Finite Group Databasebeta

← #24729

Subgroups — #24729

Order 256.

473 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 143 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×#464C212yesmaximal
2C2×#462C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic
4#2547C211yesmaximal, characteristic
5Q8×C4⋊C4C211yesmaximal, characteristic, complemented
6#2066C212yesmaximal, complemented
7#2042C212yesmaximal
8#1956C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1878C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—28no—
20#462—24no—
21C4×SD16—24no—
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#397—22no—
33#391C2212yes—
34#389—22no—
35#389C2212yes—
36#384—21no—
37#384C2211yescharacteristic
38#380—21no—
39#380—21no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16D414yes—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#124—24no—
60#124—42no—
61C4×Q8—22no—
62C4×Q8—24no—
63C4×Q8—42no—
64C4×Q8D414yes—
65C4×D4—22no—
66C4×D4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C42—21no—
83C2×C42—22no—
84C2×C42C2312yes—
85C2×C42C2311yescharacteristic
86#106—22no—
87C4⋊C8—24no—
88C4⋊C8—22no—
89#103—22no—
90#103—24no—
91#103—24no—
92#103—42no—
93#102—24no—
94#102—22no—
95C4×C8—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95D412yes—
100C24—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C2×D412yes—
111C2×Q8C2×D412yes—
112C2×Q8#4112yes—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2411yescharacteristic
132Q16—48no—
133SD16—28no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8C2×D412yes—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—82no—
157C4⋊C4C2×D412yes—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C42—24no—
162C42—22no—
163C42—22no—
164C42—44no—
165C42C2×D412yes—
166C42#4112yes—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23C22×D411yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—22no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8#12114yes—
183D4—22no—
184D4—22no—
185D4—41no—
186D4—41no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—84no—
211C2×C4#12112yes—
212C2×C4#12112yes—
213C2×C4#13512yes—
214C2×C4#13712yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4H(5,𝔽2)11yescharacteristic
220C2×C42−1+411yescharacteristic
221C8—22no—
222C8—42no—
223C8—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22C2×#12111yescharacteristic
232C22D4211yescharacteristic
233C22#54811yescharacteristic
234C22C2×#13511yescharacteristic
235C22C2×#13711yescharacteristic
236C22#57711yescharacteristic
237C22#57811yescharacteristic
238C4—24no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—82no—
244C4—82no—
245C4—81no—
246C4#46414yescomplemented
247C4C2×#12111yescharacteristic
248C4#53711yescharacteristic
249C4#54711yescharacteristic
250C4#55311yescharacteristic
251C2—82no—
252C2#241611yescharacteristic
253C2C2×#46411yescharacteristic
254C2#304611yescharacteristic
255C2#327611yescharacteristic
256C2#328411yescharacteristic
257C2#336411yescharacteristic
258C2#336911yescharacteristic
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
2. C2×#462C2semidirect
5. Q8×C4⋊C4C2semidirect
6. #2066C2semidirect
8. #1956C2semidirect
9. #1939C2semidirect
10. #1878C2semidirect
246. C4#464semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4622
C2×C4×SD162
#25472
Q8×C4⋊C42
#20662
#20422
#19562
#19392
#18782
#18022