Finite Group Databasebeta

← #18819

Subgroups — #18819

Order 256.

485 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#3237C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3116C211yesmaximal, characteristic, complemented
5#1642C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1631C212yesmaximal, complemented
8#1630C212yesmaximal, complemented
9#1555C211yesmaximal, characteristic, complemented
10#1497C211yesmaximal, characteristic, complemented
11#1488C211yesmaximal, characteristic, complemented
12#559—21no—
13#559C2212yes—
14#550—21no—
15#550C2212yes—
16#546—22no—
17#546—22no—
18#540—22no—
19#535—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532—21no—
23#532—21no—
24C2×#127C2211yescharacteristic
25#518—21no—
26C4×#41C2211yescharacteristic
27#501—21no—
28#492C2212yes—
29#491—21no—
30#486C2212yes—
31#485—22no—
32#485—21no—
33#477C2211yescharacteristic, complemented
34#477C2212yescomplemented
35#476C2211yescharacteristic, complemented
36#475C2212yescomplemented
37#462C2211yescharacteristic, complemented
38#461C2211yescharacteristic, complemented
39#460—22no—
40#452—21no—
41#450—22no—
42#441—21no—
43#440—21no—
44#432C2211yescharacteristic
45#432C2211yescharacteristic
46#431C2211yescharacteristic
47#331C2212yes—
48#331C2212yes—
49Q8⋊C8C2212yescomplemented
50Q8⋊C8C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—22no—
54C2×#41—22no—
55C2×#41D411yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4D412yes—
59C2×Q16—21no—
60C2×SD16—21no—
61C2×SD16—24no—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#127—22no—
67#127—21no—
68#127—21no—
69#127D412yes—
70#127C2312yes—
71#125—21no—
72#125—21no—
73#124—22no—
74#124—22no—
75#124—22no—
76#123—24no—
77#123—24no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8D412yes—
82#121—22no—
83#121—22no—
84#121—22no—
85#121—24no—
86#121—24no—
87#121—22no—
88#120—24no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8C2311yescharacteristic
94C4×Q8C2312yes—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4D411yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117—21no—
107#117—21no—
108#117D411yescharacteristic
109C2×C22⋊C4—24no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—21no—
113#107—22no—
114#106—21no—
115#106—22no—
116C4⋊C8—21no—
117C4⋊C8C2312yes—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—22no—
122#103—22no—
123#103—21no—
124#103—21no—
125#102—21no—
126#102—24no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130#97—21no—
131#97—21no—
132C4×C8C2312yes—
133C24—22no—
134#41—41no—
135#41—41no—
136#41—84no—
137#41—84no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8C2×D412yes—
142C2×Q8C2×D411yescharacteristic
143C2×D4—21no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—82no—
154C2×D4—82no—
155C2×D4C2×D411yescharacteristic
156C2×D4C2×D414yes—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165Q16—42no—
166SD16—42no—
167SD16—84no—
168C2×C8—22no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4C2×D411yescharacteristic
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D412yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—41no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—22no—
201C42—41no—
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2411yescharacteristic
206C23—24no—
207C23—21no—
208C23—44no—
209C23—82no—
210C23—82no—
211C23#12011yescharacteristic
212Q8—21no—
213Q8—21no—
214Q8—24no—
215Q8—42no—
216Q8—42no—
217Q8—41no—
218Q8—41no—
219D4—21no—
220D4—21no—
221D4—44no—
222D4—44no—
223D4—44no—
224D4—42no—
225D4—42no—
226D4—42no—
227D4—44no—
228D4—82no—
229D4—84no—
230D4—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4#12011yescharacteristic
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C8—42no—
254C8—41no—
255C8—41no—
256C8—42no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—41no—
261C22—44no—
262C22—41no—
263C22—44no—
264C22—41no—
265C22—44no—
266C22—164no—
267C22C2×#12011yescharacteristic
268C4—21no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—42no—
275C4—41no—
276C4—42no—
277C4—81no—
278C4—81no—
279C4#57511yescharacteristic
280C4#57511yescharacteristic
281C4#57712yes—
282C2—41no—
283C2—81no—
284C2—84no—
285C2#300511yescharacteristic
286C2#300911yescharacteristic
287C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3508C2
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
2. #3237C2semidirect
3. #3208C2semidirect
4. #3116C2semidirect
5. #1642C2semidirect
6. #1640C2semidirect
7. #1631C2semidirect
8. #1630C2semidirect
9. #1555C2semidirect
10. #1497C2semidirect
11. #1488C2semidirect
33. #477C22semidirect
34. #477C22semidirect
35. #476C22semidirect
36. #475C22semidirect
37. #462C22semidirect
38. #461C22semidirect
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
#32372
#32082
#31162
#16422
#16402
#16312
#16302
#15552
#14972
#14882