Finite Group Databasebeta

← #18939

Subgroups — #18939

Order 256.

429 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 113 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#3550C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic, complemented
3#3237C211yesmaximal, characteristic, complemented
4#3210C211yesmaximal, characteristic
5#1645C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1633C212yesmaximal, complemented
8#1630C212yesmaximal, complemented
9#1551C211yesmaximal, characteristic
10#1497C211yesmaximal, characteristic, complemented
11#1481C211yesmaximal, characteristic, complemented
12#563—22no—
13#559C2212yes—
14Q82C2212yes—
15#556—21no—
16#556—21no—
17D4×Q8—21no—
18D4×Q8—21no—
19#542—22no—
20#542—22no—
21#536—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#533C2211yescharacteristic
25#532C2211yescharacteristic
26#501—21no—
27#494C2211yescharacteristic
28#492C2211yescharacteristic
29#491—21no—
30#487C2211yescharacteristic
31#486C2211yescharacteristic
32#485—22no—
33#477—21no—
34#477C2212yes—
35#477C2212yes—
36#475C2212yes—
37#467—22no—
38#464—22no—
39#462—21no—
40#462C2212yes—
41#452—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45#333C2212yes—
46#331C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×#41D412yes—
52C22×Q8—22no—
53C2×Q16—22no—
54C2×Q16—22no—
55C2×SD16—22no—
56C22×C8—21no—
57C4⋊Q8—24no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8D411yescharacteristic
62C4⋊Q8D411yescharacteristic
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#127C2311yescharacteristic
68#126—24no—
69#126—24no—
70#125—21no—
71#125—21no—
72#125—21no—
73#124—24no—
74#124—22no—
75#124—22no—
76#124D412yes—
77#123—24no—
78#123—24no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8D412yes—
88C22⋊Q8D412yes—
89#121—21no—
90#121—21no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8C2312yes—
98C4×Q8C2312yes—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—21no—
103#117—22no—
104#117—21no—
105#117—21no—
106#117—21no—
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126#41—82no—
127C2×Q8—24no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—84no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C2×D412yes—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145Q16—82no—
146Q16—82no—
147SD16—82no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—81no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—41no—
183C42—24no—
184C42—22no—
185C42—22no—
186C42—41no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2411yescharacteristic
191C23—22no—
192C23#12011yescharacteristic
193Q8—22no—
194Q8—24no—
195Q8—22no—
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—41no—
201Q8—41no—
202Q8—84no—
203D4—42no—
204D4—41no—
205D4—42no—
206D4—41no—
207D4—82no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—84no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—41no—
228C8—41no—
229C8—42no—
230C8—42no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22C2×#12011yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4#57711yescharacteristic
250C4#57812yes—
251C4#57811yescharacteristic
252C2—41no—
253C2—82no—
254C2#301111yescharacteristic
255C2#301311yescharacteristic
256C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
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. #3249C2semidirect
3. #3237C2semidirect
5. #1645C2semidirect
6. #1640C2semidirect
7. #1633C2semidirect
8. #1630C2semidirect
10. #1497C2semidirect
11. #1481C2semidirect

Maximal subgroups

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

subgroupindex
#35502
#32492
#32372
#32102
#16452
#16402
#16332
#16302
#15512
#14972
#14812