Finite Group Databasebeta

← #25293

Subgroups — #25293

Order 256.

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

normal subgroups only

This group has 425 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×#461C211yesmaximal, characteristic
2C2×C8×Q8C211yesmaximal, characteristic, complemented
3#2568C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic
5#2066C212yesmaximal, complemented
6#2055C212yesmaximal
7#1968C212yesmaximal
8#1956C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1933C212yesmaximal, complemented
11C4×#102C211yesmaximal, characteristic
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#461—24no—
18C8×Q8—24no—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×C4×C8C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—22no—
30#396—22no—
31#391—21no—
32#391C2212yes—
33#391C2212yes—
34#390C2212yes—
35#389C2212yes—
36#384C2211yescharacteristic
37#382—21no—
38#380—21no—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42#340C2211yescharacteristic
43C22×Q8D411yescharacteristic
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—42no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C22⋊Q8—44no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#106—42no—
78C4⋊C8—24no—
79C4⋊C8D414yes—
80#103—24no—
81#103—24no—
82#103—42no—
83#102—22no—
84#102—22no—
85C4×C8—24no—
86C4×C8—22no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93C24—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8SD1614yescomplemented
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4#4112yes—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122SD16—48no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8C2×D414yes—
131C2×C8#4112yes—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4—81no—
147C22⋊C4—42no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C42—22no—
151C42—22no—
152C42—42no—
153C42C2×D412yes—
154C42#4112yes—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23C22×D411yescharacteristic
163Q8—28no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168D4—22no—
169D4—22no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—84no—
195C2×C4#12112yes—
196C2×C4#12112yes—
197C2×C4C2×SD1614yes—
198C2×C4C2×SD1612yes—
199C2×C4#13512yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C42−1+411yescharacteristic
206C8—24no—
207C8—22no—
208C8—41no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22C2×#12111yescharacteristic
217C22#54811yescharacteristic
218C22#55011yescharacteristic
219C22C22×SD1611yescharacteristic
220C22C2×#13511yescharacteristic
221C22#57711yescharacteristic
222C22#57811yescharacteristic
223C4—24no—
224C4—22no—
225C4—42no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4—82no—
230C4—81no—
231C4#46514yes—
232C4#52911yescharacteristic
233C4#54211yescharacteristic
234C4#54611yescharacteristic
235C4#54811yescharacteristic
236C2—82no—
237C2#242611yescharacteristic
238C2C2×#46511yescharacteristic
239C2#306311yescharacteristic
240C2#335911yescharacteristic
241C2#336011yescharacteristic
242C2#339211yescharacteristic
243C2#339311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C8×Q8C2semidirect
3. #2568C2semidirect
5. #2066C2semidirect
8. #1956C2semidirect
9. #1948C2semidirect
10. #1933C2semidirect
102. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×C8×Q82
#25682
#25382
#20662
#20552
#19682
#19562
#19482
#19332
C4×#1022