Finite Group Databasebeta

← #7572

Subgroups — #7572

Order 256.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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×#390C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#1270C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7C22×C22⋊C4—21no—
8C22×C22⋊C4—21no—
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8—21no—
13C2×C4×C8—21no—
14#390—21no—
15#390—21no—
16#390C412yes—
17#390C2212yes—
18#336C2212yescomplemented
19#336C2212yescomplemented
20C25—21no—
21C22×D4—21no—
22C22×D4—22no—
23C22×D4—42no—
24C22×D4—42no—
25C22×D4C814yescomplemented
26C22×D4C2×C412yes—
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—42no—
35C2×C4⋊C4C2×C412yes—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—44no—
41C2×C22⋊C4—42no—
42C2×C22⋊C4—42no—
43C2×C42—42no—
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C2×C42C2311yescharacteristic
48C4⋊C8—42no—
49C22⋊C8—24no—
50C22⋊C8—42no—
51C4×C8—42no—
52C24—24no—
53C24—22no—
54C24—42no—
55C24—42no—
56C24—42no—
57C24—42no—
58C24—42no—
59C24—42no—
60C24C22⋊C411yescharacteristic
61C2×D4—24no—
62C2×D4—22no—
63C2×D4—42no—
64C2×D4—42no—
65C2×D4—48no—
66C2×D4—48no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4C2×C814yes—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C2×C812yes—
85C22×C4M1612yes—
86C22×C4D812yes—
87C22×C4SD1612yes—
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C2×C8—24no—
92C2×C8—42no—
93C2×C8—42no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—42no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C22⋊C4—44no—
100C22⋊C4—44no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—44no—
104C42—22no—
105C42—22no—
106C42—41no—
107C42—41no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—24no—
113C23—48no—
114C23—48no—
115C23—48no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—84no—
123C23—84no—
124C23—84no—
125C23—84no—
126C23C22⋊C812yes—
127C23#10212yes—
128C23#10412yes—
129C23C2×C22⋊C411yescharacteristic
130D4—44no—
131D4—44no—
132D4—48no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—44no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—84no—
153C2×C4C22⋊C812yes—
154C2×C4#9912yes—
155C2×C4#10012yes—
156C2×C4#10212yes—
157C2×C4C22×C811yescharacteristic
158C2×C4C2×M1611yescharacteristic
159C2×C4C2×D811yescharacteristic
160C2×C4C2×SD1611yescharacteristic
161C2×C4AGL(1,ℤ/8)12yes—
162C8—44no—
163C8—82no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—44no—
171C22—44no—
172C22—44no—
173C22—816no—
174C22—82no—
175C22—82no—
176C22—82no—
177C22—82no—
178C22D4⋊C814yes—
179C22C2×C22⋊C811yescharacteristic
180C22C2×#9911yescharacteristic
181C22C2×#10011yescharacteristic
182C22C2×#10211yescharacteristic
183C22#41811yescharacteristic
184C22C2×#10411yescharacteristic
185C22#42111yescharacteristic
186C4—21no—
187C4—21no—
188C4—22no—
189C4—21no—
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4#32314yes—
195C2—24no—
196C2—84no—
197C2C2×#32311yescharacteristic
198C2C2×D4⋊C811yescharacteristic
199C2#147811yescharacteristic
200C2#150711yescharacteristic
201C2#151911yescharacteristic
202C2#152711yescharacteristic
203C2#154511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC23×D4C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. C4×C22⋊C8C2semidirect
4. #1270C2semidirect
18. #336C22semidirect
19. #336C22semidirect
25. C22×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#17972
C4×C22⋊C82
#12702