Finite Group Databasebeta

← #33538

Subgroups — #33538

Order 256.

544 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#3584C211yesmaximal, characteristic, complemented
2#3414C211yesmaximal, characteristic, complemented
3C2×#508C211yesmaximal, characteristic, complemented
4#3067C211yesmaximal, characteristic, complemented
5#2195C212yesmaximal, complemented
6#2194C212yesmaximal, complemented
7#2193C212yesmaximal, complemented
8#2192C212yesmaximal, complemented
9#2148C211yesmaximal, characteristic, complemented
10C2×#359C211yesmaximal, characteristic, complemented
11C2×#350C211yesmaximal, characteristic, complemented
12#585—21no—
13#578—21no—
14#578C2212yes—
15#577—22no—
16#577—22no—
17#576—21no—
18#576C2212yes—
19#575—22no—
20#575—21no—
21#575—21no—
22C2×#135—22no—
23C2×#135C2211yescharacteristic
24C2×#135C2211yescharacteristic
25C2×#135C2211yescharacteristic, complemented
26C2×#131C2211yescharacteristic
27#510—21no—
28#510C2212yescomplemented
29#509—21no—
30#509C2212yescomplemented
31#508—22no—
32#508—22no—
33#508—22no—
34C2×Q32C2211yescharacteristic, complemented
35C2×SD32C2211yescharacteristic, complemented
36C2×SD32C2211yescharacteristic, complemented
37C2×D16C2211yescharacteristic, complemented
38C2×M32C2211yescharacteristic, complemented
39C22×C16C2211yescharacteristic, complemented
40#473C2212yescomplemented
41#472C2212yescomplemented
42#471—22no—
43C2×#108C2211yescharacteristic, complemented
44#413—21no—
45#362C2212yescomplemented
46#361C2212yescomplemented
47#359C2212yescomplemented
48#359C2212yescomplemented
49#350C2212yescomplemented
50#350C2212yescomplemented
512−1+4—22no—
522−1+4—22no—
532−1+4—41no—
54H(5,𝔽2)—22no—
55H(5,𝔽2)—22no—
56H(5,𝔽2)—42no—
57H(5,𝔽2)—41no—
58C2×#41—24no—
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—21no—
62C2×#41—42no—
63C2×#41—42no—
64C2×#41D411yescharacteristic
65#137—22no—
66#137—24no—
67#137—24no—
68#137—22no—
69#137—22no—
70#137—41no—
71AGL(1,ℤ/8)—22no—
72AGL(1,ℤ/8)—24no—
73AGL(1,ℤ/8)—24no—
74AGL(1,ℤ/8)—22no—
75AGL(1,ℤ/8)—22no—
76AGL(1,ℤ/8)—41no—
77#135—24no—
78#135—24no—
79#135—22no—
80#135—22no—
81#135—21no—
82#135—21no—
83#135—21no—
84#135—21no—
85#135—21no—
86#135—21no—
87#135—21no—
88#135—21no—
89#135—42no—
90#135D412yes—
91#135D412yes—
92#135D412yes—
93#135C2312yes—
94C2×Q16—22no—
95C2×Q16D411yescharacteristic
96C2×Q16C2311yescharacteristic
97C2×Q16C2311yescharacteristic
98C2×SD16—24no—
99C2×SD16—21no—
100C2×SD16—21no—
101C2×SD16—21no—
102C2×D8—22no—
103C2×D8D411yescharacteristic
104C2×D8C2311yescharacteristic
105C2×D8C2311yescharacteristic
106#131—21no—
107#131—21no—
108#131D412yes—
109#131C2312yes—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16D411yescharacteristic
113C22×C8—21no—
114C22×C8C2311yescharacteristic
115Q32—22no—
116Q32—22no—
117SD32—22no—
118SD32—22no—
119SD32—22no—
120SD32—22no—
121D16—22no—
122D16—22no—
123M32—21no—
124M32C2312yes—
125C2×C16—21no—
126C2×C16—21no—
127C2×C16—21no—
128C2×C16C2311yescharacteristic
129C2×C16C2311yescharacteristic
130#108—21no—
131#108C2312yes—
132#101—22no—
133#100—22no—
134#41—21no—
135#41—21no—
136#41—24no—
137#41—24no—
138#41—22no—
139#41—22no—
140#41—22no—
141#41—22no—
142#41—44no—
143#41—44no—
144#41—44no—
145#41—44no—
146#41—42no—
147#41—42no—
148#41—42no—
149#41—42no—
150#41—41no—
151#41—41no—
152#41—41no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41D812yes—
159#41C2×D412yes—
160C2×Q8—24no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8D811yescharacteristic
167C2×D4—21no—
168C2×D4—24no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—44no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4D811yescharacteristic
181C22×C4—21no—
182C22×C4—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4C2×D411yescharacteristic
187Q16—24no—
188Q16—21no—
189Q16—21no—
190Q16—21no—
191Q16—21no—
192Q16—42no—
193Q16C2×D412yes—
194Q16C2×D412yes—
195SD16—24no—
196SD16—24no—
197SD16—22no—
198SD16—22no—
199SD16—22no—
200SD16—22no—
201SD16—42no—
202SD16—42no—
203SD16—41no—
204SD16—41no—
205D8—24no—
206D8—21no—
207D8—21no—
208D8—21no—
209D8—21no—
210D8—42no—
211D8C2×D412yes—
212D8C2×D412yes—
213M16—21no—
214M16—22no—
215M16—22no—
216M16—22no—
217M16—41no—
218M16C2×D412yes—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8C2×D411yescharacteristic
224C2×C8C2×D411yescharacteristic
225C2×C8C2×D411yescharacteristic
226C2×C8C2×D411yescharacteristic
227C2×C8C2×D411yescharacteristic
228C2×C8C2411yescharacteristic
229C16—21no—
230C16—21no—
231C16—22no—
232C23—21no—
233C23—44no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23AGL(1,ℤ/8)11yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—44no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8C2×D812yes—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—21no—
252D4—44no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—42no—
257D4—42no—
258D4—42no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—81no—
264D4—81no—
265D4C2×D812yes—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#12011yescharacteristic
282C2×C4#12011yescharacteristic
283C2×C4C2×D811yescharacteristic
284C2×C4C2×D811yescharacteristic
285C2×C4AGL(1,ℤ/8)11yescharacteristic
286C2×C4C22×D411yescharacteristic
287C8—22no—
288C8—42no—
289C8#12011yescharacteristic
290C8#12011yescharacteristic
291C8C22×D411yescharacteristic
292C8C22×D411yescharacteristic
293C22—22no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—81no—
298C22—81no—
299C22—161no—
300C22—161no—
301C22—161no—
302C22—164no—
303C22#44711yescharacteristic
304C22#44711yescharacteristic
305C22C2×AGL(1,ℤ/8)11yescharacteristic
306C4—22no—
307C4—42no—
308C4—42no—
309C4—81no—
310C4#44711yescharacteristic
311C4#44711yescharacteristic
312C4C2×#12011yescharacteristic
313C4C22×D811yescharacteristic
314C2—21no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2—82no—
319C2—82no—
320C2—161no—
321C2C2×#44711yescharacteristic
subgroupG/H
centerC4#447
commutator, Frattini, cosocleC2×C8C24
socleC2C2×#447
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3584C2semidirect
2. #3414C2semidirect
3. C2×#508C2semidirect
4. #3067C2semidirect
5. #2195C2semidirect
6. #2194C2semidirect
7. #2193C2semidirect
8. #2192C2semidirect
9. #2148C2semidirect
10. C2×#359C2semidirect
11. C2×#350C2semidirect
25. C2×#135C22semidirect
28. #510C22semidirect
30. #509C22semidirect
34. C2×Q32C22semidirect
35. C2×SD32C22semidirect
36. C2×SD32C22semidirect
37. C2×D16C22semidirect
38. C2×M32C22semidirect
39. C22×C16C22semidirect
40. #473C22semidirect
41. #472C22semidirect
43. C2×#108C22semidirect
45. #362C22semidirect
46. #361C22semidirect
47. #359C22semidirect
48. #359C22semidirect
49. #350C22semidirect
50. #350C22semidirect

Maximal subgroups

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

subgroupindex
#35842
#34142
C2×#5082
#30672
#21952
#21942
#21932
#21922
#21482
C2×#3592
C2×#3502