Finite Group Databasebeta

← #30188

Subgroups — #30188

Order 256.

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

normal subgroups only

This group has 420 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#2857C211yesmaximal, characteristic, complemented
2#2850C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2072C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2042C212yesmaximal, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1893C212yesmaximal
10#1892C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C23×C8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23#391—21no—
24#391C2211yescharacteristic
25#390C2211yescharacteristic
26#389—21no—
27#389C2212yes—
28#388—21no—
29#388—21no—
30#386—21no—
31#386—21no—
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34#380—21no—
35#380C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8—24no—
51C22×C8—42no—
52C22×C8—42no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—42no—
56#124—42no—
57C22⋊Q8—24no—
58C22⋊Q8—44no—
59#121—24no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42—42no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#103—42no—
82#103—42no—
83#102—42no—
84#102—42no—
85#95—22no—
86#95—22no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—21no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×D4—22no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4#4114yes—
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—48no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C22⋊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—41no—
156C42—41no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23#12412yes—
169C23AGL(1,ℤ/8)12yes—
170C23#13712yes—
171C23C22×D411yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—84no—
175D4—42no—
176D4—42no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4#12412yes—
210C2×C4#13512yes—
211C2×C4#13512yes—
212C2×C4C2×#4112yes—
213C2×C4H(5,𝔽2)11yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C2×C42−1+411yescharacteristic
216C8—42no—
217C8—42no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22#48914yes—
232C22C2×#12411yescharacteristic
233C22#53411yescharacteristic
234C22#53711yescharacteristic
235C22C2×#13511yescharacteristic
236C22C2×#13511yescharacteristic
237C22C2×AGL(1,ℤ/8)11yescharacteristic
238C22C2×#13711yescharacteristic
239C4—22no—
240C4—22no—
241C4—22no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—81no—
246C4—81no—
247C4—82no—
248C4—82no—
249C4#53812yes—
250C4#53912yes—
251C2—24no—
252C2—161no—
253C2#263911yescharacteristic
254C2#313111yescharacteristic
255C2C2×#48911yescharacteristic
256C2#318811yescharacteristic
257C2#319011yescharacteristic
258C2#324711yescharacteristic
259C2#324811yescharacteristic
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. #2101C2semidirect
4. #2072C2semidirect
5. #2071C2semidirect
6. #2042C2semidirect
7. #2024C2semidirect
8. #2023C2semidirect
10. #1892C2semidirect
11. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28502
#21012
#20722
#20712
#20422
#20242
#20232
#18932
#18922
C2×#3362