Finite Group Databasebeta

← #9306

Subgroups — #9306

Order 256.

245 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 75 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#3037C211yesmaximal, characteristic
2#2982C211yesmaximal, characteristic, complemented
3#1805C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic
5#1480C211yesmaximal, characteristic
6#1301C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#464—22no—
12#462C412yescomplemented
13#460C412yescomplemented
14C8×Q8—22no—
15C8×Q8—22no—
16#431—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#423C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21#417—21no—
22C2×#103—21no—
23#382C2211yescharacteristic
24#340—21no—
25#327—22no—
26D4⋊C8—22no—
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29#137—42no—
30#137—42no—
31C2×Q16—22no—
32C2×SD16—22no—
33C2×M16—21no—
34C22×C8—21no—
35C4⋊Q8C2×C411yescharacteristic
36#127C2×C411yescharacteristic
37#124—21no—
38#121—21no—
39#121—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8C2×C412yes—
43C4×Q8D412yes—
44C4×D4—21no—
45C4×D4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C42—21no—
50C2×C42C2311yescharacteristic
51C4⋊C8—24no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8C2×C412yes—
58#103—22no—
59#103—42no—
60#102—22no—
61C22⋊C8—21no—
62C22⋊C8D412yes—
63C4×C8—24no—
64C4×C8—21no—
65C4×C8D412yes—
66#95—21no—
67#95—21no—
68#41—41no—
69#41—41no—
70#41—41no—
71#41—41no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8C22⋊C412yes—
77C2×Q8D812yes—
78C2×Q8SD1612yes—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—41no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4#4111yescharacteristic
90Q16—44no—
91SD16—44no—
92M16—41no—
93M16—41no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4C22⋊C412yes—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C42—22no—
111C42—22no—
112C42C22×C411yescharacteristic
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42#4111yescharacteristic
116C23—41no—
117C23#12111yescharacteristic
118Q8—22no—
119Q8—22no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8#10214yes—
124D4—42no—
125D4—41no—
126D4—41no—
127D4—81no—
128D4—81no—
129D4—81no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4C2×C22⋊C411yescharacteristic
144C2×C4C4×D411yescharacteristic
145C2×C4C4×D411yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12311yescharacteristic
149C2×C4C2×D811yescharacteristic
150C2×C4C2×SD1611yescharacteristic
151C2×C4#13511yescharacteristic
152C2×C4#13511yescharacteristic
153C8—42no—
154C8—42no—
155C8—42no—
156C8—81no—
157C22—21no—
158C22—21no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#38611yescharacteristic
163C22#46511yescharacteristic
164C22#46611yescharacteristic
165C4—21no—
166C4—21no—
167C4—24no—
168C4—21no—
169C4—21no—
170C4—42no—
171C4—81no—
172C4—161no—
173C4C2×#10211yescharacteristic
174C4#41611yescharacteristic
175C4#46511yescharacteristic
176C4#46611yescharacteristic
177C2—21no—
178C2—21no—
179C2—161no—
180C2#188111yescharacteristic
181C2#189311yescharacteristic
182C2#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. #2982C2semidirect
3. #1805C2semidirect
6. #1301C2semidirect
7. #1286C2semidirect
12. #462C4semidirect
13. #460C4semidirect

Maximal subgroups

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

subgroupindex
#30372
#29822
#18052
#15172
#14802
#13012
#12862