Finite Group Databasebeta

← #24678

Subgroups — #24678

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×#475C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4#2355C211yesmaximal, characteristic, complemented
5#2072C212yesmaximal, complemented
6#2055C212yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1948C212yesmaximal, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1926C212yesmaximal
11#1802C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#475—24no—
19C4×SD16—24no—
20C2×#106C2212yes—
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#399—22no—
30#391C2212yes—
31#390C2212yes—
32#387—22no—
33#384C2211yescharacteristic
34#384C2212yes—
35#382—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C22⋊Q8—44no—
56C22⋊Q8—44no—
57C4×Q8—22no—
58C4×Q8D414yes—
59C4×D4—22no—
60C4×D4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42—22no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#106—22no—
81#106—24no—
82C4⋊C8—24no—
83C4⋊C8—22no—
84#103—22no—
85#103—24no—
86#102—24no—
87#102—22no—
88C4×C8—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95D412yes—
93C24—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8C2×D412yes—
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—22no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122SD16—28no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8#4112yes—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4SD1614yescomplemented
147C4⋊C4C2×D412yes—
148C22⋊C4—44no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C42—24no—
152C42—22no—
153C42—22no—
154C42—82no—
155C42C2×D412yes—
156C42#4112yes—
157C23—22no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—84no—
170D4—22no—
171D4—22no—
172D4—41no—
173D4—41no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—84no—
197C2×C4#12112yes—
198C2×C4#12112yes—
199C2×C4C2×SD1614yes—
200C2×C4C2×SD1612yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C42−1+411yescharacteristic
207C2×C42−1+411yescharacteristic
208C8—22no—
209C8—42no—
210C8—41no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22C2×#12111yescharacteristic
219C22#54711yescharacteristic
220C22#54811yescharacteristic
221C22C22×SD1611yescharacteristic
222C22C2×AGL(1,ℤ/8)11yescharacteristic
223C22#57711yescharacteristic
224C22#57811yescharacteristic
225C4—24no—
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—82no—
233C4#46014yes—
234C4#52511yescharacteristic
235C4#53611yescharacteristic
236C4#55211yescharacteristic
237C4#55611yescharacteristic
238C2—82no—
239C2#241511yescharacteristic
240C2C2×#46011yescharacteristic
241C2#304611yescharacteristic
242C2#333511yescharacteristic
243C2#335311yescharacteristic
244C2#335911yescharacteristic
245C2#337311yescharacteristic
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
1. C2×#475C2semidirect
4. #2355C2semidirect
5. #2072C2semidirect
7. #1990C2semidirect
8. #1948C2semidirect
9. #1939C2semidirect
146. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×C4×SD162
#25422
#23552
#20722
#20552
#19902
#19482
#19392
#19262
#18022