Finite Group Databasebeta

← D4×#16864641 #46848060

Subgroups — D4×#16864641 #46848060

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×#16864641C212yesmaximal, complemented
2D4×D1084C211yesmaximal, characteristic
3#31774271C211yesmaximal, characteristic, complemented
4#31774128C212yesmaximal, complemented
5#31774099C211yesmaximal, characteristic, complemented
6#31774097C211yesmaximal, characteristic, complemented
7#31774047C212yesmaximal, complemented
8#31774036C212yesmaximal, complemented
9C4×#16864641C211yesmaximal, characteristic, complemented
10D4×C271⋊C8C211yesmaximal, characteristic, complemented
11D4×Q8×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2212yes—
13C2×D4×D271—21no—
14C2×#16864641—28no—
15C2×#16864641C2212yes—
16C2×#16864641C2211yescharacteristic
17C2×#16864641C2212yescomplemented
18C2×#16864641C2212yescomplemented
19C2×#16864641C2212yescomplemented
20C22×C271⋊C8C2212yescomplemented
21#22576157C2211yescharacteristic
22#22576135—21no—
23#22576134C2212yes—
24#22576130—22no—
25C4×D1084C2211yescharacteristic
26#22576091C2211yescharacteristic, complemented
27C4⋊(C271⋊C8)C2211yescharacteristic, complemented
28#22576084C2211yescharacteristic, complemented
29#22576084C2212yescomplemented
30#22576080C2211yescharacteristic
31#22576080C2212yescomplemented
32C22⋊(C271⋊C8)C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C22×Q8×C271C2212yescomplemented
35C4⋊Q8×C271—21no—
36C4⋊Q8×C271C2211yescharacteristic, complemented
37C22⋊Q8×C271—22no—
38C22⋊Q8×C271C2212yescomplemented
39C4×Q8×C271C2211yescharacteristic, complemented
40C4×D4×C271—21no—
41C4×D4×C271C2211yescharacteristic
42C23×D271—22no—
43C2×D1084—24no—
44C2×D1084—22no—
45C2×D1084—21no—
46C2×D1084—21no—
47C2×D1084—42no—
48C2×D1084C2312yes—
49C2×D1084C2311yescharacteristic
50C2×C22⋊D271—22no—
51D4×D271—44no—
52C2×C4×D271—21no—
53#16864641—24no—
54#16864641—24no—
55#16864641—24no—
56#16864641—42no—
57#16864641D414yesdirect product factor
58C2×C271⋊C8—24no—
59C2×C271⋊C8C2312yes—
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C4⋊Dic271C2311yescharacteristic
63(C2×D271)⋊C4—21no—
64(C2×D271)⋊C4—22no—
65C2×Q8×C271—22no—
66C2×Q8×C271—24no—
67C2×Q8×C271—42no—
68C2×Q8×C271C2312yes—
69C2×Q8×C271C2311yescharacteristic
70C2×D4×C271D411yescharacteristic
71C22×C4×C271—22no—
72C22×C4×C271C2312yes—
73C4⋊C4×C271—21no—
74C4⋊C4×C271—21no—
75C4⋊C4×C271—21no—
76C4⋊C4×C271—22no—
77C4⋊C4×C271D411yescharacteristic
78C4⋊C4×C271C2311yescharacteristic
79C22⋊C4×C271—22no—
80C22⋊C4×C271D412yes—
81C42×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88D1084—22no—
89D1084—21no—
90D1084—22no—
91D1084—44no—
92D1084—42no—
93D1084—81no—
94D1084C2×D412yes—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C4×D271—42no—
98C271⋊C8—22no—
99C271⋊C8—21no—
100C271⋊C8C2×D412yescomplemented
101C23×C271C2×D412yes—
102Q8×C271—21no—
103Q8×C271—22no—
104Q8×C271—22no—
105Q8×C271—44no—
106Q8×C271C2×D412yescomplemented
107D4×C271SD1614yescomplemented
108C2×C4×C271—21no—
109C2×C4×C271—22no—
110C2×C4×C271—22no—
111C2×C4×C271—21no—
112C2×C4×C271—22no—
113C2×C4×C271—44no—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2411yescharacteristic
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—22no—
131C4×C271—21no—
132C4×C271—42no—
133C4×C271—41no—
134C4×C271C2×SD1612yescomplemented
135C4×C271C22×D411yescharacteristic
136C4×C271H(5,𝔽2)11yescharacteristic
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×Q8D27111yescharacteristic, complemented
148D42—2711no—
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×Q8C2×D27112yescomplemented
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×C8—2712no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D27111yescharacteristic, complemented
169#127—2711no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D27112yescomplemented
172#121—2712no—
173#121—5421no—
174#120—5422no—
175C4×Q8C2×D27111yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—2711no—
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—5422no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D27112yes—
192C2×Q8C22×D27111yescharacteristic
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×D4C22⋊D27111yescharacteristic
203C22×C4—22no—
204C22×C4—5421no—
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—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—2711no—
220C4⋊C4C22⋊D27111yescharacteristic
221C4⋊C4C22×D27111yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—5421no—
224C22⋊C4—5422no—
225C22⋊C4C22⋊D27112yes—
226C42—21no—
227C42C22×D27111yescharacteristic
228C23—5421no—
229C23—5422no—
230C23—10844no—
231C23—10844no—
232C23—21682no—
233C23C2×C22⋊D27112yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D27112yescomplemented
239D4—2712no—
240D4—5422no—
241D4—5421no—
242D4—5422no—
243D4—10844no—
244D4—10844no—
245D4—10842no—
246D4—21681no—
247D4#1686464114yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—5421no—
255C2×C4—10842no—
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—5422no—
266C22—10841no—
267C22—10841no—
268C22—10842no—
269C22—10842no—
270C22—10842no—
271C22—21688no—
272C22C2×#1686464114yescomplemented
273C22C22×C22⋊D27111yescharacteristic
274C4—22no—
275C4—22no—
276C4—21no—
277C4—42no—
278C4—41no—
279C4—10841no—
280C4C2×#1686464112yescomplemented
281C4C2×D4×D27111yescharacteristic
282C4#2257620511yescharacteristic
283C2—24no—
284C2—10842no—
285C2—21682no—
286C2D4×C22⋊D27111yescharacteristic
287C2C22×#1686464111yescharacteristic
288C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingD4×Q8×C271C2
socleC22×C271C22×D4
prime radicalD4×D1084C2
hypercenterD4×Q8D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#16864641C2semidirect
3. #31774271C2semidirect
4. #31774128C2semidirect
5. #31774099C2semidirect
6. #31774097C2semidirect
7. #31774047C2semidirect
8. #31774036C2semidirect
9. C4×#16864641C2semidirect
10. D4×C271⋊C8C2semidirect
11. D4×Q8×C271C2semidirect
17. C2×#16864641C22semidirect
18. C2×#16864641C22semidirect
19. C2×#16864641C22semidirect
20. C22×C271⋊C8C22semidirect
26. #22576091C22semidirect
27. C4⋊(C271⋊C8)C22semidirect
28. #22576084C22semidirect
29. #22576084C22semidirect
31. #22576080C22semidirect
32. C22⋊(C271⋊C8)C22semidirect
33. C4×C271⋊C8C22semidirect
34. C22×Q8×C271C22semidirect
36. C4⋊Q8×C271C22semidirect
38. C22⋊Q8×C271C22semidirect
39. C4×Q8×C271C22semidirect
57. #16864641D4direct (N × H)
100. C271⋊C8C2×D4semidirect
106. Q8×C271C2×D4semidirect
107. D4×C271SD16semidirect
120. C22×C271C2×SD16semidirect
134. C4×C271C2×SD16semidirect
144. C271D4×SD16semidirect
147. D4×Q8D271semidirect
157. C22×Q8C2×D271semidirect
168. C4⋊Q8C2×D271semidirect
171. C22⋊Q8C2×D271semidirect
175. C4×Q8C2×D271semidirect
238. Q8D4×D271semidirect
247. D4#16864641direct (N × H)
272. C22C2×#16864641semidirect
280. C4C2×#16864641semidirect

Maximal subgroups

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

subgroupindex
C22×#168646412
D4×D10842
#317742712
#317741282
#317740992
#317740972
#317740472
#317740362
C4×#168646412
D4×C271⋊C82
D4×Q8×C2712
D4×SD16271