—
116C4×Q8—31no—
117C4×D4—31no—
118C4×D4—61no—
119#117—61no—
120C2×C42—31no—
121C4⋊C8—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125#102—32no—
126C22⋊C8—32no—
127#97—31no—
128#97C2×D311yescharacteristic, complemented
129C23×C3—21no—
130C23×C3—22no—
131C22×D3—22no—
132C22×D3—44no—
133C22×D3—44no—
134C22×D3—44no—
135C22×D3C2×D411yescharacteristic
136C3×Q8—22no—
137C3×D4—22no—
138C3×D4—22no—
139C3×D4—22no—
140C3×D4—42no—
141C3×D4—42no—
142C2×C3×C4—21no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146#67—44no—
147#67—44no—
148#67—42no—
149#67—44no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—42no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155C2×Dic3C2×D411yescharacteristic
156D12—22no—
157D12—22no—
158D12—42no—
159D12—42no—
160C4×D3—22no—
161C4×D3—22no—
162C4×D3—22no—
163C3×C8—22no—
164C3⋊C8—22no—
165C24—62no—
166C2×Q8C22×D311yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—32no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—122no—
177C2×D4—122no—
178C2×D4—124no—
179C2×D4—124no—
180C2×D4C22×D312yes—
181C2×D4C22×D311yescharacteristic
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—61no—
186SD16—42no—
187SD16—122no—
188D8—42no—
189D8—122no—
190C2×C8—32no—
191C2×C8C22×D312yes—
192C4⋊C4—31no—
193C4⋊C4—31no—
194C4⋊C4—61no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C22⋊C4—61no—
203C42—31no—
204C42—31no—
205C42—31no—
206C42—61no—
207C42C22×D311yescharacteristic
208C22×C3—42no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—41no—
214C22×C3C22×D411yescharacteristic
215C2×D3—21no—
216C2×D3—21no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—88no—
221C2×D3AGL(1,ℤ/8)12yes—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—41no—
225C3×C4H(5,𝔽2)12yes—
226Dic3—21no—
227Dic3—21no—
228Dic3—41no—
229Dic3—41no—
230Dic3AGL(1,ℤ/8)12yes—
231C23—21no—
232C23—22no—
233C23—31no—
234C23—62no—
235C23—124no—
236C23—124no—
237C23—124no—
238Q8—22no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4—42no—
243D4—42no—
244D4—62no—
245D4—62no—
246D4—122no—
247D4—122no—
248D4—124no—
249D4—124no—
250D4—124no—
251D4—122no—
252C2×C4—21no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4D3×D411yescharacteristic
263C2×C4D3×D411yescharacteristic
264C2×C4C23×D311yescharacteristic
265C8—22no—
266C8—62no—
267C2×C3—82no—
268C2×C3—81no—
269C2×C3#53511yescharacteristic
270C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
271C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
272D3—22no—
273D3—82no—
274C22—32no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—61no—
282C22—61no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—248no—
287C22C2×D3×D411yescharacteristic
288C4—21no—
289C4—21no—
290C4—32no—
291C4—41no—
292C4—61no—
293C4—61no—
294C4—121no—
295C4—121no—
296C4#100412yes—
297C3#321711yescharacteristic, complemented, Sylow
298C2—62no—
299C2—82no—
300C2—81no—
301C2—242no—
302C2#571511yescharacteristic
303C2D3×AGL(1,ℤ/8)11yescharacteristic
304C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3217
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. D3×#127C2semidirect
2. #5680C2semidirect
3. C3×#496C2semidirect
4. #5169C2semidirect
5. #5163C2semidirect
6. #5156C2semidirect
7. #5150C2semidirect
8. #5071C2semidirect
9. #5058C2semidirect
10. #4713C2semidirect
11. #4705C2semidirect
16. C2×C3×SD16C22semidirect
17. C2×C3×D8C22semidirect
19. C3×#124C22semidirect
21. Q8×Dic3C22semidirect
22. C2×#215C22semidirect
27. D4×Dic3C22semidirect
28. C2×#213C22semidirect
35. #870C22semidirect
36. C3×#97C22semidirect
37. #832C22semidirect
38. #832C22semidirect
39. #821C22semidirect
40. #821C22semidirect
42. Dic3⋊C8C22semidirect
43. #803C22semidirect
47. #496D3semidirect
102. C2×SD16C2×D3semidirect
104. C2×D8C2×D3semidirect
112. #124C2×D3semidirect
128. #97C2×D3semidirect
297. C3#3217semidirect

Maximal subgroups

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

subgroupindex
D3×#1272
#56802
C3×#4962
#51692
#51632
#51562
#51502
#50712
#50582
#47132
#47052
#32173