Finite Group Databasebeta

← #13611

Subgroups — #13611

Order 256.

158 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 43 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#3221C211yesmaximal, characteristic
2#2135C211yesmaximal, characteristic, complemented
3#2133C211yesmaximal, characteristic, complemented
4#1413C212yesmaximal, complemented
5#1410C212yesmaximal, complemented
6#540C411yescharacteristic
7#536C2211yescharacteristic
8#496C411yescharacteristic
9#492C2211yescharacteristic
10#491C2211yescharacteristic
11#488C411yescharacteristic
12#463—21no—
13#461—21no—
14#449—21no—
15#448—21no—
16#433C411yescharacteristic
17#413—21no—
18C2×#101—21no—
19#356—21no—
20#356C2212yes—
21#356C2212yes—
22#355—21no—
23C2×#41D411yescharacteristic
24C22×Q8D411yescharacteristic
25C2×SD16—21no—
26C2×SD16—21no—
27C2×SD16—21no—
28C2×D8—21no—
29C2×M16—41no—
30C2×M16—41no—
31C4⋊Q8D411yescharacteristic
32C4⋊Q8C2311yescharacteristic
33#127C2×C411yescharacteristic
34#125C2×C411yescharacteristic
35#124C2×C411yescharacteristic
36#124D411yescharacteristic
37#123—41no—
38#123—41no—
39C22⋊Q8—41no—
40#121—21no—
41#121—41no—
42C4×D4—21no—
43#117C2×C411yescharacteristic
44C4⋊C8—21no—
45#103—21no—
46#103—21no—
47#103—21no—
48#102—21no—
49#101—22no—
50#101—22no—
51#101—41no—
52#100—41no—
53C22⋊C8—21no—
54#97C2×C411yescharacteristic
55C4×C8C2×C411yescharacteristic
56#41—81no—
57C2×Q8—21no—
58C2×Q8—81no—
59C2×Q8C22⋊C411yescharacteristic
60C2×Q8C2×D411yescharacteristic
61C2×Q8C2×D411yescharacteristic
62C2×D4—21no—
63C2×D4—21no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4C22⋊C411yescharacteristic
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—41no—
70C22×C4C22⋊C411yescharacteristic
71SD16—81no—
72SD16—81no—
73SD16—81no—
74D8—81no—
75M16—42no—
76M16—42no—
77M16—81no—
78M16—81no—
79C2×C8—21no—
80C2×C8—21no—
81C2×C8—81no—
82C2×C8—81no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C22⋊C4—21no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42C22⋊C411yescharacteristic
94C42C22×C411yescharacteristic
95C23—21no—
96C23—41no—
97C23#9911yescharacteristic
98Q8—41no—
99Q8—41no—
100Q8—41no—
101Q8—81no—
102D4—41no—
103D4—41no—
104D4—41no—
105D4—81no—
106D4—161no—
107D4—161no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—161no—
120C2×C4#9911yescharacteristic
121C2×C4C2×C22⋊C411yescharacteristic
122C8—41no—
123C8—41no—
124C8—82no—
125C8—82no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—81no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22C2×#9911yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—41no—
139C4—41no—
140C4—41no—
141C4—81no—
142C2—21no—
143C2—41no—
144C2—81no—
145C2—161no—
146C2#212711yescharacteristic
subgroupG/H
center, socleC2#2127
commutatorC42C22×C4
Frattini, cosocleC4⋊Q8C23
prime radical#3221C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ G
LECSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2135C2semidirect
3. #2133C2semidirect
4. #1413C2semidirect
5. #1410C2semidirect

Maximal subgroups

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

subgroupindex
#32212
#21352
#21332
#14132
#14102