Finite Group Databasebeta

← #9443

Subgroups — #9443

Order 256.

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

normal subgroups only

This group has 356 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
1#2879C211yesmaximal, characteristic
2#2432C211yesmaximal, characteristic
3#1782C211yesmaximal, characteristic
4#1511C212yesmaximal, complemented
5#1284C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C22×M16—21no—
8C2×C4⋊Q8C411yescharacteristic, complemented
9C2×#124C411yescharacteristic, complemented
10C2×C22⋊Q8C2211yescharacteristic
11C2×#99—21no—
12C2×#99—21no—
13#408—24no—
14#407—21no—
15#407C2212yes—
16#394—21no—
17#394—21no—
18#391C411yescharacteristic, complemented
19#390C411yescharacteristic, complemented
20#385—21no—
21#385—21no—
22#381C2212yes—
23C4×C22⋊C4C2211yescharacteristic
24#328—24no—
25C2×#41—22no—
26C2×#41—22no—
27C2×#41—42no—
28C2×#41—42no—
29C22×Q8C2×C411yescharacteristic
30C22×Q8C2×C411yescharacteristic
31C22×D4C2×C411yescharacteristic
32C22×D4D411yescharacteristic
33C22×D4D411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C2×M16—22no—
38C2×M16—22no—
39C2×M16—42no—
40C2×M16—42no—
41C22×C8—22no—
42C4⋊Q8—42no—
43#124—42no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4C2×C411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—22no—
57C2×C42C2×C411yescharacteristic
58C2×C42C2×C411yescharacteristic
59#99—42no—
60#99—42no—
61C22⋊C8—24no—
62C22⋊C8—22no—
63C22⋊C8D414yes—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—22no—
68#95—22no—
69C24C22⋊C411yescharacteristic
70C24C22⋊C411yescharacteristic
71C24C2×D411yescharacteristic
72#41—44no—
73#41—44no—
74#41—48no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×Q8—41no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4C22⋊C411yescharacteristic
106C22×C4C22⋊C411yescharacteristic
107C22×C4C22×C411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4#4112yes—
111M16—44no—
112M16—42no—
113M16—42no—
114C2×C8—24no—
115C2×C8—44no—
116C4⋊C4—22no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—41no—
130C42—41no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#9912yes—
145C23C2×C22⋊C411yescharacteristic
146C23#12112yes—
147Q8—44no—
148Q8—82no—
149D4—44no—
150D4—44no—
151D4—44no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#9912yes—
184C2×C4C4×D412yes—
185C2×C4#12011yescharacteristic
186C2×C4#12311yescharacteristic
187C8—44no—
188C22—21no—
189C22—21no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22#38611yescharacteristic
206C22C2×#9911yescharacteristic
207C22C2×#9911yescharacteristic
208C4—21no—
209C4—21no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—41no—
214C4—81no—
215C4—81no—
216C4—81no—
217C4—162no—
218C2—21no—
219C2—21no—
220C2—42no—
221C2—42no—
222C2—81no—
223C2#188411yescharacteristic
224C2#189811yescharacteristic
225C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1511C2semidirect
5. #1284C2semidirect
8. C2×C4⋊Q8C4semidirect
9. C2×#124C4semidirect
18. #391C4semidirect
19. #390C4semidirect

Maximal subgroups

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

subgroupindex
#28792
#24322
#17822
#15112
#12842