Finite Group Databasebeta

← #25347

Subgroups — #25347

Order 256.

473 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 131 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×#474C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic, complemented
3#2547C211yesmaximal, characteristic, complemented
4#2537C211yesmaximal, characteristic
5#2076C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1895C212yesmaximal, complemented
10#1809C212yesmaximal, complemented
11C4×#102C211yesmaximal, characteristic, complemented
12C22×M16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124—21no—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#474—24no—
20#436—24no—
21C2×#107C2212yes—
22C2×#106C2212yes—
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×C22⋊C8C2212yes—
28C2×C4×C8C2211yescharacteristic
29#397—22no—
30#394—22no—
31#391—21no—
32#384C2211yescharacteristic
33#380—21no—
34#380C2212yes—
35#380C2212yes—
36#380C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38#340C2211yescharacteristic
39#340C2212yes—
40#340C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8—21no—
44C22×D4—22no—
45C22×D4D411yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4C2312yes—
49C2×M16—24no—
50C2×M16—24no—
51C2×M16—42no—
52C22×C8C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57#124—44no—
58#121—24no—
59#121—42no—
60#121—44no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4D412yes—
80C2×C42—21no—
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—42no—
84C4⋊C8D414yescomplemented
85#102—22no—
86#102—24no—
87#102—22no—
88C22⋊C8—24no—
89C4×C8—22no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95—22no—
94C24—21no—
95C24C2×D412yes—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—82no—
110C2×D4#4112yes—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129M16—48no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8C2×D414yes—
138C2×C8#4112yes—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—81no—
153C22⋊C4—24no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—42no—
159C42—42no—
160C42C2×D412yes—
161C42#4112yes—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23C22×D411yescharacteristic
177Q8—84no—
178D4—22no—
179D4—22no—
180D4—41no—
181D4—41no—
182D4—44no—
183D4—88no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—84no—
209C2×C4#12112yes—
210C2×C4#12112yes—
211C2×C4AGL(1,ℤ/8)12yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4H(5,𝔽2)11yescharacteristic
218C2×C42−1+411yescharacteristic
219C8—24no—
220C8—22no—
221C8—41no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22C2×#12111yescharacteristic
237C22#54611yescharacteristic
238C22#54711yescharacteristic
239C22C2×AGL(1,ℤ/8)11yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22#57711yescharacteristic
242C22#57811yescharacteristic
243C4—22no—
244C4—42no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4—82no—
249C4—82no—
250C4#46914yes—
251C4#52911yescharacteristic
252C4#54211yescharacteristic
253C4#54611yescharacteristic
254C4#54811yescharacteristic
255C2—82no—
256C2—82no—
257C2#242711yescharacteristic
258C2C2×#46911yescharacteristic
259C2#306311yescharacteristic
260C2#332411yescharacteristic
261C2#333011yescharacteristic
262C2#335311yescharacteristic
263C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#474C2semidirect
2. C2×#436C2semidirect
3. #2547C2semidirect
5. #2076C2semidirect
6. #2034C2semidirect
7. #1990C2semidirect
8. #1948C2semidirect
9. #1895C2semidirect
10. #1809C2semidirect
11. C4×#102C2semidirect
84. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4362
#25472
#25372
#20762
#20342
#19902
#19482
#18952
#18092
C4×#1022