Finite Group Databasebeta

← #13085

Subgroups — #13085

Order 256.

362 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 63 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×#410C211yesmaximal, characteristic
2#2406C211yesmaximal, characteristic
3C2×#352C212yesmaximal, complemented
4#1785C211yesmaximal, characteristic
5#1390C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×#123C411yescharacteristic, complemented
8C2×#123C2211yescharacteristic
9C2×#117—21no—
10C22×C4⋊C4C411yescharacteristic, complemented
11C2×#101—21no—
12C2×#100C2212yes—
13#410—24no—
14C2×#99—21no—
15C2×#99C2212yes—
16#407—21no—
17#395—21no—
18#394—21no—
19#386C411yescharacteristic, complemented
20#386C2211yescharacteristic
21#385—21no—
22#385—21no—
23#381C411yescharacteristic, complemented
24#352—24no—
25C2×#41—22no—
26C2×#41—22no—
27C2×#41—41no—
28C2×#41—41no—
29C2×#41—42no—
30C22×Q8D411yescharacteristic
31C22×D4C2×C411yescharacteristic
32C22×D4D411yescharacteristic
33C22×D4C2311yescharacteristic
34C23×C4C2×C411yescharacteristic
35C23×C4C2×C411yescharacteristic
36C23×C4C2×C411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C2×M16—22no—
40#123—22no—
41#123—22no—
42#123—42no—
43#117—24no—
44#117—42no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4C2×C411yescharacteristic
58C2×C22⋊C4C2×C411yescharacteristic
59C2×C42—41no—
60#101—42no—
61#100—22no—
62#100—22no—
63#99—24no—
64#99—22no—
65#99D414yes—
66C22⋊C8—41no—
67C22⋊C8—41no—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72C24C22⋊C411yescharacteristic
73C24C22×C411yescharacteristic
74C24C2×D411yescharacteristic
75#41—44no—
76#41—44no—
77#41—44no—
78#41—44no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—41no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—22no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4C2×D412yes—
92C2×D4#4112yes—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C22⋊C411yescharacteristic
119C22×C4C2×D411yescharacteristic
120M16—42no—
121M16—42no—
122C2×C8—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C22⋊C4—22no—
130C22⋊C4—24no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C42—44no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23C2×C22⋊C411yescharacteristic
153C23C4×D412yes—
154C23#12011yescharacteristic
155C23#12311yescharacteristic
156Q8—44no—
157D4—42no—
158D4—42no—
159D4—44no—
160D4—42no—
161D4—42no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4#9912yes—
198C2×C4#9911yescharacteristic
199C2×C4#9911yescharacteristic
200C2×C4#12112yes—
201C8—82no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22#38611yescharacteristic
222C22C2×#9911yescharacteristic
223C22C2×#9911yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—22no—
228C4—42no—
229C4—41no—
230C4—81no—
231C4—81no—
232C4—84no—
233C4—81no—
234C2—21no—
235C2—21no—
236C2—41no—
237C2—42no—
238C2—41no—
239C2—81no—
240C2#189811yescharacteristic
241C2#212011yescharacteristic
242C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC22×#41C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#352C2semidirect
5. #1390C2semidirect
7. C2×#123C4semidirect
10. C22×C4⋊C4C4semidirect
19. #386C4semidirect
23. #381C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4102
#24062
C2×#3522
#17852
#13902