Finite Group Databasebeta

← #8315

Subgroups — #8315

Order 256.

434 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 97 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#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×Q8⋊C8C212yesmaximal, complemented
5C2×D4⋊C8C212yesmaximal, complemented
6#529—24no—
7#529—24no—
8C2×#126—21no—
9C2×#125C411yescharacteristic
10C2×#124C411yescharacteristic
11C2×#123—21no—
12C2×C22⋊Q8C411yescharacteristic
13C2×#121C411yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic
17C2×#117—21no—
18C22×C42C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C22⋊C8—21no—
21C2×C22⋊C8—21no—
22C2×C4×C8C2212yes—
23#336—21no—
24#336—21no—
25Q8⋊C8—24no—
26D4⋊C8—24no—
27C22×Q8C2×C411yescharacteristic
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C23×C4C2×C411yescharacteristic
32C23×C4D411yescharacteristic
33C23×C4D411yescharacteristic
34C22×C8—22no—
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⋊Q8—22no—
44#121—22no—
45#121—22no—
46C4×Q8—22no—
47C4×Q8D414yes—
48C4×D4—28no—
49C4×D4—22no—
50C4×D4D414yes—
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×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C4⋊C8—22no—
71C4⋊C8—22no—
72C22⋊C8—24no—
73C22⋊C8—42no—
74C4×C8—22no—
75C4×C8—22no—
76C24—21no—
77C24C22⋊C411yescharacteristic
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8C22⋊C412yes—
82C2×D4—24no—
83C2×D4—21no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—42no—
87C2×D4—42no—
88C2×D4C22⋊C412yes—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4C22⋊C411yescharacteristic
108C22×C4C22⋊C411yescharacteristic
109C22×C4C22⋊C411yescharacteristic
110C22×C4C22×C411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C2×C8—24no—
114C2×C8—42no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—42no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—24no—
125C4⋊C4C22⋊C412yes—
126C4⋊C4C22⋊C412yes—
127C22⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—24no—
131C22⋊C4—24no—
132C42—21no—
133C42—21no—
134C42—22no—
135C42—22no—
136C42—24no—
137C42C2×D412yes—
138C42C2×D412yes—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—22no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23#10412yes—
153C23C2×C22⋊C411yescharacteristic
154Q8—24no—
155Q8—42no—
156D4—24no—
157D4—48no—
158D4—42no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—44no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4C2×C22⋊C411yescharacteristic
187C2×C4C2×C22⋊C411yescharacteristic
188C2×C4#12012yes—
189C2×C4#12012yes—
190C2×C4#13112yes—
191C2×C4#13112yes—
192C2×C4#13512yes—
193C2×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C4#13711yescharacteristic
195C8—44no—
196C8—82no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22#37911yescharacteristic
212C22#40811yescharacteristic
213C22#40811yescharacteristic
214C22#41611yescharacteristic
215C22#41711yescharacteristic
216C22C2×#10411yescharacteristic
217C22#42111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—42no—
225C4#40812yes—
226C4#40812yes—
227C4#44912yes—
228C4#45212yes—
229C2—42no—
230C2—82no—
231C2#148211yescharacteristic
232C2#148311yescharacteristic
233C2#149411yescharacteristic
234C2#149511yescharacteristic
235C2#178011yescharacteristic
236C2#178811yescharacteristic
237C2#178911yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. C4×C22⋊C8C2semidirect
4. C2×Q8⋊C8C2semidirect
5. C2×D4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
#17972
C4×C22⋊C82
C2×Q8⋊C82
C2×D4⋊C82