Finite Group Databasebeta

← #10162

Subgroups — #10162

Order 256.

255 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 61 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#3141C211yesmaximal, characteristic
3C2×#487C211yesmaximal, characteristic, complemented
4#1997C211yesmaximal, characteristic, complemented
5#1548C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#492—21no—
12#489—22no—
13#487—22no—
14#487—21no—
15#487—21no—
16#487—22no—
17#486—21no—
18#461—22no—
19#448—22no—
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#417—22no—
23C2×#103—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#333—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8D412yes—
32C2×SD16—24no—
33C2×M16—21no—
34C2×M16D412yes—
35C22×C8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8D412yes—
44C4⋊Q8D411yescharacteristic
45#127D411yescharacteristic
46#124—21no—
47#121—21no—
48#121—21no—
49C4×D4—21no—
50C4×D4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—28no—
60#103—24no—
61#103—24no—
62#102—24no—
63C22⋊C8—21no—
64#97—21no—
65C4×C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8D412yes—
69#41—42no—
70#41—42no—
71C2×Q8—21no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—44no—
76C2×Q8—41no—
77C2×Q8—82no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—41no—
82C22×C4—22no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4#4111yescharacteristic
88SD16—84no—
89M16—22no—
90M16—22no—
91M16—42no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—42no—
96C2×C8—41no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C23—41no—
114C23#12111yescharacteristic
115Q8—44no—
116Q8—42no—
117Q8—44no—
118Q8—82no—
119Q8—82no—
120D4—42no—
121D4—42no—
122D4—42no—
123D4—82no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—82no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—82no—
133C2×C4#12011yescharacteristic
134C2×C4#12011yescharacteristic
135C2×C4#12011yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12111yescharacteristic
138C2×C4#12711yescharacteristic
139C2×C4#13512yes—
140C2×C4#13712yes—
141C8—42no—
142C8—42no—
143C8—42no—
144C8—81no—
145C22—21no—
146C22—21no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22#39211yescharacteristic
151C22#46412yes—
152C4—21no—
153C4—21no—
154C4—82no—
155C4—81no—
156C4—82no—
157C4#45212yes—
158C4#45512yes—
159C4#47112yes—
160C4#49511yescharacteristic
161C4#49711yescharacteristic
162C2—22no—
163C2—161no—
164C2#200411yescharacteristic
165C2#201511yescharacteristic
166C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
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
3. C2×#487C2semidirect
4. #1997C2semidirect
5. #1548C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32062
#31412
C2×#4872
#19972
#15482
#14922
#12782