Finite Group Databasebeta

← #18773

Subgroups — #18773

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×#555C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1646C214yesmaximal, complemented
6#1634C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10#555—22no—
11#555—22no—
12#555—22no—
13#555C2214yes—
14#533C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124—22no—
17C2×#121—21no—
18C2×#121—21no—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117—21no—
23#494C2212yescomplemented
24#487C2212yescomplemented
25#483—21no—
26#478—22no—
27#476C2212yescomplemented
28#476C2212yescomplemented
29#474C2212yescomplemented
30#462—22no—
31#461C2212yescomplemented
32#451—22no—
33#450—21no—
34C4×SD16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#418—21no—
40#333C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C22×Q8D412yes—
44C22×D4—22no—
45C22×D4D411yescharacteristic
46C22×D4D411yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C2×Q16—24no—
50C2×SD16—22no—
51C2×M16—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8D412yes—
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2312yes—
58#125—21no—
59#124—28no—
60#124—28no—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63#121—24no—
64#121—24no—
65#121—24no—
66#121—24no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×D4—21no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4D412yes—
74C4×D4D412yes—
75C4×D4C2312yes—
76C4×D4C2312yes—
77#117—24no—
78#117—24no—
79#117—24no—
80#117—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—21no—
89C2×C22⋊C4—21no—
90C2×C42—22no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#106—22no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8C2312yes—
98C4⋊C8C2312yes—
99#103—24no—
100#103—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C4×C8—21no—
107C4×C8C2312yes—
108C24—21no—
109C24—21no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—82no—
114C2×Q8C2×D414yes—
115C2×D4—28no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—42no—
121C2×D4—84no—
122C2×D4C2×D412yes—
123C2×D4C2×D412yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137Q16—84no—
138SD16—44no—
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—22no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—41no—
164C42—28no—
165C42—41no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2411yescharacteristic
170C23—22no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23#12011yescharacteristic
179Q8—44no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—22no—
184D4—22no—
185D4—22no—
186D4—22no—
187D4—42no—
188D4—42no—
189D4—88no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—41no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4#13512yes—
209C2×C4#13712yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C8—42no—
214C8—42no—
215C8—42no—
216C22—21no—
217C22—21no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—84no—
225C22—84no—
226C22#45212yes—
227C22C2×#12011yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—42no—
232C4—42no—
233C4—44no—
234C4—81no—
235C4—81no—
236C4#45212yes—
237C4C2×#13511yescharacteristic
238C4C2×#13711yescharacteristic
239C4#57512yes—
240C2—22no—
241C2—82no—
242C2—82no—
243C2C2×#45211yescharacteristic
244C2#300511yescharacteristic
245C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#555C2
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. #3209C2semidirect
3. #3101C2semidirect
4. #3077C2semidirect
5. #1646C2semidirect
6. #1634C2semidirect
7. #1548C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
23. #494C22semidirect
24. #487C22semidirect
27. #476C22semidirect
28. #476C22semidirect
29. #474C22semidirect
31. #461C22semidirect
40. #333C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
#32092
#31012
#30772
#16462
#16342
#15482
#14862
#14782