Finite Group Databasebeta

← #28563

Subgroups — #28563

Order 256.

460 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 131 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×#467C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic
3#2417C212yesmaximal
4#2086C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2056C211yesmaximal, characteristic, complemented
7#2001C211yesmaximal, characteristic, complemented
8#1935C212yesmaximal, complemented
9#1894C212yesmaximal, complemented
10#1872C212yesmaximal, complemented
11C2×#340C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2212yes—
17C2×C4×D4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19#467—24no—
20#435—24no—
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#97C2211yescharacteristic
29#398—22no—
30#396—22no—
31#391C2212yes—
32#389C2212yes—
33#386—22no—
34#380—22no—
35#380C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8C2312yes—
42C22×D4D411yescharacteristic
43C23×C4—22no—
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—42no—
48C2×M16—22no—
49C2×M16—42no—
50C22×C8—42no—
51C22×C8D414yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—42no—
60C4×D4—22no—
61C4×D4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4D411yescharacteristic
76C2×C22⋊C4D411yescharacteristic
77C2×C42C2311yescharacteristic
78#107—22no—
79C4⋊C8—22no—
80#103—24no—
81#103—24no—
82#103—42no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#97—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90C24C2×D411yescharacteristic
91C24C2×D411yescharacteristic
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×D4—22no—
99C2×D4—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—82no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4#4112yes—
119C22×C4C2411yescharacteristic
120Q16—44no—
121M16—44no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—44no—
129C2×C8C2×D412yes—
130C2×C8C2×D412yes—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—82no—
141C4⋊C4#4114yes—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—82no—
147C42—21no—
148C42—21no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—22no—
155C23—21no—
156C23—21no—
157C23#12112yes—
158C23#13714yes—
159C23C22×D411yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164D4—44no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—88no—
192C2×C4#12112yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4112yes—
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)12yes—
197C8—22no—
198C8—22no—
199C8—41no—
200C8—41no—
201C22—22no—
202C22—24no—
203C22—24no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—42no—
209C22—41no—
210C22#47014yes—
211C22C2×#12111yescharacteristic
212C22#54612yes—
213C22C2×#13712yes—
214C22#57611yescharacteristic
215C22#57811yescharacteristic
216C4—22no—
217C4—41no—
218C4—41no—
219C4—44no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—82no—
224C4#53211yescharacteristic
225C4#53411yescharacteristic
226C4#55312yes—
227C2—24no—
228C2—81no—
229C2#254311yescharacteristic
230C2C2×#47011yescharacteristic
231C2#306111yescharacteristic
232C2#331212yes—
233C2#331412yes—
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
1. C2×#467C2semidirect
4. #2086C2semidirect
5. #2075C2semidirect
6. #2056C2semidirect
7. #2001C2semidirect
8. #1935C2semidirect
9. #1894C2semidirect
10. #1872C2semidirect
11. C2×#340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4352
#24172
#20862
#20752
#20562
#20012
#19352
#18942
#18722
C2×#3402