Finite Group Databasebeta

← #24558

Subgroups — #24558

Order 256.

617 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 137 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×#480C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic, complemented
3#2847C211yesmaximal, characteristic
4C2×#380C211yesmaximal, characteristic, complemented
5#2076C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1895C212yesmaximal, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1809C212yesmaximal, complemented
12C24×C4C2211yescharacteristic
13C22×M16C2212yes—
14C2×#121C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×#120—21no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4—21no—
21#480—24no—
22C2×#107C2212yescomplemented
23C2×#106C2212yescomplemented
24C2×#102C2212yescomplemented
25C2×#102C2212yescomplemented
26#407—24no—
27C2×C22⋊C8C2212yes—
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#386—21no—
31#386—21no—
32#386—21no—
33#380—21no—
34#380—21no—
35#380C2211yescharacteristic
36#380C2212yes—
37#380C2211yescharacteristic
38#380C2212yes—
39#380C2211yescharacteristic
40C2×#95—21no—
41#340C2212yescomplemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×D4—21no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—42no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4D411yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C2×M16—24no—
63C2×M16—24no—
64C2×M16—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67#121—22no—
68#121—22no—
69#121—24no—
70#121—42no—
71#120—44no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—44no—
91#107—24no—
92#106—42no—
93#102—24no—
94#102—42no—
95C22⋊C8—22no—
96C22⋊C8—24no—
97C22⋊C8D414yescomplemented
98#95—22no—
99#95—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24—41no—
110C24—41no—
111C24C2×D411yescharacteristic
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D412yes—
161C22×C4C2×D411yescharacteristic
162C22×C4#4112yes—
163C22×C4#4114yes—
164C22×C4C2411yescharacteristic
165M16—48no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8C2×D414yes—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—44no—
184C22⋊C4—44no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—28no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23#12112yes—
221C23#12312yes—
222C23AGL(1,ℤ/8)12yes—
223C23AGL(1,ℤ/8)12yes—
224C23#13712yes—
225C23C22×D411yescharacteristic
226D4—42no—
227D4—42no—
228D4—84no—
229D4—84no—
230D4—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4#12112yes—
265C2×C4#12312yes—
266C2×C4C22×D411yescharacteristic
267C2×C4C2×#4112yes—
268C2×C4C2×#4111yescharacteristic
269C2×C4H(5,𝔽2)12yes—
270C8—24no—
271C8—42no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—416no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#46914yes—
301C22C2×#12111yescharacteristic
302C22C2×#12311yescharacteristic
303C22#53411yescharacteristic
304C22C2×AGL(1,ℤ/8)11yescharacteristic
305C22C2×AGL(1,ℤ/8)11yescharacteristic
306C22C2×#13711yescharacteristic
307C22#57511yescharacteristic
308C4—22no—
309C4—22no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—82no—
314C4—82no—
315C4—81no—
316C4#54612yes—
317C4#55112yes—
318C2—24no—
319C2—42no—
320C2—42no—
321C2—161no—
322C2#241211yescharacteristic
323C2C2×#46911yescharacteristic
324C2#305611yescharacteristic
325C2#309211yescharacteristic
326C2#309411yescharacteristic
327C2#319311yescharacteristic
328C2#319411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#480C2semidirect
2. C2×#407C2semidirect
4. C2×#380C2semidirect
5. #2076C2semidirect
6. #2034C2semidirect
7. #1895C2semidirect
8. #1874C2semidirect
9. #1853C2semidirect
10. #1823C2semidirect
11. #1809C2semidirect
22. C2×#107C22semidirect
23. C2×#106C22semidirect
24. C2×#102C22semidirect
25. C2×#102C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect
97. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4072
#28472
C2×#3802
#20762
#20342
#18952
#18742
#18532
#18232
#18092