Finite Group Databasebeta

← #18534

Subgroups — #18534

Order 256.

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

normal subgroups only

This group has 584 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
1C2×D4×Q8C211yesmaximal, characteristic, complemented
2#3141C211yesmaximal, characteristic, complemented
3#3112C211yesmaximal, characteristic, complemented
4#3109C211yesmaximal, characteristic
5#1637C214yesmaximal, complemented
6#1627C214yesmaximal, complemented
7#1556C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11D4×Q8—22no—
12D4×Q8—22no—
13D4×Q8—22no—
14D4×Q8C2214yes—
15#532C2211yescharacteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—22no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23#489—22no—
24#487C2212yescomplemented
25#486C2212yescomplemented
26#481—21no—
27#480—21no—
28#477C2212yescomplemented
29#476C2212yes—
30#460C2212yescomplemented
31#459C2212yes—
32#448—21no—
33#447—21no—
34#442—22no—
35#440—22no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#417—22no—
39C4×M16C2211yescharacteristic
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×D4D411yescharacteristic
51C23×C4—22no—
52C23×C4—21no—
53C2×SD16—22no—
54C2×D8—22no—
55C2×M16—21no—
56C4⋊Q8—24no—
57C4⋊Q8—24no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D412yes—
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2312yes—
63#127C2311yescharacteristic
64#124—21no—
65C22⋊Q8—24no—
66C22⋊Q8—28no—
67C22⋊Q8—24no—
68C22⋊Q8—28no—
69#121—21no—
70#121—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8D412yes—
74C4×Q8C2312yes—
75C4×D4—24no—
76C4×D4—24no—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4D412yes—
82C4×D4C2312yes—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D412yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93#107—22no—
94#106—22no—
95C4⋊C8C2312yes—
96C4⋊C8C2312yes—
97#103—22no—
98#103—24no—
99#102—22no—
100#102—24no—
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—24no—
112C2×Q8—28no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—48no—
118C2×Q8—44no—
119C2×Q8—416no—
120C2×Q8—48no—
121C2×Q8—48no—
122C2×Q8C2×D412yes—
123C2×D4—24no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4C2×D412yes—
130C22×C4—21no—
131C22×C4—28no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—48no—
138C22×C4—44no—
139C22×C4—41no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143SD16—44no—
144D8—44no—
145M16—42no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—28no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4C2×D414yes—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—28no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—41no—
166C22⋊C4—41no—
167C42—22no—
168C42—22no—
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2411yescharacteristic
173C23—24no—
174C23—22no—
175C23—42no—
176C23—42no—
177C23—41no—
178C23#12011yescharacteristic
179Q8—24no—
180Q8—22no—
181Q8—416no—
182Q8—44no—
183Q8—42no—
184Q8—44no—
185Q8AGL(1,ℤ/8)14yes—
186D4—24no—
187D4—22no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—82no—
192D4#13714yes—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—416no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—82no—
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4#12011yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C8—42no—
216C8—42no—
217C8—42no—
218C22—21no—
219C22—21no—
220C22—24no—
221C22—48no—
222C22—42no—
223C22—42no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#45412yes—
228C22C2×#12011yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—44no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4C2×AGL(1,ℤ/8)12yes—
239C4C2×#13712yes—
240C2—22no—
241C2—44no—
242C2—161no—
243C2#300311yescharacteristic
244C2#300411yescharacteristic
245C2C2×#45411yescharacteristic
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×D4×Q8C2semidirect
2. #3141C2semidirect
3. #3112C2semidirect
5. #1637C2semidirect
6. #1627C2semidirect
7. #1556C2semidirect
8. #1491C2semidirect
9. #1490C2semidirect
24. #487C22semidirect
25. #486C22semidirect
28. #477C22semidirect
30. #460C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
#31412
#31122
#31092
#16372
#16272
#15562
#14912
#14902