Finite Group Databasebeta

← #10489

Subgroups — #10489

Order 256.

197 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 49 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#3206C211yesmaximal, characteristic
2#1966C211yesmaximal, characteristic, complemented
3#1574C211yesmaximal, characteristic, complemented
4#1560C211yesmaximal, characteristic
5#1548C211yesmaximal, characteristic, complemented
6#1484C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#492—21no—
11#487—21no—
12#461—21no—
13#461—21no—
14#448—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#423C2211yescharacteristic
19C2×#101—22no—
20C4×M16C2211yescharacteristic
21#335—21no—
22#334—21no—
23#333—21no—
24#333—22no—
25#331—21no—
26D4⋊C8—22no—
27C2×#41—21no—
28C22×Q8D412yes—
29C2×SD16—22no—
30C2×SD16—22no—
31C2×M16—21no—
32C2×M16—21no—
33C2×M16—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8D411yescharacteristic
39#127D411yescharacteristic
40#124—21no—
41#121—21no—
42#121—21no—
43C4×D4—21no—
44C4×D4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C42C2311yescharacteristic
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8D412yes—
53C4⋊C8D411yescharacteristic
54C4⋊C8D411yescharacteristic
55#103—22no—
56#103—22no—
57#101—44no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#97—21no—
61C4×C8—21no—
62C4×C8—21no—
63C4×C8—21no—
64#41—82no—
65C2×Q8—22no—
66C2×Q8—22no—
67C2×Q8—21no—
68C2×Q8—42no—
69C2×Q8—42no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—21no—
73C2×D4—41no—
74C22×C4—22no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77C22×C4#4111yescharacteristic
78C22×C4#4111yescharacteristic
79SD16—82no—
80SD16—82no—
81M16—22no—
82M16—41no—
83M16—42no—
84M16—81no—
85C2×C8—21no—
86C2×C8—21no—
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C42C2×D411yescharacteristic
100C42C2×D411yescharacteristic
101C42C2×D411yescharacteristic
102C42#4111yescharacteristic
103C23—41no—
104C23#12311yescharacteristic
105Q8—42no—
106Q8—42no—
107Q8—82no—
108Q8—81no—
109D4—41no—
110D4—41no—
111D4—41no—
112D4—41no—
113D4—81no—
114D4—82no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—44no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—82no—
124C2×C4#12011yescharacteristic
125C2×C4#12111yescharacteristic
126C2×C4#12111yescharacteristic
127C2×C4#12111yescharacteristic
128C2×C4#12311yescharacteristic
129C2×C4#12411yescharacteristic
130C8—42no—
131C8—41no—
132C8—41no—
133C8—42no—
134C8—81no—
135C8—81no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—81no—
140C22—81no—
141C22—81no—
142C22#39411yescharacteristic
143C4—21no—
144C4—21no—
145C4—21no—
146C4—82no—
147C4—81no—
148C4#45311yescharacteristic
149C4#45511yescharacteristic
150C4#47111yescharacteristic
151C4#47112yes—
152C4#47311yescharacteristic
153C2—41no—
154C2—161no—
155C2#203711yescharacteristic
156C2#204611yescharacteristic
157C2#205211yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1966C2semidirect
3. #1574C2semidirect
5. #1548C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32062
#19662
#15742
#15602
#15482
#14842
#12992