Finite Group Databasebeta

← #81771

Subgroups — #81771

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#5727C211yesmaximal, characteristic, complemented
3#5717C211yesmaximal, characteristic
4C3×#492C211yesmaximal, characteristic, complemented
5#5171C212yesmaximal, complemented
6#5170C212yesmaximal, complemented
7#5169C212yesmaximal, complemented
8#5083C211yesmaximal, characteristic, complemented
9#5075C211yesmaximal, characteristic, complemented
10C4×#204C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12D4×SD16—31nomaximal, Sylow
13C2×D3×Q8C2212yescomplemented
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×SD16C2212yescomplemented
17C2×C3×SD16C2212yescomplemented
18C3×C4⋊Q8C2211yescharacteristic, complemented
19C3×#127C2211yescharacteristic
20#946C2212yescomplemented
21C2×#215C2212yes—
22#938—21no—
23#938C2212yes—
24#937—22no—
25C2×#214C2212yescomplemented
26D3×SD16—28no—
27C2×#204C2211yescharacteristic
28C2×D3×C8C2212yescomplemented
29#896—22no—
30#893—21no—
31#888—21no—
32C4×D12C2211yescharacteristic
33C4×D12C2211yescharacteristic
34C4×Dic6C2211yescharacteristic, complemented
35C3×C4×C8C2211yescharacteristic, complemented
36#835C2212yescomplemented
37#832C2212yescomplemented
38#821C2211yescharacteristic
39(C2×D3)⋊C8C2212yescomplemented
40#817C2211yescharacteristic, complemented
41#816C2211yescharacteristic, complemented
42C4⋊(C3⋊C8)C2211yescharacteristic, complemented
43C22×SD16—32no—
44D4×Q8—31no—
45D42—31no—
46#492D311yescharacteristic, complemented
47#465—32no—
48#461—31no—
49#460—31no—
50#450—32no—
51#448—32no—
52C4×SD16—31no—
53C8×D4—31no—
54C23×D3—22no—
55C2×C3×Q8C2312yes—
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×Dic6C2311yescharacteristic
73C3×SD16—24no—
74C3×SD16—42no—
75C2×C3×C8C2311yescharacteristic
76C2×C3×C8C2311yescharacteristic
77C3×C4⋊C4—21no—
78C3×C4⋊C4—21no—
79C3×C42C2311yescharacteristic
80#217—22no—
81#215—24no—
82#214—24no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4D412yes—
86C4⋊Dic3D411yescharacteristic
87C4⋊Dic3C2311yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90Dic3⋊C4—22no—
91C4×Dic3—21no—
92C2×C3⋊C8C2312yes—
93#204D414yescomplemented
94D3×C8—24no—
95C22×Q8—32no—
96C22×D4—32no—
97C22×D4—61no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×SD16—31no—
101C2×SD16—68no—
102C2×SD16C2×D312yescomplemented
103C2×SD16C2×D312yescomplemented
104C22×C8—32no—
105C4⋊Q8—61no—
106C4⋊Q8C2×D311yescharacteristic, complemented
107#127C2×D311yescharacteristic
108C22⋊Q8—32no—
109C22⋊Q8—62no—
110#121—32no—
111#121—61no—
112#120—62no—
113C4×Q8—31no—
114C4×D4—31no—
115C4×D4—31no—
116C4×D4—61no—
117#106—31no—
118C4⋊C8—31no—
119#103—32no—
120#103—31no—
121#102—32no—
122#102—31no—
123C22⋊C8—32no—
124C4×C8C2×D311yescharacteristic, complemented
125C23×C3—22no—
126C22×D3—21no—
127C22×D3—44no—
128C22×D3—44no—
129C22×D3—82no—
130C22×D3C2×D412yes—
131C3×Q8—22no—
132C3×Q8—22no—
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—42no—
136C3×D4—81no—
137C2×C3×C4—22no—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141#67—44no—
142#67—44no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3C2×D412yes—
147D12—21no—
148D12SD1614yescomplemented
149D12C2×D412yes—
150C4×D3—22no—
151C4×D3—22no—
152C4×D3—44no—
153C4×D3—42no—
154Dic6—21no—
155Dic6—44no—
156Dic6C2×D412yescomplemented
157C3×C8—21no—
158C3×C8C2×D412yescomplemented
159C3⋊C8—22no—
160C24—62no—
161C2×Q8—31no—
162C2×Q8—62no—
163C2×Q8—64no—
164C2×Q8—122no—
165C2×Q8C22×D312yes—
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—42no—
182SD16—64no—
183SD16—64no—
184C2×C8—32no—
185C2×C8—64no—
186C2×C8C22×D311yescharacteristic
187C2×C8C22×D311yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—31no—
191C4⋊C4—31no—
192C4⋊C4—31no—
193C4⋊C4—62no—
194C4⋊C4—61no—
195C22⋊C4—32no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—61no—
199C42—61no—
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—21no—
213C3×C4—42no—
214C3×C4C2×SD1612yescomplemented
215C3×C4C22×D411yescharacteristic
216C3×C4H(5,𝔽2)11yescharacteristic
217Dic3—22no—
218Dic3—22no—
219Dic3—41no—
220Dic3—41no—
221C23—22no—
222C23—32no—
223C23—61no—
224C23—124no—
225C23—124no—
226C23—242no—
227Q8—22no—
228Q8—22no—
229Q8—32no—
230Q8—61no—
231Q8—124no—
232D4—22no—
233D4—22no—
234D4—34no—
235D4—32no—
236D4—42no—
237D4—61no—
238D4—81no—
239D4—124no—
240D4—124no—
241C2×C4—22no—
242C2×C4—32no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—124no—
249C2×C4—122no—
250C2×C4D3×D411yescharacteristic
251C2×C4D3×D411yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—21no—
254C8—62no—
255C8D3×D412yescomplemented
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—42no—
275C4—62no—
276C4—62no—
277C4—121no—
278C4—121no—
279C4D3×SD1612yescomplemented
280C4C2×D3×D411yescharacteristic
281C4#100411yescharacteristic
282C3D4×SD1611yescharacteristic, complemented, Sylow
283C2—64no—
284C2—82no—
285C2—122no—
286C2#571711yescharacteristic
287C2C2×D3×SD1611yescharacteristic
288C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#492C2
socleC22×C3C22×D4
prime radical#5717C2
hypercenter#492D3
hypocenterC3D4×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×D3×SD16C2semidirect
2. #5727C2semidirect
4. C3×#492C2semidirect
5. #5171C2semidirect
6. #5170C2semidirect
7. #5169C2semidirect
8. #5083C2semidirect
9. #5075C2semidirect
10. C4×#204C2semidirect
11. C8×D12C2semidirect
13. C2×D3×Q8C22semidirect
16. C2×C3×SD16C22semidirect
17. C2×C3×SD16C22semidirect
18. C3×C4⋊Q8C22semidirect
20. #946C22semidirect
25. C2×#214C22semidirect
28. C2×D3×C8C22semidirect
34. C4×Dic6C22semidirect
35. C3×C4×C8C22semidirect
36. #835C22semidirect
37. #832C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #817C22semidirect
41. #816C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
46. #492D3semidirect
93. #204D4semidirect
102. C2×SD16C2×D3semidirect
103. C2×SD16C2×D3semidirect
106. C4⋊Q8C2×D3semidirect
124. C4×C8C2×D3semidirect
148. D12SD16semidirect
156. Dic6C2×D4semidirect
158. C3×C8C2×D4semidirect
211. C2×D3C2×SD16semidirect
214. C3×C4C2×SD16semidirect
255. C8D3×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
#57272
#57172
C3×#4922
#51712
#51702
#51692
#50832
#50752
C4×#2042
C8×D122
D4×SD163