Finite Group Databasebeta

← #18480

Subgroups — #18480

Order 256.

541 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#3481C211yesmaximal, characteristic, complemented
2#3147C211yesmaximal, characteristic, complemented
3#3106C212yesmaximal, complemented
4#1632C212yesmaximal, complemented
5#1630C211yesmaximal, characteristic, complemented
6#1629C211yesmaximal, characteristic, complemented
7#1622C212yesmaximal, complemented
8#1620C211yesmaximal, characteristic, complemented
9#1619C211yesmaximal, characteristic, complemented
10#1543C211yesmaximal, characteristic, complemented
11#1476C212yesmaximal, complemented
12C2×#135—22no—
13#559C2211yescharacteristic
14#554C2212yes—
15#553—22no—
16#551—22no—
17#546—22no—
18D42C2211yescharacteristic
19#532C2211yescharacteristic
20#532C2211yescharacteristic, complemented
21#529—22no—
22#525—22no—
23C4×#41C2212yes—
24C2×C4×D4—22no—
25#495—22no—
26#494C2211yescharacteristic, complemented
27#493C2211yescharacteristic
28#492C2211yescharacteristic
29#492C2211yescharacteristic, complemented
30#482—22no—
31#477C2212yescomplemented
32#476C2212yes—
33#460C2212yescomplemented
34#459C2212yes—
35#449—22no—
36C4×SD16—22no—
37C4×D8—22no—
38#431C2212yescomplemented
39C2×C4×C8C2211yescharacteristic, complemented
40#333C2212yescomplemented
41#331C2211yescharacteristic
42#331C2211yescharacteristic
43Q8⋊C8C2212yescomplemented
44Q8⋊C8C2212yescomplemented
45D4⋊C8C2212yes—
46D4⋊C8C2212yes—
47C2×#41—21no—
48C2×#41D412yes—
49C2×#41D412yes—
50C22×D4—22no—
51C23×C4—22no—
52#135—48no—
53C2×Q16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×D8—22no—
58C2×D8—22no—
59C22×C8—21no—
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62#127C2311yescharacteristic
63#127C2311yescharacteristic
64#126—24no—
65#125—22no—
66#124—22no—
67#124—21no—
68#124D412yes—
69#123—22no—
70#123—24no—
71#123—22no—
72C22⋊Q8—24no—
73C22⋊Q8—22no—
74#121—24no—
75#121—22no—
76#121—22no—
77#121—22no—
78#120—22no—
79#120—22no—
80C4×Q8D412yes—
81C4×Q8C2312yes—
82C4×D4—24no—
83C4×D4—24no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4D412yes—
91C4×D4C2312yes—
92#117—22no—
93#117—22no—
94#117D412yes—
95C2×C4⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C42—22no—
98C2×C42C2311yescharacteristic
99#107—22no—
100#106—22no—
101C4⋊C8C2312yes—
102C4⋊C8C2312yes—
103#103—22no—
104#102—22no—
105#102—22no—
106#102—22no—
107C22⋊C8—22no—
108C4×C8—21no—
109C4×C8—22no—
110C4×C8C2311yescharacteristic
111C4×C8C2311yescharacteristic
112C24—22no—
113#41—24no—
114#41—42no—
115#41—44no—
116#41—44no—
117C2×Q8—21no—
118C2×Q8C2×D412yes—
119C2×Q8C2×D412yes—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—41no—
129C2×D4C2×D412yes—
130C2×D4C2×D412yes—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—84no—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141Q16—44no—
142SD16—44no—
143SD16—44no—
144SD16—44no—
145D8—44no—
146D8—44no—
147C2×C8—24no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—42no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42C2×D411yescharacteristic
173C42C2×D412yes—
174C42C2411yescharacteristic
175C23—22no—
176C23—22no—
177C23—48no—
178C23—41no—
179C23—82no—
180C23#12011yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—42no—
185D4—22no—
186D4—42no—
187D4—42no—
188D4—44no—
189D4—48no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—42no—
194D4—82no—
195D4—82no—
196D4#13514yescomplemented
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—41no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4—82no—
213C2×C4#12012yes—
214C2×C4#12011yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D412yes—
217C8—22no—
218C8—22no—
219C8—42no—
220C8—42no—
221C22—22no—
222C22—24no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—44no—
227C22—42no—
228C22—88no—
229C22—82no—
230C22—81no—
231C22C2×#12011yescharacteristic
232C4—22no—
233C4—21no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—81no—
239C4#45312yes—
240C4C2×#13512yes—
241C4#57512yes—
242C2—41no—
243C2—44no—
244C2—82no—
245C2—161no—
246C2#300212yes—
247C2C2×#45311yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC42C24
socleC22C2×#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3481C2semidirect
2. #3147C2semidirect
3. #3106C2semidirect
4. #1632C2semidirect
5. #1630C2semidirect
6. #1629C2semidirect
7. #1622C2semidirect
8. #1620C2semidirect
9. #1619C2semidirect
10. #1543C2semidirect
11. #1476C2semidirect
20. #532C22semidirect
26. #494C22semidirect
29. #492C22semidirect
31. #477C22semidirect
33. #460C22semidirect
38. #431C22semidirect
39. C2×C4×C8C22semidirect
40. #333C22semidirect
43. Q8⋊C8C22semidirect
44. Q8⋊C8C22semidirect
196. D4#135semidirect

Maximal subgroups

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

subgroupindex
#34812
#31472
#31062
#16322
#16302
#16292
#16222
#16202
#16192
#15432
#14762