Finite Group Databasebeta

← #8422

Subgroups — #8422

Order 256.

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

normal subgroups only

This group has 378 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#1987C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic, complemented
4#1549C211yesmaximal, characteristic, complemented
5#1547C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic, complemented
8#556C412yescomplemented
9#555—21no—
10#554—21no—
11#553—22no—
12#548—22no—
13#547C412yescomplemented
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—
26C4×C4⋊C4C2211yescharacteristic
27#340—22no—
28#339—21no—
29#332—21no—
30#332—22no—
31#331—21no—
32#327—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
35C2×#41—22no—
36C2×#41—21no—
37C2×#41D411yescharacteristic
38C22×Q8—21no—
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—24no—
50#124—22no—
51#124—21no—
52#123—24no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57#121—22no—
58#121—22no—
59#121—24no—
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—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D412yes—
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—
93C22⋊C8—21no—
94C4×C8—21no—
95#95—21no—
96#41—21no—
97#41—21no—
98#41—21no—
99#41—21no—
100#41—41no—
101#41—41no—
102#41—41no—
103#41—41no—
104#41—84no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—82no—
109C2×Q8C22⋊C411yescharacteristic
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—24no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C22⋊C411yescharacteristic
116C2×D4C22⋊C412yes—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127M16—41no—
128M16—41no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—41no—
143C4⋊C4C22⋊C411yescharacteristic
144C4⋊C4C22⋊C414yes—
145C4⋊C4C22⋊C411yescharacteristic
146C4⋊C4C22⋊C412yes—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—22no—
155C42—24no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—41no—
161C42C22×C411yescharacteristic
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C23—21no—
166C23—22no—
167C23#12011yescharacteristic
168Q8—21no—
169Q8—21no—
170Q8—44no—
171Q8—41no—
172Q8—41no—
173Q8—82no—
174D4—21no—
175D4—21no—
176D4—42no—
177D4—41no—
178D4—42no—
179D4—41no—
180D4—42no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185D4—84no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—82no—
203C2×C4—84no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4C2×C22⋊C411yescharacteristic
207C2×C4C2×C22⋊C411yescharacteristic
208C2×C4C2×C22⋊C411yescharacteristic
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4#12011yescharacteristic
212C2×C4#13511yescharacteristic
213C2×C4#13511yescharacteristic
214C2×C4AGL(1,ℤ/8)11yescharacteristic
215C2×C4#13711yescharacteristic
216C8—81no—
217C8—81no—
218C8—81no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—42no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22—42no—
227C22#37911yescharacteristic
228C22#44911yescharacteristic
229C22#45211yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—44no—
236C4—41no—
237C4—42no—
238C4—82no—
239C4#41611yescharacteristic
240C4#41711yescharacteristic
241C4#44911yescharacteristic
242C4#45211yescharacteristic
243C4#45411yescharacteristic
244C4#45611yescharacteristic
245C2—21no—
246C2—21no—
247C2—81no—
248C2—82no—
249C2#178811yescharacteristic
250C2#179311yescharacteristic
251C2#179511yescharacteristic
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. #1987C2semidirect
3. #1806C2semidirect
4. #1549C2semidirect
5. #1547C2semidirect
6. #1502C2semidirect
7. #1477C2semidirect
8. #556C4semidirect
13. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35112
#19872
#18062
#15492
#15472
#15022
#14772