Finite Group Databasebeta

← #24215

Subgroups — #24215

Order 256.

821 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 821 conjugacy classes of subgroups — too many to draw. Only its 151 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×#124C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3C2×#448C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic
5C4×#121C211yesmaximal, characteristic, complemented
6#2042C212yesmaximal, complemented
7#2012C212yesmaximal, complemented
8#1983C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic, complemented
10#1787C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×SD16C2212yescomplemented
15C22×D8C2212yes—
16C2×#124—28no—
17C2×#124—21no—
18C2×#124—21no—
19C2×#124C2212yes—
20C2×#124C2212yes—
21C2×#121C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic, complemented
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C42C2211yescharacteristic
27#463—28no—
28#448—24no—
29#447—24no—
30C2×C4⋊C8C2212yescomplemented
31C2×#103C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#103C2212yescomplemented
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#390C2211yescharacteristic, complemented
40#389C2211yescharacteristic, complemented
41#388—21no—
42#386—21no—
43#380C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic, complemented
45C4×C22⋊C4—21no—
46#336C2212yescomplemented
47C25—21no—
48C22×Q8—24no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2312yes—
54C22×D4—21no—
55C22×D4—24no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—42no—
60C22×D4C2311yescharacteristic
61C22×D4C2312yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—42no—
70C2×D8—24no—
71C2×D8—24no—
72C2×D8—42no—
73C22×C8C2312yes—
74C22×C8C2312yes—
75#124—28no—
76#124—28no—
77#124—24no—
78#124—24no—
79#124—24no—
80#124D418yescomplemented
81#121—22no—
82#121D414yes—
83C4×D4—24no—
84C4×D4—44no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—28no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—24no—
101C2×C42—21no—
102C2×C42D411yescharacteristic
103C2×C42D411yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C4⋊C8—24no—
107#103—24no—
108#103—24no—
109#103—42no—
110#102—24no—
111#102—24no—
112#102—42no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118C24—24no—
119C24—21no—
120C24—22no—
121C24—42no—
122C24—42no—
123C24—42no—
124C24—42no—
125C24—42no—
126C24—42no—
127C24C2×D411yescharacteristic
128C2×Q8—24no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—48no—
135C2×Q8—48no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8C2×D414yes—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4C2×D414yes—
156C2×D4C2×D412yes—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4#4112yes—
182C22×C4C2411yescharacteristic
183SD16—48no—
184D8—48no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4C2×D412yes—
196C22⋊C4—28no—
197C22⋊C4—28no—
198C22⋊C4—28no—
199C22⋊C4—28no—
200C22⋊C4—24no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—22no—
208C42—42no—
209C42C2×D414yes—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—22no—
216C23—48no—
217C23—48no—
218C23—48no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23—84no—
232C23—84no—
233C23—84no—
234C23—84no—
235C23#12112yes—
236C23#13512yes—
237C23AGL(1,ℤ/8)12yes—
238C23C22×D411yescharacteristic
239Q8—24no—
240Q8—24no—
241Q8—48no—
242Q8—42no—
243Q8—42no—
244D4—24no—
245D4—24no—
246D4—24no—
247D4—48no—
248D4—48no—
249D4—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—84no—
282C2×C4#12012yes—
283C2×C4#12012yes—
284C2×C4#12112yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C22×D412yes—
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)12yes—
289C8—42no—
290C8—42no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—816no—
304C22—81no—
305C22—82no—
306C22—81no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22—82no—
311C22—82no—
312C22#46314yescomplemented
313C22C2×#12011yescharacteristic
314C22C2×#12111yescharacteristic
315C22#53511yescharacteristic
316C22C2×#13511yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22#57511yescharacteristic
319C22#57511yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—41no—
325C4—41no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4D4212yes—
330C4#54612yes—
331C2—24no—
332C2—82no—
333C2—84no—
334C2#240811yescharacteristic
335C2#300211yescharacteristic
336C2#300611yescharacteristic
337C2C2×#46311yescharacteristic
338C2#304011yescharacteristic
339C2#321111yescharacteristic
340C2#321311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×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
1. C22×#124C2semidirect
2. C2×#463C2semidirect
3. C2×#448C2semidirect
5. C4×#121C2semidirect
6. #2042C2semidirect
7. #2012C2semidirect
8. #1983C2semidirect
9. #1797C2semidirect
10. #1787C2semidirect
11. #1786C2semidirect
14. C22×SD16C22semidirect
23. C2×C4×D4C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
39. #390C22semidirect
40. #389C22semidirect
44. C4×C4⋊C4C22semidirect
46. #336C22semidirect
80. #124D4semidirect
312. C22#463semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#4632
C2×#4482
C2×#4472
C4×#1212
#20422
#20122
#19832
#17972
#17872
#17862