Finite Group Databasebeta

← #8330

Subgroups — #8330

Order 256.

454 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 89 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×#529C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic, complemented
3C2×#328C212yesmaximal, complemented
4C2×#327C212yesmaximal
5#1434C211yesmaximal, characteristic, complemented
6#529—24no—
7#529—24no—
8C2×#126—21no—
9C2×#125C411yescharacteristic, complemented
10C2×#124C411yescharacteristic, complemented
11C2×#123—21no—
12C2×C22⋊Q8C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C2×C4×Q8C411yescharacteristic, complemented
15C2×C4×D4—21no—
16C2×C4×D4C411yescharacteristic, complemented
17C2×#117—21no—
18C22×C42C2211yescharacteristic
19C2×C22⋊C8C2212yes—
20C2×#95—21no—
21C2×#95—21no—
22C2×#95C2212yes—
23#336—21no—
24#336—21no—
25#328—24no—
26#327—24no—
27C22×Q8C2×C411yescharacteristic
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C23×C4—22no—
32C23×C4D411yescharacteristic
33C23×C4D411yescharacteristic
34C23×C4C2311yescharacteristic
35C22×C8—22no—
36#126—28no—
37#125—22no—
38#125—22no—
39#124—22no—
40#124—22no—
41#123—28no—
42C22⋊Q8—22no—
43C22⋊Q8D414yes—
44#121—22no—
45#121D414yes—
46C4×Q8—22no—
47C4×Q8—22no—
48C4×D4—28no—
49C4×D4—22no—
50C4×D4—22no—
51#117—28no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2×C411yescharacteristic
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C42—21no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42C2×C411yescharacteristic
67C2×C42C2×C411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C22⋊C8—22no—
71C22⋊C8—22no—
72#95—24no—
73#95—22no—
74#95—22no—
75#95—42no—
76C24—21no—
77C24C2×D411yescharacteristic
78C2×Q8—21no—
79C2×Q8—41no—
80C2×Q8—41no—
81C2×Q8C22⋊C412yes—
82C2×D4—24no—
83C2×D4—21no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—42no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4C22⋊C412yes—
90C22×C4—24no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—24no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4C22⋊C411yescharacteristic
113C22×C4C22⋊C411yescharacteristic
114C22×C4C22⋊C411yescharacteristic
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22×C411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—24no—
129C4⋊C4C22⋊C412yes—
130C4⋊C4C22⋊C412yes—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C42—24no—
137C42—21no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—22no—
142C42—22no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—22no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23C2×C22⋊C411yescharacteristic
155C23#12012yes—
156C23#13512yes—
157C23AGL(1,ℤ/8)11yescharacteristic
158C23#13711yescharacteristic
159Q8—42no—
160Q8—42no—
161D4—42no—
162D4—42no—
163D4—44no—
164D4—44no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#10412yes—
203C2×C4C2×C22⋊C411yescharacteristic
204C2×C4C2×C22⋊C411yescharacteristic
205C2×C4#12012yes—
206C8—82no—
207C22—22no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22#37911yescharacteristic
223C22#41011yescharacteristic
224C22#41011yescharacteristic
225C22#41611yescharacteristic
226C22#41711yescharacteristic
227C22C2×#10411yescharacteristic
228C22#42111yescharacteristic
229C22#44912yes—
230C22#45212yes—
231C4—22no—
232C4—21no—
233C4—21no—
234C4—22no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—42no—
241C4—84no—
242C2—22no—
243C2—22no—
244C2—82no—
245C2#150011yescharacteristic
246C2#150911yescharacteristic
247C2#151211yescharacteristic
248C2#152111yescharacteristic
249C2#178211yescharacteristic
250C2#178811yescharacteristic
251C2#178911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1817C2semidirect
3. C2×#328C2semidirect
5. #1434C2semidirect
9. C2×#125C4semidirect
10. C2×#124C4semidirect
14. C2×C4×Q8C4semidirect
16. C2×C4×D4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
#18172
C2×#3282
C2×#3272
#14342