Finite Group Databasebeta

← #23929

Subgroups — #23929

Order 256.

912 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#537C212yesmaximal, complemented
2C2×#452C212yesmaximal, complemented
3C2×#449C212yesmaximal, complemented
4#2001C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C211yesmaximal, characteristic, complemented
7#1874C212yesmaximal, complemented
8#1872C212yesmaximal, complemented
9#1780C211yesmaximal, characteristic, complemented
10C22×#41C2212yes—
11C22×#41C2211yescharacteristic
12C22×Q16C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2211yescharacteristic
15#537—28no—
16#537—28no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2212yes—
20C2×#121—22no—
21C2×#121—22no—
22C2×#121—22no—
23C2×#121C2212yes—
24C2×#121C2212yes—
25C22×C4⋊C4C2212yes—
26#452—28no—
27#449—28no—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33C2×C22⋊C8C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39C2×#41—24no—
40C2×#41—24no—
41C2×#41—24no—
42C2×#41—24no—
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—41no—
46C2×#41—41no—
47C2×#41—42no—
48C2×#41—42no—
49C22×Q8D411yescharacteristic
50C22×Q8C2312yes—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4D411yescharacteristic
55C22×D4D412yes—
56C22×D4C2312yes—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C2×Q16—24no—
62C2×Q16—41no—
63C2×SD16—24no—
64C2×SD16—24no—
65C2×SD16—42no—
66C2×D8—24no—
67C2×D8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2312yes—
70C22×C8C2311yescharacteristic
71C22⋊Q8—216no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8D418yescomplemented
76#121—216no—
77#121—216no—
78#121—216no—
79#121—24no—
80#121—24no—
81#121—24no—
82#121D418yescomplemented
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94#103—24no—
95#103—24no—
96#102—24no—
97#102—24no—
98C22⋊C8—24no—
99C22⋊C8—24no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102C24—22no—
103C24C2×D411yescharacteristic
104C24C2×D412yes—
105#41—44no—
106#41—48no—
107#41—44no—
108#41—48no—
109#41—48no—
110#41—48no—
111#41—48no—
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8C2×D414yes—
120C2×D4—28no—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—28no—
124C2×D4—28no—
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—42no—
132C2×D4—41no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—82no—
136C2×D4—82no—
137C2×D4—82no—
138C2×D4—82no—
139C2×D4C2×D414yes—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—28no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—28no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D412yes—
159C22×C4C2×D414yes—
160C22×C4C2×D414yes—
161C22×C4C2411yescharacteristic
162Q16—44no—
163SD16—48no—
164D8—44no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—21no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—28no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4C2×D414yes—
178C4⋊C4C2×D414yes—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—28no—
182C22⋊C4—28no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—24no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23#12014yes—
202C23C22×D411yescharacteristic
203Q8—24no—
204Q8—24no—
205Q8—44no—
206Q8—42no—
207Q8—42no—
208D4—24no—
209D4—24no—
210D4—48no—
211D4—44no—
212D4—416no—
213D4—42no—
214D4—42no—
215D4—88no—
216D4—88no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—416no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—88no—
239C2×C4#12014yes—
240C2×C4C22×D412yes—
241C2×C4C22×D412yes—
242C2×C4H(5,𝔽2)11yescharacteristic
243C2×C4H(5,𝔽2)11yescharacteristic
244C8—41no—
245C8—42no—
246C8—41no—
247C22—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—44no—
260C22—42no—
261C22—44no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22C2×#12012yes—
267C22#53411yescharacteristic
268C22#57611yescharacteristic
269C22#57712yes—
270C22#57811yescharacteristic
271C4—24no—
272C4—41no—
273C4—44no—
274C4—44no—
275C4—44no—
276C4D4211yescharacteristic
277C4D4212yes—
278C4D4211yescharacteristic
279C2—44no—
280C2—81no—
281C2—84no—
282C2#240311yescharacteristic
283C2#301212yes—
284C2#301312yes—
285C2#320311yescharacteristic
286C2#320411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#537C2semidirect
2. C2×#452C2semidirect
3. C2×#449C2semidirect
4. #2001C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1874C2semidirect
8. #1872C2semidirect
9. #1780C2semidirect
75. C22⋊Q8D4semidirect
82. #121D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4522
C2×#4492
#20012
#20002
#19992
#18742
#18722
#17802