Finite Group Databasebeta

← #18525

Subgroups — #18525

Order 256.

524 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#554C211yesmaximal, characteristic, complemented
2#3152C211yesmaximal, characteristic, complemented
3#3112C211yesmaximal, characteristic, complemented
4#3109C211yesmaximal, characteristic
5#1632C214yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1558C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, characteristic, complemented
10C2×#137—22no—
11#554—22no—
12#554—22no—
13#554—22no—
14#554C2214yes—
15#532C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—22no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#497—22no—
25#494C2212yescomplemented
26#492C2212yescomplemented
27#481—21no—
28#480—21no—
29#477C2212yescomplemented
30#476C2212yes—
31#460C2212yescomplemented
32#459C2212yes—
33#448—21no—
34#447—21no—
35#442—22no—
36#440—22no—
37#432C2211yescharacteristic
38#432C2211yescharacteristic
39C4×M16C2211yescharacteristic
40#333C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8D411yescharacteristic
45C22×Q8D412yes—
46C22×D4D411yescharacteristic
47C23×C4—22no—
48C23×C4—21no—
49#137—48no—
50C2×Q16—24no—
51C2×SD16—24no—
52C2×SD16—22no—
53C2×D8—22no—
54C2×M16—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8D412yes—
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2312yes—
60#127C2311yescharacteristic
61#125—24no—
62#125—24no—
63#124—21no—
64C22⋊Q8—24no—
65C22⋊Q8—28no—
66C22⋊Q8—24no—
67C22⋊Q8—28no—
68#121—21no—
69#121—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8D412yes—
73C4×Q8C2312yes—
74C4×D4—24no—
75C4×D4—24no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4D412yes—
81C4×D4C2312yes—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95#107—22no—
96#106—22no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#102—22no—
101#102—22no—
102#102—22no—
103C22⋊C8—21no—
104C22⋊C8—21no—
105#97—21no—
106#97—21no—
107#97—21no—
108C4×C8C2312yes—
109C24—21no—
110#41—44no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8—44no—
116C2×Q8C2×D412yes—
117C2×Q8C2×D414yes—
118C2×D4—24no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—41no—
124C2×D4C2×D412yes—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—28no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4—41no—
134C22×C4—84no—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138Q16—48no—
139SD16—48no—
140SD16—44no—
141D8—44no—
142M16—24no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—28no—
147C4⋊C4—28no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—28no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C42—22no—
166C42—22no—
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2411yescharacteristic
171C23—24no—
172C23—22no—
173C23—42no—
174C23—42no—
175C23—41no—
176C23#12011yescharacteristic
177Q8—22no—
178Q8—22no—
179Q8—44no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—24no—
184D4—22no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—82no—
189D4#13714yescomplemented
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—88no—
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4#12011yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C8—24no—
213C8—42no—
214C8—42no—
215C22—21no—
216C22—21no—
217C22—24no—
218C22—48no—
219C22—42no—
220C22—42no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22#45312yes—
225C22C2×#12011yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—44no—
234C4—81no—
235C4C2×#13712yes—
236C4#57512yes—
237C2—22no—
238C2—44no—
239C2—161no—
240C2#300411yescharacteristic
241C2#300511yescharacteristic
242C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3109C2
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. C2×#554C2semidirect
2. #3152C2semidirect
3. #3112C2semidirect
5. #1632C2semidirect
6. #1622C2semidirect
7. #1558C2semidirect
8. #1491C2semidirect
9. #1490C2semidirect
25. #494C22semidirect
26. #492C22semidirect
29. #477C22semidirect
31. #460C22semidirect
40. #333C22semidirect
41. Q8⋊C8C22semidirect
189. D4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#31522
#31122
#31092
#16322
#16222
#15582
#14912
#14902