Finite Group Databasebeta

← #8220

Subgroups — #8220

Order 256.

362 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C4×#121C211yesmaximal, characteristic
2C4×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×#327C212yesmaximal
4#1434C211yesmaximal, characteristic, complemented
5#1294C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic, complemented
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11C2×C4×C8—21no—
12#390C411yescharacteristic, complemented
13#389C411yescharacteristic, complemented
14#388—21no—
15#386—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#327C418yescomplemented
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4—22no—
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32#121—22no—
33#121C2×C414yes—
34C4×D4—42no—
35C4×D4—42no—
36C4×D4—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42—21no—
47C2×C42—42no—
48C2×C42C2×C411yescharacteristic
49C2×C42C2×C411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C22⋊C8—22no—
53C22⋊C8C2×C414yescomplemented
54C4×C8—42no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—22no—
59#95—42no—
60#95—42no—
61#95C2×C414yescomplemented
62C24—21no—
63C24C2×D411yescharacteristic
64C2×D4—24no—
65C2×D4—21no—
66C2×D4—41no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4C4212yes—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—24no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4D812yes—
97C22×C4SD1612yes—
98C22×C4C22×C411yescharacteristic
99C22×C4C22×C412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4#4112yes—
102C2×C8—24no—
103C2×C8—42no—
104C2×C8—42no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—42no—
109C4⋊C4C4212yes—
110C22⋊C4—24no—
111C22⋊C4—42no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C42—22no—
117C42—22no—
118C42—41no—
119C42—41no—
120C42—42no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23C2×C22⋊C411yescharacteristic
133C23C4×D412yes—
134C23C2×D811yescharacteristic
135C23C2×SD1611yescharacteristic
136C23#13512yes—
137D4—42no—
138D4—42no—
139D4—44no—
140D4—44no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—44no—
171C2×C4—84no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#9912yes—
177C2×C4#10212yes—
178C2×C4#10212yes—
179C2×C4C2×C4211yescharacteristic
180C2×C4#11711yescharacteristic
181C2×C4C4×D412yes—
182C8—44no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—22no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22C4×C22⋊C411yescharacteristic
199C22C2×#9911yescharacteristic
200C22#41011yescharacteristic
201C22C2×#10211yescharacteristic
202C22#41611yescharacteristic
203C22#42111yescharacteristic
204C22#42111yescharacteristic
205C22C4×D812yes—
206C22C4×SD1612yes—
207C4—22no—
208C4—21no—
209C4—21no—
210C4—22no—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—41no—
215C4—84no—
216C4—81no—
217C4—81no—
218C2—22no—
219C2—22no—
220C2—82no—
221C2#151611yescharacteristic
222C2#151711yescharacteristic
223C2#151911yescharacteristic
224C2#152111yescharacteristic
225C2C4×#9911yescharacteristic
226C2#175911yescharacteristic
227C2C4×#10211yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C2×C42
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊C8C2semidirect
4. #1434C2semidirect
5. #1294C2semidirect
8. C2×C4×D4C4semidirect
12. #390C4semidirect
13. #389C4semidirect
17. C4×C4⋊C4C4semidirect
23. #327C4semidirect
53. C22⋊C8C2×C4semidirect
61. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
C4×C22⋊C82
C2×#3272
#14342
#12942