Finite Group Databasebeta

← #18540

Subgroups — #18540

Order 256.

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

normal subgroups only

This group has 573 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
2#3165C211yesmaximal, characteristic, complemented
3#3113C211yesmaximal, characteristic, complemented
4#3110C211yesmaximal, characteristic
5#1651C214yesmaximal, complemented
6#1649C214yesmaximal, complemented
7#1556C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, characteristic
10C23×Q8—21no—
11D4×Q8—22no—
12D4×Q8—22no—
13D4×Q8—22no—
14D4×Q8C2214yes—
15#533C2211yescharacteristic
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#501—22no—
24#500C2212yescomplemented
25#498C2212yescomplemented
26#484—21no—
27#483—21no—
28#475C2212yescomplemented
29#474C2212yes—
30#462C2212yescomplemented
31#461C2212yes—
32#451—21no—
33#450—21no—
34#441—22no—
35#440—22no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#428—22no—
39C4×M16C2211yescharacteristic
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—48no—
48C22×Q8D411yescharacteristic
49C22×D4D411yescharacteristic
50C23×C4—22no—
51C23×C4—21no—
52C2×Q16—22no—
53C2×SD16—22no—
54C2×M16—21no—
55C4⋊Q8—24no—
56C4⋊Q8—24no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8D412yes—
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2312yes—
63#125—21no—
64C22⋊Q8—24no—
65C22⋊Q8—28no—
66C22⋊Q8—24no—
67C22⋊Q8—28no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8D412yes—
75C4×Q8C2312yes—
76C4×D4—24no—
77C4×D4—24no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4D412yes—
81C4×D4C2312yes—
82#117—21no—
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—24no—
94#107—22no—
95#106—24no—
96#106—22no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—22no—
102#102—22no—
103C22⋊C8—21no—
104C22⋊C8—21no—
105#97—21no—
106#97—21no—
107#97—21no—
108C4×C8C2312yes—
109C24—21no—
110C2×Q8—24no—
111C2×Q8—28no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
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×D4C2×D412yes—
126C22×C4—21no—
127C22×C4—28no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138Q16—44no—
139SD16—44no—
140M16—24no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—28no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C4⋊C4C2×D414yes—
161C22⋊C4—28no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—41no—
165C42—22no—
166C42—22no—
167C42—41no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—24no—
173C23—22no—
174C23—42no—
175C23—42no—
176C23#12011yescharacteristic
177Q8—24no—
178Q8—22no—
179Q8—416no—
180Q8—44no—
181Q8—44no—
182Q8—42no—
183Q8—42no—
184Q8AGL(1,ℤ/8)14yes—
185D4—24no—
186D4—22no—
187D4#13714yes—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—416no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C8—24no—
211C8—42no—
212C8—42no—
213C22—21no—
214C22—21no—
215C22—24no—
216C22—48no—
217C22—42no—
218C22—42no—
219C22#45612yes—
220C22C2×#12011yescharacteristic
221C4—24no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—44no—
227C4—42no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4C2×AGL(1,ℤ/8)12yes—
232C4C2×#13712yes—
233C2—22no—
234C2—44no—
235C2#300311yescharacteristic
236C2#300411yescharacteristic
237C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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. #3165C2semidirect
3. #3113C2semidirect
5. #1651C2semidirect
6. #1649C2semidirect
7. #1556C2semidirect
8. #1491C2semidirect
24. #500C22semidirect
25. #498C22semidirect
28. #475C22semidirect
30. #462C22semidirect
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
#31652
#31132
#31102
#16512
#16492
#15562
#14912
#14902