Finite Group Databasebeta

← #18600

Subgroups — #18600

Order 256.

472 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 117 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#3501C211yesmaximal, characteristic, complemented
2#3216C211yesmaximal, characteristic, complemented
3#3182C211yesmaximal, characteristic, complemented
4#3100C211yesmaximal, characteristic
5#1649C212yesmaximal, complemented
6#1644C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1622C212yesmaximal, complemented
9#1554C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic, complemented
11#1485C211yesmaximal, characteristic, complemented
12#554—21no—
13#554—22no—
14#554—22no—
15#554C2212yes—
16D4×Q8—21no—
17D4×Q8C2212yes—
18#544—21no—
19#543—22no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C22⋊Q8—22no—
24#518—21no—
25C2×C4×D4C2211yescharacteristic
26#500C2211yescharacteristic, complemented
27#497—21no—
28#492C2211yescharacteristic, complemented
29#489—21no—
30#487C2211yescharacteristic, complemented
31#486C2211yescharacteristic, complemented
32#480—21no—
33#474C2212yes—
34#468—22no—
35#462C2212yescomplemented
36#460C2212yescomplemented
37#459C2212yes—
38#452—22no—
39#447—21no—
40#440—22no—
41C4×D8—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#333C2212yescomplemented
46#332C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8—22no—
53C22×D4D411yescharacteristic
54C23×C4—21no—
55C2×Q16—22no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×D8—22no—
59C2×M16—21no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#127C2311yescharacteristic
68#125—22no—
69#125—22no—
70#125—21no—
71#125—21no—
72#124—21no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8D412yes—
84C22⋊Q8D412yes—
85#121—21no—
86#121—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8D412yes—
90C4×Q8C2312yes—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D412yes—
100C4×D4C2312yes—
101#117—21no—
102#117—21no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4D411yescharacteristic
106C2×C22⋊C4—24no—
107C2×C22⋊C4—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#103—22no—
114#103—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119C22⋊C8—21no—
120C22⋊C8—21no—
121#97—21no—
122#97—21no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—21no—
127#41—82no—
128C2×Q8—24no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—41no—
142C2×D4C2×D412yes—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4—82no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153Q16—82no—
154SD16—44no—
155SD16—82no—
156D8—44no—
157M16—42no—
158C2×C8—21no—
159C2×C8—22no—
160C2×C8—21no—
161C2×C8—22no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4C2×D412yes—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C42—22no—
190C42—22no—
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C42C2×D411yescharacteristic
194C42C2411yescharacteristic
195C23—24no—
196C23—22no—
197C23—41no—
198C23—81no—
199C23—81no—
200C23#12011yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—84no—
208D4—24no—
209D4—22no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—82no—
214D4#13714yes—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—81no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4#12011yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—22no—
238C8—41no—
239C8—42no—
240C8—42no—
241C22—21no—
242C22—24no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—84no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22C2×#12011yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—81no—
262C4C2×#13712yes—
263C4#57612yes—
264C2—41no—
265C2—44no—
266C2—161no—
267C2#300411yescharacteristic
268C2#300811yescharacteristic
269C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3100C2
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. #3501C2semidirect
2. #3216C2semidirect
3. #3182C2semidirect
5. #1649C2semidirect
6. #1644C2semidirect
7. #1627C2semidirect
8. #1622C2semidirect
9. #1554C2semidirect
10. #1490C2semidirect
11. #1485C2semidirect
26. #500C22semidirect
28. #492C22semidirect
30. #487C22semidirect
31. #486C22semidirect
35. #462C22semidirect
36. #460C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35012
#32162
#31822
#31002
#16492
#16442
#16272
#16222
#15542
#14902
#14852