Finite Group Databasebeta

← #24006

Subgroups — #24006

Order 256.

750 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#484C212yesmaximal, complemented
3C2×#380C211yesmaximal, characteristic
4#2036C214yesmaximal, complemented
5#1872C212yesmaximal, complemented
6#1852C212yesmaximal, complemented
7#1787C212yesmaximal, complemented
8#1779C211yesmaximal, characteristic
9C23×Q8C2211yescharacteristic
10C24×C4C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—28no—
14C2×C22⋊Q8C2212yes—
15C2×C22⋊Q8C2212yes—
16C22×C4⋊C4—21no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4—21no—
21C22×C22⋊C4—21no—
22#484—28no—
23C2×#107C2214yescomplemented
24C2×#103C2214yescomplemented
25C2×#103C2214yescomplemented
26C2×C22⋊C8C2212yes—
27C2×C22⋊C8C2212yes—
28C2×C22⋊C8C2212yes—
29#380—21no—
30#380—21no—
31#380C2212yes—
32#380C2212yes—
33C2×#95—21no—
34#340C2214yescomplemented
35C25D411yescharacteristic
36C22×Q8—24no—
37C22×Q8—21no—
38C22×Q8—24no—
39C22×Q8—42no—
40C22×Q8C2312yes—
41C23×C4—21no—
42C23×C4—21no—
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4—42no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C22×C8C2314yes—
55C22⋊Q8—28no—
56C22⋊Q8—28no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8D418yes—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—44no—
79#107—28no—
80#103—28no—
81#103—44no—
82C22⋊C8—24no—
83C22⋊C8—24no—
84C22⋊C8—42no—
85#95—22no—
86#95—22no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—41no—
92C24—41no—
93C24Q1614yes—
94C24C2×D411yescharacteristic
95C24C2×D411yescharacteristic
96C24C2×D411yescharacteristic
97C2×Q8—24no—
98C2×Q8—22no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—416no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C2×D414yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—416no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D414yes—
141C22×C4C2×D411yescharacteristic
142C22×C4#4114yes—
143C22×C4C2411yescharacteristic
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—24no—
147C4⋊C4—24no—
148C4⋊C4—28no—
149C4⋊C4—28no—
150C4⋊C4—22no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4C2×D414yes—
156C22⋊C4—28no—
157C22⋊C4—28no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—28no—
175C23—28no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—44no—
180C23—44no—
181C23#12012yes—
182C23#12312yes—
183C23C2×Q1612yes—
184C23C2×Q1612yes—
185C23C2×Q1612yes—
186C23AGL(1,ℤ/8)12yes—
187C23#13712yes—
188C23C22×D411yescharacteristic
189Q8—28no—
190Q8—48no—
191Q8—44no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—48no—
207C2×C4—48no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4#12012yes—
222C2×C4#12312yes—
223C2×C4C22×D412yes—
224C2×C4C2×#4112yes—
225C2×C4H(5,𝔽2)12yes—
226C8—44no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—416no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22#45114yes—
248C22#48414yescomplemented
249C22C2×#12011yescharacteristic
250C22C2×#12311yescharacteristic
251C22#53411yescharacteristic
252C22C22×Q1611yescharacteristic
253C22C2×AGL(1,ℤ/8)11yescharacteristic
254C22C2×#13711yescharacteristic
255C22#57511yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—44no—
260C4—44no—
261C4—82no—
262C4—82no—
263C4#54614yes—
264C2—24no—
265C2—24no—
266C2—42no—
267C2#240511yescharacteristic
268C2C2×#45111yescharacteristic
269C2#300611yescharacteristic
270C2C2×#48411yescharacteristic
271C2#309511yescharacteristic
272C2#318911yescharacteristic
273C2#319311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
4. #2036C2semidirect
5. #1872C2semidirect
6. #1852C2semidirect
7. #1787C2semidirect
23. C2×#107C22semidirect
24. C2×#103C22semidirect
25. C2×#103C22semidirect
34. #340C22semidirect
248. C22#484semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4842
C2×#3802
#20362
#18722
#18522
#17872
#17792