Finite Group Databasebeta

← #24816

Subgroups — #24816

Order 256.

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

normal subgroups only

This group has 447 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×#474C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic, complemented
4D4×C4⋊C4C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2068C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1939C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal
10#1804C212yesmaximal
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2212yes—
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4—21no—
18#474—24no—
19C4×SD16—24no—
20C2×#107C2212yes—
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—22no—
30#391—21no—
31#391—21no—
32#391C2212yes—
33#389C2212yes—
34#385—22no—
35#384C2211yescharacteristic
36#382C2212yes—
37#380—21no—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C2×SD16Q814yescomplemented
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4—42no—
61C4×D4D414yes—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—44no—
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42C2312yes—
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—22no—
81C4⋊C8—22no—
82C4⋊C8—42no—
83#103—22no—
84#103—42no—
85#102—22no—
86#102—24no—
87C4×C8—22no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95D412yes—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8C2×Q812yes—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×D412yes—
108C2×D4C2×Q812yes—
109C2×D4#4112yes—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126SD16—28no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8C2×Q812yes—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—81no—
150C4⋊C4C2×D412yes—
151C22⋊C4—24no—
152C22⋊C4—44no—
153C42—22no—
154C42—22no—
155C42—42no—
156C42C2×D412yes—
157C42#4112yes—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23C22×D411yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—84no—
171D4—22no—
172D4—22no—
173D4—22no—
174D4C22⋊Q814yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—48no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—88no—
200C2×C4#12112yes—
201C2×C4#12112yes—
202C2×C4#13512yes—
203C2×C4AGL(1,ℤ/8)12yes—
204C2×C4C22×D411yescharacteristic
205C2×C4C22×Q811yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C2×C4H(5,𝔽2)11yescharacteristic
209C2×C42−1+411yescharacteristic
210C8—22no—
211C8—41no—
212C8—42no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22C2×#12111yescharacteristic
221C22#54611yescharacteristic
222C22D4×Q811yescharacteristic
223C22C2×#13511yescharacteristic
224C22C2×AGL(1,ℤ/8)11yescharacteristic
225C22#57711yescharacteristic
226C22#57811yescharacteristic
227C4—22no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—82no—
233C4—81no—
234C4—82no—
235C4#46314yes—
236C4C2×C22⋊Q811yescharacteristic
237C4#53711yescharacteristic
238C4#55211yescharacteristic
239C4#55411yescharacteristic
240C2—44no—
241C2#241711yescharacteristic
242C2C2×#46311yescharacteristic
243C2#304611yescharacteristic
244C2#330211yescharacteristic
245C2#332411yescharacteristic
246C2#338111yescharacteristic
247C2#338611yescharacteristic
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
3. #2568C2semidirect
5. #2085C2semidirect
6. #2068C2semidirect
7. #1990C2semidirect
8. #1939C2semidirect
11. #1803C2semidirect
50. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×C4×SD162
#25682
D4×C4⋊C42
#20852
#20682
#19902
#19392
#18772
#18042
#18032