Finite Group Databasebeta

← #10137

Subgroups — #10137

Order 256.

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

normal subgroups only

This group has 242 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#3209C211yesmaximal, characteristic, complemented
3C2×#489C211yesmaximal, characteristic
4#1556C211yesmaximal, characteristic, complemented
5#1531C211yesmaximal, characteristic, complemented
6#1469C211yesmaximal, characteristic
7#1302C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#124C2211yescharacteristic
11#497—21no—
12#494—21no—
13#489—22no—
14#489—21no—
15#489—22no—
16#487—21no—
17#464—22no—
18#462—22no—
19#451—21no—
20#451—22no—
21#450—21no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25C2×#103—21no—
26C2×#102—21no—
27C2×#97C2211yescharacteristic
28#332—22no—
29#330—22no—
30#323—22no—
31C22×Q8—21no—
32C22×Q8D411yescharacteristic
33C22×D4D411yescharacteristic
34C2×Q16—24no—
35C2×Q16—22no—
36C2×SD16—22no—
37C22×C8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D412yes—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#125—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48#124—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C42C2311yescharacteristic
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8D412yes—
62#103—24no—
63#103—24no—
64#103—22no—
65#102—24no—
66#102—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8D412yes—
70#97—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75C24—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—22no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8—41no—
83C2×Q8—82no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—42no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4#4111yescharacteristic
92Q16—84no—
93Q16—82no—
94SD16—82no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—42no—
99C2×C8—41no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C23—41no—
121C23—42no—
122C23—81no—
123C23—81no—
124C23#12111yescharacteristic
125Q8—42no—
126Q8—44no—
127Q8—42no—
128Q8—42no—
129Q8—42no—
130Q8—82no—
131D4—42no—
132D4—42no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—82no—
141C2×C4—81no—
142C2×C4#12011yescharacteristic
143C2×C4#12011yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12711yescharacteristic
148C8—42no—
149C8—42no—
150C8—42no—
151C8—81no—
152C22—21no—
153C22—21no—
154C22—81no—
155C22—81no—
156C22—81no—
157C22—162no—
158C22#39211yescharacteristic
159C22#47312yes—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—21no—
164C4—21no—
165C4—81no—
166C4—81no—
167C4—82no—
168C4—81no—
169C4#45512yes—
170C2—22no—
171C2—161no—
172C2#200411yescharacteristic
173C2#200811yescharacteristic
174C2#202011yescharacteristic
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. #3209C2semidirect
4. #1556C2semidirect
5. #1531C2semidirect
7. #1302C2semidirect

Maximal subgroups

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

subgroupindex
#32152
#32092
C2×#4892
#15562
#15312
#14692
#13022