Finite Group Databasebeta

← #18874

Subgroups — #18874

Order 256.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#3508C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic, complemented
5#1642C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1621C212yesmaximal, complemented
8#1620C212yesmaximal, complemented
9C2×#331C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×#135—21no—
13#559—21no—
14#559C2212yes—
15#550—21no—
16#550C2212yes—
17#546—22no—
18#546—22no—
19#540—22no—
20#535—21no—
21#533C2211yescharacteristic
22#532—21no—
23#532—21no—
24#532C2211yescharacteristic
25C2×#127C2211yescharacteristic
26#518—21no—
27C4×#41C2211yescharacteristic
28#493C2212yes—
29#486C2212yes—
30#476—21no—
31#475C2212yescomplemented
32#465—22no—
33#464—21no—
34#463—21no—
35#462C2211yescharacteristic, complemented
36#461—21no—
37#461C2211yescharacteristic, complemented
38#460C2211yescharacteristic, complemented
39#460C2212yescomplemented
40#459—22no—
41#459C2211yescharacteristic, complemented
42#447—22no—
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#331C2212yes—
48#331C2212yes—
49Q8⋊C8C2212yescomplemented
50Q8⋊C8C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—21no—
56C2×#41D411yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4D412yes—
60#135—44no—
61C2×Q16—21no—
62C2×SD16—21no—
63C2×SD16—21no—
64C2×SD16—22no—
65C2×D8—21no—
66C2×D8—24no—
67C22×C8—21no—
68C22×C8—21no—
69C4⋊Q8D411yescharacteristic
70C4⋊Q8D411yescharacteristic
71C4⋊Q8C2311yescharacteristic
72#127—22no—
73#127—21no—
74#127—21no—
75#127D412yes—
76#127C2311yescharacteristic
77#127C2312yes—
78#125—21no—
79#124—22no—
80#124—22no—
81#124—22no—
82#124—21no—
83#123—24no—
84#123—24no—
85C22⋊Q8—21no—
86C22⋊Q8D412yes—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—24no—
91#121—24no—
92#121—22no—
93#121—21no—
94#121—21no—
95#120—24no—
96C4×Q8—21no—
97C4×Q8D411yescharacteristic
98C4×Q8C2311yescharacteristic
99C4×Q8C2312yes—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4D411yescharacteristic
111C4×D4C2311yescharacteristic
112#117—21no—
113#117—21no—
114#117D411yescharacteristic
115C2×C22⋊C4—24no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118#106—22no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8C2312yes—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125#103—22no—
126#103—21no—
127#103—21no—
128#102—24no—
129#102—21no—
130#102—21no—
131#102—22no—
132C22⋊C8—21no—
133C4×C8C2312yes—
134C24—22no—
135#41—22no—
136#41—22no—
137#41—42no—
138#41—42no—
139#41—82no—
140#41—82no—
141#41—84no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C2×D412yes—
145C2×Q8C2×D411yescharacteristic
146C2×D4—21no—
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—82no—
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D414yes—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174Q16—42no—
175SD16—42no—
176SD16—42no—
177SD16—82no—
178D8—42no—
179D8—84no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—42no—
185C2×C8—81no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D412yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—22no—
212C42C2×D411yescharacteristic
213C42C2×D411yescharacteristic
214C42C2×D411yescharacteristic
215C42C2411yescharacteristic
216C23—24no—
217C23—21no—
218C23—44no—
219C23—41no—
220C23—82no—
221C23—82no—
222C23#12011yescharacteristic
223Q8—21no—
224Q8—21no—
225Q8—22no—
226Q8—22no—
227Q8—42no—
228Q8—41no—
229Q8—41no—
230D4—21no—
231D4—21no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—41no—
238D4—42no—
239D4—41no—
240D4—42no—
241D4—41no—
242D4—42no—
243D4—41no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
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—82no—
265C2×C4#12011yescharacteristic
266C2×C4#12011yescharacteristic
267C2×C4#12011yescharacteristic
268C2×C4#13512yes—
269C2×C4AGL(1,ℤ/8)12yes—
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C8—42no—
274C8—41no—
275C8—41no—
276C8—42no—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—44no—
281C22—41no—
282C22—44no—
283C22—41no—
284C22—44no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—164no—
289C22#44912yes—
290C22C2×#12011yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—81no—
300C4#44912yes—
301C4C2×#13511yescharacteristic
302C4C2×AGL(1,ℤ/8)11yescharacteristic
303C4#57611yescharacteristic
304C4#57711yescharacteristic
305C2—22no—
306C2—81no—
307C2—84no—
308C2—161no—
309C2C2×#44911yescharacteristic
310C2#301211yescharacteristic
311C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3508C2semidirect
2. #3246C2semidirect
3. #3205C2semidirect
4. #3035C2semidirect
5. #1642C2semidirect
6. #1640C2semidirect
7. #1621C2semidirect
8. #1620C2semidirect
9. C2×#331C2semidirect
10. #1481C2semidirect
11. #1477C2semidirect
31. #475C22semidirect
35. #462C22semidirect
37. #461C22semidirect
38. #460C22semidirect
39. #460C22semidirect
41. #459C22semidirect
49. Q8⋊C8C22semidirect
50. Q8⋊C8C22semidirect
51. Q8⋊C8C22semidirect
52. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32462
#32052
#30352
#16422
#16402
#16212
#16202
C2×#3312
#14812
#14772