Finite Group Databasebeta

← #81442

Subgroups — #81442

Order 384.

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

normal subgroups only

This group has 468 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
1D3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic
3C3×#483C211yesmaximal, characteristic, complemented
4#5046C211yesmaximal, characteristic, complemented
5#4971C211yesmaximal, characteristic, complemented
6#4864C212yesmaximal, complemented
7#4863C212yesmaximal, complemented
8#4807C212yesmaximal, complemented
9#4806C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3187—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic, complemented
14C22×D12—21no—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19#946C2211yescharacteristic, complemented
20C2×#215C2212yescomplemented
21C2×(C2×D3)⋊C4—21no—
22#924C2211yescharacteristic
23C2×#204C2212yescomplemented
24C2×D3×C8C2212yescomplemented
25#897C2211yescharacteristic, complemented
26#896—21no—
27#895—21no—
28#895—21no—
29#895C2212yes—
30#895C2212yes—
31D3×C4⋊C4—21no—
32D3×C4⋊C4C2211yescharacteristic, complemented
33D3×C22⋊C4—21no—
34#878—21no—
35C3×#106C2212yescomplemented
36C3×#103C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#830C2211yescharacteristic
39#821C2212yescomplemented
40(C2×D3)⋊C8C2212yescomplemented
41#810C2212yescomplemented
42#809C2212yescomplemented
43#808C2212yescomplemented
44C2×C22⋊Q8—31no—
45C2×#121—31no—
46#483—31no—
47#483D311yescharacteristic, complemented
48#481—32no—
49#465—32no—
50#465—32no—
51#450—31no—
52#450—31no—
53#448—32no—
54C2×C22⋊C8—31no—
55C23×D3—21no—
56C23×D3D411yescharacteristic
57C2×C3×Q8C2311yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C22×Dic3D411yescharacteristic
61D3×Q8—42no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12—21no—
65C2×D12—42no—
66C2×D12—44no—
67C2×D12C2312yes—
68C2×C4×D3—21no—
69C2×C4×D3—21no—
70C2×C4×D3—42no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C2×Dic6—21no—
76C2×Dic6C2311yescharacteristic
77C2×C3×C8C2312yes—
78C3×C4⋊C4—21no—
79C3×C4⋊C4—21no—
80C3×C4⋊C4C2312yes—
81C3×C4⋊C4C2311yescharacteristic
82C3×C22⋊C4—21no—
83#217—21no—
84#215—42no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93Dic3⋊C4—21no—
94C2×C3⋊C8C2312yes—
95#204—42no—
96D3×C8—42no—
97C22×Q8—31no—
98C22×D4—31no—
99C22×D4—61no—
100C23×C4—31no—
101C2×SD16—32no—
102C2×SD16—32no—
103C22×C8—32no—
104C22⋊Q8—31no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109C22⋊Q8C2×D311yescharacteristic
110#121—32no—
111#121—32no—
112#121—61no—
113#121—61no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4C2×D311yescharacteristic
117C2×C22⋊C4—61no—
118C2×C22⋊C4—61no—
119#106—32no—
120#106C2×D312yescomplemented
121#103—32no—
122#103C2×D312yescomplemented
123#102—32no—
124#102—32no—
125C22⋊C8—31no—
126C22⋊C8—32no—
127C22⋊C8C2×D311yescharacteristic
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C22×D3—41no—
131C22×D3—41no—
132C22×D3—42no—
133C22×D3—42no—
134C22×D3—42no—
135C22×D3SD1614yescomplemented
136C22×D3C2×D411yescharacteristic
137C22×D3C2×D411yescharacteristic
138C3×Q8—22no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—41no—
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151D12—22no—
152D12—22no—
153D12—42no—
154D12—44no—
155D12—44no—
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—41no—
159C4×D3—41no—
160C4×D3—42no—
161C4×D3—42no—
162Dic6—22no—
163Dic6—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166C24—31no—
167C24—61no—
168C2×Q8—31no—
169C2×Q8—61no—
170C2×Q8—122no—
171C2×Q8C22×D311yescharacteristic
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—61no—
176C2×D4—122no—
177C2×D4—124no—
178C22×C4—21no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4—122no—
187C22×C4C22×D311yescharacteristic
188SD16—122no—
189SD16—122no—
190C2×C8—32no—
191C2×C8—122no—
192C2×C8C22×D312yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—31no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4C22×D311yescharacteristic
201C4⋊C4C22×D312yes—
202C22⋊C4—21no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—61no—
206C22⋊C4—61no—
207C22⋊C4—61no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3C2×SD1612yes—
211C22×C3C22×D411yescharacteristic
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—21no—
216C2×D3—44no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—84no—
221C2×D3C2×SD1612yes—
222C2×D3C2×SD1612yes—
223C3×C4—21no—
224C3×C4—41no—
225C3×C4—42no—
226C3×C4—41no—
227C3×C4H(5,𝔽2)12yes—
228Dic3—21no—
229Dic3—21no—
230Dic3—41no—
231Dic3—41no—
232C23—31no—
233C23—31no—
234C23—34no—
235C23—62no—
236C23—121no—
237C23—121no—
238C23—122no—
239C23—122no—
240C23—122no—
241C23D3×D411yescharacteristic
242Q8—22no—
243Q8—62no—
244Q8—122no—
245D4—62no—
246D4—62no—
247D4—122no—
248D4—124no—
249D4—124no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—42no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—121no—
266C2×C4D3×D411yescharacteristic
267C2×C4C23×D311yescharacteristic
268C8—22no—
269C8—62no—
270C2×C3—22no—
271C2×C3#53411yescharacteristic
272C2×C3C22×SD1611yescharacteristic
273C2×C3#57511yescharacteristic
274D3—22no—
275D3—22no—
276D3—82no—
277C22—21no—
278C22—21no—
279C22—32no—
280C22—32no—
281C22—61no—
282C22—61no—
283C22—61no—
284C22—61no—
285C22—124no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22—244no—
290C22D3×SD1612yes—
291C22C2×D3×D411yescharacteristic
292C4—21no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—61no—
297C4—61no—
298C4—121no—
299C4—121no—
300C4#100912yes—
301C3#318711yescharacteristic, complemented, Sylow
302C2—22no—
303C2—62no—
304C2—62no—
305C2—242no—
306C2#565311yescharacteristic
307C2C2×D3×SD1611yescharacteristic
308C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#483C2
socleC22×C3C22×D4
prime radicalC2×#895C2
hypercenter#483D3
hypocenterC3#3187
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. D3×C22⋊Q8C2semidirect
3. C3×#483C2semidirect
4. #5046C2semidirect
5. #4971C2semidirect
6. #4864C2semidirect
7. #4863C2semidirect
8. #4807C2semidirect
9. #4806C2semidirect
10. #4731C2semidirect
11. D3×C22⋊C8C2semidirect
13. C2×D3×Q8C22semidirect
19. #946C22semidirect
20. C2×#215C22semidirect
23. C2×#204C22semidirect
24. C2×D3×C8C22semidirect
25. #897C22semidirect
32. D3×C4⋊C4C22semidirect
35. C3×#106C22semidirect
36. C3×#103C22semidirect
39. #821C22semidirect
40. (C2×D3)⋊C8C22semidirect
41. #810C22semidirect
42. #809C22semidirect
43. #808C22semidirect
47. #483D3semidirect
120. #106C2×D3semidirect
122. #103C2×D3semidirect
135. C22×D3SD16semidirect
301. C3#3187semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
C2×#8952
C3×#4832
#50462
#49712
#48642
#48632
#48072
#48062
#47312
D3×C22⋊C82
#31873