Finite Group Databasebeta

← #24748

Subgroups — #24748

Order 256.

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

normal subgroups only

This group has 488 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×#461C212yesmaximal
2C2×#461C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic, complemented
5#2349C211yesmaximal, characteristic
6#2068C212yesmaximal, complemented
7#2043C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1877C212yesmaximal
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×C4×D4C2211yescharacteristic
19#461—28no—
20#461—24no—
21#440—24no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2211yescharacteristic
29C2×#103C2212yes—
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#398—21no—
33#398—21no—
34#391C2212yes—
35#389—22no—
36#389C2212yes—
37#385—22no—
38#382C2211yescharacteristic
39#380—21no—
40C4×C4⋊C4C2211yescharacteristic
41C2×#95—21no—
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—22no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16D414yes—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C4⋊Q8—24no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—42no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4—42no—
67C4×D4D414yes—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D412yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C42—21no—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83#107—22no—
84C4⋊C8—24no—
85C4⋊C8—22no—
86#103—24no—
87#103—24no—
88#103—22no—
89#103—24no—
90#103—42no—
91#102—22no—
92#97—22no—
93#95—21no—
94#95—22no—
95#95—21no—
96#95—22no—
97#95—22no—
98#95D411yescharacteristic
99#95D411yescharacteristic
100C24—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8C2×D412yes—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×D412yes—
117C2×D4C2×D412yes—
118C2×D4#4112yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4C2411yescharacteristic
134SD16—28no—
135SD16—48no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8C2×D412yes—
145C4⋊C4—24no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—44no—
160C42—22no—
161C42—22no—
162C42—42no—
163C42C2×D412yes—
164C42#4112yes—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23C22×D411yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178Q8—42no—
179D4—22no—
180D4—22no—
181D4—22no—
182D4#12114yes—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4—88no—
205C2×C4#12112yes—
206C2×C4#12112yes—
207C2×C4#13714yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C4H(5,𝔽2)11yescharacteristic
213C2×C42−1+411yescharacteristic
214C8—22no—
215C8—42no—
216C8—41no—
217C22—24no—
218C22—44no—
219C22—44no—
220C22—42no—
221C22—42no—
222C22C2×#12111yescharacteristic
223C22D4211yescharacteristic
224C22#54811yescharacteristic
225C22C2×#13712yes—
226C22#57712yes—
227C4—24no—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—82no—
234C4—82no—
235C4—81no—
236C4C2×#12111yescharacteristic
237C4#53711yescharacteristic
238C4#54711yescharacteristic
239C4#55311yescharacteristic
240C2—44no—
241C2#241611yescharacteristic
242C2#303911yescharacteristic
243C2#304211yescharacteristic
244C2#326612yes—
245C2#337212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2546C2semidirect
6. #2068C2semidirect
7. #2043C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4402
#25462
#23492
#20682
#20432
#19602
#19402
#18772
#18052