Finite Group Databasebeta

← #80406

Subgroups — #80406

Order 384.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×D3×SD16C212yesmaximal, complemented
2#5717C211yesmaximal, characteristic
3Q8×D12C211yesmaximal, characteristic, complemented
4C3×#460C211yesmaximal, characteristic, complemented
5#5170C212yesmaximal, complemented
6#5076C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4778C212yesmaximal, complemented
11#4776C212yesmaximal, complemented
12D4×SD16—31nomaximal, Sylow
13C2×D3×Q8C2212yescomplemented
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×SD16C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic, complemented
19C2×#215C2211yescharacteristic
20#938—21no—
21#938C2212yes—
22#937—22no—
23C2×#214C2212yescomplemented
24C2×#214C2212yescomplemented
25D3×SD16—28no—
26C2×#204C2212yes—
27C2×D3×C8C2212yescomplemented
28#897—22no—
29#897C2212yescomplemented
30C4×D12—21no—
31C4×D12C2211yescharacteristic
32C4×D12C2211yescharacteristic
33#869—21no—
34#869C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#835C2211yescharacteristic, complemented
38(C2×D3)⋊C8C2212yescomplemented
39#816C2212yescomplemented
40#809C2211yescharacteristic
41#808C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C22×SD16—32no—
44D4×Q8—31no—
45D42—31no—
46#492—31no—
47#465—32no—
48#461—31no—
49#460D311yescharacteristic, complemented
50#450—32no—
51#448—32no—
52C4×SD16—31no—
53C8×D4—31no—
54C23×D3—22no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×#67—22no—
60C2×#67—22no—
61D3×Q8—24no—
62D3×Q8—42no—
63D3×D4—24no—
64D3×D4—42no—
65D3×D4—44no—
66C2×D12D411yescharacteristic
67C2×D12C2311yescharacteristic
68C2×C4×D3—22no—
69C2×C4×D3—21no—
70C2×C4×D3C2312yes—
71C2×Dic6—22no—
72C2×Dic6C2312yes—
73C3×SD16—24no—
74C2×C3×C8C2312yes—
75C3×C4⋊C4—21no—
76C3×C4⋊C4C2311yescharacteristic
77C3×C42—21no—
78C3×C42C2311yescharacteristic
79#217—22no—
80#215D414yescomplemented
81#214—24no—
82#214—42no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4D412yes—
86C4⋊Dic3—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3—22no—
89C4⋊Dic3D411yescharacteristic
90C4⋊Dic3C2311yescharacteristic
91C2×C3⋊C8C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93#204—24no—
94D3×C8—24no—
95C22×Q8—32no—
96C22×D4—32no—
97C22×D4—61no—
98C2×SD16—31no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×SD16—68no—
103C2×SD16C2×D312yescomplemented
104C22×C8—32no—
105C4⋊Q8—31no—
106C4⋊Q8—61no—
107#127C2×D311yescharacteristic
108C22⋊Q8—32no—
109C22⋊Q8—62no—
110#121—32no—
111#121—61no—
112#120—62no—
113C4×Q8C2×D311yescharacteristic, complemented
114C4×D4—31no—
115C4×D4—31no—
116C4×D4—61no—
117#106—31no—
118C4⋊C8C2×D311yescharacteristic, complemented
119#103—31no—
120#103—32no—
121#102—31no—
122#102C2×D312yescomplemented
123C22⋊C8—32no—
124C4×C8—31no—
125C23×C3—22no—
126C22×D3—21no—
127C22×D3—44no—
128C22×D3—44no—
129C22×D3—82no—
130C22×D3C2×D412yes—
131C3×Q8—21no—
132C3×Q8C2×D412yescomplemented
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—42no—
136C3×D4—81no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—44no—
143#67—44no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3C2×D412yes—
147D12—21no—
148D12SD1614yescomplemented
149D12C2×D412yes—
150C4×D3—22no—
151C4×D3—22no—
152C4×D3—44no—
153C4×D3—42no—
154Dic6—22no—
155Dic6—22no—
156Dic6—44no—
157C3×C8—22no—
158C3⋊C8—21no—
159C3⋊C8C2×D412yescomplemented
160C24—62no—
161C2×Q8—32no—
162C2×Q8—62no—
163C2×Q8—64no—
164C2×Q8—122no—
165C2×Q8C22×D311yescharacteristic
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—31no—
169C2×D4—31no—
170C2×D4—62no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—122no—
174C2×D4—124no—
175C2×D4C22×D312yes—
176C22×C4—32no—
177C22×C4—62no—
178C22×C4—61no—
179SD16—24no—
180SD16—34no—
181SD16—64no—
182SD16—64no—
183SD16—122no—
184C2×C8—31no—
185C2×C8—31no—
186C2×C8—64no—
187C2×C8C22×D312yes—
188C4⋊C4—21no—
189C4⋊C4—31no—
190C4⋊C4—31no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—32no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—61no—
199C42—21no—
200C42C22×D311yescharacteristic
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—88no—
211C2×D3C2×SD1614yescomplemented
212C3×C4—22no—
213C3×C4—21no—
214C3×C4—41no—
215C3×C4C2×SD1612yescomplemented
216C3×C4C22×D411yescharacteristic
217C3×C4H(5,𝔽2)11yescharacteristic
218Dic3—22no—
219Dic3—42no—
220Dic3—41no—
221C23—22no—
222C23—32no—
223C23—61no—
224C23—124no—
225C23—124no—
226C23—242no—
227Q8—21no—
228Q8—62no—
229Q8—62no—
230Q8—124no—
231Q8D3×D412yescomplemented
232D4—22no—
233D4—22no—
234D4—34no—
235D4—32no—
236D4—42no—
237D4—61no—
238D4—81no—
239D4—124no—
240D4—124no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—32no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—124no—
249C2×C4—122no—
250C2×C4D3×D411yescharacteristic
251C2×C4D3×D411yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—22no—
254C8—32no—
255C8—61no—
256C2×C3—82no—
257C2×C3D4211yescharacteristic
258C2×C3C22×SD1611yescharacteristic
259C2×C3#57711yescharacteristic
260D3—24no—
261D3—42no—
262C22—34no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—62no—
267C22—62no—
268C22—62no—
269C22—121no—
270C22—121no—
271C22—248no—
272C22C2×D3×D411yescharacteristic
273C4—21no—
274C4—22no—
275C4—41no—
276C4—62no—
277C4—121no—
278C4—122no—
279C4D3×SD1612yescomplemented
280C4C2×D3×D411yescharacteristic
281C4#100911yescharacteristic
282C3D4×SD1611yescharacteristic, complemented, Sylow
283C2—64no—
284C2—82no—
285C2—122no—
286C2#560911yescharacteristic
287C2C2×D3×SD1611yescharacteristic
288C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radical#5717C2
hypercenter#460D3
hypocenterC3D4×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×D3×SD16C2semidirect
3. Q8×D12C2semidirect
4. C3×#460C2semidirect
5. #5170C2semidirect
6. #5076C2semidirect
7. C4×#215C2semidirect
8. #4841C2semidirect
9. #4834C2semidirect
10. #4778C2semidirect
11. #4776C2semidirect
13. C2×D3×Q8C22semidirect
16. C2×C3×SD16C22semidirect
18. C3×C4×Q8C22semidirect
23. C2×#214C22semidirect
24. C2×#214C22semidirect
27. C2×D3×C8C22semidirect
29. #897C22semidirect
34. #869C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×#102C22semidirect
37. #835C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #816C22semidirect
41. #808C22semidirect
42. C4×C3⋊C8C22semidirect
49. #460D3semidirect
80. #215D4semidirect
103. C2×SD16C2×D3semidirect
113. C4×Q8C2×D3semidirect
118. C4⋊C8C2×D3semidirect
122. #102C2×D3semidirect
132. C3×Q8C2×D4semidirect
148. D12SD16semidirect
159. C3⋊C8C2×D4semidirect
211. C2×D3C2×SD16semidirect
215. C3×C4C2×SD16semidirect
231. Q8D3×D4semidirect
279. C4D3×SD16semidirect
282. C3D4×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×D3×SD162
#57172
Q8×D122
C3×#4602
#51702
#50762
C4×#2152
#48412
#48342
#47782
#47762
D4×SD163