Finite Group Databasebeta

← #18752

Subgroups — #18752

Order 256.

510 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 125 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#3206C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic, complemented
5#1644C214yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C2×#137—21no—
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#492C2212yescomplemented
25#487C2212yescomplemented
26#480—21no—
27#474C2212yes—
28#464—22no—
29#462C2212yescomplemented
30#461—22no—
31#460C2212yescomplemented
32#459C2212yes—
33#448—22no—
34#447—21no—
35C4×D8—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#333C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8D411yescharacteristic
45C22×Q8D412yes—
46C22×D4D411yescharacteristic
47C23×C4—22no—
48C23×C4—21no—
49#137—44no—
50C2×Q16—22no—
51C2×SD16—22no—
52C2×SD16—24no—
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—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—24no—
101#103—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109C24—21no—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—82no—
118C2×Q8C2×D412yes—
119C2×Q8C2×D414yes—
120C2×D4—24no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4C2×D412yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—28no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—84no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141Q16—44no—
142SD16—44no—
143SD16—84no—
144D8—44no—
145M16—42no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—28no—
150C4⋊C4—28no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—28no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—41no—
167C22⋊C4—41no—
168C42—22no—
169C42—22no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—24no—
175C23—22no—
176C23—42no—
177C23—42no—
178C23—41no—
179C23#12011yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—44no—
183Q8—42no—
184Q8—41no—
185Q8—41no—
186Q8—82no—
187D4—24no—
188D4—22no—
189D4—42no—
190D4—41no—
191D4—41no—
192D4—42no—
193D4—82no—
194D4#13714yescomplemented
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—88no—
210C2×C4—82no—
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#13512yes—
215C2×C4AGL(1,ℤ/8)12yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C8—42no—
220C8—42no—
221C8—42no—
222C22—21no—
223C22—21no—
224C22—24no—
225C22—48no—
226C22—42no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22#44912yes—
232C22C2×#12011yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—44no—
240C4—81no—
241C4#44912yes—
242C4C2×#13511yescharacteristic
243C4C2×AGL(1,ℤ/8)11yescharacteristic
244C4C2×#13712yes—
245C2—22no—
246C2—44no—
247C2—161no—
248C2C2×#44911yescharacteristic
249C2#300411yescharacteristic
250C2#301911yescharacteristic
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. C2×#554C2semidirect
2. #3206C2semidirect
4. #3037C2semidirect
5. #1644C2semidirect
6. #1622C2semidirect
7. #1548C2semidirect
8. #1486C2semidirect
9. #1479C2semidirect
24. #492C22semidirect
25. #487C22semidirect
29. #462C22semidirect
31. #460C22semidirect
40. #333C22semidirect
41. Q8⋊C8C22semidirect
194. D4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#32062
#31002
#30372
#16442
#16222
#15482
#14862
#14792