Finite Group Databasebeta

← #9111

Subgroups — #9111

Order 256.

492 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 93 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×C22⋊C8C211yesmaximal, characteristic
2#2848C211yesmaximal, characteristic
3C2×#328C212yesmaximal, complemented
4C22⋊#95C211yesmaximal, characteristic
5#1294C212yesmaximal, complemented
6C24×C4C2211yescharacteristic
7C23×C8—21no—
8C2×C22⋊Q8C411yescharacteristic, complemented
9C2×C22⋊Q8C411yescharacteristic, complemented
10C2×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C4—21no—
12C2×C22⋊C8—21no—
13C2×C22⋊C8—24no—
14C2×C22⋊C8C2212yes—
15#385—21no—
16#385—21no—
17#385—21no—
18#380C411yescharacteristic, complemented
19#380C411yescharacteristic, complemented
20#380C2211yescharacteristic
21#379—21no—
22C2×#95—21no—
23C2×#95C2212yes—
24#328—24no—
25C25D411yescharacteristic
26C22×Q8C2×C411yescharacteristic
27C23×C4—22no—
28C23×C4—22no—
29C23×C4—22no—
30C23×C4—42no—
31C23×C4—42no—
32C23×C4C2×C411yescharacteristic
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—24no—
39C22×C8—22no—
40C22×C8—42no—
41C22×C8—42no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44C22⋊Q8—42no—
45C22⋊Q8—42no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4C2×C411yescharacteristic
50C2×C4⋊C4C2×C411yescharacteristic
51C2×C4⋊C4C2×C411yescharacteristic
52C2×C22⋊C4—24no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C22⋊C8—24no—
62C22⋊C8—22no—
63C22⋊C8D414yes—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—22no—
68#95—22no—
69#95—42no—
70C24—22no—
71C24—22no—
72C24—22no—
73C24—22no—
74C24—41no—
75C24—41no—
76C24C22⋊C411yescharacteristic
77C24C22⋊C411yescharacteristic
78C24SD1612yes—
79C24Q1612yes—
80C24C2×D411yescharacteristic
81C2×Q8—22no—
82C2×Q8—41no—
83C2×Q8—41no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—24no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—48no—
109C22×C4—48no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C22×C411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4#4112yes—
122C2×C8—24no—
123C2×C8—48no—
124C2×C8—42no—
125C2×C8—42no—
126C4⋊C4—22no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C22⋊C4—42no—
134C22⋊C4—48no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—24no—
149C23—24no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—24no—
157C23—24no—
158C23—42no—
159C23—42no—
160C23—44no—
161C23—44no—
162C23—44no—
163C23—44no—
164C23#9912yes—
165C23#10312yes—
166C23#10312yes—
167C23#10412yes—
168C23C2×C22⋊C411yescharacteristic
169C23#12112yes—
170C23C2×SD1611yescharacteristic
171C23C2×Q1611yescharacteristic
172C23#13512yes—
173Q8—82no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
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—41no—
206C2×C4—41no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#9912yes—
212C2×C4C4×D412yes—
213C2×C4#12011yescharacteristic
214C2×C4#12311yescharacteristic
215C8—44no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—22no—
229C22—21no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—48no—
234C22—48no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22#32814yescomplemented
240C22#38611yescharacteristic
241C22C2×#9911yescharacteristic
242C22C2×#9911yescharacteristic
243C22C2×#10311yescharacteristic
244C22#41611yescharacteristic
245C22C2×#10411yescharacteristic
246C22#42111yescharacteristic
247C22#46512yes—
248C22#46712yes—
249C4—21no—
250C4—22no—
251C4—21no—
252C4—22no—
253C4—41no—
254C4—84no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—81no—
259C2—22no—
260C2—24no—
261C2—22no—
262C2—42no—
263C2C2×#32811yescharacteristic
264C2#150111yescharacteristic
265C2#151611yescharacteristic
266C2#152011yescharacteristic
267C2#187111yescharacteristic
268C2#189211yescharacteristic
269C2#189811yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
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
3. C2×#328C2semidirect
5. #1294C2semidirect
8. C2×C22⋊Q8C4semidirect
9. C2×C22⋊Q8C4semidirect
18. #380C4semidirect
19. #380C4semidirect
239. C22#328semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C82
#28482
C2×#3282
C22⋊#952
#12942