Finite Group Databasebeta

← #18635

Subgroups — #18635

Order 256.

502 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 123 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#3209C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1641C214yesmaximal, complemented
6#1640C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10#559—21no—
11#559C2212yes—
12#550—21no—
13#550C2212yes—
14#546—22no—
15#546—22no—
16#540—22no—
17#535—21no—
18#533C2211yescharacteristic
19#532—21no—
20#532—21no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#127C2211yescharacteristic
23#518—21no—
24C4×#41C2211yescharacteristic
25#494—21no—
26#487—21no—
27#486—22no—
28#486C2214yes—
29#477—21no—
30#476—21no—
31#475C2212yescomplemented
32#474C2212yescomplemented
33#462C2214yescomplemented
34#451—22no—
35#431C2211yescharacteristic
36#428—21no—
37#423C2211yescharacteristic
38#417—21no—
39C2×#102—22no—
40C2×C4×C8C2211yescharacteristic
41#331C2214yes—
42Q8⋊C8C2212yescomplemented
43Q8⋊C8C2214yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41—22no—
46C2×#41—22no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D412yes—
52C2×Q16—24no—
53C2×M16—21no—
54C22×C8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8C2312yes—
61#127—22no—
62#127—21no—
63#127—21no—
64#127D412yes—
65#127C2312yes—
66#125—21no—
67#124—22no—
68#124—22no—
69#124—22no—
70#123—24no—
71#123—24no—
72C22⋊Q8—21no—
73C22⋊Q8D412yes—
74#121—22no—
75#121—22no—
76#121—22no—
77#121—24no—
78#121—24no—
79#121—22no—
80#120—24no—
81C4×Q8—21no—
82C4×Q8D411yescharacteristic
83C4×Q8C2311yescharacteristic
84C4×Q8C2312yes—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4D411yescharacteristic
94C4×D4C2311yescharacteristic
95#117—21no—
96#117—21no—
97#117D411yescharacteristic
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C22⋊C4—24no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103#107—22no—
104#106—22no—
105C4⋊C8—21no—
106C4⋊C8C2312yes—
107C4⋊C8C2312yes—
108#103—22no—
109#103—24no—
110#102—28no—
111#102—22no—
112C22⋊C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8C2312yes—
116C24—22no—
117#41—22no—
118#41—22no—
119#41—84no—
120#41—84no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—41no—
124C2×Q8—82no—
125C2×Q8C2×D411yescharacteristic
126C2×Q8C2×D412yes—
127C2×D4—21no—
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—44no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—82no—
139C2×D4C2×D411yescharacteristic
140C2×D4C2×D414yes—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150Q16—84no—
151M16—42no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—42no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D411yescharacteristic
169C4⋊C4C2×D411yescharacteristic
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—21no—
175C22⋊C4—22no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—22no—
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2411yescharacteristic
186C23—24no—
187C23—21no—
188C23—44no—
189C23—82no—
190C23—82no—
191C23#12011yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—24no—
195Q8—42no—
196Q8—44no—
197Q8—82no—
198D4—21no—
199D4—21no—
200D4—44no—
201D4—42no—
202D4—42no—
203D4—42no—
204D4—44no—
205D4—44no—
206D4—44no—
207D4—82no—
208D4—84no—
209D4—84no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—82no—
224C2×C4#12011yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4#13512yes—
228C2×C4#13712yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C8—42no—
233C8—42no—
234C8—42no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—44no—
239C22—41no—
240C22—44no—
241C22—41no—
242C22—44no—
243C22—164no—
244C22#45212yes—
245C22C2×#12011yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4—42no—
253C4—82no—
254C4#45212yes—
255C4#45512yes—
256C4C2×#13511yescharacteristic
257C4C2×#13711yescharacteristic
258C4#57612yes—
259C2—22no—
260C2—81no—
261C2—84no—
262C2C2×#45211yescharacteristic
263C2#300811yescharacteristic
264C2C2×#45511yescharacteristic
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. #3209C2semidirect
3. C2×#486C2semidirect
4. #3079C2semidirect
5. #1641C2semidirect
6. #1640C2semidirect
7. #1545C2semidirect
8. #1493C2semidirect
9. #1476C2semidirect
31. #475C22semidirect
32. #474C22semidirect
33. #462C22semidirect
42. Q8⋊C8C22semidirect
43. Q8⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32092
C2×#4862
#30792
#16412
#16402
#15452
#14932
#14762