Finite Group Databasebeta

← #18906

Subgroups — #18906

Order 256.

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

normal subgroups only

This group has 451 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#3507C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic
4C2×#462C211yesmaximal, characteristic, complemented
5#1648C214yesmaximal, complemented
6#1640C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic
8#1481C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C22×Q16—21no—
11#559C2212yes—
12#557C2212yes—
13#556—21no—
14#556—22no—
15#554—21no—
16#552—22no—
17#539—22no—
18#536—21no—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22#529—22no—
23#520—21no—
24C2×C4×Q8C2211yescharacteristic
25#500C2212yes—
26#498—21no—
27#487—21no—
28#486C2212yes—
29#483—22no—
30#480—22no—
31#475C2214yes—
32#474—22no—
33#462—22no—
34#462C2214yes—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103—21no—
39#332C2212yes—
40#331C2212yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—21no—
45C22×Q8D411yescharacteristic
46C2×Q16—24no—
47C2×Q16—44no—
48C22×C8—21no—
49C4⋊Q8—22no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D411yescharacteristic
54C4⋊Q8D411yescharacteristic
55C4⋊Q8C2312yes—
56C4⋊Q8C2311yescharacteristic
57#127C2311yescharacteristic
58#126—24no—
59#126—24no—
60#125—22no—
61#125—24no—
62#125—21no—
63#124—22no—
64#124—22no—
65#124—21no—
66#123—24no—
67#123—24no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—24no—
71C22⋊Q8—21no—
72C22⋊Q8D412yes—
73#121—21no—
74#121D412yes—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8D412yes—
81C4×Q8C2312yes—
82C4×Q8C2312yes—
83C4×D4—24no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—21no—
87C4×D4D412yes—
88#117—22no—
89#117—24no—
90#117—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98#107—24no—
99#106—24no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8C2312yes—
103C4⋊C8C2312yes—
104#103—24no—
105#103—24no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111#41—42no—
112#41—42no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—41no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8C2×D412yes—
123C2×Q8C2×D412yes—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C2×D412yes—
128C22×C4—21no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136Q16—48no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—42no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C42—24no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—22no—
175C23#12011yescharacteristic
176Q8—22no—
177Q8—22no—
178Q8—24no—
179Q8—41no—
180Q8—41no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—81no—
185Q8—82no—
186D4—42no—
187D4—44no—
188D4—41no—
189D4—41no—
190D4—42no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4#13512yes—
208C2×C4AGL(1,ℤ/8)12yes—
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C8—42no—
213C8—42no—
214C8—42no—
215C22—21no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22#44912yes—
221C22C2×#12011yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4#44912yes—
232C4C2×#13511yescharacteristic
233C4C2×AGL(1,ℤ/8)11yescharacteristic
234C4#57812yes—
235C2—22no—
236C2—82no—
237C2C2×#44911yescharacteristic
238C2#301111yescharacteristic
239C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3229C2semidirect
4. C2×#462C2semidirect
5. #1648C2semidirect
6. #1640C2semidirect
8. #1481C2semidirect
9. #1479C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32292
#32262
C2×#4622
#16482
#16402
#15402
#14812
#14792