Finite Group Databasebeta

← #10098

Subgroups — #10098

Order 256.

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

normal subgroups only

This group has 256 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#3215C211yesmaximal, characteristic
2#3139C211yesmaximal, characteristic, complemented
3#2640C211yesmaximal, characteristic
4#1558C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic
6#1506C211yesmaximal, characteristic, complemented
7#1300C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#124C2211yescharacteristic
11#497—21no—
12#491—22no—
13#489—21no—
14#487—21no—
15#487—21no—
16#464—22no—
17#451—21no—
18#450—21no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#419—22no—
22C4×M16C2211yescharacteristic
23#397—22no—
24#393—22no—
25#391—21no—
26#390—21no—
27#376C2211yescharacteristic
28#333—22no—
29#329—22no—
30#328—22no—
31C22×Q8D411yescharacteristic
32C22×Q8D412yes—
33C22×D4D411yescharacteristic
34C2×Q16—22no—
35C2×SD16—22no—
36C2×M16—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#125—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—24no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60C4⋊C8—21no—
61C4⋊C8D412yes—
62#103—24no—
63#103—24no—
64#103—22no—
65#102—22no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68#97—21no—
69#97—21no—
70#97D412yes—
71C4×C8—21no—
72#95—24no—
73#95—21no—
74#95D412yes—
75C24—21no—
76C2×Q8—22no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—42no—
83C2×Q8—44no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—42no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93Q16—82no—
94SD16—82no—
95M16—42no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—84no—
115C42—42no—
116C42—41no—
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C42#4111yescharacteristic
121C23—42no—
122C23—42no—
123C23—41no—
124C23—42no—
125C23#12011yescharacteristic
126Q8—44no—
127Q8—42no—
128Q8—42no—
129Q8—81no—
130Q8—81no—
131Q8—82no—
132D4—42no—
133D4—81no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4#12011yescharacteristic
147C2×C4#12011yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12711yescharacteristic
152C8—42no—
153C8—42no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—84no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22#39211yescharacteristic
162C22#45512yes—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—82no—
169C4—82no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4#45512yes—
174C2—22no—
175C2—161no—
176C2#200411yescharacteristic
177C2#200411yescharacteristic
178C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #3139C2semidirect
4. #1558C2semidirect
6. #1506C2semidirect
7. #1300C2semidirect

Maximal subgroups

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

subgroupindex
#32152
#31392
#26402
#15582
#15272
#15062
#13002