Finite Group Databasebeta

← #13234

Subgroups — #13234

Order 256.

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

normal subgroups only

This group has 232 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#3058C211yesmaximal, characteristic
2#2134C212yesmaximal, complemented
3#2122C212yesmaximal, complemented
4#1813C211yesmaximal, characteristic, complemented
5#1757C211yesmaximal, characteristic, complemented
6C22×D8C411yescharacteristic, complemented
7C2×#131C2211yescharacteristic
8#535C2212yes—
9#469—21no—
10#469—21no—
11#466—21no—
12#466—21no—
13C2×#108—21no—
14#427C411yescharacteristic, complemented
15#418C412yescomplemented
16#413C2212yes—
17#413C2211yescharacteristic
18#410C2211yescharacteristic
19#405—21no—
20#355—24no—
21AΣL(1,𝔽16)—24no—
22#344—21no—
23#338—21no—
24C2×#41C2311yescharacteristic
25C22×D4C2×C412yes—
26C2×D8—21no—
27C2×D8—21no—
28C2×D8—41no—
29C2×D8—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
40#127—22no—
41#124—22no—
42#121—22no—
43#121—22no—
44#120—22no—
45#120—22no—
46#117—21no—
47#117C2×C412yes—
48#108—41no—
49#108—41no—
50#107—21no—
51#106—21no—
52#102—22no—
53#102—22no—
54#101—22no—
55#101D412yes—
56#100—21no—
57#100—22no—
58#99—21no—
59#99D412yes—
60#97—41no—
61C4×C8—41no—
62C24—22no—
63#41—41no—
64#41—41no—
65C2×Q8C2×D411yescharacteristic
66C2×D4—22no—
67C2×D4—22no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4C2×D411yescharacteristic
73C2×D4#4111yescharacteristic
74C2×D4#4111yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78D8—42no—
79D8—41no—
80D8—41no—
81M16—22no—
82M16—41no—
83M16—41no—
84M16—41no—
85M16—41no—
86M16—42no—
87M16—42no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—41no—
95C2×C8—42no—
96C2×C8C22⋊C411yescharacteristic
97C2×C8C22⋊C411yescharacteristic
98C2×C8C22⋊C411yescharacteristic
99C2×C8C22⋊C411yescharacteristic
100C4⋊C4—22no—
101C4⋊C4—41no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—22no—
107C42—41no—
108C23—42no—
109C23—82no—
110C23—162no—
111C23C4×D411yescharacteristic
112C23#12111yescharacteristic
113C23#12411yescharacteristic
114Q8—41no—
115D4—41no—
116D4—41no—
117D4—41no—
118D4—42no—
119D4—42no—
120D4—82no—
121D4—162no—
122D4—162no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4C2×C22⋊C411yescharacteristic
132C2×C4C4×D411yescharacteristic
133C2×C4#12111yescharacteristic
134C2×C4#12711yescharacteristic
135C8—21no—
136C8—21no—
137C8—21no—
138C8—42no—
139C8—82no—
140C8#9911yescharacteristic
141C8#9911yescharacteristic
142C22—21no—
143C22—21no—
144C22—21no—
145C22—41no—
146C22—41no—
147C22—41no—
148C22—82no—
149C22—82no—
150C22—82no—
151C22—164no—
152C22#39011yescharacteristic
153C4—21no—
154C4—21no—
155C4—21no—
156C4—82no—
157C4—82no—
158C4C2×#9911yescharacteristic
159C4#41011yescharacteristic
160C2—21no—
161C2—41no—
162C2—41no—
163C2—41no—
164C2—162no—
165C2#196411yescharacteristic
subgroupG/H
center, socleC2#1964
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3058C2
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
2. #2134C2semidirect
3. #2122C2semidirect
4. #1813C2semidirect
5. #1757C2semidirect
6. C22×D8C4semidirect
14. #427C4semidirect
15. #418C4semidirect

Maximal subgroups

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

subgroupindex
#30582
#21342
#21222
#18132
#17572