Finite Group Databasebeta

← D4×#16864391 #46822583

Subgroups — D4×#16864391 #46822583

Order 34432.

546 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#16864391C212yesmaximal, complemented
2D4×Dic538C211yesmaximal, characteristic, complemented
3D4×D1076C211yesmaximal, characteristic
4#31770978C211yesmaximal, characteristic, complemented
5#31770834C212yesmaximal, complemented
6#31770811C211yesmaximal, characteristic, complemented
7#31770806C211yesmaximal, characteristic, complemented
8#31770758C212yesmaximal, complemented
9#31770746C212yesmaximal, complemented
10C4×#16864391C211yesmaximal, characteristic, complemented
11C8×D4×C269C211yesmaximal, characteristic, complemented
12C22×Dic538C2212yescomplemented
13C22×D1076C2212yes—
14C2×D4×D269—21no—
15C2×#16864391—28no—
16C2×#16864391C2212yes—
17C2×#16864391C2211yescharacteristic
18C2×#16864391C2212yescomplemented
19C2×#16864391C2212yescomplemented
20C2×#16864391C2212yescomplemented
21#22575453C2211yescharacteristic, complemented
22#22575452—21no—
23#22575450C2211yescharacteristic
24#22575432C2212yescomplemented
25#22575430—22no—
26#22575428—21no—
27#22575427C2212yes—
28#22575423—22no—
29C4×Dic538C2211yescharacteristic, complemented
30C4×D1076C2211yescharacteristic
31D4×Dic269—21no—
32#22575385C2211yescharacteristic, complemented
33#22575378C2211yescharacteristic, complemented
34#22575378C2212yescomplemented
35#22575375C2211yescharacteristic
36#22575375C2212yescomplemented
37C22×C8×C269C2212yescomplemented
38C4×D4×C269C2211yescharacteristic
39C4⋊C8×C269C2211yescharacteristic, complemented
40C22⋊C8×C269C2212yescomplemented
41C4×C8×C269C2211yescharacteristic, complemented
42C23×D269—22no—
43C2×Dic538—24no—
44C2×Dic538—22no—
45C2×Dic538—42no—
46C2×Dic538C2312yes—
47C2×Dic538C2311yescharacteristic
48C2×D1076—24no—
49C2×D1076—22no—
50C2×D1076—21no—
51C2×D1076—21no—
52C2×D1076—42no—
53C2×D1076C2312yes—
54C2×D1076C2311yescharacteristic
55C2×C22⋊D269—22no—
56D4×D269—44no—
57C22×Dic269—22no—
58C2×C4×D269—21no—
59#16864391—24no—
60#16864391—24no—
61#16864391—24no—
62#16864391—42no—
63#16864391D414yesdirect product factor
64C4⋊Dic269—21no—
65C4⋊Dic269—21no—
66C4⋊Dic269C2311yescharacteristic
67C4⋊Dic269C2311yescharacteristic
68#16864381—21no—
69#16864381—22no—
70C22⋊Dic269—22no—
71#16864379—21no—
72#16864379—22no—
73C4×Dic269—21no—
74C2×D4×C269D411yescharacteristic
75C22×C4×C269C2312yes—
76C2×C8×C269—24no—
77C2×C8×C269C2311yescharacteristic
78C2×C8×C269C2312yes—
79C2×C8×C269C2311yescharacteristic
80C4⋊C4×C269D411yescharacteristic
81C22⋊C4×C269D412yes—
82C42×C269C2311yescharacteristic
83C22×D269—21no—
84C22×D269—22no—
85C22×D269—44no—
86C22×D269—44no—
87C22×D269—82no—
88Dic538—21no—
89Dic538—22no—
90Dic538—22no—
91Dic538—44no—
92Dic538C2×D412yescomplemented
93D1076—22no—
94D1076—21no—
95D1076—22no—
96D1076—44no—
97D1076—42no—
98D1076—81no—
99D1076C2×D412yes—
100C22⋊D269—44no—
101C2×Dic269—21no—
102C2×Dic269—21no—
103C2×Dic269—21no—
104C2×Dic269—22no—
105C2×Dic269—44no—
106C4×D269—42no—
107C23×C269C2×D412yes—
108D4×C269SD1614yescomplemented
109C2×C4×C269—22no—
110C2×C4×C269—22no—
111C2×C4×C269C2×D411yescharacteristic
112C2×C4×C269C2×D412yes—
113C2×C4×C269C2×D411yescharacteristic
114C2×C4×C269C2411yescharacteristic
115C8×C269—22no—
116C8×C269—21no—
117C8×C269C2×D412yescomplemented
118C22×C269—22no—
119C22×C269—22no—
120C22×C269C2×SD1614yescomplemented
121C22×C269C22×D411yescharacteristic
122C2×D269—22no—
123C2×D269—41no—
124C2×D269—41no—
125C2×D269—42no—
126C2×D269—42no—
127C2×D269—42no—
128C2×D269—88no—
129C4×C269—22no—
130C4×C269—21no—
131C4×C269C2×SD1612yescomplemented
132C4×C269C22×D411yescharacteristic
133C4×C269H(5,𝔽2)11yescharacteristic
134Dic269—22no—
135Dic269—42no—
136Dic269—41no—
137Dic269—41no—
138C2×C269—24no—
139C2×C269D4211yescharacteristic
140C2×C269C22×SD1611yescharacteristic
141C2×C269#57711yescharacteristic
142D269—42no—
143D269—82no—
144C269D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—2691nomaximal, Sylow
146C22×SD16—2692no—
147D4×Q8—2691no—
148D42—2691no—
149#492—2691no—
150#465—2692no—
151#461—2691no—
152#460—2691no—
153#450—2692no—
154#448—2692no—
155C4×SD16—2691no—
156C8×D4D26911yescharacteristic, complemented
157C22×Q8—2692no—
158C22×D4—2692no—
159C22×D4—5381no—
160C2×SD16—2691no—
161C2×SD16—2692no—
162C2×SD16—2692no—
163C2×SD16—2692no—
164C2×SD16—2692no—
165C2×SD16—5388no—
166C22×C8C2×D26912yescomplemented
167C4⋊Q8—2691no—
168C4⋊Q8—5381no—
169#127—2691no—
170C22⋊Q8—2692no—
171C22⋊Q8—5382no—
172#121—2692no—
173#121—5381no—
174#120—5382no—
175C4×Q8—2691no—
176C4×D4—2691no—
177C4×D4—5381no—
178C4×D4C2×D26911yescharacteristic
179#106—2691no—
180C4⋊C8C2×D26911yescharacteristic, complemented
181#103—2691no—
182#103—2692no—
183#102—2691no—
184#102—2692no—
185C22⋊C8C2×D26912yescomplemented
186C4×C8C2×D26911yescharacteristic, complemented
187C24—5382no—
188C2×Q8—2691no—
189C2×Q8—2692no—
190C2×Q8—5384no—
191C2×Q8—5382no—
192C2×Q8—10762no—
193C2×D4—2691no—
194C2×D4—2692no—
195C2×D4—5384no—
196C2×D4—5382no—
197C2×D4—5382no—
198C2×D4—5381no—
199C2×D4—5381no—
200C2×D4—10762no—
201C2×D4—10764no—
202C2×D4D107611yescharacteristic
203C22×C4—5382no—
204C22×C4—5381no—
205C22×C4C22×D26912yes—
206SD16—2694no—
207SD16—5384no—
208SD16—5384no—
209SD16—5384no—
210SD16—10762no—
211C2×C8—24no—
212C2×C8C22×D26911yescharacteristic
213C2×C8C22×D26912yes—
214C2×C8C22×D26911yescharacteristic
215C4⋊C4—2691no—
216C4⋊C4—2691no—
217C4⋊C4—5381no—
218C4⋊C4—5381no—
219C4⋊C4—5382no—
220C4⋊C4—5381no—
221C4⋊C4D107611yescharacteristic
222C22⋊C4—5382no—
223C22⋊C4—5381no—
224C22⋊C4—5382no—
225C22⋊C4D107612yes—
226C42—5381no—
227C42C22×D26911yescharacteristic
228C23—5381no—
229C23—5382no—
230C23—10764no—
231C23—10764no—
232C23—21522no—
233C23C2×D107612yes—
234Q8—2692no—
235Q8—5381no—
236Q8—5382no—
237Q8—5382no—
238Q8—10764no—
239D4—2692no—
240D4—5382no—
241D4—5381no—
242D4—5382no—
243D4—10764no—
244D4—10764no—
245D4—10762no—
246D4—21521no—
247D4#1686439114yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—5381no—
251C2×C4—5381no—
252C2×C4—5381no—
253C2×C4—5382no—
254C2×C4—10764no—
255C2×C4—10762no—
256C2×C4C2×D107611yescharacteristic
257C2×C4C2×D107612yes—
258C2×C4C2×D107611yescharacteristic
259C2×C4C23×D26911yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D26912yescomplemented
263C22—22no—
264C22—22no—
265C22—5382no—
266C22—10761no—
267C22—10761no—
268C22—10762no—
269C22—10762no—
270C22—10762no—
271C22—21528no—
272C22C2×#1686439114yescomplemented
273C22C22×D107611yescharacteristic
274C4—21no—
275C4—22no—
276C4—5382no—
277C4—10761no—
278C4—10762no—
279C4—10761no—
280C4C2×#1686439112yescomplemented
281C4C2×D4×D26911yescharacteristic
282C4#2257549911yescharacteristic
283C2—24no—
284C2—10762no—
285C2—21522no—
286C2D4×D107611yescharacteristic
287C2C22×#1686439111yescharacteristic
288C2#3177141711yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC8×D4×C269C2
socleC22×C269C22×D4
prime radicalD4×D1076C2
hypercenterC8×D4D269
hypocenterC269D4×SD16
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#16864391C2semidirect
2. D4×Dic538C2semidirect
4. #31770978C2semidirect
5. #31770834C2semidirect
6. #31770811C2semidirect
7. #31770806C2semidirect
8. #31770758C2semidirect
9. #31770746C2semidirect
10. C4×#16864391C2semidirect
11. C8×D4×C269C2semidirect
12. C22×Dic538C22semidirect
18. C2×#16864391C22semidirect
19. C2×#16864391C22semidirect
20. C2×#16864391C22semidirect
21. #22575453C22semidirect
24. #22575432C22semidirect
29. C4×Dic538C22semidirect
32. #22575385C22semidirect
33. #22575378C22semidirect
34. #22575378C22semidirect
36. #22575375C22semidirect
37. C22×C8×C269C22semidirect
39. C4⋊C8×C269C22semidirect
40. C22⋊C8×C269C22semidirect
41. C4×C8×C269C22semidirect
63. #16864391D4direct (N × H)
92. Dic538C2×D4semidirect
108. D4×C269SD16semidirect
117. C8×C269C2×D4semidirect
120. C22×C269C2×SD16semidirect
131. C4×C269C2×SD16semidirect
144. C269D4×SD16semidirect
156. C8×D4D269semidirect
166. C22×C8C2×D269semidirect
180. C4⋊C8C2×D269semidirect
185. C22⋊C8C2×D269semidirect
186. C4×C8C2×D269semidirect
247. D4#16864391direct (N × H)
262. C8D4×D269semidirect
272. C22C2×#16864391semidirect
280. C4C2×#16864391semidirect

Maximal subgroups

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

subgroupindex
C22×#168643912
D4×Dic5382
D4×D10762
#317709782
#317708342
#317708112
#317708062
#317707582
#317707462
C4×#168643912
C8×D4×C2692
D4×SD16269