Finite Group Databasebeta

← #9382

Subgroups — #9382

Order 256.

231 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 59 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#3234C211yesmaximal, characteristic
2#2336C211yesmaximal, characteristic, complemented
3#1980C211yesmaximal, characteristic
4#1528C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic
6#1288C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10#501C412yescomplemented
11#490C412yescomplemented
12#481—21no—
13#480—21no—
14#478—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C2×#104—21no—
18C2×#104—21no—
19C4×M16C2211yescharacteristic
20#389—21no—
21#387—21no—
22#387—21no—
23#387—21no—
24#387—21no—
25#383C2211yescharacteristic
26C4×C4⋊C4—21no—
27C4×C4⋊C4C2211yescharacteristic
28#330—22no—
29#327—21no—
30#327—21no—
31C2×#41—21no—
32C2×M16—21no—
33C4⋊Q8C2×C411yescharacteristic
34#127C2×C411yescharacteristic
35#125—21no—
36#125—21no—
37#125C2×C412yes—
38#125D412yes—
39#124—21no—
40#121—21no—
41#121—21no—
42C4×D4—21no—
43C4×D4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C42—21no—
52C2×C42—21no—
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42C2311yescharacteristic
56#107—22no—
57#106—22no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60#104—42no—
61#104—42no—
62#102—22no—
63#102—22no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66#97—21no—
67#97—21no—
68#97C2×C412yes—
69C4×C8—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95D411yescharacteristic
78#95D411yescharacteristic
79#41—41no—
80#41—41no—
81#41—41no—
82#41—41no—
83C2×Q8—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98M16—22no—
99M16—41no—
100C2×C8—22no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—81no—
114C4⋊C4C22⋊C412yes—
115C4⋊C4C22⋊C412yes—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C42—41no—
119C42—41no—
120C42—42no—
121C42—81no—
122C42C22×C411yescharacteristic
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C42#4111yescharacteristic
126C23—41no—
127C23#12011yescharacteristic
128Q8—41no—
129Q8—41no—
130D4—42no—
131D4—41no—
132D4—41no—
133D4—81no—
134D4—81no—
135D4—81no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4C2×C22⋊C411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4C4×D411yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12311yescharacteristic
167C8—42no—
168C8—42no—
169C8—81no—
170C22—21no—
171C22—21no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22#38611yescharacteristic
176C22#45311yescharacteristic
177C22#45411yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—42no—
185C4—42no—
186C4—42no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—81no—
191C2—21no—
192C2—21no—
193C2—161no—
194C2#188311yescharacteristic
195C2#190611yescharacteristic
196C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ 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. #2336C2semidirect
4. #1528C2semidirect
6. #1288C2semidirect
7. #1285C2semidirect
10. #501C4semidirect
11. #490C4semidirect

Maximal subgroups

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

subgroupindex
#32342
#23362
#19802
#15282
#15142
#12882
#12852