Finite Group Databasebeta

← D4×#14578941 #34111832

Subgroups — D4×#14578941 #34111832

Order 20096.

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×#14578941C212yesmaximal, complemented
2D4×D628C211yesmaximal, characteristic
3#23757647C211yesmaximal, characteristic, complemented
4#23757504C212yesmaximal, complemented
5#23757475C211yesmaximal, characteristic, complemented
6#23757473C211yesmaximal, characteristic, complemented
7#23757423C212yesmaximal, complemented
8#23757412C212yesmaximal, complemented
9C4×#14578941C211yesmaximal, characteristic, complemented
10D4×#11929434C211yesmaximal, characteristic, complemented
11D4×Q8×C157C211yesmaximal, characteristic, complemented
12C22×D628C2212yes—
13C2×D4×D157—21no—
14C2×#14578941—28no—
15C2×#14578941C2212yes—
16C2×#14578941C2211yescharacteristic
17C2×#14578941C2212yescomplemented
18C2×#14578941C2212yescomplemented
19C2×#14578941C2212yescomplemented
20C22×#11929434C2212yescomplemented
21#19097768C2211yescharacteristic
22#19097746—21no—
23#19097745C2212yes—
24#19097741—22no—
25C4×D628C2211yescharacteristic
26#19097702C2211yescharacteristic, complemented
27C4⋊#11929434C2211yescharacteristic, complemented
28#19097695C2211yescharacteristic, complemented
29#19097695C2212yescomplemented
30#19097691C2211yescharacteristic
31#19097691C2212yescomplemented
32C22⋊#11929434C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
34C22×Q8×C157C2212yescomplemented
35C4⋊Q8×C157—21no—
36C4⋊Q8×C157C2211yescharacteristic, complemented
37C22⋊Q8×C157—22no—
38C22⋊Q8×C157C2212yescomplemented
39C4×Q8×C157C2211yescharacteristic, complemented
40C4×D4×C157—21no—
41C4×D4×C157C2211yescharacteristic
42C23×D157—22no—
43C2×D628—24no—
44C2×D628—22no—
45C2×D628—21no—
46C2×D628—21no—
47C2×D628—42no—
48C2×D628C2312yes—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—22no—
51D4×D157—44no—
52C2×C4×D157—21no—
53#14578941—24no—
54#14578941—24no—
55#14578941—24no—
56#14578941—42no—
57#14578941D414yesdirect product factor
58C2×#11929434—24no—
59C2×#11929434C2311yescharacteristic
60C2×#11929434C2312yes—
61C2×#11929434C2311yescharacteristic
62C4⋊Dic157C2311yescharacteristic
63#14578931—21no—
64#14578931—22no—
65C2×Q8×C157—22no—
66C2×Q8×C157—24no—
67C2×Q8×C157—42no—
68C2×Q8×C157C2312yes—
69C2×Q8×C157C2311yescharacteristic
70C2×D4×C157D411yescharacteristic
71C22×C4×C157—22no—
72C22×C4×C157C2312yes—
73C4⋊C4×C157—21no—
74C4⋊C4×C157—21no—
75C4⋊C4×C157—21no—
76C4⋊C4×C157—22no—
77C4⋊C4×C157D411yescharacteristic
78C4⋊C4×C157C2311yescharacteristic
79C22⋊C4×C157—22no—
80C22⋊C4×C157D412yes—
81C42×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—44no—
86C22×D157—44no—
87C22×D157—82no—
88D628—22no—
89D628—21no—
90D628—22no—
91D628—44no—
92D628—42no—
93D628—81no—
94D628C2×D412yes—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C4×D157—42no—
98#11929434—22no—
99#11929434—21no—
100#11929434C2×D412yescomplemented
101C23×C157C2×D412yes—
102Q8×C157—21no—
103Q8×C157—22no—
104Q8×C157—22no—
105Q8×C157—44no—
106Q8×C157C2×D412yescomplemented
107D4×C157SD1614yescomplemented
108C2×C4×C157—21no—
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—21no—
112C2×C4×C157—22no—
113C2×C4×C157—44no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2411yescharacteristic
118C22×C157—22no—
119C22×C157—22no—
120C22×C157C2×SD1614yescomplemented
121C22×C157C22×D411yescharacteristic
122C2×D157—22no—
123C2×D157—41no—
124C2×D157—41no—
125C2×D157—42no—
126C2×D157—42no—
127C2×D157—42no—
128C2×D157—88no—
129C4×C157—22no—
130C4×C157—22no—
131C4×C157—21no—
132C4×C157—42no—
133C4×C157—41no—
134C4×C157C2×SD1612yescomplemented
135C4×C157C22×D411yescharacteristic
136C4×C157H(5,𝔽2)11yescharacteristic
137Dic157—41no—
138C2×C157—24no—
139C2×C157D4211yescharacteristic
140C2×C157C22×SD1611yescharacteristic
141C2×C157#57711yescharacteristic
142D157—42no—
143D157—82no—
144C157D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1571nomaximal, Sylow
146C22×SD16—1572no—
147D4×Q8D15711yescharacteristic, complemented
148D42—1571no—
149#492—1571no—
150#465—1572no—
151#461—1571no—
152#460—1571no—
153#450—1572no—
154#448—1572no—
155C4×SD16—1571no—
156C8×D4—1571no—
157C22×Q8C2×D15712yescomplemented
158C22×D4—3141no—
159C22×D4—1572no—
160C2×SD16—3148no—
161C2×SD16—1572no—
162C2×SD16—1572no—
163C2×SD16—1572no—
164C2×SD16—1572no—
165C2×SD16—1571no—
166C22×C8—1572no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D15711yescharacteristic, complemented
169#127—1571no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D15712yescomplemented
172#121—3141no—
173#121—1572no—
174#120—3142no—
175C4×Q8C2×D15711yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1571no—
178C4×D4C2×D15711yescharacteristic
179#106—1571no—
180C4⋊C8—1571no—
181#103—1571no—
182#103—1572no—
183#102—1571no—
184#102—1572no—
185C22⋊C8—1572no—
186C4×C8—1571no—
187C24—3142no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D15712yes—
192C2×Q8C22×D15711yescharacteristic
193C2×D4—3144no—
194C2×D4—3142no—
195C2×D4—3142no—
196C2×D4—3141no—
197C2×D4—3141no—
198C2×D4—6282no—
199C2×D4—6284no—
200C2×D4—1572no—
201C2×D4—1571no—
202C2×D4C22⋊D15711yescharacteristic
203C22×C4—22no—
204C22×C4—3141no—
205C22×C4C22×D15712yes—
206SD16—3144no—
207SD16—3144no—
208SD16—3144no—
209SD16—6282no—
210SD16—1574no—
211C2×C8—3144no—
212C2×C8—1571no—
213C2×C8—1572no—
214C2×C8—1571no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1571no—
220C4⋊C4C22⋊D15711yescharacteristic
221C4⋊C4C22×D15711yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—3141no—
224C22⋊C4—3142no—
225C22⋊C4C22⋊D15712yes—
226C42—21no—
227C42C22×D15711yescharacteristic
228C23—3141no—
229C23—3142no—
230C23—6284no—
231C23—6284no—
232C23—12562no—
233C23C2×C22⋊D15712yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D15712yescomplemented
239D4—3142no—
240D4—3141no—
241D4—3142no—
242D4—6284no—
243D4—6284no—
244D4—6282no—
245D4—1572no—
246D4—12561no—
247D4#1457894114yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—3141no—
255C2×C4—6282no—
256C2×C4C2×C22⋊D15711yescharacteristic
257C2×C4C2×C22⋊D15712yes—
258C2×C4C2×C22⋊D15711yescharacteristic
259C2×C4C23×D15711yescharacteristic
260C8—3142no—
261C8—3141no—
262C8—1572no—
263C22—22no—
264C22—22no—
265C22—3142no—
266C22—6281no—
267C22—6281no—
268C22—6282no—
269C22—6282no—
270C22—6282no—
271C22—12568no—
272C22C2×#1457894114yescomplemented
273C22C22×C22⋊D15711yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—6281no—
280C4C2×#1457894112yescomplemented
281C4C2×D4×D15711yescharacteristic
282C4#1909781611yescharacteristic
283C2—24no—
284C2—6282no—
285C2—12562no—
286C2D4×C22⋊D15711yescharacteristic
287C2C22×#1457894111yescharacteristic
288C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingD4×Q8×C157C2
socleC22×C157C22×D4
prime radicalD4×D628C2
hypercenterD4×Q8D157
hypocenterC157D4×SD16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14578941C2semidirect
3. #23757647C2semidirect
4. #23757504C2semidirect
5. #23757475C2semidirect
6. #23757473C2semidirect
7. #23757423C2semidirect
8. #23757412C2semidirect
9. C4×#14578941C2semidirect
10. D4×#11929434C2semidirect
11. D4×Q8×C157C2semidirect
17. C2×#14578941C22semidirect
18. C2×#14578941C22semidirect
19. C2×#14578941C22semidirect
20. C22×#11929434C22semidirect
26. #19097702C22semidirect
27. C4⋊#11929434C22semidirect
28. #19097695C22semidirect
29. #19097695C22semidirect
31. #19097691C22semidirect
32. C22⋊#11929434C22semidirect
33. C4×#11929434C22semidirect
34. C22×Q8×C157C22semidirect
36. C4⋊Q8×C157C22semidirect
38. C22⋊Q8×C157C22semidirect
39. C4×Q8×C157C22semidirect
57. #14578941D4direct (N × H)
100. #11929434C2×D4semidirect
106. Q8×C157C2×D4semidirect
107. D4×C157SD16semidirect
120. C22×C157C2×SD16semidirect
134. C4×C157C2×SD16semidirect
144. C157D4×SD16semidirect
147. D4×Q8D157semidirect
157. C22×Q8C2×D157semidirect
168. C4⋊Q8C2×D157semidirect
171. C22⋊Q8C2×D157semidirect
175. C4×Q8C2×D157semidirect
238. Q8D4×D157semidirect
247. D4#14578941direct (N × H)
272. C22C2×#14578941semidirect
280. C4C2×#14578941semidirect

Maximal subgroups

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

subgroupindex
C22×#145789412
D4×D6282
#237576472
#237575042
#237574752
#237574732
#237574232
#237574122
C4×#145789412
D4×#119294342
D4×Q8×C1572
D4×SD16157