Finite Group Databasebeta

← D4×#63219 #14540482

Subgroups — D4×#63219 #14540482

Order 2176.

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×#63219C212yesmaximal, complemented
2D4×Dic34C211yesmaximal, characteristic, complemented
3D4×D68C211yesmaximal, characteristic
4#11763182C211yesmaximal, characteristic, complemented
5#11763038C212yesmaximal, complemented
6#11763015C211yesmaximal, characteristic, complemented
7#11763010C211yesmaximal, characteristic, complemented
8#11762962C212yesmaximal, complemented
9#11762950C212yesmaximal, complemented
10C4×#63219C211yesmaximal, characteristic, complemented
11C8×D4×C17C211yesmaximal, characteristic, complemented
12C2×D4×D17—21no—
13C22×D68C2212yes—
14C22×Dic34C2212yescomplemented
15C22×C8×C17C2212yescomplemented
16C4×D4×C17C2211yescharacteristic
17D4×Dic17—21no—
18#10587616C2212yes—
19#10587610C2212yescomplemented
20C2×#63219—28no—
21C2×#63219C2212yes—
22C2×#63219C2211yescharacteristic
23C2×#63219C2212yescomplemented
24C2×#63219C2212yescomplemented
25C2×#63219C2212yescomplemented
26#10587581—21no—
27#10587574—21no—
28#10587568—22no—
29#10587564—22no—
30#10587560C2211yescharacteristic
31C4×D68C2211yescharacteristic
32#10587555C2211yescharacteristic, complemented
33C4×Dic34C2211yescharacteristic, complemented
34C17×C4⋊C8C2211yescharacteristic, complemented
35C17×C22⋊C8C2212yescomplemented
36C4×C8×C17C2211yescharacteristic, complemented
37#10587492C2211yescharacteristic
38#10587492C2212yescomplemented
39#10587488C2211yescharacteristic, complemented
40#10587487C2211yescharacteristic, complemented
41#10587487C2212yescomplemented
42C23×D17—22no—
43C2×D4×C17D411yescharacteristic
44C22×C4×C17C2312yes—
45C2×C22⋊D17—22no—
46C22×Dic17—22no—
47D4×D17—44no—
48C2×D68—24no—
49C2×D68—22no—
50C2×D68—21no—
51C2×D68—21no—
52C2×D68—42no—
53C2×D68C2312yes—
54C2×D68C2311yescharacteristic
55C2×C4×D17—21no—
56C2×Dic34—24no—
57C2×Dic34—22no—
58C2×Dic34—42no—
59C2×Dic34C2312yes—
60C2×Dic34C2311yescharacteristic
61C2×C8×C17—24no—
62C2×C8×C17C2312yes—
63C2×C8×C17C2311yescharacteristic
64C2×C8×C17C2311yescharacteristic
65C4⋊C4×C17D411yescharacteristic
66C22⋊C4×C17D412yes—
67C42×C17C2311yescharacteristic
68C22⋊Dic17—22no—
69#63227—21no—
70#63227—22no—
71C4⋊Dic17—21no—
72C4⋊Dic17—21no—
73C4⋊Dic17C2311yescharacteristic
74C4⋊Dic17C2311yescharacteristic
75#63225—21no—
76#63225—22no—
77C4×Dic17—21no—
78#63219—24no—
79#63219—24no—
80#63219—24no—
81#63219—42no—
82#63219D414yesdirect product factor
83C23×C17C2×D412yes—
84C22×D17—21no—
85C22×D17—22no—
86C22×D17—44no—
87C22×D17—44no—
88C22×D17—82no—
89D4×C17SD1614yescomplemented
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17C2×D411yescharacteristic
93C2×C4×C17C2×D412yes—
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2411yescharacteristic
96C22⋊D17—44no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—21no—
101C2×Dic17—44no—
102D68—22no—
103D68—21no—
104D68—22no—
105D68—44no—
106D68—42no—
107D68—81no—
108D68C2×D412yes—
109C4×D17—42no—
110Dic34—21no—
111Dic34—22no—
112Dic34—22no—
113Dic34—44no—
114Dic34C2×D412yescomplemented
115C8×C17—22no—
116C8×C17—21no—
117C8×C17C2×D412yescomplemented
118D4×SD16—171nomaximal, Sylow
119C22×C17—22no—
120C22×C17—22no—
121C22×C17C2×SD1614yescomplemented
122C22×C17C22×D411yescharacteristic
123C2×D17—22no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—88no—
130C4×C17—22no—
131C4×C17—21no—
132C4×C17C2×SD1612yescomplemented
133C4×C17C22×D411yescharacteristic
134C4×C17H(5,𝔽2)11yescharacteristic
135Dic17—22no—
136Dic17—42no—
137Dic17—41no—
138Dic17—41no—
139C22×SD16—172no—
140D4×Q8—171no—
141D42—171no—
142#492—171no—
143#465—172no—
144#461—171no—
145#460—171no—
146#450—172no—
147#448—172no—
148C4×SD16—171no—
149C8×D4D1711yescharacteristic, complemented
150C2×C17—24no—
151C2×C17D4211yescharacteristic
152C2×C17C22×SD1611yescharacteristic
153C2×C17#57711yescharacteristic
154D17—42no—
155D17—82no—
156C22×Q8—172no—
157C22×D4—172no—
158C22×D4—341no—
159C2×SD16—171no—
160C2×SD16—172no—
161C2×SD16—172no—
162C2×SD16—172no—
163C2×SD16—172no—
164C2×SD16—348no—
165C22×C8C2×D1712yescomplemented
166C4⋊Q8—171no—
167C4⋊Q8—341no—
168#127—171no—
169C22⋊Q8—172no—
170C22⋊Q8—342no—
171#121—172no—
172#121—341no—
173#120—342no—
174C4×Q8—171no—
175C4×D4—171no—
176C4×D4—341no—
177C4×D4C2×D1711yescharacteristic
178#106—171no—
179C4⋊C8C2×D1711yescharacteristic, complemented
180#103—171no—
181#103—172no—
182#102—171no—
183#102—172no—
184C22⋊C8C2×D1712yescomplemented
185C4×C8C2×D1711yescharacteristic, complemented
186C17D4×SD1611yescharacteristic, complemented, Sylow
187C24—342no—
188C2×Q8—171no—
189C2×Q8—172no—
190C2×Q8—344no—
191C2×Q8—342no—
192C2×Q8—682no—
193C2×D4—171no—
194C2×D4—172no—
195C2×D4—344no—
196C2×D4—342no—
197C2×D4—342no—
198C2×D4—341no—
199C2×D4—341no—
200C2×D4—682no—
201C2×D4—684no—
202C2×D4D6811yescharacteristic
203C22×C4—342no—
204C22×C4—341no—
205C22×C4C22×D1712yes—
206SD16—174no—
207SD16—344no—
208SD16—344no—
209SD16—344no—
210SD16—682no—
211C2×C8—24no—
212C2×C8C22×D1711yescharacteristic
213C2×C8C22×D1712yes—
214C2×C8C22×D1711yescharacteristic
215C4⋊C4—171no—
216C4⋊C4—171no—
217C4⋊C4—341no—
218C4⋊C4—341no—
219C4⋊C4—342no—
220C4⋊C4—341no—
221C4⋊C4D6811yescharacteristic
222C22⋊C4—342no—
223C22⋊C4—341no—
224C22⋊C4—342no—
225C22⋊C4D6812yes—
226C42—341no—
227C42C22×D1711yescharacteristic
228C23—341no—
229C23—342no—
230C23—684no—
231C23—684no—
232C23—1362no—
233C23C2×D6812yes—
234Q8—172no—
235Q8—341no—
236Q8—342no—
237Q8—342no—
238Q8—684no—
239D4—172no—
240D4—342no—
241D4—341no—
242D4—342no—
243D4—684no—
244D4—684no—
245D4—682no—
246D4—1361no—
247D4#6321914yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4—341no—
253C2×C4—342no—
254C2×C4—684no—
255C2×C4—682no—
256C2×C4C2×D6811yescharacteristic
257C2×C4C2×D6812yes—
258C2×C4C2×D6811yescharacteristic
259C2×C4C23×D1711yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D1712yescomplemented
263C22—22no—
264C22—22no—
265C22—342no—
266C22—681no—
267C22—681no—
268C22—682no—
269C22—682no—
270C22—682no—
271C22—1368no—
272C22C2×#6321914yescomplemented
273C22C22×D6811yescharacteristic
274C4—21no—
275C4—22no—
276C4—342no—
277C4—681no—
278C4—681no—
279C4—682no—
280C4C2×#6321912yescomplemented
281C4C2×D4×D1711yescharacteristic
282C4#1058769611yescharacteristic
283C2—24no—
284C2—682no—
285C2—1362no—
286C2D4×D6811yescharacteristic
287C2C22×#6321911yescharacteristic
288C2#1176362111yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC8×D4×C17C2
socleC22×C17C22×D4
prime radicalD4×D68C2
hypercenterC8×D4D17
hypocenterC17D4×SD16
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#63219C2semidirect
2. D4×Dic34C2semidirect
4. #11763182C2semidirect
5. #11763038C2semidirect
6. #11763015C2semidirect
7. #11763010C2semidirect
8. #11762962C2semidirect
9. #11762950C2semidirect
10. C4×#63219C2semidirect
11. C8×D4×C17C2semidirect
14. C22×Dic34C22semidirect
15. C22×C8×C17C22semidirect
19. #10587610C22semidirect
23. C2×#63219C22semidirect
24. C2×#63219C22semidirect
25. C2×#63219C22semidirect
32. #10587555C22semidirect
33. C4×Dic34C22semidirect
34. C17×C4⋊C8C22semidirect
35. C17×C22⋊C8C22semidirect
36. C4×C8×C17C22semidirect
38. #10587492C22semidirect
39. #10587488C22semidirect
40. #10587487C22semidirect
41. #10587487C22semidirect
82. #63219D4direct (N × H)
89. D4×C17SD16semidirect
114. Dic34C2×D4semidirect
117. C8×C17C2×D4semidirect
121. C22×C17C2×SD16semidirect
132. C4×C17C2×SD16semidirect
149. C8×D4D17semidirect
165. C22×C8C2×D17semidirect
179. C4⋊C8C2×D17semidirect
184. C22⋊C8C2×D17semidirect
185. C4×C8C2×D17semidirect
186. C17D4×SD16semidirect
247. D4#63219direct (N × H)
262. C8D4×D17semidirect
272. C22C2×#63219semidirect
280. C4C2×#63219semidirect

Maximal subgroups

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

subgroupindex
C22×#632192
D4×Dic342
D4×D682
#117631822
#117630382
#117630152
#117630102
#117629622
#117629502
C4×#632192
C8×D4×C172
D4×SD1617