Finite Group Databasebeta

← #22049

Subgroups — #22049

Order 256.

606 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#450C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#390C211yesmaximal, characteristic
4#1973C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1925C212yesmaximal, complemented
7#1880C212yesmaximal, complemented
8#1872C211yesmaximal, characteristic, complemented
9#1852C211yesmaximal, characteristic, complemented
10#1786C211yesmaximal, characteristic
11#1749C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic, complemented
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4—21no—
18C22×C22⋊C4—21no—
19C22×C42C2211yescharacteristic
20#450C418yescomplemented
21C2×#107C2212yescomplemented
22C2×#103C2212yescomplemented
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#97C2212yescomplemented
29#391C2211yescharacteristic, complemented
30#390—21no—
31#390—21no—
32#390C2212yes—
33#390C2212yes—
34#389C2211yescharacteristic, complemented
35#387—21no—
36#385—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic, complemented
40C4×C22⋊C4—21no—
41#340C2212yescomplemented
42#340C2212yescomplemented
43#336C2212yescomplemented
44C25—21no—
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4C2×C414yes—
51C22×D4C2312yes—
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4C2311yescharacteristic
56C2×SD16—42no—
57C2×SD16—42no—
58C2×SD16C2×C418yescomplemented
59C22×C8C2312yes—
60C22×C8C2312yes—
61C22⋊Q8—22no—
62C22⋊Q8C2×C414yes—
63C4×Q8—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—44no—
80C2×C22⋊C4—44no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—42no—
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#107—42no—
89#103—42no—
90#102—42no—
91#102—42no—
92#102C2×C418yescomplemented
93C22⋊C8—22no—
94C22⋊C8C2×C414yes—
95#97—42no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24—24no—
102C24—22no—
103C24—42no—
104C24—42no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24—42no—
109C24C2×D411yescharacteristic
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8C22×C412yes—
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—48no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—48no—
124C2×D4C22×C414yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C22×C412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4#4112yes—
148C22×C4C2411yescharacteristic
149SD16—48no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8C22×C414yes—
156C4⋊C4—24no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4C22×C412yes—
166C22⋊C4—24no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C42—22no—
173C42—22no—
174C42—41no—
175C42—41no—
176C42—42no—
177C42—42no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—24no—
183C23—48no—
184C23—48no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—48no—
192C23—84no—
193C23—84no—
194C23—84no—
195C23—84no—
196C23C4×D412yes—
197C23AGL(1,ℤ/8)12yes—
198C23#13712yes—
199C23C22×D411yescharacteristic
200Q8—24no—
201Q8—42no—
202D4—24no—
203D4—24no—
204D4—48no—
205D4—42no—
206D4—42no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—84no—
240C2×C4C4×D412yes—
241C2×C4AGL(1,ℤ/8)12yes—
242C2×C4C23×C411yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4H(5,𝔽2)12yes—
245C2×C4H(5,𝔽2)12yes—
246C8—24no—
247C8—42no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—816no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22#44014yes—
263C22C2×C4×D411yescharacteristic
264C22#53411yescharacteristic
265C22#53511yescharacteristic
266C22C2×AGL(1,ℤ/8)11yescharacteristic
267C22C2×AGL(1,ℤ/8)11yescharacteristic
268C22C2×#13711yescharacteristic
269C22#57511yescharacteristic
270C4—22no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—42no—
277C4—81no—
278C4—81no—
279C4—82no—
280C4#51812yes—
281C4#53812yes—
282C2—24no—
283C2—84no—
284C2#232111yescharacteristic
285C2C2×#44011yescharacteristic
286C2C4×AGL(1,ℤ/8)11yescharacteristic
287C2#319311yescharacteristic
288C2#319411yescharacteristic
289C2#321511yescharacteristic
290C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Q8C2semidirect
4. #1973C2semidirect
5. #1936C2semidirect
6. #1925C2semidirect
7. #1880C2semidirect
8. #1872C2semidirect
9. #1852C2semidirect
11. #1749C2semidirect
15. C2×C4×Q8C22semidirect
20. #450C4semidirect
21. C2×#107C22semidirect
22. C2×#103C22semidirect
28. C2×#97C22semidirect
29. #391C22semidirect
34. #389C22semidirect
39. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #336C22semidirect
58. C2×SD16C2×C4semidirect
92. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C4×C22⋊Q82
C2×#3902
#19732
#19362
#19252
#18802
#18722
#18522
#17862
#17492