Finite Group Databasebeta

← #25498

Subgroups — #25498

Order 256.

566 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 149 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×#533C211yesmaximal, characteristic
2C2×#467C212yesmaximal
3#2855C211yesmaximal, characteristic
4#2057C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#2016C211yesmaximal, characteristic, complemented
8#1892C212yesmaximal
9#1878C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C23×C8C2212yes—
14#533—28no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2212yes—
21C2×C4×Q8—21no—
22C2×C4×Q8C2212yes—
23C2×#117—21no—
24C22×C42C2211yescharacteristic
25#467—28no—
26C2×#107C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#387—21no—
36#387—21no—
37#385—21no—
38#381—21no—
39#380C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2212yes—
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×Q8C2312yes—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—42no—
54C22×C8—24no—
55C22×C8—24no—
56C22×C8—42no—
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—24no—
60C4⋊Q8—24no—
61#125—28no—
62C22⋊Q8—28no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
65C22⋊Q8D418yes—
66C4×Q8—28no—
67C4×Q8—24no—
68C4×Q8—24no—
69#117—28no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C42—22no—
86C2×C42—24no—
87C2×C42D411yescharacteristic
88C2×C42D411yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#107—24no—
92#103—24no—
93#103—24no—
94#103—24no—
95#103—42no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103C24C2×D411yescharacteristic
104C2×Q8—24no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8C2×D414yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4Q1614yescomplemented
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D414yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139Q16—48no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—48no—
146C2×C8#4114yes—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—28no—
150C4⋊C4—24no—
151C4⋊C4—28no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4C2×D414yes—
161C22⋊C4—28no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—28no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—22no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23#12012yes—
174C23C2×Q1612yes—
175C23#13712yes—
176C23C22×D411yescharacteristic
177Q8—24no—
178Q8—24no—
179Q8—48no—
180Q8—42no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4#12012yes—
218C2×C4C2×Q1612yes—
219C2×C4C2×Q1612yes—
220C2×C4#13512yes—
221C2×C4C22×D412yes—
222C2×C4C2×#4112yes—
223C2×C4H(5,𝔽2)11yescharacteristic
224C2×C42−1+411yescharacteristic
225C8—24no—
226C8—42no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22#45114yes—
234C22C2×#12011yescharacteristic
235C22#53211yescharacteristic
236C22#53711yescharacteristic
237C22C22×Q1611yescharacteristic
238C22C2×#13511yescharacteristic
239C22C2×#13711yescharacteristic
240C22#57511yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—44no—
246C4—41no—
247C4—44no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4#54612yes—
253C4#54712yes—
254C2—24no—
255C2#243011yescharacteristic
256C2C2×#45111yescharacteristic
257C2#300211yescharacteristic
258C2#317011yescharacteristic
259C2#318111yescharacteristic
260C2#324811yescharacteristic
261C2#324911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
4. #2057C2semidirect
5. #2043C2semidirect
6. #2024C2semidirect
7. #2016C2semidirect
9. #1878C2semidirect
10. #1863C2semidirect
131. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4672
#28552
#20572
#20432
#20242
#20162
#18922
#18782
#18632
C2×#3362