Finite Group Databasebeta

← #18656

Subgroups — #18656

Order 256.

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

normal subgroups only

This group has 480 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
1#3511C211yesmaximal, characteristic
2#3226C211yesmaximal, characteristic, complemented
3C2×#498C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1654C214yesmaximal, complemented
6#1652C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×#137—21no—
11#556C2212yes—
12#555—21no—
13#554—21no—
14#553—22no—
15#548—22no—
16#547C2212yes—
17#542—22no—
18#537—21no—
19#533—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#520—21no—
25C4×#41C2211yescharacteristic
26#500—21no—
27#498—22no—
28#498C2214yes—
29#486—21no—
30#480—22no—
31#474C2214yes—
32#462—21no—
33#462C2212yes—
34#461—21no—
35#461C2212yes—
36#431C2211yescharacteristic
37C2×#106—22no—
38#423C2211yescharacteristic
39#419—21no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yes—
42Q8⋊C8C2212yes—
43D4⋊C8C2214yes—
44D4⋊C8C2212yes—
45C2×#41—22no—
46C2×#41—21no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×Q8—21no—
50#137—44no—
51C2×Q16—22no—
52C2×SD16—22no—
53C2×M16—21no—
54C22×C8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D412yes—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8C2312yes—
63C4⋊Q8C2312yes—
64#127D411yescharacteristic
65#126—24no—
66#125—22no—
67#124—24no—
68#124—22no—
69#124—21no—
70#123—24no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75#121—22no—
76#121—22no—
77#121—24no—
78#121—22no—
79#121—21no—
80#121D412yes—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8D411yescharacteristic
84C4×Q8C2311yescharacteristic
85C4×D4—22no—
86C4×D4—24no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4D411yescharacteristic
92C4×D4C2311yescharacteristic
93C4×D4C2312yes—
94#117—22no—
95#117—22no—
96#117—21no—
97#117D411yescharacteristic
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D412yes—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#107—24no—
106#106—28no—
107C4⋊C8—21no—
108C4⋊C8C2312yes—
109C4⋊C8C2312yes—
110#103—22no—
111#103—22no—
112#102—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117#41—22no—
118#41—22no—
119#41—42no—
120#41—42no—
121#41—84no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—82no—
130C2×Q8C2×D411yescharacteristic
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C2×D412yes—
137C2×D4C2×D411yescharacteristic
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148Q16—44no—
149SD16—44no—
150M16—42no—
151C2×C8—24no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4C2×D411yescharacteristic
168C4⋊C4C2×D411yescharacteristic
169C4⋊C4C2×D414yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—22no—
177C22⋊C4—21no—
178C22⋊C4—22no—
179C42—24no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2411yescharacteristic
187C23—21no—
188C23—22no—
189C23#12011yescharacteristic
190Q8—21no—
191Q8—21no—
192Q8—44no—
193Q8—42no—
194Q8—42no—
195Q8—42no—
196Q8—82no—
197Q8—82no—
198D4—21no—
199D4—21no—
200D4—24no—
201D4—42no—
202D4—44no—
203D4—42no—
204D4—42no—
205D4—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—84no—
221C2×C4—82no—
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#12011yescharacteristic
225C2×C4#13512yes—
226C2×C4AGL(1,ℤ/8)12yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C8—24no—
231C8—42no—
232C8—42no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22#44912yes—
242C22C2×#12011yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—44no—
249C4—41no—
250C4—82no—
251C4#44912yes—
252C4#45612yes—
253C4C2×#13511yescharacteristic
254C4C2×AGL(1,ℤ/8)11yescharacteristic
255C4#57712yes—
256C2—22no—
257C2—81no—
258C2—82no—
259C2C2×#44911yescharacteristic
260C2#300911yescharacteristic
261C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3511C2
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. #3226C2semidirect
3. C2×#498C2semidirect
4. #3039C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1549C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#32262
C2×#4982
#30392
#16542
#16522
#15492
#14922
#14762