Finite Group Databasebeta

← #8423

Subgroups — #8423

Order 256.

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

normal subgroups only

This group has 386 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#3511C211yesmaximal, characteristic
2#1981C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic, complemented
4#1548C211yesmaximal, characteristic, complemented
5#1547C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic, complemented
8#556—21no—
9#555C412yescomplemented
10#554C412yescomplemented
11#553—22no—
12#548—22no—
13#547—21no—
14#542—22no—
15#537—21no—
16#533—21no—
17#532—21no—
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#520—21no—
21C4×#41C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—21no—
26C2×#103—22no—
27C4×C4⋊C4C2211yescharacteristic
28#339—21no—
29#333—22no—
30#332—21no—
31#331—21no—
32#327—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
35C2×#41—22no—
36C2×#41—21no—
37C2×#41D411yescharacteristic
38C22×Q8D412yes—
39C2×M16—21no—
40C22×C8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8C2×C412yes—
44C4⋊Q8D412yes—
45C4⋊Q8D411yescharacteristic
46#127D411yescharacteristic
47#126—24no—
48#125—22no—
49#124—22no—
50#124—24no—
51#124—21no—
52#123—24no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57#121—22no—
58#121—24no—
59#121—22no—
60#121—22no—
61#121—21no—
62#121D412yes—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8C2×C411yescharacteristic
66C4×Q8D411yescharacteristic
67C4×D4—22no—
68C4×D4—24no—
69C4×D4—22no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4C2×C411yescharacteristic
74C4×D4C2×C412yes—
75C4×D4D411yescharacteristic
76#117—22no—
77#117—22no—
78#117—21no—
79#117D411yescharacteristic
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
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#103—44no—
94C22⋊C8—21no—
95C4×C8—21no—
96#95—21no—
97#41—21no—
98#41—21no—
99#41—21no—
100#41—21no—
101#41—41no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—82no—
106#41—82no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8C22⋊C411yescharacteristic
112C2×Q8C22⋊C414yes—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C22⋊C411yescharacteristic
119C2×D4C22⋊C412yes—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130M16—41no—
131M16—41no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—41no—
147C4⋊C4C22⋊C411yescharacteristic
148C4⋊C4C22⋊C411yescharacteristic
149C4⋊C4C22⋊C412yes—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—22no—
156C22⋊C4—21no—
157C22⋊C4—22no—
158C42—24no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—41no—
163C42—41no—
164C42C22×C411yescharacteristic
165C42C2×D411yescharacteristic
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C23—21no—
169C23—22no—
170C23#12011yescharacteristic
171Q8—21no—
172Q8—21no—
173Q8—44no—
174Q8—41no—
175Q8—41no—
176Q8—42no—
177Q8—42no—
178D4—21no—
179D4—21no—
180D4—42no—
181D4—41no—
182D4—42no—
183D4—41no—
184D4—42no—
185D4—41no—
186D4—41no—
187D4—41no—
188D4—41no—
189D4—84no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—82no—
207C2×C4—84no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4C2×C22⋊C411yescharacteristic
211C2×C4C2×C22⋊C411yescharacteristic
212C2×C4C2×C22⋊C411yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#13511yescharacteristic
217C2×C4#13511yescharacteristic
218C2×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C4#13711yescharacteristic
220C8—81no—
221C8—81no—
222C8—81no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—42no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—42no—
231C22#37911yescharacteristic
232C22#44911yescharacteristic
233C22#45211yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—44no—
239C4—41no—
240C4—41no—
241C4—42no—
242C4—82no—
243C4#41611yescharacteristic
244C4#41711yescharacteristic
245C4#44911yescharacteristic
246C4#45211yescharacteristic
247C4#45311yescharacteristic
248C4#45511yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—82no—
253C2#178811yescharacteristic
254C2#179211yescharacteristic
255C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3511C2
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. #1981C2semidirect
3. #1806C2semidirect
4. #1548C2semidirect
5. #1547C2semidirect
6. #1502C2semidirect
7. #1477C2semidirect
9. #555C4semidirect
10. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35112
#19812
#18062
#15482
#15472
#15022
#14772