Finite Group Databasebeta

← #9530

Subgroups — #9530

Order 256.

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

normal subgroups only

This group has 214 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#3110C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic, complemented
3#1559C211yesmaximal, characteristic, complemented
4#1486C211yesmaximal, characteristic
5#1466C211yesmaximal, characteristic
6#1307C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#484C412yes—
11#474—21no—
12#461—21no—
13#450C412yes—
14#446—22no—
15#446—21no—
16C8×Q8—21no—
17#440—22no—
18#433—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24#333—21no—
25#332—21no—
26D4⋊C8—21no—
27D4⋊C8—21no—
28#323—22no—
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C2×SD16—22no—
32C2×M16—21no—
33C4⋊Q8D411yescharacteristic
34C4⋊Q8D411yescharacteristic
35#125—21no—
36C22⋊Q8—21no—
37C22⋊Q8C2×C412yes—
38C4×Q8—21no—
39C4×Q8D412yes—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4—21no—
44#117—21no—
45C2×C4⋊C4C2×C411yescharacteristic
46C2×C22⋊C4—21no—
47C2×C42C2311yescharacteristic
48#107—22no—
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57#103—22no—
58#102—22no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8C2×C412yes—
62C22⋊C8D412yes—
63#97—22no—
64#97—21no—
65#97—21no—
66#97—21no—
67C4×C8—22no—
68C4×C8D411yescharacteristic
69C4×C8D411yescharacteristic
70C24—21no—
71C2×Q8—21no—
72C2×Q8C22⋊C412yes—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—42no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—81no—
80C22×C4—81no—
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4#4111yescharacteristic
84SD16—44no—
85M16—81no—
86C2×C8—22no—
87C2×C8—21no—
88C2×C8—22no—
89C2×C8—21no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—22no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4C22⋊C412yes—
100C22⋊C4—22no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C42—22no—
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C42#4111yescharacteristic
108C23—22no—
109C23—41no—
110C23—41no—
111C23—81no—
112C23—81no—
113C23C4×D411yescharacteristic
114Q8—42no—
115Q8—41no—
116Q8—41no—
117D4—22no—
118D4—22no—
119D4—41no—
120D4—41no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—82no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4C2×C22⋊C411yescharacteristic
133C2×C4C4×D411yescharacteristic
134C2×C4#12011yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12311yescharacteristic
138C8—41no—
139C8—42no—
140C8—41no—
141C8—42no—
142C8—81no—
143C8—81no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—82no—
149C22—82no—
150C22—81no—
151C22—81no—
152C22#38611yescharacteristic
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—42no—
160C4—81no—
161C4#47111yescharacteristic
162C4#47211yescharacteristic
163C2—41no—
164C2—82no—
165C2#189111yescharacteristic
166C2#190511yescharacteristic
167C2#190711yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ 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. #2985C2semidirect
3. #1559C2semidirect
6. #1307C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#31102
#29852
#15592
#14862
#14662
#13072
#12872