Finite Group Databasebeta

← #25320

Subgroups — #25320

Order 256.

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

normal subgroups only

This group has 447 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×#487C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4#2365C211yesmaximal, characteristic, complemented
5#2072C212yesmaximal, complemented
6#2055C212yesmaximal
7#1988C211yesmaximal, characteristic, complemented
8#1939C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1803C212yesmaximal, complemented
11C4×#102C211yesmaximal, characteristic
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8—21no—
17C2×C4×Q8C2212yes—
18C2×C4×D4C2211yescharacteristic
19#487—24no—
20#461—24no—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#399—22no—
30#391C2212yes—
31#390C2212yes—
32#389—21no—
33#387—22no—
34#384C2212yes—
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—42no—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8D414yes—
57C22⋊Q8—44no—
58C22⋊Q8—44no—
59C4×Q8—24no—
60C4×Q8—42no—
61C4×D4—22no—
62C4×D4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42C2312yes—
81C2×C42C2311yescharacteristic
82C4⋊C8—22no—
83C4⋊C8—42no—
84#103—24no—
85#103—24no—
86#103—24no—
87#102—22no—
88#102—22no—
89C4×C8—22no—
90#95—22no—
91#95—21no—
92#95—22no—
93#95D412yes—
94C24—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8C2×D414yes—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4#4112yes—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123SD16—48no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4SD1614yescomplemented
148C4⋊C4#4112yes—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C42—22no—
153C42—22no—
154C42—44no—
155C42—42no—
156C42C2×D412yes—
157C42#4112yes—
158C23—22no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—24no—
168Q8—42no—
169Q8—42no—
170Q8—84no—
171D4—22no—
172D4—22no—
173D4—41no—
174D4—41no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—84no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4C2×SD1614yes—
201C2×C4C2×SD1612yes—
202C2×C4AGL(1,ℤ/8)12yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C42−1+411yescharacteristic
209C8—41no—
210C8—42no—
211C8—41no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22C2×#12111yescharacteristic
220C22#54611yescharacteristic
221C22#54811yescharacteristic
222C22C22×SD1611yescharacteristic
223C22C2×AGL(1,ℤ/8)11yescharacteristic
224C22#57811yescharacteristic
225C22#57811yescharacteristic
226C4—24no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—44no—
231C4—81no—
232C4—81no—
233C4—82no—
234C4#46014yes—
235C4#52911yescharacteristic
236C4#54211yescharacteristic
237C4#54611yescharacteristic
238C4#54711yescharacteristic
239C2—82no—
240C2#242711yescharacteristic
241C2C2×#46011yescharacteristic
242C2#304411yescharacteristic
243C2#329811yescharacteristic
244C2#333011yescharacteristic
245C2#335911yescharacteristic
246C2#336711yescharacteristic
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×#487C2semidirect
4. #2365C2semidirect
5. #2072C2semidirect
7. #1988C2semidirect
8. #1939C2semidirect
10. #1803C2semidirect
147. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4612
#25422
#23652
#20722
#20552
#19882
#19392
#18802
#18032
C4×#1022