Finite Group Databasebeta

← #22046

Subgroups — #22046

Order 256.

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

normal subgroups only

This group has 648 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×#447C211yesmaximal, characteristic
2C4×#121C211yesmaximal, characteristic, complemented
3C2×#390C211yesmaximal, characteristic, complemented
4#1972C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1926C212yesmaximal, complemented
7#1879C212yesmaximal, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1786C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C2×#121C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic, complemented
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4—21no—
19C22×C22⋊C4—21no—
20C22×C42C2211yescharacteristic
21#447C418yescomplemented
22C2×#106C2212yescomplemented
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yescomplemented
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×#97C2212yescomplemented
30#390—21no—
31#390—21no—
32#390C2212yes—
33#390C2212yes—
34#390C2211yescharacteristic, complemented
35#389C2211yescharacteristic, complemented
36#388—21no—
37#386—21no—
38#380C2211yescharacteristic
39C4×C4⋊C4C2211yescharacteristic, complemented
40C4×C22⋊C4—21no—
41#340C2212yescomplemented
42#340C2212yescomplemented
43#336C2212yescomplemented
44C25—21no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4C2×C414yescomplemented
51C22×D4C2311yescharacteristic
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×D8—42no—
59C2×D8—42no—
60C2×D8C2×C418yescomplemented
61C22×C8C2312yes—
62C22×C8C2312yes—
63#121—22no—
64#121C2×C414yes—
65C4×D4—42no—
66C4×D4—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—44no—
85C2×C42—21no—
86C2×C42—42no—
87C2×C42D411yescharacteristic
88C2×C42D411yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#106—42no—
92#102—42no—
93#102—42no—
94#102—42no—
95#102C2×C418yescomplemented
96C22⋊C8—22no—
97C22⋊C8C2×C414yescomplemented
98#97—42no—
99#95—21no—
100#95—21no—
101#95—21no—
102C24—24no—
103C24—21no—
104C24—22no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24—42no—
109C24—42no—
110C24—42no—
111C24C2×D411yescharacteristic
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—48no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4C22×C414yes—
129C2×D4C22×C412yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C22×C412yes—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4#4112yes—
155C22×C4C2411yescharacteristic
156D8—48no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8C22×C414yes—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4C22×C412yes—
171C22⋊C4—24no—
172C22⋊C4—42no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C22⋊C4—44no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—44no—
180C42—22no—
181C42—22no—
182C42—41no—
183C42—41no—
184C42—42no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—24no—
191C23—48no—
192C23—48no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23—42no—
205C23—48no—
206C23—84no—
207C23—84no—
208C23—84no—
209C23—84no—
210C23C4×D412yes—
211C23AGL(1,ℤ/8)12yes—
212C23AGL(1,ℤ/8)12yes—
213C23C22×D411yescharacteristic
214D4—24no—
215D4—24no—
216D4—24no—
217D4—48no—
218D4—48no—
219D4—42no—
220D4—42no—
221D4—42no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4C4×D412yes—
254C2×C4AGL(1,ℤ/8)12yes—
255C2×C4C23×C411yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4H(5,𝔽2)12yes—
258C2×C4H(5,𝔽2)12yes—
259C8—24no—
260C8—42no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—44no—
271C22—42no—
272C22—44no—
273C22—816no—
274C22—81no—
275C22—82no—
276C22—81no—
277C22—82no—
278C22—81no—
279C22—82no—
280C22—81no—
281C22—82no—
282C22#44214yes—
283C22C2×C4×D411yescharacteristic
284C22#53411yescharacteristic
285C22#53511yescharacteristic
286C22C2×AGL(1,ℤ/8)11yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22C2×AGL(1,ℤ/8)11yescharacteristic
289C22#57511yescharacteristic
290C4—22no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—81no—
297C4—81no—
298C4—82no—
299C4#51812yes—
300C4#53812yes—
301C2—24no—
302C2—82no—
303C2—84no—
304C2#232111yescharacteristic
305C2C2×#44211yescharacteristic
306C2C4×AGL(1,ℤ/8)11yescharacteristic
307C2#319211yescharacteristic
308C2#319411yescharacteristic
309C2#321311yescharacteristic
310C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#121C2semidirect
3. C2×#390C2semidirect
4. #1972C2semidirect
5. #1936C2semidirect
6. #1926C2semidirect
7. #1879C2semidirect
8. #1874C2semidirect
9. #1853C2semidirect
10. #1786C2semidirect
11. #1749C2semidirect
16. C2×C4×D4C22semidirect
21. #447C4semidirect
22. C2×#106C22semidirect
26. C2×#102C22semidirect
29. C2×#97C22semidirect
34. #390C22semidirect
35. #389C22semidirect
39. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #336C22semidirect
50. C22×D4C2×C4semidirect
60. C2×D8C2×C4semidirect
95. #102C2×C4semidirect
97. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C4×#1212
C2×#3902
#19722
#19362
#19262
#18792
#18742
#18532
#17862
#17492