Finite Group Databasebeta

← #24730

Subgroups — #24730

Order 256.

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

normal subgroups only

This group has 509 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×#461C212yesmaximal
2C2×#461C211yesmaximal, characteristic
3C2×C4×SD16C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic, complemented
5D4×C4⋊C4C211yesmaximal, characteristic
6#2066C212yesmaximal, complemented
7#2043C212yesmaximal, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1877C212yesmaximal
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4—21no—
20#461—28no—
21#461—24no—
22C4×SD16—24no—
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#102C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#398—22no—
34#391C2212yes—
35#389—21no—
36#389—21no—
37#389C2212yes—
38#385—22no—
39#384C2211yescharacteristic
40#380—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—22no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16D414yes—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C4⋊Q8—24no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—42no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4—42no—
67C4×D4D414yes—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#106—22no—
88C4⋊C8—24no—
89C4⋊C8—22no—
90#103—24no—
91#103—24no—
92#103—22no—
93#103—24no—
94#103—42no—
95#102—22no—
96C4×C8—22no—
97#95—22no—
98#95—22no—
99#95—22no—
100#95D412yes—
101C24—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4C2×D412yes—
118C2×D4C2×D412yes—
119C2×D4#4112yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—82no—
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—28no—
137SD16—48no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8C2×D412yes—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—81no—
161C4⋊C4SD1614yescomplemented
162C4⋊C4C2×D412yes—
163C22⋊C4—24no—
164C22⋊C4—44no—
165C42—22no—
166C42—22no—
167C42—42no—
168C42C2×D412yes—
169C42#4112yes—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23C22×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—41no—
181Q8—41no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186D4—22no—
187D4—22no—
188D4—22no—
189D4#12114yes—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—48no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—88no—
214C2×C4#12112yes—
215C2×C4#12112yes—
216C2×C4C2×SD1614yes—
217C2×C4C2×SD1612yes—
218C2×C4#13712yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4H(5,𝔽2)11yescharacteristic
224C2×C42−1+411yescharacteristic
225C8—22no—
226C8—42no—
227C8—41no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22C2×#12111yescharacteristic
236C22D4211yescharacteristic
237C22#54811yescharacteristic
238C22C22×SD1611yescharacteristic
239C22C2×#13711yescharacteristic
240C22#57711yescharacteristic
241C22#57811yescharacteristic
242C4—24no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—82no—
248C4—82no—
249C4—81no—
250C4#46114yescomplemented
251C4C2×#12111yescharacteristic
252C4#53711yescharacteristic
253C4#54711yescharacteristic
254C4#55311yescharacteristic
255C2—44no—
256C2#241611yescharacteristic
257C2C2×#46111yescharacteristic
258C2#304611yescharacteristic
259C2#327611yescharacteristic
260C2D4×SD1611yescharacteristic
261C2#335911yescharacteristic
262C2#337211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2546C2semidirect
6. #2066C2semidirect
7. #2043C2semidirect
8. #1956C2semidirect
9. #1939C2semidirect
11. #1803C2semidirect
161. C4⋊C4SD16semidirect
250. C4#461semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×C4×SD162
#25462
D4×C4⋊C42
#20662
#20432
#19562
#19392
#18772
#18032