Finite Group Databasebeta

← #29645

Subgroups — #29645

Order 256.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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×#533C211yesmaximal, characteristic
2C2×#485C212yesmaximal
3#2855C211yesmaximal, characteristic
4#2059C212yesmaximal, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2026C212yesmaximal, complemented
8#2024C211yesmaximal, characteristic, complemented
9#2024C211yesmaximal, characteristic, complemented
10#1825C212yesmaximal
11#1817C211yesmaximal, characteristic
12#533—28no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×#117C2212yes—
22C22×C42C2211yescharacteristic
23#485—28no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×C22⋊C8C2212yes—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#387—21no—
34#387—21no—
35#385—21no—
36#381—21no—
37#380C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8—21no—
47C22×Q8C2311yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55#125—24no—
56#125—24no—
57C22⋊Q8—28no—
58C22⋊Q8—28no—
59C22⋊Q8—24no—
60C4×Q8—28no—
61C4×Q8—28no—
62#117—24no—
63#117D418yes—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C42—22no—
80C2×C42—24no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#107—24no—
86#106—24no—
87#103—24no—
88#103—24no—
89C22⋊C8—24no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24C2×D411yescharacteristic
98C2×Q8—24no—
99C2×Q8—24no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D414yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C4⋊C4—28no—
137C4⋊C4—28no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4C2×D414yes—
150C4⋊C4C2×D414yes—
151C22⋊C4—28no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C42—22no—
155C42—22no—
156C42—28no—
157C42—22no—
158C42—22no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23#12012yes—
166C23C22×D411yescharacteristic
167Q8—48no—
168Q8—48no—
169Q8—42no—
170Q8—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4#12012yes—
206C2×C4#13512yes—
207C2×C4#13712yes—
208C2×C4C22×D412yes—
209C2×C42−1+412yes—
210C2×C42−1+411yescharacteristic
211C2×C42−1+411yescharacteristic
212C8—42no—
213C8—42no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22C2×#12011yescharacteristic
222C22#53611yescharacteristic
223C22#53611yescharacteristic
224C22C2×#13511yescharacteristic
225C22C2×#13711yescharacteristic
226C22#57511yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—41no—
232C4—44no—
233C4—41no—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4#53612yes—
240C4#53612yes—
241C2—42no—
242C2#261411yescharacteristic
243C2#300211yescharacteristic
244C2#300711yescharacteristic
245C2#323111yescharacteristic
246C2#323311yescharacteristic
247C2#323711yescharacteristic
248C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2059C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
7. #2026C2semidirect
8. #2024C2semidirect
9. #2024C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4852
#28552
#20592
#20582
#20572
#20262
#20242
#18252
#18172