Finite Group Databasebeta

← #11900

Subgroups — #11900

Order 256.

226 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 59 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#2563C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1351C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#99—21no—
8C2×C22⋊C8—21no—
9#393C412yescomplemented
10#393C2212yes—
11#390—21no—
12#382—21no—
13#380—21no—
14#380—21no—
15#380—21no—
16C4×C4⋊C4C411yescharacteristic, complemented
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8C2×C412yes—
21C22×D4D411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C22×C8—41no—
25C4⋊Q8—41no—
26C4⋊Q8—41no—
27C4×D4—41no—
28C4×D4—41no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4D411yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4C2×C411yescharacteristic
38C2×C22⋊C4C2311yescharacteristic
39C2×C42—21no—
40C2×C42C2×C411yescharacteristic
41#99—42no—
42C22⋊C8—22no—
43C22⋊C8—41no—
44#95—22no—
45#95—22no—
46#95—41no—
47#95C2×C412yes—
48C24C22⋊C411yescharacteristic
49C24C2×D411yescharacteristic
50C2×Q8—42no—
51C2×Q8—82no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4SD1612yes—
55C2×D4Q1612yes—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C2×C8—42no—
71C2×C8—82no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—82no—
78C22⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—82no—
83C22⋊C4—82no—
84C42—21no—
85C42—21no—
86C42—42no—
87C23—21no—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23—41no—
95C23#10312yes—
96C23C2×C22⋊C411yescharacteristic
97C23C2×SD1611yescharacteristic
98C23C2×Q1611yescharacteristic
99Q8—84no—
100D4—42no—
101D4—42no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—81no—
120C2×C4—81no—
121C2×C4—82no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4#9911yescharacteristic
125C2×C4#9911yescharacteristic
126C2×C4#10312yes—
127C8—82no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—22no—
132C22—21no—
133C22—42no—
134C22—41no—
135C22—41no—
136C22—41no—
137C22—41no—
138C22—41no—
139C22—41no—
140C22—81no—
141C22—81no—
142C22C2×#9911yescharacteristic
143C22C2×#10311yescharacteristic
144C22#42111yescharacteristic
145C4—21no—
146C4—22no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—82no—
151C4—82no—
152C4—162no—
153C4#35412yes—
154C2—21no—
155C2—21no—
156C2—41no—
157C2—42no—
158C2—41no—
159C2#152011yescharacteristic
160C2#212211yescharacteristic
161C2C2×#35411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1351C2semidirect
5. C2×C4⋊Q8C4semidirect
9. #393C4semidirect
16. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#25632
#17992
#14592
#13512