Finite Group Databasebeta

← #13233

Subgroups — #13233

Order 256.

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

normal subgroups only

This group has 216 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#3059C211yesmaximal, characteristic
2#2135C212yesmaximal
3#2123C212yesmaximal
4#1813C211yesmaximal, characteristic
5#1757C211yesmaximal, characteristic
6C22×Q16C411yescharacteristic, complemented
7C2×#131C2211yescharacteristic
8#536C2212yes—
9#470—21no—
10#470—21no—
11#467—21no—
12#467—21no—
13C2×#108—21no—
14#427C411yescharacteristic, complemented
15#419C412yescomplemented
16#413C2212yes—
17#413C2211yescharacteristic
18#410C2211yescharacteristic
19#405—21no—
20#356—24no—
21#352—24no—
22#344—21no—
23#338—21no—
24C2×#41C2311yescharacteristic
25C22×Q8C2×C412yes—
26C2×Q16—21no—
27C2×Q16—21no—
28C2×Q16—41no—
29C2×Q16—41no—
30#131—41no—
31#131—41no—
32C2×M16—21no—
33C2×M16—22no—
34C2×M16C2×C411yescharacteristic
35C2×M16D411yescharacteristic
36C2×M16D411yescharacteristic
37C22×C8C2×C411yescharacteristic
38C22×C8D411yescharacteristic
39C22×C8D411yescharacteristic
40C4⋊Q8—22no—
41#124—22no—
42#123—22no—
43#123—22no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46#117—21no—
47#117C2×C412yes—
48#108—41no—
49#108—41no—
50#107—21no—
51#106—21no—
52#103—22no—
53#103—22no—
54#101—22no—
55#101D412yes—
56#100—21no—
57#100—22no—
58#99—21no—
59#99D412yes—
60#97—41no—
61C4×C8—41no—
62#41—41no—
63#41—41no—
64C2×Q8—22no—
65C2×Q8—22no—
66C2×Q8—42no—
67C2×Q8—42no—
68C2×Q8C2×D411yescharacteristic
69C2×D4C2×D411yescharacteristic
70C2×D4#4111yescharacteristic
71C2×D4#4111yescharacteristic
72C22×C4—22no—
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76Q16—42no—
77Q16—41no—
78Q16—41no—
79M16—22no—
80M16—41no—
81M16—41no—
82M16—41no—
83M16—41no—
84M16—42no—
85M16—42no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—41no—
93C2×C8—42no—
94C2×C8C22⋊C411yescharacteristic
95C2×C8C22⋊C411yescharacteristic
96C2×C8C22⋊C411yescharacteristic
97C2×C8C22⋊C411yescharacteristic
98C4⋊C4—22no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—22no—
107C42—41no—
108C23C4×D411yescharacteristic
109C23#12111yescharacteristic
110C23#12411yescharacteristic
111Q8—41no—
112Q8—42no—
113Q8—42no—
114Q8—82no—
115D4—41no—
116D4—41no—
117D4—41no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—82no—
128C2×C4C2×C22⋊C411yescharacteristic
129C2×C4C4×D411yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12711yescharacteristic
132C8—21no—
133C8—21no—
134C8—21no—
135C8—42no—
136C8—82no—
137C8#9911yescharacteristic
138C8#9911yescharacteristic
139C22—21no—
140C22—21no—
141C22—21no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22#39011yescharacteristic
146C4—21no—
147C4—21no—
148C4—21no—
149C4—82no—
150C4—82no—
151C4—82no—
152C4C2×#9911yescharacteristic
153C4#41011yescharacteristic
154C2—21no—
155C2—41no—
156C2—41no—
157C2—41no—
158C2#196411yescharacteristic
subgroupG/H
center, socleC2#1964
commutatorC22×C4C22×C4
Frattini, prime radical, cosocleC2×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×Q16C4semidirect
14. #427C4semidirect
15. #419C4semidirect

Maximal subgroups

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

subgroupindex
#30592
#21352
#21232
#18132
#17572