Finite Group Databasebeta

← #9330

Subgroups — #9330

Order 256.

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

normal subgroups only

This group has 241 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#3112C211yesmaximal, characteristic
2#2320C211yesmaximal, characteristic, complemented
3#1980C211yesmaximal, characteristic
4#1505C211yesmaximal, characteristic
5#1491C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#480—21no—
11#477C412yescomplemented
12#460C412yescomplemented
13#448—21no—
14#440—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C2×#104—22no—
18C4×M16C2211yescharacteristic
19#391—21no—
20#389—21no—
21#387—22no—
22#383—21no—
23#382—21no—
24C4×C4⋊C4C2211yescharacteristic
25#376C2211yescharacteristic
26#327—21no—
27#327—21no—
28Q8⋊C8—22no—
29C2×#41—21no—
30C22×Q8D411yescharacteristic
31C2×SD16—22no—
32C2×M16—21no—
33C4⋊Q8C2×C411yescharacteristic
34#127C2×C411yescharacteristic
35#124—21no—
36#121—21no—
37#121—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8C2×C412yes—
41C4×Q8D412yes—
42C4×D4—21no—
43C4×D4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42—21no—
50C2×C42D412yes—
51C2×C42C2311yescharacteristic
52#107—22no—
53C4⋊C8—21no—
54C4⋊C8C2×C412yes—
55#104—44no—
56#103—22no—
57#102—22no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#97—21no—
61#97—21no—
62#97—21no—
63C4×C8—21no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—21no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
71#41—42no—
72#41—42no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—42no—
76C2×Q8C22⋊C412yes—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90SD16—44no—
91M16—22no—
92M16—41no—
93C2×C8—22no—
94C2×C8—41no—
95C2×C8—41no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—42no—
102C4⋊C4—82no—
103C4⋊C4C22⋊C412yes—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C42—22no—
107C42—22no—
108C42—42no—
109C42—42no—
110C42C22×C411yescharacteristic
111C42C2×D411yescharacteristic
112C42#4111yescharacteristic
113C42#4111yescharacteristic
114C23—41no—
115C23#12011yescharacteristic
116Q8—22no—
117Q8—22no—
118Q8—42no—
119Q8—42no—
120D4—42no—
121D4—42no—
122D4—81no—
123D4—81no—
124D4—81no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—82no—
145C2×C4C2×C22⋊C411yescharacteristic
146C2×C4C4×D411yescharacteristic
147C2×C4C4×D411yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12311yescharacteristic
151C8—42no—
152C8—42no—
153C8—81no—
154C22—21no—
155C22—21no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22#38611yescharacteristic
160C22#45311yescharacteristic
161C22#45411yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—42no—
169C4—81no—
170C4—82no—
171C4—81no—
172C4—81no—
173C4#42112yes—
174C2—21no—
175C2—21no—
176C2—161no—
177C2#188311yescharacteristic
178C2#189711yescharacteristic
179C2#190511yescharacteristic
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. #2320C2semidirect
5. #1491C2semidirect
6. #1300C2semidirect
7. #1285C2semidirect
11. #477C4semidirect
12. #460C4semidirect

Maximal subgroups

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

subgroupindex
#31122
#23202
#19802
#15052
#14912
#13002
#12852