Finite Group Databasebeta

← #8328

Subgroups — #8328

Order 256.

810 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#121C211yesmaximal, characteristic
2#1779C211yesmaximal, characteristic, complemented
3C2×#327C214yesmaximal, complemented
4C22⋊#95C211yesmaximal, characteristic, complemented
5C23×D4—21no—
6C23×D4C411yescharacteristic, complemented
7C24×C4C2211yescharacteristic
8C2×#121—21no—
9C2×#121—21no—
10C2×#121—28no—
11C2×#121C412yescomplemented
12C2×#121C2212yes—
13C22×C4⋊C4C411yescharacteristic, complemented
14C22×C22⋊C4—21no—
15C22×C22⋊C4—21no—
16C2×C22⋊C8—21no—
17C2×C22⋊C8—21no—
18C2×C22⋊C8C2212yescomplemented
19C2×#95—21no—
20C2×#95C2212yescomplemented
21#327—28no—
22C25—21no—
23C25D411yescharacteristic
24C22×D4—28no—
25C22×D4—22no—
26C22×D4—22no—
27C22×D4—24no—
28C22×D4—21no—
29C22×D4—42no—
30C22×D4—42no—
31C22×D4—44no—
32C22×D4—42no—
33C22×D4—42no—
34C22×D4C2×C412yes—
35C23×C4—21no—
36C23×C4—24no—
37C23×C4—22no—
38C23×C4—42no—
39C23×C4—42no—
40C23×C4C2×C411yescharacteristic
41C23×C4C2×C411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C22×C8—22no—
47#121—28no—
48#121—28no—
49#121—24no—
50#121—24no—
51#121—24no—
52#121D418yes—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4C2×C412yes—
56C2×C22⋊C4—28no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4—42no—
62C22⋊C8—24no—
63C22⋊C8—42no—
64C22⋊C8—42no—
65#95—24no—
66#95—42no—
67C24—22no—
68C24—22no—
69C24—22no—
70C24—22no—
71C24—24no—
72C24—22no—
73C24—41no—
74C24—41no—
75C24—42no—
76C24—42no—
77C24—42no—
78C24—42no—
79C24—42no—
80C24—42no—
81C24C22⋊C411yescharacteristic
82C24C22⋊C411yescharacteristic
83C24D812yes—
84C24SD1612yes—
85C24C2×D411yescharacteristic
86C2×D4—24no—
87C2×D4—28no—
88C2×D4—28no—
89C2×D4—22no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—416no—
93C2×D4—42no—
94C2×D4—416no—
95C2×D4—416no—
96C2×D4—44no—
97C2×D4—44no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—44no—
101C2×D4—42no—
102C2×D4C22⋊C414yescomplemented
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—416no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C22⋊C411yescharacteristic
133C22×C4C22⋊C411yescharacteristic
134C22×C4C22×C411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D414yes—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4C22⋊C414yescomplemented
143C22⋊C4—28no—
144C22⋊C4—28no—
145C22⋊C4—48no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—24no—
155C23—24no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—24no—
163C23—24no—
164C23—24no—
165C23—42no—
166C23—42no—
167C23—44no—
168C23—44no—
169C23—44no—
170C23—44no—
171C23—48no—
172C23—48no—
173C23—48no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—84no—
181C23—84no—
182C23—84no—
183C23—84no—
184C23#9912yes—
185C23#10212yes—
186C23#10212yes—
187C23#10412yes—
188C23C2×C22⋊C411yescharacteristic
189C23#12012yes—
190C23C2×D811yescharacteristic
191C23C2×SD1611yescharacteristic
192C23AGL(1,ℤ/8)12yes—
193D4—44no—
194D4—48no—
195D4—44no—
196D4—416no—
197D4—48no—
198D4—48no—
199C2×C4—28no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4#9912yes—
226C2×C4C2×C22⋊C412yes—
227C2×C4#12012yes—
228C8—82no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—22no—
237C22—21no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—48no—
247C22—48no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—44no—
253C22—44no—
254C22—44no—
255C22—816no—
256C22—82no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22#32714yescomplemented
261C22#37911yescharacteristic
262C22C2×#9911yescharacteristic
263C22C2×#9911yescharacteristic
264C22C2×#10211yescharacteristic
265C22#41811yescharacteristic
266C22C2×#10411yescharacteristic
267C22#42111yescharacteristic
268C22#44712yes—
269C22#45012yes—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—44no—
275C4—84no—
276C2—22no—
277C2—24no—
278C2—22no—
279C2—42no—
280C2—84no—
281C2C2×#32711yescharacteristic
282C2#150711yescharacteristic
283C2#151011yescharacteristic
284C2#151911yescharacteristic
285C2#178111yescharacteristic
286C2#178611yescharacteristic
287C2#178911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×#121C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1779C2semidirect
3. C2×#327C2semidirect
4. C22⋊#95C2semidirect
6. C23×D4C4semidirect
11. C2×#121C4semidirect
13. C22×C4⋊C4C4semidirect
18. C2×C22⋊C8C22semidirect
20. C2×#95C22semidirect
102. C2×D4C22⋊C4semidirect
142. C4⋊C4C22⋊C4semidirect
260. C22#327semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
#17792
C2×#3272
C22⋊#952