Finite Group Databasebeta

← #24972

Subgroups — #24972

Order 256.

385 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#2538C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2035C212yesmaximal, complemented
5#1954C211yesmaximal, characteristic
6#1948C211yesmaximal, characteristic, complemented
7#1873C212yesmaximal, complemented
8#1834C211yesmaximal, characteristic, complemented
9#1825C212yesmaximal, complemented
10#1803C211yesmaximal, characteristic, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C2×C22⋊Q8C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2211yescharacteristic
19C2×#103C2211yescharacteristic
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2212yes—
23C2×C4×C8C2211yescharacteristic
24#396—22no—
25#390—21no—
26#390—21no—
27#389—21no—
28#388—22no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#380C2212yes—
32#380C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4D411yescharacteristic
43C23×C4C2312yes—
44C22×C8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C22×C8C2312yes—
47C22⋊Q8—24no—
48C22⋊Q8—42no—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×D4—22no—
52C4×D4—22no—
53#117—24no—
54#117—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4D412yes—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C2311yescharacteristic
76#106—42no—
77C4⋊C8—22no—
78#103—22no—
79#103—22no—
80#103—42no—
81C22⋊C8—42no—
82C22⋊C8—42no—
83C4×C8—22no—
84#95—22no—
85#95—21no—
86#95—22no—
87C24C2×D412yes—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8#4112yes—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C4⋊C4—81no—
138C4⋊C4#4112yes—
139C22⋊C4—24no—
140C22⋊C4—44no—
141C22⋊C4—44no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—44no—
145C42—42no—
146C42—42no—
147C42#4112yes—
148C42#4112yes—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23C22×D411yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161D4—44no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4#12312yes—
190C2×C4#12312yes—
191C2×C4#13512yes—
192C2×C4#13712yes—
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4H(5,𝔽2)11yescharacteristic
198C2×C4H(5,𝔽2)11yescharacteristic
199C8—42no—
200C8—41no—
201C8—41no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22C2×#12311yescharacteristic
210C22#54611yescharacteristic
211C22#54611yescharacteristic
212C22C2×#13511yescharacteristic
213C22C2×#13711yescharacteristic
214C22#57711yescharacteristic
215C22#57711yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—81no—
221C4—81no—
222C4—82no—
223C4—82no—
224C4#48514yes—
225C4#52911yescharacteristic
226C4#54011yescharacteristic
227C4#55311yescharacteristic
228C4#55511yescharacteristic
229C2—82no—
230C2#242011yescharacteristic
231C2C2×#48511yescharacteristic
232C2#309911yescharacteristic
233C2#330411yescharacteristic
234C2#330411yescharacteristic
235C2#332911yescharacteristic
236C2#333111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
3. #2077C2semidirect
4. #2035C2semidirect
6. #1948C2semidirect
7. #1873C2semidirect
8. #1834C2semidirect
9. #1825C2semidirect
10. #1803C2semidirect
11. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
#25382
#23542
#20772
#20352
#19542
#19482
#18732
#18342
#18252
#18032
C4×#1032