Finite Group Databasebeta

← #28115

Subgroups — #28115

Order 256.

539 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 139 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×#481C211yesmaximal, characteristic
2C2×#448C211yesmaximal, characteristic
3C4×#121C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1927C212yesmaximal, complemented
8#1877C212yesmaximal, complemented
9#1824C211yesmaximal, characteristic, complemented
10#1797C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2212yes—
14C2×#121C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#481—24no—
20#448—24no—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#391—21no—
30#391—21no—
31#391C2212yes—
32#391C2212yes—
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#388—21no—
36#386—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4C2211yescharacteristic
39C4×C22⋊C4—21no—
40C2×#95—21no—
41C2×#95—21no—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×Q8C2312yes—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C2×SD16—24no—
60C2×SD16—24no—
61C2×SD16—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64#121—22no—
65#121D414yes—
66C4×D4—42no—
67C4×D4—44no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C42—21no—
82C2×C42—42no—
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87#106—24no—
88C4⋊C8—42no—
89#103—24no—
90#103—24no—
91#103—42no—
92#102—24no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95—21no—
100#95—21no—
101#95—21no—
102C24—21no—
103C24C2×D411yescharacteristic
104C2×Q8—24no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—48no—
111C2×Q8—48no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8#4114yes—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C2×D412yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4SD1614yescomplemented
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D412yes—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4#4112yes—
151C22×C4C2411yescharacteristic
152SD16—48no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4C2×D412yes—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—22no—
173C42—22no—
174C42—41no—
175C42—41no—
176C42—42no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23#12112yes—
189C23C2×SD1612yes—
190C23#13712yes—
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—24no—
194Q8—48no—
195Q8—42no—
196Q8—42no—
197D4—24no—
198D4—48no—
199D4—42no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4#12112yes—
235C2×C4C2×SD1612yes—
236C2×C4C2×SD1612yes—
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4112yes—
239C2×C4C2×#4111yescharacteristic
240C2×C4H(5,𝔽2)12yes—
241C8—42no—
242C8—42no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22#46114yes—
257C22C2×#12111yescharacteristic
258C22#52511yescharacteristic
259C22#53411yescharacteristic
260C22C22×SD1611yescharacteristic
261C22C2×#13711yescharacteristic
262C22#57511yescharacteristic
263C22#57511yescharacteristic
264C4—22no—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—42no—
270C4—44no—
271C4—81no—
272C4—81no—
273C4—82no—
274C4#54612yes—
275C4#55112yes—
276C2—24no—
277C2—82no—
278C2#253411yescharacteristic
279C2C2×#46111yescharacteristic
280C2#304011yescharacteristic
281C2#310311yescharacteristic
282C2#311711yescharacteristic
283C2#318711yescharacteristic
284C2#319611yescharacteristic
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
4. C2×#391C2semidirect
5. #2066C2semidirect
6. #2055C2semidirect
7. #1927C2semidirect
8. #1877C2semidirect
9. #1824C2semidirect
10. #1797C2semidirect
11. #1787C2semidirect
143. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4482
C4×#1212
C2×#3912
#20662
#20552
#19272
#18772
#18242
#17972
#17872