Finite Group Databasebeta

← #18751

Subgroups — #18751

Order 256.

493 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 121 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×#559C211yesmaximal, characteristic
2#3208C211yesmaximal, characteristic
3#3104C211yesmaximal, characteristic, complemented
4#3078C211yesmaximal, characteristic, complemented
5#1640C214yesmaximal, complemented
6#1630C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic
8#1487C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10#559—22no—
11#559—22no—
12#559—22no—
13#559C2214yes—
14#533C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124—21no—
17C2×#124—21no—
18C2×#123—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4—21no—
23#492C2212yes—
24#486C2212yes—
25#484—21no—
26#479—22no—
27#477C2212yescomplemented
28#477C2212yescomplemented
29#475C2212yescomplemented
30#462C2212yescomplemented
31#460—22no—
32#451—21no—
33#450—22no—
34C4×Q16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#331C2214yes—
41Q8⋊C8C2214yescomplemented
42Q8⋊C8C2214yescomplemented
43C22×Q8D411yescharacteristic
44C22×Q8D411yescharacteristic
45C22×D4—21no—
46C22×D4—21no—
47C22×D4D412yes—
48C23×C4—22no—
49C2×Q16—22no—
50C2×SD16—24no—
51C2×M16—21no—
52C4⋊Q8C2311yescharacteristic
53C4⋊Q8C2311yescharacteristic
54#127—21no—
55#127—21no—
56#127D412yes—
57#127C2312yes—
58#125—21no—
59#124—24no—
60#124—24no—
61#124—24no—
62#124—24no—
63#123—28no—
64#123—28no—
65C22⋊Q8—21no—
66C22⋊Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8D412yes—
74C4×Q8D412yes—
75C4×Q8C2312yes—
76C4×Q8C2312yes—
77C4×D4—24no—
78C4×D4—24no—
79C4×D4—24no—
80C4×D4—24no—
81#117—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#107—22no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—22no—
100#103—22no—
101#103—22no—
102#103—22no—
103#102—24no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C24—22no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—42no—
116C2×Q8C2×D412yes—
117C2×Q8C2×D412yes—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—44no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—82no—
127C2×D4C2×D414yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—28no—
133C22×C4—84no—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137Q16—44no—
138SD16—84no—
139M16—42no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—28no—
158C22⋊C4—28no—
159C22⋊C4—28no—
160C22⋊C4—41no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—41no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2411yescharacteristic
170C23—24no—
171C23—44no—
172C23—44no—
173C23—44no—
174C23#12011yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—22no—
178Q8—22no—
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—42no—
183D4—44no—
184D4—44no—
185D4—42no—
186D4—42no—
187D4—44no—
188D4—44no—
189D4—82no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—88no—
203C2×C4#12011yescharacteristic
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#13512yes—
207C2×C4#13712yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C8—42no—
212C8—42no—
213C8—42no—
214C22—21no—
215C22—21no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—88no—
220C22#45212yes—
221C22C2×#12011yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4#45212yes—
232C4C2×#13511yescharacteristic
233C4C2×#13711yescharacteristic
234C4#57512yes—
235C2—22no—
236C2—84no—
237C2C2×#45211yescharacteristic
238C2#300511yescharacteristic
239C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#127C22
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
3. #3104C2semidirect
4. #3078C2semidirect
5. #1640C2semidirect
6. #1630C2semidirect
8. #1487C2semidirect
9. #1479C2semidirect
27. #477C22semidirect
28. #477C22semidirect
29. #475C22semidirect
30. #462C22semidirect
41. Q8⋊C8C22semidirect
42. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
#32082
#31042
#30782
#16402
#16302
#15452
#14872
#14792