Finite Group Databasebeta

← #19146286

Subgroups — #19146286

Order 5504.

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×SD16×D43C212yesmaximal, complemented
2Q8×D172C211yesmaximal, characteristic, complemented
3#14583534C211yesmaximal, characteristic
4#14583271C211yesmaximal, characteristic, complemented
5#14583127C212yesmaximal, complemented
6#14583105C211yesmaximal, characteristic, complemented
7#14583051C212yesmaximal, complemented
8#14583036C212yesmaximal, complemented
9C4×#10621884C211yesmaximal, characteristic, complemented
10#14582964C211yesmaximal, characteristic, complemented
11C43×#460C211yesmaximal, characteristic, complemented
12C2×Q8×D43C2212yescomplemented
13C2×D4×D43—21no—
14C2×D4×D43C2212yes—
15C2×#10621873C2212yes—
16C2×#10621883C2212yescomplemented
17C2×#10621883C2212yescomplemented
18C2×#10621884C2211yescharacteristic
19SD16×D43—28no—
20C2×C8×D43C2212yescomplemented
21#13063325—21no—
22#13063325C2211yescharacteristic, complemented
23#13063305—22no—
24#13063305C2212yescomplemented
25#13063296—21no—
26#13063296C2212yes—
27#13063293—22no—
28C4×D172—21no—
29C4×D172C2211yescharacteristic
30C4×D172C2211yescharacteristic
31#13063256C2211yescharacteristic, complemented
32#13063250C2212yescomplemented
33#13063249C2211yescharacteristic, complemented
34#13063245C2211yescharacteristic
35(C2×D43)⋊C8C2212yescomplemented
36C4×C43⋊C8C2211yescharacteristic, complemented
37C2×SD16×C43C2212yescomplemented
38#127×C43C2211yescharacteristic
39C4×Q8×C43C2211yescharacteristic, complemented
40C4⋊C8×C43C2211yescharacteristic, complemented
41#102×C43C2212yescomplemented
42C23×D43—22no—
43C2×Q8×C43C2311yescharacteristic
44C2×D4×C43—21no—
45C2×D4×C43—21no—
46C2×D4×C43C2312yes—
47C2×C22⋊D43—22no—
48C2×C22⋊D43—22no—
49Q8×D43—24no—
50Q8×D43—42no—
51D4×D43—24no—
52D4×D43—42no—
53D4×D43—44no—
54C2×D172D411yescharacteristic
55C2×D172C2311yescharacteristic
56C2×C4×D43—21no—
57C2×C4×D43—22no—
58C2×C4×D43C2312yes—
59C2×Dic86—22no—
60C2×Dic86C2312yes—
61SD16×C43—24no—
62C2×C8×C43C2312yes—
63C4⋊C4×C43—21no—
64C4⋊C4×C43C2311yescharacteristic
65C42×C43—21no—
66C42×C43C2311yescharacteristic
67C22⋊Dic43—22no—
68#10621884D414yescomplemented
69#10621883—24no—
70#10621883—42no—
71(C2×D43)⋊C4—21no—
72(C2×D43)⋊C4—22no—
73(C2×D43)⋊C4D412yes—
74C4⋊Dic43—21no—
75C4⋊Dic43—21no—
76C4⋊Dic43—22no—
77C4⋊Dic43D411yescharacteristic
78C4⋊Dic43C2311yescharacteristic
79C2×C43⋊C8C2311yescharacteristic
80C2×C43⋊C8C2311yescharacteristic
81#10621873—24no—
82C8×D43—24no—
83C23×C43—22no—
84C22×D43—21no—
85C22×D43—44no—
86C22×D43—44no—
87C22×D43—82no—
88C22×D43C2×D412yes—
89Q8×C43—21no—
90Q8×C43C2×D412yescomplemented
91D4×C43—22no—
92D4×C43—22no—
93D4×C43—42no—
94D4×C43—81no—
95C2×C4×C43—21no—
96C2×C4×C43—21no—
97C2×C4×C43C2×D411yescharacteristic
98C2×C4×C43C2×D411yescharacteristic
99C2×C4×C43C2411yescharacteristic
100C22⋊D43—44no—
101C22⋊D43—44no—
102C2×Dic43—22no—
103C2×Dic43—21no—
104C2×Dic43C2×D412yes—
105D172—21no—
106D172SD1614yescomplemented
107D172C2×D412yes—
108C4×D43—22no—
109C4×D43—22no—
110C4×D43—44no—
111C4×D43—42no—
112Dic86—22no—
113Dic86—22no—
114Dic86—44no—
115C8×C43—22no—
116C43⋊C8—21no—
117C43⋊C8C2×D412yescomplemented
118C22×C43—42no—
119C22×C43—42no—
120C22×C43—42no—
121C22×C43C22×D411yescharacteristic
122C2×D43—22no—
123C2×D43—22no—
124C2×D43—22no—
125C2×D43—41no—
126C2×D43—41no—
127C2×D43—88no—
128C2×D43C2×SD1614yescomplemented
129C4×C43—22no—
130C4×C43—21no—
131C4×C43—41no—
132C4×C43C2×SD1612yescomplemented
133C4×C43C22×D411yescharacteristic
134C4×C43H(5,𝔽2)11yescharacteristic
135Dic43—22no—
136Dic43—42no—
137Dic43—41no—
138D4×SD16—431nomaximal, Sylow
139C2×C43—82no—
140C2×C43D4211yescharacteristic
141C2×C43C22×SD1611yescharacteristic
142C2×C43#57711yescharacteristic
143D43—24no—
144D43—42no—
145C22×SD16—432no—
146D4×Q8—431no—
147D42—431no—
148#492—431no—
149#465—432no—
150#461—431no—
151#460D4311yescharacteristic, complemented
152#450—432no—
153#448—432no—
154C4×SD16—431no—
155C8×D4—431no—
156C43D4×SD1611yescharacteristic, complemented, Sylow
157C22×Q8—432no—
158C22×D4—432no—
159C22×D4—861no—
160C2×SD16—431no—
161C2×SD16—432no—
162C2×SD16—432no—
163C2×SD16—432no—
164C2×SD16—868no—
165C2×SD16C2×D4312yescomplemented
166C22×C8—432no—
167C4⋊Q8—431no—
168C4⋊Q8—861no—
169#127C2×D4311yescharacteristic
170C22⋊Q8—432no—
171C22⋊Q8—862no—
172#121—432no—
173#121—861no—
174#120—862no—
175C4×Q8C2×D4311yescharacteristic, complemented
176C4×D4—431no—
177C4×D4—431no—
178C4×D4—861no—
179#106—431no—
180C4⋊C8C2×D4311yescharacteristic, complemented
181#103—431no—
182#103—432no—
183#102—431no—
184#102C2×D4312yescomplemented
185C22⋊C8—432no—
186C4×C8—431no—
187C24—862no—
188C2×Q8—432no—
189C2×Q8—862no—
190C2×Q8—864no—
191C2×Q8—1722no—
192C2×Q8C22×D4311yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—431no—
196C2×D4—431no—
197C2×D4—862no—
198C2×D4—864no—
199C2×D4—862no—
200C2×D4—1722no—
201C2×D4—1724no—
202C2×D4C22×D4312yes—
203C22×C4—432no—
204C22×C4—862no—
205C22×C4—861no—
206SD16—24no—
207SD16—434no—
208SD16—864no—
209SD16—864no—
210SD16—1722no—
211C2×C8—431no—
212C2×C8—431no—
213C2×C8—864no—
214C2×C8C22×D4312yes—
215C4⋊C4—21no—
216C4⋊C4—431no—
217C4⋊C4—431no—
218C4⋊C4—862no—
219C4⋊C4—861no—
220C4⋊C4—861no—
221C4⋊C4C22×D4311yescharacteristic
222C22⋊C4—432no—
223C22⋊C4—862no—
224C22⋊C4—862no—
225C22⋊C4—861no—
226C42—21no—
227C42C22×D4311yescharacteristic
228C23—22no—
229C23—432no—
230C23—861no—
231C23—3442no—
232C23—1724no—
233C23—1724no—
234Q8—21no—
235Q8—862no—
236Q8—862no—
237Q8—1724no—
238Q8D4×D4312yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—434no—
244D4—432no—
245D4—861no—
246D4—1724no—
247D4—1724no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—432no—
251C2×C4—862no—
252C2×C4—862no—
253C2×C4—861no—
254C2×C4—862no—
255C2×C4—1724no—
256C2×C4—1722no—
257C2×C4D4×D4311yescharacteristic
258C2×C4D4×D4311yescharacteristic
259C2×C4C23×D4311yescharacteristic
260C8—22no—
261C8—432no—
262C8—861no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—434no—
267C22—862no—
268C22—862no—
269C22—862no—
270C22—3448no—
271C22—1721no—
272C22—1721no—
273C22C2×D4×D4311yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—862no—
278C4—1721no—
279C4—1722no—
280C4SD16×D4312yescomplemented
281C4C2×D4×D4311yescharacteristic
282C4#1306337111yescharacteristic
283C2—82no—
284C2—864no—
285C2—1722no—
286C2#1458353311yescharacteristic
287C2C2×SD16×D4311yescharacteristic
288C2#1458370911yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#460C2
socleC22×C43C22×D4
hypercenter#460D43
hypocenterC43D4×SD16
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D43C2semidirect
2. Q8×D172C2semidirect
4. #14583271C2semidirect
5. #14583127C2semidirect
6. #14583105C2semidirect
7. #14583051C2semidirect
8. #14583036C2semidirect
9. C4×#10621884C2semidirect
10. #14582964C2semidirect
11. C43×#460C2semidirect
12. C2×Q8×D43C22semidirect
16. C2×#10621883C22semidirect
17. C2×#10621883C22semidirect
20. C2×C8×D43C22semidirect
22. #13063325C22semidirect
24. #13063305C22semidirect
31. #13063256C22semidirect
32. #13063250C22semidirect
33. #13063249C22semidirect
35. (C2×D43)⋊C8C22semidirect
36. C4×C43⋊C8C22semidirect
37. C2×SD16×C43C22semidirect
39. C4×Q8×C43C22semidirect
40. C4⋊C8×C43C22semidirect
41. #102×C43C22semidirect
68. #10621884D4semidirect
90. Q8×C43C2×D4semidirect
106. D172SD16semidirect
117. C43⋊C8C2×D4semidirect
128. C2×D43C2×SD16semidirect
132. C4×C43C2×SD16semidirect
151. #460D43semidirect
156. C43D4×SD16semidirect
165. C2×SD16C2×D43semidirect
175. C4×Q8C2×D43semidirect
180. C4⋊C8C2×D43semidirect
184. #102C2×D43semidirect
238. Q8D4×D43semidirect
280. C4SD16×D43semidirect