Finite Group Databasebeta

← #27587

Subgroups — #27587

Order 256.

458 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 121 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#2297C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2096C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1854C212yesmaximal, complemented
10#1817C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal
12C22×#41C2211yescharacteristic
13C2×#117—21no—
14C2×#117C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2212yes—
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#390C2211yescharacteristic
26#390C2211yescharacteristic
27#386—21no—
28#386—21no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382—21no—
32#380—21no—
33#380C2212yes—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4—21no—
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—24no—
46C2×#41—42no—
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56#117—42no—
57#117—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—24no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—42no—
80#106—42no—
81#103—24no—
82#103—42no—
83#102—24no—
84#102—42no—
85C22⋊C8—24no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92C24—21no—
93C24C2×D411yescharacteristic
94#41—48no—
95#41—48no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8#4112yes—
100C2×D4—24no—
101C2×D4—22no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4#4112yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4#4114yes—
130C22×C4C2411yescharacteristic
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—42no—
158C42—44no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23#12312yes—
173C23C22×D411yescharacteristic
174Q8—24no—
175Q8—42no—
176D4—24no—
177D4—48no—
178D4—42no—
179C2×C4—28no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—44no—
195C2×C4—48no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4#12312yes—
211C2×C4#13512yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4#13712yes—
214C2×C4C2×#4112yes—
215C2×C4C2×#4111yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4H(5,𝔽2)12yes—
218C8—42no—
219C8—42no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22C2×#12311yescharacteristic
235C22#52511yescharacteristic
236C22#53511yescharacteristic
237C22C2×#13511yescharacteristic
238C22C2×AGL(1,ℤ/8)11yescharacteristic
239C22C2×#13711yescharacteristic
240C22#57511yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—81no—
251C4#55112yes—
252C4#55112yes—
253C2—42no—
254C2—82no—
255C2#248211yescharacteristic
256C2#309111yescharacteristic
257C2#309211yescharacteristic
258C2#310711yescharacteristic
259C2#311511yescharacteristic
260C2#320711yescharacteristic
261C2#321611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2285C2semidirect
3. #2096C2semidirect
4. #2093C2semidirect
5. #2072C2semidirect
6. #2071C2semidirect
7. #1928C2semidirect
8. #1926C2semidirect
9. #1854C2semidirect
10. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#22972
#22852
#20962
#20932
#20722
#20712
#19282
#19262
#18542
#18172
#17882