Finite Group Databasebeta

← #24209

Subgroups — #24209

Order 256.

757 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 167 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×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#461C212yesmaximal, complemented
3C2×#448C212yesmaximal
4C4×#121C211yesmaximal, characteristic
5#2043C212yesmaximal, complemented
6#2013C212yesmaximal, complemented
7#1981C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic, complemented
9#1787C212yesmaximal, complemented
10C23×Q8C2212yes—
11C22×SD16C2214yes—
12C2×C4⋊Q8—28no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2212yes—
17C2×#121C2211yescharacteristic
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4—21no—
21C22×C4⋊C4—21no—
22C22×C42C2211yescharacteristic
23#461—28no—
24#448—28no—
25C2×C4⋊C8C2212yes—
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×#103C2214yes—
29C2×C22⋊C8C2212yes—
30#390C2211yescharacteristic
31#389C2211yescharacteristic
32#388—21no—
33#386—21no—
34#380C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#336C2212yes—
38C22×Q8—28no—
39C22×Q8—22no—
40C22×Q8—24no—
41C22×Q8—24no—
42C22×Q8—44no—
43C22×Q8C2314yes—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—22no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×SD16—28no—
52C2×SD16—28no—
53C2×SD16—44no—
54C22×C8C2314yes—
55C4⋊Q8—28no—
56C4⋊Q8—28no—
57C4⋊Q8—28no—
58C4⋊Q8—24no—
59C4⋊Q8D418yescomplemented
60#121—22no—
61#121D414yes—
62C4×D4—24no—
63C4×D4—44no—
64C2×C4⋊C4—28no—
65C2×C4⋊C4—28no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C42—24no—
80C2×C42—21no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8—24no—
86#103—28no—
87#103—28no—
88#103—44no—
89C22⋊C8—24no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—28no—
96C2×Q8—24no—
97C2×Q8—24no—
98C2×Q8—24no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—416no—
102C2×Q8—416no—
103C2×Q8—44no—
104C2×Q8—44no—
105C2×Q8C2×D418yes—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4C2×D412yes—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—28no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4SD1614yescomplemented
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139SD16—416no—
140C2×C8—24no—
141C2×C8—24no—
142C2×C8—24no—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—28no—
146C4⋊C4—28no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4C2×D412yes—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—42no—
161C42C2×D414yes—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23#12112yes—
174C23C2×SD1612yes—
175C23#13712yes—
176C23C22×D411yescharacteristic
177Q8—28no—
178Q8—28no—
179Q8—416no—
180Q8—44no—
181Q8—44no—
182D4—24no—
183D4—48no—
184D4—42no—
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—22no—
199C2×C4—416no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—84no—
211C2×C4#12012yes—
212C2×C4#12012yes—
213C2×C4#12112yes—
214C2×C4C2×SD1612yes—
215C2×C4C2×SD1612yes—
216C2×C4#13712yes—
217C2×C4#13712yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D412yes—
220C2×C4C2×#4111yescharacteristic
221C2×C4H(5,𝔽2)12yes—
222C8—44no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22#46114yescomplemented
237C22C2×#12011yescharacteristic
238C22C2×#12111yescharacteristic
239C22#53511yescharacteristic
240C22C22×SD1611yescharacteristic
241C22C2×#13711yescharacteristic
242C22C2×#13711yescharacteristic
243C22C2×#13711yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—48no—
249C4—42no—
250C4—81no—
251C4—81no—
252C4#44814yescomplemented
253C4D4212yes—
254C4#54612yes—
255C2—24no—
256C2—82no—
257C2#240811yescharacteristic
258C2C2×#44811yescharacteristic
259C2#300411yescharacteristic
260C2C2×#46111yescharacteristic
261C2#303911yescharacteristic
262C2#320611yescharacteristic
263C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Q8C2semidirect
2. C2×#461C2semidirect
5. #2043C2semidirect
6. #2013C2semidirect
7. #1981C2semidirect
8. #1797C2semidirect
9. #1787C2semidirect
59. C4⋊Q8D4semidirect
130. C22×C4SD16semidirect
236. C22#461semidirect
252. C4#448semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4612
C2×#4482
C4×#1212
#20432
#20132
#19812
#17972
#17872