Finite Group Databasebeta

← #9417

Subgroups — #9417

Order 256.

221 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 57 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#3184C211yesmaximal, characteristic
2#2980C211yesmaximal, characteristic, complemented
3#1536C211yesmaximal, characteristic, complemented
4#1514C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic, complemented
6#1291C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9#532C2211yescharacteristic
10#498C411yescharacteristic, complemented
11#493C411yescharacteristic, complemented
12#486C411yescharacteristic, complemented
13#486C411yescharacteristic, complemented
14#469—22no—
15#442—22no—
16#435—22no—
17#435—22no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#408—22no—
22C4×M16C2211yescharacteristic
23#383C2211yescharacteristic
24#329—22no—
25#327—22no—
26#327—22no—
27C2×#41—21no—
28C2×#41—21no—
29C2×D8—22no—
30C2×M16—21no—
31C2×M16—22no—
32C22×C8—22no—
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#127C2×C411yescharacteristic
37#124—21no—
38#124—21no—
39#121—21no—
40#121—21no—
41#121—21no—
42#121D412yes—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4D412yes—
47C2×C42C2311yescharacteristic
48#106—22no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52#102—22no—
53#102—22no—
54C22⋊C8—24no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57C22⋊C8D412yes—
58#97—21no—
59#97—21no—
60#97—21no—
61#97D412yes—
62C4×C8C2×C411yescharacteristic
63C4×C8C2×C411yescharacteristic
64#95—21no—
65#95—21no—
66#95—21no—
67#41—81no—
68#41—81no—
69#41—82no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×D4—22no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—41no—
79C2×D4C22⋊C412yes—
80C22×C4—22no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4#4111yescharacteristic
86D8—44no—
87M16—42no—
88M16—82no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—41no—
94C2×C8—82no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4C22⋊C412yes—
99C22⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C42C22×C411yescharacteristic
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C42#4111yescharacteristic
107C23—22no—
108C23—41no—
109C23#12111yescharacteristic
110Q8—81no—
111Q8—81no—
112D4—42no—
113D4—42no—
114D4—81no—
115D4—81no—
116D4—81no—
117D4—81no—
118D4—82no—
119D4—82no—
120C2×C4—22no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—41no—
127C2×C4—82no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4C2×C22⋊C411yescharacteristic
132C2×C4C4×D411yescharacteristic
133C2×C4C4×D411yescharacteristic
134C2×C4#12011yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12311yescharacteristic
137C8—22no—
138C8—42no—
139C8—42no—
140C8—41no—
141C8—81no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22—81no—
149C22—81no—
150C22—81no—
151C22#38611yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—81no—
160C4—161no—
161C2—41no—
162C2—82no—
163C2—161no—
164C2#188411yescharacteristic
165C2#190311yescharacteristic
166C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3184C2
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. #2980C2semidirect
3. #1536C2semidirect
4. #1514C2semidirect
5. #1514C2semidirect
6. #1291C2semidirect
7. #1288C2semidirect
10. #498C4semidirect
11. #493C4semidirect
12. #486C4semidirect
13. #486C4semidirect

Maximal subgroups

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

subgroupindex
#31842
#29802
#15362
#15142
#12912
#12882