Finite Group Databasebeta

← #8426

Subgroups — #8426

Order 256.

418 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#3508C211yesmaximal, characteristic
2#1982C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic, complemented
4#1550C211yesmaximal, characteristic, complemented
5#1545C211yesmaximal, characteristic, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic, complemented
8#559—21no—
9#559C412yescomplemented
10#550—21no—
11#550C412yescomplemented
12#546—22no—
13#546—22no—
14#540—22no—
15#535—21no—
16#533C2211yescharacteristic
17#532—21no—
18#532—21no—
19C2×#127C2211yescharacteristic
20#518—21no—
21C4×#41C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—21no—
26C2×#102—22no—
27C4×C4⋊C4C2211yescharacteristic
28#339—21no—
29#333—21no—
30#332—21no—
31#331—22no—
32#328—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41D411yescharacteristic
38C22×D4—21no—
39C22×D4—21no—
40C22×D4D412yes—
41C2×M16—21no—
42C22×C8—21no—
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#127—22no—
46#127—21no—
47#127—21no—
48#127C2×C412yes—
49#127D412yes—
50#125—21no—
51#124—22no—
52#124—22no—
53#124—22no—
54#123—24no—
55#123—24no—
56C22⋊Q8—21no—
57C22⋊Q8D412yes—
58#121—22no—
59#121—22no—
60#121—22no—
61#121—24no—
62#121—24no—
63#121—22no—
64#120—24no—
65C4×Q8—21no—
66C4×Q8C2×C411yescharacteristic
67C4×Q8C2×C412yes—
68C4×Q8D411yescharacteristic
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4C2×C411yescharacteristic
78C4×D4D411yescharacteristic
79#117—21no—
80#117—21no—
81#117D411yescharacteristic
82C2×C4⋊C4—21no—
83C2×C22⋊C4—24no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92#104—42no—
93#102—44no—
94C22⋊C8—21no—
95C4×C8—21no—
96#95—21no—
97C24—22no—
98#41—21no—
99#41—21no—
100#41—21no—
101#41—21no—
102#41—84no—
103#41—82no—
104#41—82no—
105C2×Q8—21no—
106C2×Q8C22⋊C411yescharacteristic
107C2×Q8C22⋊C412yes—
108C2×D4—21no—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—82no—
123C2×D4C22⋊C411yescharacteristic
124C2×D4C22⋊C414yes—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134M16—41no—
135M16—41no—
136C2×C8—42no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4C22⋊C411yescharacteristic
150C4⋊C4C22⋊C411yescharacteristic
151C4⋊C4C22⋊C412yes—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—24no—
155C22⋊C4—24no—
156C22⋊C4—21no—
157C22⋊C4—22no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C42—21no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42C22×C411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C23—24no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—82no—
175C23—82no—
176C23#12011yescharacteristic
177Q8—21no—
178Q8—21no—
179Q8—42no—
180Q8—41no—
181Q8—41no—
182D4—21no—
183D4—21no—
184D4—44no—
185D4—44no—
186D4—41no—
187D4—41no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—42no—
194D4—82no—
195D4—84no—
196D4—84no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4C2×C22⋊C411yescharacteristic
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4C2×C22⋊C411yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#13511yescharacteristic
222C2×C4#13511yescharacteristic
223C2×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C4#13711yescharacteristic
225C8—81no—
226C8—81no—
227C8—81no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—44no—
232C22—41no—
233C22—44no—
234C22—41no—
235C22—44no—
236C22—162no—
237C22—162no—
238C22#37911yescharacteristic
239C22#44911yescharacteristic
240C22#45211yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—41no—
245C4—42no—
246C4—41no—
247C4—42no—
248C4—41no—
249C4—82no—
250C4#41611yescharacteristic
251C4#41711yescharacteristic
252C4#44911yescharacteristic
253C4#45211yescharacteristic
254C4#45311yescharacteristic
255C4#45511yescharacteristic
256C2—21no—
257C2—21no—
258C2—81no—
259C2—84no—
260C2#178811yescharacteristic
261C2#179211yescharacteristic
262C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3508C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1982C2semidirect
3. #1806C2semidirect
4. #1550C2semidirect
5. #1545C2semidirect
6. #1503C2semidirect
7. #1477C2semidirect
9. #559C4semidirect
11. #550C4semidirect

Maximal subgroups

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

subgroupindex
#35082
#19822
#18062
#15502
#15452
#15032
#14772