Finite Group Databasebeta

← D4×#16864642 #46848056

Subgroups — D4×#16864642 #46848056

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×#16864642C212yesmaximal, complemented
2D4×Dic542C211yesmaximal, characteristic, complemented
3#31774269C211yesmaximal, characteristic, complemented
4#31774124C212yesmaximal, complemented
5#31774102C211yesmaximal, characteristic, complemented
6#31774094C211yesmaximal, characteristic, complemented
7#31774048C212yesmaximal, complemented
8#31774035C212yesmaximal, complemented
9C4×#16864642C211yesmaximal, characteristic, complemented
10D4×C271⋊C8C211yesmaximal, characteristic, complemented
11D42×C271C211yesmaximal, characteristic
12C22×Dic542C2212yescomplemented
13C2×#16864642—28no—
14C2×#16864642C2212yes—
15C2×#16864642C2211yescharacteristic
16C2×#16864642C2212yescomplemented
17C2×#16864642C2212yescomplemented
18C2×#16864642C2212yescomplemented
19C22×C271⋊C8C2212yescomplemented
20#22576160C2211yescharacteristic, complemented
21#22576159—21no—
22#22576139C2212yescomplemented
23#22576137—22no—
24C4×Dic542C2211yescharacteristic, complemented
25D4×Dic271—21no—
26#22576091C2211yescharacteristic, complemented
27C4⋊(C271⋊C8)C2211yescharacteristic, complemented
28#22576083C2211yescharacteristic, complemented
29#22576083C2212yescomplemented
30#22576081C2211yescharacteristic
31#22576081C2212yescomplemented
32C22⋊(C271⋊C8)C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C22×D4×C271—21no—
35C22×D4×C271C2212yes—
36#127×C271C2211yescharacteristic
37#121×C271—21no—
38#121×C271C2212yes—
39#120×C271—22no—
40C4×D4×C271C2211yescharacteristic
41C4×D4×C271C2211yescharacteristic
42C2×Dic542—24no—
43C2×Dic542—22no—
44C2×Dic542—42no—
45C2×Dic542C2312yes—
46C2×Dic542C2311yescharacteristic
47C22×Dic271—22no—
48#16864642—24no—
49#16864642—24no—
50#16864642—24no—
51#16864642—42no—
52#16864642D414yesdirect product factor
53C2×C271⋊C8—24no—
54C2×C271⋊C8C2312yes—
55C2×C271⋊C8C2311yescharacteristic
56C2×C271⋊C8C2311yescharacteristic
57C4⋊Dic271—21no—
58C4⋊Dic271—21no—
59C4⋊Dic271C2311yescharacteristic
60Dic271⋊C4—21no—
61Dic271⋊C4—22no—
62C22⋊Dic271—22no—
63C4×Dic271—21no—
64C24×C271—22no—
65C2×D4×C271—24no—
66C2×D4×C271—22no—
67C2×D4×C271—22no—
68C2×D4×C271—21no—
69C2×D4×C271—21no—
70C2×D4×C271—42no—
71C2×D4×C271—44no—
72C2×D4×C271D411yescharacteristic
73C2×D4×C271C2312yes—
74C2×D4×C271C2311yescharacteristic
75C22×C4×C271—21no—
76C22×C4×C271C2312yes—
77C4⋊C4×C271D411yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C22⋊C4×C271—21no—
80C22⋊C4×C271—22no—
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83Dic542—21no—
84Dic542—22no—
85Dic542—22no—
86Dic542—44no—
87Dic542C2×D412yescomplemented
88C2×Dic271—21no—
89C2×Dic271—21no—
90C2×Dic271—22no—
91C2×Dic271—44no—
92C271⋊C8—22no—
93C271⋊C8—21no—
94C271⋊C8C2×D412yescomplemented
95C23×C271—21no—
96C23×C271—22no—
97C23×C271—44no—
98C23×C271—44no—
99C23×C271—82no—
100C23×C271C2×D412yes—
101D4×C271—22no—
102D4×C271—21no—
103D4×C271—22no—
104D4×C271—44no—
105D4×C271—44no—
106D4×C271—42no—
107D4×C271—81no—
108D4×C271SD1614yescomplemented
109D4×C271C2×D412yes—
110C2×C4×C271—22no—
111C2×C4×C271—22no—
112C2×C4×C271—21no—
113C2×C4×C271—42no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2411yescharacteristic
118C22×C271—22no—
119C22×C271—22no—
120C22×C271—22no—
121C22×C271—41no—
122C22×C271—41no—
123C22×C271—42no—
124C22×C271—42no—
125C22×C271—42no—
126C22×C271—88no—
127C22×C271C2×SD1614yescomplemented
128C22×C271C22×D411yescharacteristic
129C4×C271—22no—
130C4×C271—21no—
131C4×C271—41no—
132C4×C271C2×SD1612yescomplemented
133C4×C271C22×D411yescharacteristic
134C4×C271H(5,𝔽2)11yescharacteristic
135Dic271—22no—
136Dic271—42no—
137Dic271—41no—
138C2×C271—24no—
139C2×C271—42no—
140C2×C271—82no—
141C2×C271D4211yescharacteristic
142C2×C271C22×SD1611yescharacteristic
143C2×C271#57711yescharacteristic
144C271D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—2711nomaximal, Sylow
146C22×SD16—2712no—
147D4×Q8—2711no—
148D42D27111yescharacteristic
149#492—2711no—
150#465—2712no—
151#461—2711no—
152#460—2711no—
153#450—2712no—
154#448—2712no—
155C4×SD16—2711no—
156C8×D4—2711no—
157C22×Q8—2712no—
158C22×D4—21no—
159C22×D4C2×D27112yes—
160C2×SD16—2711no—
161C2×SD16—2712no—
162C2×SD16—2712no—
163C2×SD16—2712no—
164C2×SD16—2712no—
165C2×SD16—5428no—
166C22×C8—2712no—
167C4⋊Q8—2711no—
168C4⋊Q8—5421no—
169#127C2×D27111yescharacteristic
170C22⋊Q8—2712no—
171C22⋊Q8—5422no—
172#121—21no—
173#121C2×D27112yes—
174#120—22no—
175C4×Q8—2711no—
176C4×D4—5421no—
177C4×D4C2×D27111yescharacteristic
178C4×D4C2×D27111yescharacteristic
179#106—2711no—
180C4⋊C8—2711no—
181#103—2711no—
182#103—2712no—
183#102—2711no—
184#102—2712no—
185C22⋊C8—2712no—
186C4×C8—2711no—
187C24—22no—
188C2×Q8—2711no—
189C2×Q8—2712no—
190C2×Q8—5424no—
191C2×Q8—5422no—
192C2×Q8—10842no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D27111yescharacteristic
201C2×D4C22×D27112yes—
202C2×D4C22×D27111yescharacteristic
203C22×C4—21no—
204C22×C4—5422no—
205C22×C4C22×D27112yes—
206SD16—2714no—
207SD16—5424no—
208SD16—5424no—
209SD16—5424no—
210SD16—10842no—
211C2×C8—2711no—
212C2×C8—2711no—
213C2×C8—2712no—
214C2×C8—5424no—
215C4⋊C4—2711no—
216C4⋊C4—5421no—
217C4⋊C4—5421no—
218C4⋊C4—5422no—
219C4⋊C4—5421no—
220C4⋊C4C22⋊D27111yescharacteristic
221C4⋊C4C22×D27111yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—5422no—
225C22⋊C4C22⋊D27112yes—
226C42—5421no—
227C42C22×D27111yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D27112yes—
234Q8—2712no—
235Q8—5421no—
236Q8—5422no—
237Q8—5422no—
238Q8—10844no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1686464214yesdirect product factor
247D4D4×D27112yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—5421no—
253C2×C4—5421no—
254C2×C4—5422no—
255C2×C4—10844no—
256C2×C4C2×C22⋊D27111yescharacteristic
257C2×C4C2×C22⋊D27112yes—
258C2×C4C2×C22⋊D27111yescharacteristic
259C2×C4C23×D27111yescharacteristic
260C8—2712no—
261C8—5422no—
262C8—5421no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1686464214yescomplemented
273C22C22×C22⋊D27111yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—5422no—
278C4—10842no—
279C4—10841no—
280C4C2×#1686464212yescomplemented
281C4C2×D4×D27111yescharacteristic
282C4#2257620511yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D27111yescharacteristic
287C2C22×#1686464211yescharacteristic
288C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting, prime radicalD42×C271C2
socleC22×C271C22×D4
hypercenterD42D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#16864642C2semidirect
2. D4×Dic542C2semidirect
3. #31774269C2semidirect
4. #31774124C2semidirect
5. #31774102C2semidirect
6. #31774094C2semidirect
7. #31774048C2semidirect
8. #31774035C2semidirect
9. C4×#16864642C2semidirect
10. D4×C271⋊C8C2semidirect
12. C22×Dic542C22semidirect
16. C2×#16864642C22semidirect
17. C2×#16864642C22semidirect
18. C2×#16864642C22semidirect
19. C22×C271⋊C8C22semidirect
20. #22576160C22semidirect
22. #22576139C22semidirect
24. C4×Dic542C22semidirect
26. #22576091C22semidirect
27. C4⋊(C271⋊C8)C22semidirect
28. #22576083C22semidirect
29. #22576083C22semidirect
31. #22576081C22semidirect
32. C22⋊(C271⋊C8)C22semidirect
33. C4×C271⋊C8C22semidirect
52. #16864642D4direct (N × H)
87. Dic542C2×D4semidirect
94. C271⋊C8C2×D4semidirect
108. D4×C271SD16semidirect
127. C22×C271C2×SD16semidirect
132. C4×C271C2×SD16semidirect
144. C271D4×SD16semidirect
246. D4#16864642direct (N × H)
272. C22C2×#16864642semidirect
280. C4C2×#16864642semidirect

Maximal subgroups

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

subgroupindex
C22×#168646422
D4×Dic5422
#317742692
#317741242
#317741022
#317740942
#317740482
#317740352
C4×#168646422
D4×C271⋊C82
D42×C2712
D4×SD16271