Finite Group Databasebeta

← #25857

Subgroups — #25857

Order 256.

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

normal subgroups only

This group has 695 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×#532C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic, complemented
5#2042C214yesmaximal, complemented
6#1957C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1788C214yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic, complemented
10C22×#41C2212yes—
11#532—24no—
12#532—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124C2212yes—
17C2×C22⋊Q8—21no—
18C2×#121—21no—
19C2×#121—21no—
20C2×C4×D4—21no—
21C2×C4×D4—21no—
22C22×C42C2211yescharacteristic
23#487—24no—
24#486—24no—
25C2×#103C2214yes—
26C2×#103C2214yes—
27C2×#102C2214yes—
28C2×#102C2214yes—
29C2×C22⋊C8C2212yes—
30C2×C4×C8C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#391—21no—
33#389—21no—
34#389C2212yes—
35#384C2211yescharacteristic
36#380—21no—
37#380C2212yes—
38#336C2212yes—
39C2×#41—24no—
40C2×#41—24no—
41C2×#41—28no—
42C2×#41—44no—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×D4—22no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4C2312yes—
49C23×C4—22no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#127—22no—
60#127D414yes—
61#124—24no—
62#124—24no—
63C22⋊Q8—44no—
64#121—28no—
65#121—28no—
66C4×D4—28no—
67C4×D4—28no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#103—28no—
86#103—44no—
87#102—28no—
88#102—44no—
89C22⋊C8—24no—
90C4×C8—22no—
91C4×C8—22no—
92#95—22no—
93#95—22no—
94C24—22no—
95C24C2×D411yescharacteristic
96#41—416no—
97#41—416no—
98C2×Q8—22no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C2×D414yes—
105C2×D4—28no—
106C2×D4—24no—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4C2×D414yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4#4114yes—
139C22×C4C2411yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C22⋊C4—44no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42C2×D412yes—
163C42C2×D412yes—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—24no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23#12412yes—
178C23C22×D411yescharacteristic
179Q8—28no—
180Q8—44no—
181Q8—84no—
182D4—28no—
183D4—416no—
184D4—44no—
185D4—48no—
186D4—48no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—416no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#12012yes—
210C2×C4#12012yes—
211C2×C4#12412yes—
212C2×C4#13512yes—
213C2×C4#13512yes—
214C2×C4AGL(1,ℤ/8)12yes—
215C2×C4#13712yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4H(5,𝔽2)12yes—
220C8—42no—
221C8—42no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22C2×#12011yescharacteristic
237C22C2×#12411yescharacteristic
238C22#53511yescharacteristic
239C22C2×#13511yescharacteristic
240C22C2×#13511yescharacteristic
241C22C2×AGL(1,ℤ/8)11yescharacteristic
242C22C2×#13711yescharacteristic
243C4—22no—
244C4—44no—
245C4—41no—
246C4—41no—
247C4—82no—
248C4—82no—
249C4#44914yes—
250C4#45214yes—
251C4#54612yes—
252C4#54612yes—
253C2—42no—
254C2—84no—
255C2#243511yescharacteristic
256C2C2×#44911yescharacteristic
257C2C2×#45211yescharacteristic
258C2#313111yescharacteristic
259C2#314111yescharacteristic
260C2#320711yescharacteristic
261C2#321011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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
2. C2×#487C2semidirect
3. C2×#486C2semidirect
4. #2857C2semidirect
5. #2042C2semidirect
6. #1957C2semidirect
7. #1956C2semidirect
8. #1788C2semidirect
9. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4872
C2×#4862
#28572
#20422
#19572
#19562
#17882
C4×C22⋊C82