Finite Group Databasebeta

← #10283

Subgroups — #10283

Order 256.

313 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 65 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#3227C211yesmaximal, characteristic
2#3206C211yesmaximal, characteristic
3#2868C211yesmaximal, characteristic, complemented
4#1962C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic, complemented
6#1548C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8—22no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#498—21no—
13#492—21no—
14#487—21no—
15#486—21no—
16#481—22no—
17#476—22no—
18#461—22no—
19#448—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23C2×#104—22no—
24#384—22no—
25#384—22no—
26C43C2211yescharacteristic
27#333—22no—
28#332—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8D412yes—
32C2×SD16—24no—
33C2×M16—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—48no—
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46#121—21no—
47#121—21no—
48C4×D4—21no—
49C4×D4—21no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D412yes—
55C2×C42—21no—
56C2×C42—21no—
57C2×C42D412yes—
58C2×C42C2311yescharacteristic
59#106—24no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63C4⋊C8D412yes—
64#104—44no—
65#103—24no—
66#102—24no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C4×C8—21no—
70C4×C8—21no—
71#41—42no—
72#41—42no—
73C2×Q8—22no—
74C2×Q8—21no—
75C2×Q8—22no—
76C2×Q8—44no—
77C2×Q8—44no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91SD16—84no—
92M16—42no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—416no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C42—24no—
110C42—24no—
111C42—22no—
112C42SD1614yescomplemented
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42#4111yescharacteristic
117C23—41no—
118C23#12011yescharacteristic
119Q8—44no—
120Q8—42no—
121Q8—82no—
122Q8—82no—
123Q8—82no—
124D4—42no—
125D4—42no—
126D4—42no—
127D4—82no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—44no—
134C2×C4—44no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12711yescharacteristic
146C2×C4C2×SD1612yes—
147C2×C4#13712yes—
148C8—42no—
149C8—42no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#39211yescharacteristic
157C22#44812yes—
158C4—28no—
159C4—21no—
160C4—21no—
161C4—82no—
162C4—82no—
163C4—81no—
164C4#45312yes—
165C4#45612yes—
166C4#46112yes—
167C4#49211yescharacteristic
168C4#49711yescharacteristic
169C2—22no—
170C2—161no—
171C2#200211yescharacteristic
172C2#200511yescharacteristic
173C2#201311yescharacteristic
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. #2868C2semidirect
5. #1549C2semidirect
6. #1548C2semidirect
7. #1276C2semidirect
112. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#32272
#32062
#28682
#19622
#15492
#15482
#12762