Finite Group Databasebeta

← #10097

Subgroups — #10097

Order 256.

271 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 51 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#3236C211yesmaximal, characteristic, complemented
2#3138C211yesmaximal, characteristic
3#2641C211yesmaximal, characteristic
4#1555C211yesmaximal, characteristic
5#1528C211yesmaximal, characteristic, complemented
6#1506C211yesmaximal, characteristic, complemented
7#1300C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#491—21no—
13#490—22no—
14#486—21no—
15#486—21no—
16#484—21no—
17#483—21no—
18#479—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#418—22no—
22C4×M16C2211yescharacteristic
23#394—22no—
24#394—22no—
25#390—21no—
26#390—21no—
27#376C2211yescharacteristic
28#331—22no—
29#330—22no—
30#328—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D412yes—
34C2×M16—21no—
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37#127—21no—
38#127—21no—
39#127—21no—
40#127—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44#125—21no—
45#125—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59#107—22no—
60#106—22no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63#103—22no—
64#103—22no—
65#102—24no—
66#102—24no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69#97—21no—
70#97—21no—
71#97D412yes—
72C4×C8—21no—
73#95—21no—
74#95D412yes—
75C24—22no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—44no—
81C2×D4—44no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—82no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93M16—42no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—84no—
113C22⋊C4—84no—
114C42—42no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42#4111yescharacteristic
120C23—44no—
121C23—44no—
122C23—42no—
123C23—44no—
124C23#12011yescharacteristic
125Q8—42no—
126Q8—81no—
127D4—44no—
128D4—82no—
129D4—82no—
130D4—82no—
131D4—82no—
132D4—82no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4#12011yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12711yescharacteristic
151C8—42no—
152C8—42no—
153C8—81no—
154C22—21no—
155C22—21no—
156C22—88no—
157C22—82no—
158C22—82no—
159C22—82no—
160C22#39211yescharacteristic
161C22#45512yes—
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—82no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C4#45412yes—
173C2—22no—
174C2—162no—
175C2#200311yescharacteristic
176C2#200411yescharacteristic
177C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
1. #3236C2semidirect
5. #1528C2semidirect
6. #1506C2semidirect
7. #1300C2semidirect

Maximal subgroups

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

subgroupindex
#32362
#31382
#26412
#15552
#15282
#15062
#13002