Finite Group Databasebeta

← #9415

Subgroups — #9415

Order 256.

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

normal subgroups only

This group has 210 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#3182C211yesmaximal, characteristic
2#2980C211yesmaximal, characteristic
3#1537C211yesmaximal, characteristic, complemented
4#1515C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic
6#1291C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#500C411yescharacteristic, complemented
11#492C411yescharacteristic, complemented
12#487C411yescharacteristic, complemented
13#486C411yescharacteristic, complemented
14#468—22no—
15#440—22no—
16#435—22no—
17#435—22no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#408—22no—
22C4×M16C2211yescharacteristic
23#383C2211yescharacteristic
24#330—22no—
25#328—22no—
26#327—22no—
27C2×#41—21no—
28C2×SD16—22no—
29C2×M16—21no—
30C2×M16—22no—
31C22×C8—22no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#125—21no—
37#124—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40#121—21no—
41#121D412yes—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×D4—21no—
45C4×D4D412yes—
46#117—21no—
47C2×C42C2311yescharacteristic
48#107—22no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52#103—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—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×D4—22no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4C22⋊C412yes—
76C22×C4—22no—
77C22×C4—41no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4#4111yescharacteristic
81SD16—44no—
82M16—42no—
83M16—82no—
84C2×C8—21no—
85C2×C8—21no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—82no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4C22⋊C412yes—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—41no—
101C42—41no—
102C42C22×C411yescharacteristic
103C42C2×D411yescharacteristic
104C42C2×D411yescharacteristic
105C42#4111yescharacteristic
106C23—22no—
107C23#12111yescharacteristic
108Q8—42no—
109Q8—81no—
110Q8—81no—
111D4—42no—
112D4—81no—
113D4—81no—
114D4—82no—
115C2×C4—22no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4C2×C22⋊C411yescharacteristic
127C2×C4C4×D411yescharacteristic
128C2×C4C4×D411yescharacteristic
129C2×C4#12011yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12311yescharacteristic
132C8—22no—
133C8—42no—
134C8—42no—
135C8—41no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—42no—
141C22—42no—
142C22—42no—
143C22#38611yescharacteristic
144C4—21no—
145C4—21no—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—21no—
150C4—42no—
151C4—81no—
152C4—81no—
153C4—161no—
154C2—41no—
155C2—82no—
156C2#188411yescharacteristic
157C2#190211yescharacteristic
158C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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
3. #1537C2semidirect
4. #1515C2semidirect
6. #1291C2semidirect
7. #1288C2semidirect
10. #500C4semidirect
11. #492C4semidirect
12. #487C4semidirect
13. #486C4semidirect

Maximal subgroups

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

subgroupindex
#31822
#29802
#15372
#15152
#15142
#12912
#12882