Finite Group Databasebeta

← D4×#13083561 #26031948

Subgroups — D4×#13083561 #26031948

Order 11392.

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×#13083561C212yesmaximal, complemented
2D4×D356C211yesmaximal, characteristic
3#19151972C211yesmaximal, characteristic, complemented
4#19151829C212yesmaximal, complemented
5#19151800C211yesmaximal, characteristic, complemented
6#19151798C211yesmaximal, characteristic, complemented
7#19151748C212yesmaximal, complemented
8#19151737C212yesmaximal, complemented
9C4×#13083561C211yesmaximal, characteristic, complemented
10D4×#10623494C211yesmaximal, characteristic, complemented
11D4×Q8×C89C211yesmaximal, characteristic, complemented
12C22×D356C2212yes—
13C2×D4×D89—21no—
14C2×#13083561—28no—
15C2×#13083561C2212yes—
16C2×#13083561C2211yescharacteristic
17C2×#13083561C2212yescomplemented
18C2×#13083561C2212yescomplemented
19C2×#13083561C2212yescomplemented
20C22×#10623494C2212yescomplemented
21#15668009C2211yescharacteristic
22#15667987—21no—
23#15667986C2212yes—
24#15667982—22no—
25C4×D356C2211yescharacteristic
26#15667943C2211yescharacteristic, complemented
27C4⋊#10623494C2211yescharacteristic, complemented
28#15667936C2211yescharacteristic, complemented
29#15667936C2212yescomplemented
30#15667932C2211yescharacteristic
31#15667932C2212yescomplemented
32C22⋊#10623494C2212yescomplemented
33C4×#10623494C2211yescharacteristic, complemented
34C22×Q8×C89C2212yescomplemented
35C4⋊Q8×C89—21no—
36C4⋊Q8×C89C2211yescharacteristic, complemented
37C22⋊Q8×C89—22no—
38C22⋊Q8×C89C2212yescomplemented
39C4×Q8×C89C2211yescharacteristic, complemented
40C4×D4×C89—21no—
41C4×D4×C89C2211yescharacteristic
42C23×D89—22no—
43C2×D356—24no—
44C2×D356—22no—
45C2×D356—21no—
46C2×D356—21no—
47C2×D356—42no—
48C2×D356C2312yes—
49C2×D356C2311yescharacteristic
50C2×C22⋊D89—22no—
51D4×D89—44no—
52C2×C4×D89—21no—
53#13083561—24no—
54#13083561—24no—
55#13083561—24no—
56#13083561—42no—
57#13083561D414yesdirect product factor
58C2×#10623494—24no—
59C2×#10623494C2312yes—
60C2×#10623494C2311yescharacteristic
61C2×#10623494C2311yescharacteristic
62C4⋊Dic89C2311yescharacteristic
63#13083551—21no—
64#13083551—22no—
65C2×Q8×C89—22no—
66C2×Q8×C89—24no—
67C2×Q8×C89—42no—
68C2×Q8×C89C2312yes—
69C2×Q8×C89C2311yescharacteristic
70C2×D4×C89D411yescharacteristic
71C22×C4×C89—22no—
72C22×C4×C89C2312yes—
73C4⋊C4×C89—21no—
74C4⋊C4×C89—21no—
75C4⋊C4×C89—21no—
76C4⋊C4×C89—22no—
77C4⋊C4×C89D411yescharacteristic
78C4⋊C4×C89C2311yescharacteristic
79C22⋊C4×C89—22no—
80C22⋊C4×C89D412yes—
81C42×C89—21no—
82C42×C89C2311yescharacteristic
83C23×C89C2×D412yes—
84C22×D89—21no—
85C22×D89—22no—
86C22×D89—44no—
87C22×D89—44no—
88C22×D89—82no—
89Q8×C89—21no—
90Q8×C89—22no—
91Q8×C89—22no—
92Q8×C89—44no—
93Q8×C89C2×D412yescomplemented
94D4×C89SD1614yescomplemented
95C2×C4×C89—21no—
96C2×C4×C89—22no—
97C2×C4×C89—22no—
98C2×C4×C89—21no—
99C2×C4×C89—22no—
100C2×C4×C89—44no—
101C2×C4×C89C2×D411yescharacteristic
102C2×C4×C89C2×D412yes—
103C2×C4×C89C2×D411yescharacteristic
104C2×C4×C89C2411yescharacteristic
105C22⋊D89—44no—
106C2×Dic89—21no—
107D356—22no—
108D356—21no—
109D356—22no—
110D356—44no—
111D356—42no—
112D356—81no—
113D356C2×D412yes—
114C4×D89—42no—
115#10623494—22no—
116#10623494—21no—
117#10623494C2×D412yescomplemented
118C22×C89—22no—
119C22×C89—22no—
120C22×C89C2×SD1614yescomplemented
121C22×C89C22×D411yescharacteristic
122C2×D89—22no—
123C2×D89—41no—
124C2×D89—41no—
125C2×D89—42no—
126C2×D89—42no—
127C2×D89—42no—
128C2×D89—88no—
129C4×C89—22no—
130C4×C89—22no—
131C4×C89—21no—
132C4×C89—42no—
133C4×C89—41no—
134C4×C89C2×SD1612yescomplemented
135C4×C89C22×D411yescharacteristic
136C4×C89H(5,𝔽2)11yescharacteristic
137Dic89—41no—
138C2×C89—24no—
139C2×C89D4211yescharacteristic
140C2×C89C22×SD1611yescharacteristic
141C2×C89#57711yescharacteristic
142D89—42no—
143D89—82no—
144D4×SD16—891nomaximal, Sylow
145C89D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—892no—
147D4×Q8D8911yescharacteristic, complemented
148D42—891no—
149#492—891no—
150#465—892no—
151#461—891no—
152#460—891no—
153#450—892no—
154#448—892no—
155C4×SD16—891no—
156C8×D4—891no—
157C22×Q8C2×D8912yescomplemented
158C22×D4—892no—
159C22×D4—1781no—
160C2×SD16—891no—
161C2×SD16—892no—
162C2×SD16—892no—
163C2×SD16—892no—
164C2×SD16—892no—
165C2×SD16—1788no—
166C22×C8—892no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D8911yescharacteristic, complemented
169#127—891no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D8912yescomplemented
172#121—892no—
173#121—1781no—
174#120—1782no—
175C4×Q8C2×D8911yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—891no—
178C4×D4C2×D8911yescharacteristic
179#106—891no—
180C4⋊C8—891no—
181#103—891no—
182#103—892no—
183#102—891no—
184#102—892no—
185C22⋊C8—892no—
186C4×C8—891no—
187C24—1782no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D8912yes—
192C2×Q8C22×D8911yescharacteristic
193C2×D4—891no—
194C2×D4—892no—
195C2×D4—3562no—
196C2×D4—3564no—
197C2×D4—1784no—
198C2×D4—1782no—
199C2×D4—1782no—
200C2×D4—1781no—
201C2×D4—1781no—
202C2×D4C22⋊D8911yescharacteristic
203C22×C4—22no—
204C22×C4—1781no—
205C22×C4C22×D8912yes—
206SD16—894no—
207SD16—3562no—
208SD16—1784no—
209SD16—1784no—
210SD16—1784no—
211C2×C8—891no—
212C2×C8—891no—
213C2×C8—892no—
214C2×C8—1784no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—891no—
220C4⋊C4C22⋊D8911yescharacteristic
221C4⋊C4C22×D8911yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—1781no—
224C22⋊C4—1782no—
225C22⋊C4C22⋊D8912yes—
226C42—21no—
227C42C22×D8911yescharacteristic
228C23—3564no—
229C23—3564no—
230C23—1782no—
231C23—1781no—
232C23—7122no—
233C23C2×C22⋊D8912yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D8912yescomplemented
239D4—892no—
240D4—3564no—
241D4—3564no—
242D4—3562no—
243D4—1782no—
244D4—1782no—
245D4—1781no—
246D4—7121no—
247D4#1308356114yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—3562no—
255C2×C4—1781no—
256C2×C4C2×C22⋊D8911yescharacteristic
257C2×C4C2×C22⋊D8912yes—
258C2×C4C2×C22⋊D8911yescharacteristic
259C2×C4C23×D8911yescharacteristic
260C8—892no—
261C8—1782no—
262C8—1781no—
263C22—22no—
264C22—22no—
265C22—3561no—
266C22—3561no—
267C22—3562no—
268C22—3562no—
269C22—3562no—
270C22—1782no—
271C22—7128no—
272C22C2×#1308356114yescomplemented
273C22C22×C22⋊D8911yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—3561no—
280C4C2×#1308356112yescomplemented
281C4C2×D4×D8911yescharacteristic
282C4#1566805711yescharacteristic
283C2—24no—
284C2—3562no—
285C2—7122no—
286C2D4×C22⋊D8911yescharacteristic
287C2C22×#1308356111yescharacteristic
288C2#1915240611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
FittingD4×Q8×C89C2
socleC22×C89C22×D4
prime radicalD4×D356C2
hypercenterD4×Q8D89
hypocenterC89D4×SD16
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13083561C2semidirect
3. #19151972C2semidirect
4. #19151829C2semidirect
5. #19151800C2semidirect
6. #19151798C2semidirect
7. #19151748C2semidirect
8. #19151737C2semidirect
9. C4×#13083561C2semidirect
10. D4×#10623494C2semidirect
11. D4×Q8×C89C2semidirect
17. C2×#13083561C22semidirect
18. C2×#13083561C22semidirect
19. C2×#13083561C22semidirect
20. C22×#10623494C22semidirect
26. #15667943C22semidirect
27. C4⋊#10623494C22semidirect
28. #15667936C22semidirect
29. #15667936C22semidirect
31. #15667932C22semidirect
32. C22⋊#10623494C22semidirect
33. C4×#10623494C22semidirect
34. C22×Q8×C89C22semidirect
36. C4⋊Q8×C89C22semidirect
38. C22⋊Q8×C89C22semidirect
39. C4×Q8×C89C22semidirect
57. #13083561D4direct (N × H)
93. Q8×C89C2×D4semidirect
94. D4×C89SD16semidirect
117. #10623494C2×D4semidirect
120. C22×C89C2×SD16semidirect
134. C4×C89C2×SD16semidirect
145. C89D4×SD16semidirect
147. D4×Q8D89semidirect
157. C22×Q8C2×D89semidirect
168. C4⋊Q8C2×D89semidirect
171. C22⋊Q8C2×D89semidirect
175. C4×Q8C2×D89semidirect
238. Q8D4×D89semidirect
247. D4#13083561direct (N × H)
272. C22C2×#13083561semidirect
280. C4C2×#13083561semidirect

Maximal subgroups

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

subgroupindex
C22×#130835612
D4×D3562
#191519722
#191518292
#191518002
#191517982
#191517482
#191517372
C4×#130835612
D4×#106234942
D4×Q8×C892
D4×SD1689