Finite Group Databasebeta

← #11878

Subgroups — #11878

Order 256.

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

normal subgroups only

This group has 236 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#2547C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1344C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×#124—21no—
7C2×C4×D4C2211yescharacteristic
8C2×#99—21no—
9C2×C22⋊C8—21no—
10#397C412yescomplemented
11#397C2212yes—
12#391—21no—
13#380—21no—
14#380—21no—
15#380—21no—
16C4×C4⋊C4C411yescharacteristic, complemented
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8—21no—
21C22×D4D411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C22×C8—41no—
25C4⋊Q8—41no—
26C4⋊Q8—41no—
27#124—42no—
28#124—42no—
29C4×D4—41no—
30C4×D4—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4C2×C411yescharacteristic
40C2×C22⋊C4C2311yescharacteristic
41C2×C42—21no—
42C2×C42C2×C411yescharacteristic
43#99—42no—
44C22⋊C8—22no—
45C22⋊C8—41no—
46#95—22no—
47#95—22no—
48#95—41no—
49#95C2×C412yes—
50C24C22⋊C411yescharacteristic
51C24C2×D411yescharacteristic
52C2×Q8—42no—
53C2×Q8—42no—
54C2×Q8—42no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4D812yes—
58C2×D4SD1612yes—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C2×C8—42no—
74C2×C8—82no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—82no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—44no—
84C22⋊C4—42no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C42—21no—
89C42—21no—
90C42—42no—
91C23—21no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23—41no—
99C23#10212yes—
100C23C2×C22⋊C411yescharacteristic
101C23C2×D811yescharacteristic
102C23C2×SD1611yescharacteristic
103Q8—84no—
104D4—42no—
105D4—42no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4#9911yescharacteristic
129C2×C4#9911yescharacteristic
130C2×C4#10212yes—
131C8—82no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—22no—
136C22—21no—
137C22—42no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22—41no—
142C22—41no—
143C22—41no—
144C22—81no—
145C22—81no—
146C22C2×#9911yescharacteristic
147C22C2×#10211yescharacteristic
148C22#42111yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—41no—
153C4—41no—
154C4—82no—
155C4—82no—
156C4—162no—
157C4#35412yes—
158C2—21no—
159C2—21no—
160C2—41no—
161C2—42no—
162C2—41no—
163C2#151911yescharacteristic
164C2#212311yescharacteristic
165C2C2×#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. #1344C2semidirect
5. C2×C4⋊Q8C4semidirect
10. #397C4semidirect
16. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#25472
#17992
#14592
#13442