Finite Group Databasebeta

← #18777

Subgroups — #18777

Order 256.

569 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 133 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#3229C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic
4#3078C211yesmaximal, characteristic, complemented
5#1651C214yesmaximal, complemented
6#1637C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic
9#1479C211yesmaximal, characteristic, complemented
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#498C2212yescomplemented
24#487C2212yescomplemented
25#483—21no—
26#483—22no—
27#479—22no—
28#477C2212yescomplemented
29#476C2212yes—
30#475—22no—
31#475C2212yescomplemented
32#461C2212yes—
33#450—21no—
34C4×SD16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—44no—
48C22×Q8—44no—
49C22×Q8D411yescharacteristic
50C22×D4D411yescharacteristic
51C23×C4—22no—
52C23×C4—21no—
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—22no—
94#106—22no—
95#106—22no—
96#106—24no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8C2312yes—
100C4⋊C8C2312yes—
101#103—22no—
102#103—22no—
103#103—22no—
104#103—24no—
105#102—22no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111C2×Q8—28no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—48no—
119C2×Q8—48no—
120C2×Q8—416no—
121C2×Q8—48no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8SD1614yescomplemented
125C2×Q8C2×D412yes—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4C2×D412yes—
129C22×C4—21no—
130C22×C4—28no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—48no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142SD16—44no—
143M16—42no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C4⋊C4—28no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D414yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—28no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—41no—
168C42—22no—
169C42—22no—
170C42—41no—
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2411yescharacteristic
175C23—24no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23#12011yescharacteristic
180Q8—22no—
181Q8—24no—
182Q8—416no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—44no—
187Q8—42no—
188Q8C2×SD1614yes—
189D4—24no—
190D4—22no—
191D4#13714yes—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—416no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4#12011yescharacteristic
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4C2×SD1612yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C8—42no—
217C8—42no—
218C8—42no—
219C22—21no—
220C22—21no—
221C22—24no—
222C22—48no—
223C22—42no—
224C22—42no—
225C22#45012yes—
226C22C2×#12011yescharacteristic
227C4—24no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—44no—
232C4—42no—
233C4—42no—
234C4—81no—
235C4—81no—
236C4#45012yes—
237C4C22×SD1611yescharacteristic
238C4C2×AGL(1,ℤ/8)11yescharacteristic
239C4C2×#13712yes—
240C2—22no—
241C2—44no—
242C2C2×#45011yescharacteristic
243C2#300411yescharacteristic
244C2#302011yescharacteristic
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. #3229C2semidirect
4. #3078C2semidirect
5. #1651C2semidirect
6. #1637C2semidirect
7. #1546C2semidirect
9. #1479C2semidirect
23. #498C22semidirect
24. #487C22semidirect
28. #477C22semidirect
31. #475C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
124. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
#32292
#31012
#30782
#16512
#16372
#15462
#14862
#14792