Finite Group Databasebeta

← #24609

Subgroups — #24609

Order 256.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#483C211yesmaximal, characteristic
2C2×#432C211yesmaximal, characteristic
3#2855C211yesmaximal, characteristic
4C2×#389C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2068C212yesmaximal, complemented
7#1943C212yesmaximal, complemented
8#1935C212yesmaximal, complemented
9#1872C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1823C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17#483—24no—
18#432—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2212yes—
27#391C2211yescharacteristic
28#391C2211yescharacteristic
29#389—21no—
30#389—21no—
31#389C2212yes—
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2212yes—
35#387—21no—
36#387—21no—
37#385—21no—
38#381—21no—
39#380C2211yescharacteristic
40C2×#95—21no—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—24no—
81C4⋊C8—24no—
82#103—24no—
83#103—24no—
84C22⋊C8—22no—
85C22⋊C8D414yes—
86#97—24no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24C2×D411yescharacteristic
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—21no—
100C2×Q8—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4#4112yes—
128C22×C4C2411yescharacteristic
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8C2×D414yes—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4#4114yes—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23#12112yes—
163C23AGL(1,ℤ/8)12yes—
164C23#13712yes—
165C23C22×D411yescharacteristic
166Q8—42no—
167Q8—42no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—44no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—84no—
205C2×C4#12112yes—
206C2×C4#13712yes—
207C2×C4C22×D411yescharacteristic
208C2×C4C2×#4112yes—
209C2×C4C2×#4111yescharacteristic
210C2×C42−1+412yes—
211C8—24no—
212C8—42no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22#46814yes—
220C22C2×#12111yescharacteristic
221C22#52511yescharacteristic
222C22#53611yescharacteristic
223C22C2×AGL(1,ℤ/8)11yescharacteristic
224C22C2×#13711yescharacteristic
225C22C2×#13711yescharacteristic
226C22#57511yescharacteristic
227C4—22no—
228C4—22no—
229C4—44no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4—82no—
237C4—81no—
238C4#54712yes—
239C4#55212yes—
240C2—24no—
241C2#241311yescharacteristic
242C2C2×#46811yescharacteristic
243C2#305611yescharacteristic
244C2#311311yescharacteristic
245C2#311411yescharacteristic
246C2#323611yescharacteristic
247C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #2088C2semidirect
6. #2068C2semidirect
7. #1943C2semidirect
8. #1935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4322
#28552
C2×#3892
#20882
#20682
#19432
#19352
#18722
#18522
#18232