Finite Group Databasebeta

← #8517

Subgroups — #8517

Order 256.

520 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 75 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×#537C211yesmaximal, characteristic
2#1785C211yesmaximal, characteristic, complemented
3#1782C211yesmaximal, characteristic, complemented
4#1511C212yesmaximal, complemented
5#1510C212yesmaximal, complemented
6C22×#41C411yescharacteristic, complemented
7C22×#41C2211yescharacteristic
8#537—24no—
9#537—24no—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8C411yescharacteristic, complemented
12C2×C22⋊Q8C2211yescharacteristic
13C2×#121—21no—
14C2×#121—21no—
15C2×#121—21no—
16C2×#121C411yescharacteristic, complemented
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C411yescharacteristic, complemented
19C2×#101—21no—
20C2×#100—21no—
21C2×#99—21no—
22C2×#99—21no—
23#407C2212yes—
24#381C2212yes—
25#328—24no—
26#327—24no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—22no—
31C2×#41—42no—
32C2×#41—42no—
33C2×#41—42no—
34C2×#41—42no—
35C22×Q8C2×C411yescharacteristic
36C22×Q8D411yescharacteristic
37C22×D4—21no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4C2×C411yescharacteristic
41C22×D4D411yescharacteristic
42C22×D4D411yescharacteristic
43C22×D4D411yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4C2×C411yescharacteristic
47C23×C4C2×C411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×M16—22no—
50C22⋊Q8—28no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8D414yes—
55#121—28no—
56#121—28no—
57#121—28no—
58#121—22no—
59#121—22no—
60#121—22no—
61#121D414yes—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4C2×C411yescharacteristic
68C2×C4⋊C4C2×C411yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74#101—42no—
75#100—42no—
76#99—42no—
77#99—42no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80#95—22no—
81#95—22no—
82C24—21no—
83C24C22⋊C411yescharacteristic
84C24C22⋊C411yescharacteristic
85C24C2×D411yescharacteristic
86#41—48no—
87#41—48no—
88#41—48no—
89#41—48no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8C22⋊C412yes—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—82no—
115C2×D4—82no—
116C2×D4C22⋊C412yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4C22⋊C411yescharacteristic
137C22×C4C22⋊C411yescharacteristic
138C22×C4C22×C411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D412yes—
142M16—42no—
143M16—42no—
144C2×C8—42no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—24no—
150C4⋊C4C22⋊C412yes—
151C4⋊C4C22⋊C412yes—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—24no—
155C22⋊C4—24no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#9912yes—
177C23C2×C22⋊C411yescharacteristic
178C23#12012yes—
179Q8—44no—
180Q8—42no—
181Q8—42no—
182D4—44no—
183D4—44no—
184D4—42no—
185D4—42no—
186D4—44no—
187D4—48no—
188D4—84no—
189D4—84no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#9912yes—
218C2×C4C2×C22⋊C411yescharacteristic
219C2×C4C2×C22⋊C411yescharacteristic
220C2×C4#12012yes—
221C8—82no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—42no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22—82no—
244C22#37911yescharacteristic
245C22C2×#9911yescharacteristic
246C22C2×#9911yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—41no—
254C4—42no—
255C4—162no—
256C2—21no—
257C2—21no—
258C2—42no—
259C2—42no—
260C2—81no—
261C2—82no—
262C2#178111yescharacteristic
263C2#179511yescharacteristic
264C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#537C2
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. #1785C2semidirect
3. #1782C2semidirect
4. #1511C2semidirect
5. #1510C2semidirect
6. C22×#41C4semidirect
11. C2×C22⋊Q8C4semidirect
16. C2×#121C4semidirect
18. C22×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
#17852
#17822
#15112
#15102