Finite Group Databasebeta

← #18615

Subgroups — #18615

Order 256.

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

normal subgroups only

This group has 465 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#3506C211yesmaximal, characteristic
2#3237C211yesmaximal, characteristic, complemented
3#3182C211yesmaximal, characteristic, complemented
4#3101C211yesmaximal, characteristic, complemented
5#1650C212yesmaximal, complemented
6#1642C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1631C212yesmaximal, complemented
9#1553C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic, complemented
11#1485C211yesmaximal, characteristic, complemented
12#559—22no—
13#556C2212yes—
14#555—21no—
15#553—22no—
16#550C2212yes—
17#547—21no—
18#538—22no—
19#535—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#525—22no—
24#520—21no—
25C2×C4×D4C2211yescharacteristic
26#501—21no—
27#500C2211yescharacteristic, complemented
28#492C2211yescharacteristic, complemented
29#491—21no—
30#487C2211yescharacteristic, complemented
31#486C2211yescharacteristic, complemented
32#485—22no—
33#483—21no—
34#477C2212yescomplemented
35#476C2212yes—
36#475C2212yescomplemented
37#468—22no—
38#461C2212yes—
39#450—21no—
40#440—22no—
41C4×SD16—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#332C2212yescomplemented
46#331C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—22no—
52C2×#41—21no—
53C22×D4D411yescharacteristic
54C23×C4—21no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×M16—21no—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#127—22no—
64#127C2311yescharacteristic
65#126—24no—
66#125—21no—
67#125—21no—
68#124—24no—
69#124—22no—
70#124—21no—
71#123—24no—
72#123—24no—
73C22⋊Q8—22no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8D412yes—
78#121—22no—
79#121—22no—
80#121—24no—
81#121—21no—
82#121—22no—
83#121D412yes—
84#120—24no—
85C4×Q8—22no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8C2312yes—
90C4×D4—24no—
91C4×D4—24no—
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4D412yes—
97C4×D4D412yes—
98C4×D4C2312yes—
99#117—22no—
100#117—21no—
101#117—22no—
102#117—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4D411yescharacteristic
105C2×C22⋊C4—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—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—44no—
128#41—84no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C2×D412yes—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—42no—
141C2×D4C2×D412yes—
142C2×D4C2×D412yes—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—82no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152SD16—44no—
153SD16—44no—
154M16—42no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—21no—
158C2×C8—22no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—41no—
185C42—24no—
186C42—22no—
187C42—41no—
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2411yescharacteristic
192C23—22no—
193C23—22no—
194C23—42no—
195C23—81no—
196C23—81no—
197C23#12011yescharacteristic
198Q8—24no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—81no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4—42no—
207D4—44no—
208D4—42no—
209D4—42no—
210D4—84no—
211D4—84no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—84no—
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C8—22no—
233C8—41no—
234C8—42no—
235C8—42no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22C2×#12011yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C4#57512yes—
258C4#57712yes—
259C2—41no—
260C2—82no—
261C2—82no—
262C2#300511yescharacteristic
263C2#300911yescharacteristic
264C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3506C2
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. #3182C2semidirect
4. #3101C2semidirect
5. #1650C2semidirect
6. #1642C2semidirect
7. #1638C2semidirect
8. #1631C2semidirect
9. #1553C2semidirect
10. #1490C2semidirect
11. #1485C2semidirect
27. #500C22semidirect
28. #492C22semidirect
30. #487C22semidirect
31. #486C22semidirect
34. #477C22semidirect
36. #475C22semidirect
45. #332C22semidirect
46. #331C22semidirect
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
#35062
#32372
#31822
#31012
#16502
#16422
#16382
#16312
#15532
#14902
#14852