Finite Group Databasebeta

← D4×#16864389 #46822581

Subgroups — D4×#16864389 #46822581

Order 34432.

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×#16864389C212yesmaximal, complemented
2D4×D1076C211yesmaximal, characteristic
3#31770979C211yesmaximal, characteristic, complemented
4#31770836C212yesmaximal, complemented
5#31770807C211yesmaximal, characteristic, complemented
6#31770805C211yesmaximal, characteristic, complemented
7#31770755C212yesmaximal, complemented
8#31770744C212yesmaximal, complemented
9C4×#16864389C211yesmaximal, characteristic, complemented
10D4×#14524130C211yesmaximal, characteristic, complemented
11D4×Q8×C269C211yesmaximal, characteristic, complemented
12C22×D1076C2212yes—
13C2×D4×D269—21no—
14C2×#16864389—28no—
15C2×#16864389C2212yes—
16C2×#16864389C2211yescharacteristic
17C2×#16864389C2212yescomplemented
18C2×#16864389C2212yescomplemented
19C2×#16864389C2212yescomplemented
20C22×#14524130C2212yescomplemented
21#22575450C2211yescharacteristic
22#22575428—21no—
23#22575427C2212yes—
24#22575423—22no—
25C4×D1076C2211yescharacteristic
26#22575384C2211yescharacteristic, complemented
27C4⋊#14524130C2211yescharacteristic, complemented
28#22575377C2211yescharacteristic, complemented
29#22575377C2212yescomplemented
30#22575373C2211yescharacteristic
31#22575373C2212yescomplemented
32C22⋊#14524130C2212yescomplemented
33C4×#14524130C2211yescharacteristic, complemented
34C22×Q8×C269C2212yescomplemented
35C4⋊Q8×C269—21no—
36C4⋊Q8×C269C2211yescharacteristic, complemented
37C22⋊Q8×C269—22no—
38C22⋊Q8×C269C2212yescomplemented
39C4×Q8×C269C2211yescharacteristic, complemented
40C4×D4×C269—21no—
41C4×D4×C269C2211yescharacteristic
42C23×D269—22no—
43C2×D1076—24no—
44C2×D1076—22no—
45C2×D1076—21no—
46C2×D1076—21no—
47C2×D1076—42no—
48C2×D1076C2312yes—
49C2×D1076C2311yescharacteristic
50C2×C22⋊D269—22no—
51D4×D269—44no—
52C2×C4×D269—21no—
53#16864389—24no—
54#16864389—24no—
55#16864389—24no—
56#16864389—42no—
57#16864389D414yesdirect product factor
58C2×#14524130—24no—
59C2×#14524130C2311yescharacteristic
60C2×#14524130C2312yes—
61C2×#14524130C2311yescharacteristic
62C4⋊Dic269C2311yescharacteristic
63#16864379—21no—
64#16864379—22no—
65C2×Q8×C269—22no—
66C2×Q8×C269—24no—
67C2×Q8×C269—42no—
68C2×Q8×C269C2312yes—
69C2×Q8×C269C2311yescharacteristic
70C2×D4×C269D411yescharacteristic
71C22×C4×C269—22no—
72C22×C4×C269C2312yes—
73C4⋊C4×C269—21no—
74C4⋊C4×C269—21no—
75C4⋊C4×C269—21no—
76C4⋊C4×C269—22no—
77C4⋊C4×C269D411yescharacteristic
78C4⋊C4×C269C2311yescharacteristic
79C22⋊C4×C269—22no—
80C22⋊C4×C269D412yes—
81C42×C269—21no—
82C42×C269C2311yescharacteristic
83C22×D269—21no—
84C22×D269—22no—
85C22×D269—44no—
86C22×D269—44no—
87C22×D269—82no—
88D1076—22no—
89D1076—21no—
90D1076—22no—
91D1076—44no—
92D1076—42no—
93D1076—81no—
94D1076C2×D412yes—
95C22⋊D269—44no—
96C2×Dic269—21no—
97C4×D269—42no—
98#14524130—22no—
99#14524130—21no—
100#14524130C2×D412yescomplemented
101C23×C269C2×D412yes—
102Q8×C269—21no—
103Q8×C269—22no—
104Q8×C269—22no—
105Q8×C269—44no—
106Q8×C269C2×D412yescomplemented
107D4×C269SD1614yescomplemented
108C2×C4×C269—21no—
109C2×C4×C269—22no—
110C2×C4×C269—22no—
111C2×C4×C269—21no—
112C2×C4×C269—22no—
113C2×C4×C269—44no—
114C2×C4×C269C2×D411yescharacteristic
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D411yescharacteristic
117C2×C4×C269C2411yescharacteristic
118C22×C269—22no—
119C22×C269—22no—
120C22×C269C2×SD1614yescomplemented
121C22×C269C22×D411yescharacteristic
122C2×D269—22no—
123C2×D269—41no—
124C2×D269—41no—
125C2×D269—42no—
126C2×D269—42no—
127C2×D269—42no—
128C2×D269—88no—
129C4×C269—22no—
130C4×C269—22no—
131C4×C269—21no—
132C4×C269—42no—
133C4×C269—41no—
134C4×C269C2×SD1612yescomplemented
135C4×C269C22×D411yescharacteristic
136C4×C269H(5,𝔽2)11yescharacteristic
137Dic269—41no—
138C2×C269—24no—
139C2×C269D4211yescharacteristic
140C2×C269C22×SD1611yescharacteristic
141C2×C269#57711yescharacteristic
142D269—42no—
143D269—82no—
144C269D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—2691nomaximal, Sylow
146C22×SD16—2692no—
147D4×Q8D26911yescharacteristic, complemented
148D42—2691no—
149#492—2691no—
150#465—2692no—
151#461—2691no—
152#460—2691no—
153#450—2692no—
154#448—2692no—
155C4×SD16—2691no—
156C8×D4—2691no—
157C22×Q8C2×D26912yescomplemented
158C22×D4—2692no—
159C22×D4—5381no—
160C2×SD16—2691no—
161C2×SD16—2692no—
162C2×SD16—2692no—
163C2×SD16—2692no—
164C2×SD16—2692no—
165C2×SD16—5388no—
166C22×C8—2692no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D26911yescharacteristic, complemented
169#127—2691no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D26912yescomplemented
172#121—2692no—
173#121—5381no—
174#120—5382no—
175C4×Q8C2×D26911yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—2691no—
178C4×D4C2×D26911yescharacteristic
179#106—2691no—
180C4⋊C8—2691no—
181#103—2691no—
182#103—2692no—
183#102—2691no—
184#102—2692no—
185C22⋊C8—2692no—
186C4×C8—2691no—
187C24—5382no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D26912yes—
192C2×Q8C22×D26911yescharacteristic
193C2×D4—2691no—
194C2×D4—2692no—
195C2×D4—5384no—
196C2×D4—5382no—
197C2×D4—5382no—
198C2×D4—5381no—
199C2×D4—5381no—
200C2×D4—10762no—
201C2×D4—10764no—
202C2×D4C22⋊D26911yescharacteristic
203C22×C4—22no—
204C22×C4—5381no—
205C22×C4C22×D26912yes—
206SD16—2694no—
207SD16—5384no—
208SD16—5384no—
209SD16—5384no—
210SD16—10762no—
211C2×C8—2691no—
212C2×C8—2691no—
213C2×C8—2692no—
214C2×C8—5384no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—2691no—
220C4⋊C4C22⋊D26911yescharacteristic
221C4⋊C4C22×D26911yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—5381no—
224C22⋊C4—5382no—
225C22⋊C4C22⋊D26912yes—
226C42—21no—
227C42C22×D26911yescharacteristic
228C23—5381no—
229C23—5382no—
230C23—10764no—
231C23—10764no—
232C23—21522no—
233C23C2×C22⋊D26912yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D26912yescomplemented
239D4—2692no—
240D4—5382no—
241D4—5381no—
242D4—5382no—
243D4—10764no—
244D4—10764no—
245D4—10762no—
246D4—21521no—
247D4#1686438914yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—5381no—
255C2×C4—10762no—
256C2×C4C2×C22⋊D26911yescharacteristic
257C2×C4C2×C22⋊D26912yes—
258C2×C4C2×C22⋊D26911yescharacteristic
259C2×C4C23×D26911yescharacteristic
260C8—2692no—
261C8—5382no—
262C8—5381no—
263C22—22no—
264C22—22no—
265C22—5382no—
266C22—10761no—
267C22—10761no—
268C22—10762no—
269C22—10762no—
270C22—10762no—
271C22—21528no—
272C22C2×#1686438914yescomplemented
273C22C22×C22⋊D26911yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—10761no—
280C4C2×#1686438912yescomplemented
281C4C2×D4×D26911yescharacteristic
282C4#2257549811yescharacteristic
283C2—24no—
284C2—10762no—
285C2—21522no—
286C2D4×C22⋊D26911yescharacteristic
287C2C22×#1686438911yescharacteristic
288C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingD4×Q8×C269C2
socleC22×C269C22×D4
prime radicalD4×D1076C2
hypercenterD4×Q8D269
hypocenterC269D4×SD16
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#16864389C2semidirect
3. #31770979C2semidirect
4. #31770836C2semidirect
5. #31770807C2semidirect
6. #31770805C2semidirect
7. #31770755C2semidirect
8. #31770744C2semidirect
9. C4×#16864389C2semidirect
10. D4×#14524130C2semidirect
11. D4×Q8×C269C2semidirect
17. C2×#16864389C22semidirect
18. C2×#16864389C22semidirect
19. C2×#16864389C22semidirect
20. C22×#14524130C22semidirect
26. #22575384C22semidirect
27. C4⋊#14524130C22semidirect
28. #22575377C22semidirect
29. #22575377C22semidirect
31. #22575373C22semidirect
32. C22⋊#14524130C22semidirect
33. C4×#14524130C22semidirect
34. C22×Q8×C269C22semidirect
36. C4⋊Q8×C269C22semidirect
38. C22⋊Q8×C269C22semidirect
39. C4×Q8×C269C22semidirect
57. #16864389D4direct (N × H)
100. #14524130C2×D4semidirect
106. Q8×C269C2×D4semidirect
107. D4×C269SD16semidirect
120. C22×C269C2×SD16semidirect
134. C4×C269C2×SD16semidirect
144. C269D4×SD16semidirect
147. D4×Q8D269semidirect
157. C22×Q8C2×D269semidirect
168. C4⋊Q8C2×D269semidirect
171. C22⋊Q8C2×D269semidirect
175. C4×Q8C2×D269semidirect
238. Q8D4×D269semidirect
247. D4#16864389direct (N × H)
272. C22C2×#16864389semidirect
280. C4C2×#16864389semidirect

Maximal subgroups

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

subgroupindex
C22×#168643892
D4×D10762
#317709792
#317708362
#317708072
#317708052
#317707552
#317707442
C4×#168643892
D4×#145241302
D4×Q8×C2692
D4×SD16269