Finite Group Databasebeta

← #8381

Subgroups — #8381

Order 256.

600 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 85 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×#546C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1786C212yesmaximal, complemented
4#1520C212yesmaximal, complemented
5#1459C211yesmaximal, characteristic, complemented
6C22×#41C411yescharacteristic, complemented
7C23×D4C2211yescharacteristic
8#546—24no—
9#546—24no—
10C2×#124C411yescharacteristic, complemented
11C2×#123—21no—
12C2×C22⋊Q8C2211yescharacteristic
13C2×#121—21no—
14C2×#121C411yescharacteristic, complemented
15C2×#120—21no—
16C2×C4×D4C411yescharacteristic, complemented
17C2×C4×D4C2211yescharacteristic
18C22×C22⋊C4—21no—
19C2×#102—24no—
20C2×#99—21no—
21C2×C22⋊C8C2212yes—
22#380C2212yes—
23#342—21no—
24#328—24no—
25#324—21no—
26#323—21no—
27C25—21no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—44no—
31C22×Q8C2×C411yescharacteristic
32C22×D4—21no—
33C22×D4—21no—
34C22×D4—21no—
35C22×D4—22no—
36C22×D4—44no—
37C22×D4C2×C411yescharacteristic
38C22×D4C2×C411yescharacteristic
39C22×D4D411yescharacteristic
40C22×D4D414yes—
41C23×C4—21no—
42C23×C4—21no—
43C23×C4C2×C411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8—22no—
46#124—22no—
47#124—22no—
48#123—28no—
49C22⋊Q8—22no—
50C22⋊Q8D414yes—
51#121—28no—
52#121—22no—
53#121—22no—
54#120—28no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C22⋊C4—28no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D411yescharacteristic
71C2×C22⋊C4D411yescharacteristic
72C2×C42C2×C411yescharacteristic
73#102—48no—
74#99—42no—
75C22⋊C8—22no—
76C22⋊C8—22no—
77#95—22no—
78C24—21no—
79C24—21no—
80C24—24no—
81C24—44no—
82C24—42no—
83C24—41no—
84C24—41no—
85C24—82no—
86C24C22⋊C411yescharacteristic
87C24C2×D411yescharacteristic
88#41—48no—
89#41—44no—
90#41—44no—
91C2×Q8—21no—
92C2×Q8—42no—
93C2×Q8C22⋊C412yes—
94C2×D4—24no—
95C2×D4—28no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—24no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—24no—
105C2×D4—42no—
106C2×D4—44no—
107C2×D4—48no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—82no—
111C2×D4C22⋊C412yes—
112C2×D4C22⋊C412yes—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—82no—
124C22×C4C22⋊C411yescharacteristic
125C22×C4C22⋊C411yescharacteristic
126C22×C4C22⋊C411yescharacteristic
127C22×C4C22×C411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C2×C8—44no—
132C2×C8—42no—
133C4⋊C4—21no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4C22⋊C412yes—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4—24no—
144C22⋊C4—22no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C42—21no—
148C42—21no—
149C23—24no—
150C23—22no—
151C23—22no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—41no—
156C23—41no—
157C23—48no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—44no—
168C23—44no—
169C23—84no—
170C23—88no—
171C23—84no—
172C23—84no—
173C23C2×C22⋊C411yescharacteristic
174C23#12012yes—
175C23#13512yes—
176C23AGL(1,ℤ/8)11yescharacteristic
177C23#13711yescharacteristic
178Q8—42no—
179Q8—42no—
180D4—24no—
181D4—22no—
182D4—24no—
183D4—48no—
184D4—42no—
185D4—48no—
186D4—42no—
187D4—42no—
188D4—84no—
189D4#9914yescomplemented
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—48no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4C2×C22⋊C411yescharacteristic
213C2×C4C2×C22⋊C411yescharacteristic
214C2×C4#12012yes—
215C8—82no—
216C22—21no—
217C22—22no—
218C22—21no—
219C22—22no—
220C22—24no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—44no—
225C22—44no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—42no—
230C22—41no—
231C22—42no—
232C22—41no—
233C22—42no—
234C22—88no—
235C22—88no—
236C22—81no—
237C22—81no—
238C22—82no—
239C22—82no—
240C22#37911yescharacteristic
241C22#41611yescharacteristic
242C22#41711yescharacteristic
243C22#44912yes—
244C22#45212yes—
245C4—22no—
246C4—21no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4—162no—
253C4C2×#9912yes—
254C2—22no—
255C2—21no—
256C2—21no—
257C2—22no—
258C2—44no—
259C2—81no—
260C2—82no—
261C2#178111yescharacteristic
262C2#178811yescharacteristic
263C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1799C2semidirect
3. #1786C2semidirect
4. #1520C2semidirect
5. #1459C2semidirect
6. C22×#41C4semidirect
10. C2×#124C4semidirect
14. C2×#121C4semidirect
16. C2×C4×D4C4semidirect
189. D4#99semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
#17992
#17862
#15202
#14592