Finite Group Databasebeta

← #9784

Subgroups — #9784

Order 256.

255 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 71 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#1996C212yesmaximal
3#1807C211yesmaximal, characteristic
4#1752C211yesmaximal, characteristic
5#1520C212yesmaximal
6C22×Q16C411yescharacteristic, complemented
7C23×C8C2211yescharacteristic
8C2×C22⋊Q8C2212yes—
9#467—21no—
10#467—21no—
11#467—21no—
12#467—21no—
13C2×#107C411yescharacteristic, complemented
14C2×#106—21no—
15C2×#103C412yescomplemented
16C2×C22⋊C8C2211yescharacteristic
17C2×#97—21no—
18#380C2212yes—
19C4×C22⋊C4C2211yescharacteristic
20#342—24no—
21#340—21no—
22#336—21no—
23#328—24no—
24C22×Q8C2×C412yes—
25C23×C4C2311yescharacteristic
26C2×Q16—21no—
27C2×Q16—21no—
28C2×Q16—42no—
29C22×C8—21no—
30C22×C8—42no—
31C22×C8C2×C411yescharacteristic
32C22×C8C2×C411yescharacteristic
33C22×C8D411yescharacteristic
34C22×C8D411yescharacteristic
35C22×C8D411yescharacteristic
36C22×C8D411yescharacteristic
37C22⋊Q8—22no—
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40C22⋊Q8—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4C2×C412yes—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4D412yes—
47C2×C42—21no—
48#107—21no—
49#107—21no—
50#106—42no—
51#103—22no—
52#103—22no—
53C22⋊C8—21no—
54C22⋊C8D412yes—
55#97—42no—
56#95—21no—
57#95—22no—
58C24C2×D411yescharacteristic
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×C4—42no—
69C22×C4C22×C411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4#4111yescharacteristic
74C22×C4#4111yescharacteristic
75Q16—42no—
76Q16—42no—
77C2×C8—22no—
78C2×C8—22no—
79C2×C8—22no—
80C2×C8—22no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8C22⋊C411yescharacteristic
87C2×C8C22⋊C411yescharacteristic
88C2×C8C22⋊C411yescharacteristic
89C2×C8C22⋊C411yescharacteristic
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C22⋊C4—24no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—44no—
100C42—41no—
101C42—41no—
102C23—41no—
103C23—41no—
104C23C4×D411yescharacteristic
105C23#12111yescharacteristic
106C23#12711yescharacteristic
107C23#13714yes—
108Q8—42no—
109Q8—42no—
110Q8—82no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4C2×C22⋊C411yescharacteristic
131C2×C4C4×D411yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12411yescharacteristic
134C8—21no—
135C8—21no—
136C8—21no—
137C8—42no—
138C8#9912yes—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—44no—
146C22#39011yescharacteristic
147C22#44112yes—
148C22#47012yes—
149C22#49712yes—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—82no—
154C4—82no—
155C4—82no—
156C4—162no—
157C4C2×#9911yescharacteristic
158C4#41011yescharacteristic
159C2—22no—
160C2—22no—
161C2—22no—
162C2#196411yescharacteristic
163C2#197111yescharacteristic
164C2#197611yescharacteristic
subgroupG/H
center, socleC22#390
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
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
13. 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
#19962
#18072
#17522
#15202