Finite Group Databasebeta

← #8497

Subgroups — #8497

Order 256.

410 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 71 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#3543C211yesmaximal, characteristic
2#1961C211yesmaximal, characteristic, complemented
3#1555C211yesmaximal, characteristic, complemented
4#1532C212yesmaximal, complemented
5#1512C212yesmaximal, complemented
6#560—22no—
7#559C411yescharacteristic, complemented
8#550C412yescomplemented
9#546—22no—
10D42—22no—
11D42C411yescharacteristic, complemented
12#540—22no—
13#535—22no—
14#534—22no—
15C2×#127C2211yescharacteristic
16#529C2212yes—
17#432C2212yescomplemented
18C2×#102—22no—
19C2×#97C2211yescharacteristic, complemented
20#382C2211yescharacteristic, complemented
21#331—22no—
22#330—22no—
23#329—22no—
24#328—22no—
25#327—22no—
26C2×#41—22no—
27C22×D4—22no—
28C22×D4—22no—
29C22×D4—22no—
30C22×D4D412yes—
31C22×C8—21no—
32#127—22no—
33#127—22no—
34#127C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#127D412yes—
37#126—22no—
38#125D412yes—
39#124—22no—
40#124D412yes—
41#123—22no—
42#123—24no—
43#123—22no—
44#123—22no—
45C22⋊Q8D412yes—
46#121—24no—
47#121—24no—
48#121—24no—
49#121—24no—
50#121—22no—
51#121—22no—
52#121D412yes—
53#120—22no—
54#120—24no—
55#120—24no—
56#120—22no—
57C4×Q8C2×C412yes—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4C2×C412yes—
62#117—22no—
63C2×C4⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67#102—44no—
68C22⋊C8—22no—
69#97—22no—
70#97—22no—
71#95—22no—
72C24—22no—
73C24—22no—
74#41—84no—
75C2×Q8C22⋊C412yes—
76C2×D4—22no—
77C2×D4—24no—
78C2×D4—24no—
79C2×D4—24no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—24no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—84no—
87C2×D4—84no—
88C2×D4—82no—
89C2×D4—82no—
90C2×D4C22⋊C412yes—
91C2×D4C22⋊C412yes—
92C2×D4C22⋊C412yes—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—41no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99C2×C8—22no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—42no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—42no—
108C4⋊C4C22⋊C412yes—
109C4⋊C4C22⋊C412yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C42—22no—
120C42C22×C411yescharacteristic
121C42C2×D412yes—
122C42C2×D411yescharacteristic
123C23—22no—
124C23—22no—
125C23—22no—
126C23—42no—
127C23—42no—
128C23—84no—
129C23—82no—
130C23—84no—
131C23#12011yescharacteristic
132Q8—42no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—44no—
137D4—44no—
138D4—42no—
139D4—42no—
140D4—82no—
141D4—84no—
142D4—84no—
143D4—84no—
144D4—84no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—42no—
151C2×C4—44no—
152C2×C4—82no—
153C2×C4—84no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4C2×C22⋊C412yes—
157C2×C4C2×C22⋊C411yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12012yes—
160C8—42no—
161C8—82no—
162C22—22no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—42no—
167C22—44no—
168C22—44no—
169C22—42no—
170C22—162no—
171C22—162no—
172C22—164no—
173C22#37911yescharacteristic
174C22#45311yescharacteristic
175C22#45511yescharacteristic
176C4—21no—
177C4—21no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—42no—
183C4—161no—
184C2—21no—
185C2—21no—
186C2—82no—
187C2—82no—
188C2—82no—
189C2#179211yescharacteristic
190C2#179512yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3543C2
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. #1961C2semidirect
3. #1555C2semidirect
4. #1532C2semidirect
5. #1512C2semidirect
7. #559C4semidirect
8. #550C4semidirect
11. D42C4semidirect
17. #432C22semidirect
19. C2×#97C22semidirect
20. #382C22semidirect

Maximal subgroups

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

subgroupindex
#35432
#19612
#15552
#15322
#15122