Finite Group Databasebeta

← #9817

Subgroups — #9817

Order 256.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#467C211yesmaximal, characteristic
2#1810C211yesmaximal, characteristic
3#1752C211yesmaximal, characteristic
4#1508C212yesmaximal
5#1472C212yesmaximal
6C22×Q16C411yescharacteristic, complemented
7C23×C8C2211yescharacteristic
8C2×C22⋊Q8C2212yes—
9#467—21no—
10#467—21no—
11#467—21no—
12#467—21no—
13C2×#108—21no—
14C2×#107C411yescharacteristic, complemented
15C2×#103C412yescomplemented
16#407C2212yes—
17C2×C22⋊C8C2211yescharacteristic
18C2×#97—21no—
19C4×C22⋊C4C2211yescharacteristic
20#341—21no—
21#336—21no—
22#328—24no—
23#324—24no—
24C22×Q8C2×C412yes—
25C23×C4C2311yescharacteristic
26C2×Q16—21no—
27C2×Q16—21no—
28C2×Q16—42no—
29C2×M16—22no—
30C22×C8—21no—
31C22×C8—42no—
32C22×C8C2×C411yescharacteristic
33C22×C8C2×C411yescharacteristic
34C22×C8D411yescharacteristic
35C22×C8D411yescharacteristic
36C22×C8D411yescharacteristic
37C22×C8D411yescharacteristic
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40C22⋊Q8—22no—
41C22⋊Q8—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4C2×C412yes—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C42—21no—
47#108—42no—
48#107—21no—
49#107—21no—
50#103—22no—
51#103—22no—
52C22⋊C8—21no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8D412yes—
56#97—42no—
57#95D412yes—
58C24#4111yescharacteristic
59C2×Q8—22no—
60C2×Q8—22no—
61C2×Q8—42no—
62C2×Q8—42no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4#4111yescharacteristic
74Q16—42no—
75Q16—42no—
76M16—42no—
77M16—42no—
78C2×C8—22no—
79C2×C8—22no—
80C2×C8—22no—
81C2×C8—22no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—42no—
88C2×C8C22⋊C411yescharacteristic
89C2×C8C22⋊C411yescharacteristic
90C2×C8C22⋊C411yescharacteristic
91C2×C8C22⋊C411yescharacteristic
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C42—41no—
100C42—41no—
101C23—22no—
102C23—22no—
103C23—41no—
104C23C4×D411yescharacteristic
105C23#12111yescharacteristic
106C23#12411yescharacteristic
107Q8—42no—
108Q8—42no—
109Q8—82no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4—82no—
126C2×C4C2×C22⋊C411yescharacteristic
127C2×C4C4×D411yescharacteristic
128C2×C4#12111yescharacteristic
129C2×C4#12711yescharacteristic
130C8—21no—
131C8—21no—
132C8—21no—
133C8—42no—
134C8—82no—
135C8#9912yes—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—44no—
144C22#39011yescharacteristic
145C22#44412yes—
146C22#47312yes—
147C4—21no—
148C4—21no—
149C4—21no—
150C4—82no—
151C4—82no—
152C4—82no—
153C4C2×#9911yescharacteristic
154C4#41011yescharacteristic
155C2—22no—
156C2—22no—
157C2—41no—
158C2#196411yescharacteristic
159C2#197511yescharacteristic
160C2#197911yescharacteristic
subgroupG/H
center, socleC22#390
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×Q16C4semidirect
14. C2×#107C4semidirect
15. C2×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
#18102
#17522
#15082
#14722