Finite Group Databasebeta

← #18949

Subgroups — #18949

Order 256.

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

normal subgroups only

This group has 423 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#3246C211yesmaximal, characteristic
3#3237C211yesmaximal, characteristic, complemented
4#3231C211yesmaximal, characteristic, complemented
5#1651C212yesmaximal, complemented
6#1650C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1637C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic
11#1481C211yesmaximal, characteristic, complemented
12#563—22no—
13#559—21no—
14Q82—21no—
15#556—21no—
16#556C2212yes—
17D4×Q8—21no—
18D4×Q8C2212yes—
19#542—22no—
20#542—22no—
21#536—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#533C2211yescharacteristic
25#532C2211yescharacteristic
26#501—21no—
27#500C2211yescharacteristic
28#498C2211yescharacteristic
29#491—21no—
30#487C2211yescharacteristic
31#487C2211yescharacteristic
32#485—22no—
33#485—22no—
34#479—22no—
35#477C2212yes—
36#476C2212yes—
37#475—21no—
38#475C2212yes—
39#465—22no—
40#461C2212yes—
41#460—21no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45#332C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C2×#41—21no—
52C22×Q8—22no—
53C2×SD16—22no—
54C22×C8—21no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8D411yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65#127D411yescharacteristic
66#126—24no—
67#126—24no—
68#125—21no—
69#125—21no—
70#125D412yes—
71#124—24no—
72#124—22no—
73#124—22no—
74#124—21no—
75#123—24no—
76#123—24no—
77C22⋊Q8—24no—
78C22⋊Q8—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—22no—
83C22⋊Q8—24no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8D412yes—
87#121—21no—
88#121D412yes—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8C2312yes—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4C2312yes—
101#117—21no—
102#117—22no—
103#117—21no—
104#117D412yes—
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#106—22no—
109#106—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—21no—
133C2×Q8—21no—
134C2×Q8—84no—
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
145SD16—82no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—81no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C42—24no—
182C42—22no—
183C42—22no—
184C42—41no—
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2411yescharacteristic
189C23—22no—
190C23#12011yescharacteristic
191Q8—24no—
192Q8—42no—
193Q8—42no—
194Q8—41no—
195Q8—42no—
196Q8—41no—
197Q8—41no—
198Q8—41no—
199Q8—81no—
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—81no—
205D4—82no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—84no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C8—41no—
226C8—41no—
227C8—42no—
228C8—42no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22C2×#12011yescharacteristic
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—81no—
246C4—81no—
247C4#57712yes—
248C4#57711yescharacteristic
249C4#57811yescharacteristic
250C2—41no—
251C2—82no—
252C2#300911yescharacteristic
253C2#301311yescharacteristic
254C2#302011yescharacteristic
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
3. #3237C2semidirect
4. #3231C2semidirect
5. #1651C2semidirect
6. #1650C2semidirect
7. #1638C2semidirect
8. #1637C2semidirect
9. #1552C2semidirect
11. #1481C2semidirect

Maximal subgroups

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

subgroupindex
#35502
#32462
#32372
#32312
#16512
#16502
#16382
#16372
#15522
#14962
#14812