Finite Group Databasebeta

← #32722

Subgroups — #32722

Order 256.

406 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 115 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#2857C211yesmaximal, characteristic, complemented
2#2854C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2035C212yesmaximal, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2000C212yesmaximal
11#1797C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#123—21no—
15C2×C22⋊Q8C2212yes—
16C2×#121C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#391C2212yes—
26#390C2211yescharacteristic
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#388—21no—
33#388—21no—
34#386—21no—
35#384C2211yescharacteristic
36#380—21no—
37#380—21no—
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2312yes—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×SD16—42no—
50C2×SD16—42no—
51C2×SD16—42no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—42no—
55#123—44no—
56C22⋊Q8—42no—
57C22⋊Q8—42no—
58#121—42no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—42no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#106—42no—
83C4⋊C8—42no—
84#103—42no—
85#103—42no—
86#103—42no—
87#102—42no—
88C22⋊C8—42no—
89#95—22no—
90#95—22no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24—21no—
95C24C2×D411yescharacteristic
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4SD1614yescomplemented
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130SD16—48no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C42—41no—
160C42—41no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23C2×SD1612yes—
175C23C22×D411yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180D4—42no—
181D4—42no—
182D4—84no—
183C2×C4—21no—
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—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4C2×SD1612yes—
215C2×C4C2×SD1612yes—
216C2×C4#13712yes—
217C2×C4H(5,𝔽2)11yescharacteristic
218C2×C4H(5,𝔽2)12yes—
219C2×C4H(5,𝔽2)12yes—
220C2×C42−1+411yescharacteristic
221C8—42no—
222C8—42no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22#53411yescharacteristic
238C22#53511yescharacteristic
239C22#53711yescharacteristic
240C22C22×SD1611yescharacteristic
241C22C2×#13711yescharacteristic
242C22#57511yescharacteristic
243C22#57511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—81no—
251C4—82no—
252C4—81no—
253C4—82no—
254C4#56012yes—
255C4#56212yes—
256C2—42no—
257C2—161no—
258C2#281311yescharacteristic
259C2#318711yescharacteristic
260C2#319611yescharacteristic
261C2#320611yescharacteristic
262C2#321911yescharacteristic
263C2#324611yescharacteristic
264C2#325211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. #2857C2semidirect
3. #2100C2semidirect
4. #2094C2semidirect
5. #2066C2semidirect
6. #2055C2semidirect
7. #2043C2semidirect
8. #2035C2semidirect
9. #2013C2semidirect
11. #1797C2semidirect
123. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28572
#28542
#21002
#20942
#20662
#20552
#20432
#20352
#20132
#20002
#17972