Finite Group Databasebeta

← D4×#16864643 #46848062

Subgroups — D4×#16864643 #46848062

Order 34688.

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×#16864643C212yesmaximal, complemented
2D4×Dic542C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic
4#31774270C211yesmaximal, characteristic, complemented
5#31774126C212yesmaximal, complemented
6#31774103C211yesmaximal, characteristic, complemented
7#31774098C211yesmaximal, characteristic, complemented
8#31774050C212yesmaximal, complemented
9#31774038C212yesmaximal, complemented
10C4×#16864643C211yesmaximal, characteristic, complemented
11C8×D4×C271C211yesmaximal, characteristic, complemented
12C22×Dic542C2212yescomplemented
13C22×D1084C2212yes—
14C2×D4×D271—21no—
15C2×#16864643—28no—
16C2×#16864643C2212yes—
17C2×#16864643C2211yescharacteristic
18C2×#16864643C2212yescomplemented
19C2×#16864643C2212yescomplemented
20C2×#16864643C2212yescomplemented
21#22576160C2211yescharacteristic, complemented
22#22576159—21no—
23#22576157C2211yescharacteristic
24#22576139C2212yescomplemented
25#22576137—22no—
26#22576135—21no—
27#22576134C2212yes—
28#22576130—22no—
29C4×Dic542C2211yescharacteristic, complemented
30C4×D1084C2211yescharacteristic
31D4×Dic271—21no—
32#22576092C2211yescharacteristic, complemented
33#22576085C2211yescharacteristic, complemented
34#22576085C2212yescomplemented
35#22576082C2211yescharacteristic
36#22576082C2212yescomplemented
37C22×C8×C271C2212yescomplemented
38C4×D4×C271C2211yescharacteristic
39C4⋊C8×C271C2211yescharacteristic, complemented
40C22⋊C8×C271C2212yescomplemented
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43C2×Dic542—24no—
44C2×Dic542—22no—
45C2×Dic542—42no—
46C2×Dic542C2312yes—
47C2×Dic542C2311yescharacteristic
48C2×D1084—24no—
49C2×D1084—22no—
50C2×D1084—21no—
51C2×D1084—21no—
52C2×D1084—42no—
53C2×D1084C2312yes—
54C2×D1084C2311yescharacteristic
55C2×C22⋊D271—22no—
56D4×D271—44no—
57C22×Dic271—22no—
58C2×C4×D271—21no—
59#16864643—24no—
60#16864643—24no—
61#16864643—24no—
62#16864643—42no—
63#16864643D414yesdirect product factor
64C4⋊Dic271—21no—
65C4⋊Dic271—21no—
66C4⋊Dic271C2311yescharacteristic
67C4⋊Dic271C2311yescharacteristic
68Dic271⋊C4—21no—
69Dic271⋊C4—22no—
70C22⋊Dic271—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4—22no—
73C4×Dic271—21no—
74C2×D4×C271D411yescharacteristic
75C22×C4×C271C2312yes—
76C2×C8×C271—24no—
77C2×C8×C271C2312yes—
78C2×C8×C271C2311yescharacteristic
79C2×C8×C271C2311yescharacteristic
80C4⋊C4×C271D411yescharacteristic
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88Dic542—21no—
89Dic542—22no—
90Dic542—22no—
91Dic542—44no—
92Dic542C2×D412yescomplemented
93D1084—22no—
94D1084—21no—
95D1084—22no—
96D1084—44no—
97D1084—42no—
98D1084—81no—
99D1084C2×D412yes—
100C22⋊D271—44no—
101C2×Dic271—21no—
102C2×Dic271—21no—
103C2×Dic271—21no—
104C2×Dic271—22no—
105C2×Dic271—44no—
106C4×D271—42no—
107C23×C271C2×D412yes—
108D4×C271SD1614yescomplemented
109C2×C4×C271—22no—
110C2×C4×C271—22no—
111C2×C4×C271C2×D411yescharacteristic
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2×D411yescharacteristic
114C2×C4×C271C2411yescharacteristic
115C8×C271—22no—
116C8×C271—21no—
117C8×C271C2×D412yescomplemented
118C22×C271—22no—
119C22×C271—22no—
120C22×C271C2×SD1614yescomplemented
121C22×C271C22×D411yescharacteristic
122C2×D271—22no—
123C2×D271—41no—
124C2×D271—41no—
125C2×D271—42no—
126C2×D271—42no—
127C2×D271—42no—
128C2×D271—88no—
129C4×C271—22no—
130C4×C271—21no—
131C4×C271C2×SD1612yescomplemented
132C4×C271C22×D411yescharacteristic
133C4×C271H(5,𝔽2)11yescharacteristic
134Dic271—22no—
135Dic271—42no—
136Dic271—41no—
137Dic271—41no—
138C2×C271—24no—
139C2×C271D4211yescharacteristic
140C2×C271C22×SD1611yescharacteristic
141C2×C271#57711yescharacteristic
142D271—42no—
143D271—82no—
144C271D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—2711nomaximal, Sylow
146C22×SD16—2712no—
147D4×Q8—2711no—
148D42—2711no—
149#492—2711no—
150#465—2712no—
151#461—2711no—
152#460—2711no—
153#450—2712no—
154#448—2712no—
155C4×SD16—2711no—
156C8×D4D27111yescharacteristic, complemented
157C22×Q8—2712no—
158C22×D4—2712no—
159C22×D4—5421no—
160C2×SD16—2711no—
161C2×SD16—2712no—
162C2×SD16—2712no—
163C2×SD16—2712no—
164C2×SD16—2712no—
165C2×SD16—5428no—
166C22×C8C2×D27112yescomplemented
167C4⋊Q8—2711no—
168C4⋊Q8—5421no—
169#127—2711no—
170C22⋊Q8—2712no—
171C22⋊Q8—5422no—
172#121—2712no—
173#121—5421no—
174#120—5422no—
175C4×Q8—2711no—
176C4×D4—2711no—
177C4×D4—5421no—
178C4×D4C2×D27111yescharacteristic
179#106—2711no—
180C4⋊C8C2×D27111yescharacteristic, complemented
181#103—2711no—
182#103—2712no—
183#102—2711no—
184#102—2712no—
185C22⋊C8C2×D27112yescomplemented
186C4×C8C2×D27111yescharacteristic, complemented
187C24—5422no—
188C2×Q8—2711no—
189C2×Q8—2712no—
190C2×Q8—5424no—
191C2×Q8—5422no—
192C2×Q8—10842no—
193C2×D4—2711no—
194C2×D4—2712no—
195C2×D4—5424no—
196C2×D4—5422no—
197C2×D4—5422no—
198C2×D4—5421no—
199C2×D4—5421no—
200C2×D4—10842no—
201C2×D4—10844no—
202C2×D4D108411yescharacteristic
203C22×C4—5422no—
204C22×C4—5421no—
205C22×C4C22×D27112yes—
206SD16—2714no—
207SD16—5424no—
208SD16—5424no—
209SD16—5424no—
210SD16—10842no—
211C2×C8—24no—
212C2×C8C22×D27111yescharacteristic
213C2×C8C22×D27112yes—
214C2×C8C22×D27111yescharacteristic
215C4⋊C4—2711no—
216C4⋊C4—2711no—
217C4⋊C4—5421no—
218C4⋊C4—5421no—
219C4⋊C4—5422no—
220C4⋊C4—5421no—
221C4⋊C4D108411yescharacteristic
222C22⋊C4—5422no—
223C22⋊C4—5421no—
224C22⋊C4—5422no—
225C22⋊C4D108412yes—
226C42—5421no—
227C42C22×D27111yescharacteristic
228C23—5421no—
229C23—5422no—
230C23—10844no—
231C23—10844no—
232C23—21682no—
233C23C2×D108412yes—
234Q8—2712no—
235Q8—5421no—
236Q8—5422no—
237Q8—5422no—
238Q8—10844no—
239D4—2712no—
240D4—5422no—
241D4—5421no—
242D4—5422no—
243D4—10844no—
244D4—10844no—
245D4—10842no—
246D4—21681no—
247D4#1686464314yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—5421no—
251C2×C4—5421no—
252C2×C4—5421no—
253C2×C4—5422no—
254C2×C4—10844no—
255C2×C4—10842no—
256C2×C4C2×D108411yescharacteristic
257C2×C4C2×D108412yes—
258C2×C4C2×D108411yescharacteristic
259C2×C4C23×D27111yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D27112yescomplemented
263C22—22no—
264C22—22no—
265C22—5422no—
266C22—10841no—
267C22—10841no—
268C22—10842no—
269C22—10842no—
270C22—10842no—
271C22—21688no—
272C22C2×#1686464314yescomplemented
273C22C22×D108411yescharacteristic
274C4—22no—
275C4—21no—
276C4—5422no—
277C4—10842no—
278C4—10841no—
279C4—10841no—
280C4C2×#1686464312yescomplemented
281C4C2×D4×D27111yescharacteristic
282C4#2257620611yescharacteristic
283C2—24no—
284C2—10842no—
285C2—21682no—
286C2D4×D108411yescharacteristic
287C2C22×#1686464311yescharacteristic
288C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC8×D4×C271C2
socleC22×C271C22×D4
prime radicalD4×D1084C2
hypercenterC8×D4D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#16864643C2semidirect
2. D4×Dic542C2semidirect
4. #31774270C2semidirect
5. #31774126C2semidirect
6. #31774103C2semidirect
7. #31774098C2semidirect
8. #31774050C2semidirect
9. #31774038C2semidirect
10. C4×#16864643C2semidirect
11. C8×D4×C271C2semidirect
12. C22×Dic542C22semidirect
18. C2×#16864643C22semidirect
19. C2×#16864643C22semidirect
20. C2×#16864643C22semidirect
21. #22576160C22semidirect
24. #22576139C22semidirect
29. C4×Dic542C22semidirect
32. #22576092C22semidirect
33. #22576085C22semidirect
34. #22576085C22semidirect
36. #22576082C22semidirect
37. C22×C8×C271C22semidirect
39. C4⋊C8×C271C22semidirect
40. C22⋊C8×C271C22semidirect
41. C4×C8×C271C22semidirect
63. #16864643D4direct (N × H)
92. Dic542C2×D4semidirect
108. D4×C271SD16semidirect
117. C8×C271C2×D4semidirect
120. C22×C271C2×SD16semidirect
131. C4×C271C2×SD16semidirect
144. C271D4×SD16semidirect
156. C8×D4D271semidirect
166. C22×C8C2×D271semidirect
180. C4⋊C8C2×D271semidirect
185. C22⋊C8C2×D271semidirect
186. C4×C8C2×D271semidirect
247. D4#16864643direct (N × H)
262. C8D4×D271semidirect
272. C22C2×#16864643semidirect
280. C4C2×#16864643semidirect

Maximal subgroups

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

subgroupindex
C22×#168646432
D4×Dic5422
D4×D10842
#317742702
#317741262
#317741032
#317740982
#317740502
#317740382
C4×#168646432
C8×D4×C2712
D4×SD16271