Finite Group Databasebeta

← #8321

Subgroups — #8321

Order 256.

355 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 81 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#3461C211yesmaximal, characteristic
2#1984C211yesmaximal, characteristic, complemented
3#1477C212yesmaximal, complemented
4#1476C212yesmaximal, complemented
5#1461C211yesmaximal, characteristic, complemented
6#544C411yescharacteristic
7#542—22no—
8#541—22no—
9#539—22no—
10#537C411yescharacteristic
11#536C412yes—
12#533—22no—
13#529—22no—
14#529—22no—
15C4×#41C2212yes—
16C2×C4×Q8C2211yescharacteristic
17#431C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4×C8C2211yescharacteristic
20#336—22no—
21Q8⋊C8—22no—
22Q8⋊C8—22no—
23D4⋊C8—22no—
24D4⋊C8—22no—
25#324—22no—
26C2×#41C2×C412yes—
27C22×Q8C2×C411yescharacteristic
28C22×C8—21no—
29C22×C8—21no—
30C4⋊Q8—22no—
31C4⋊Q8—22no—
32#126—24no—
33#126—24no—
34#125—22no—
35#125—24no—
36#125—22no—
37#124—22no—
38#124—22no—
39#123—24no—
40#123—24no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46#121—22no—
47#121—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×Q8D412yes—
52C4×Q8D412yes—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4D412yes—
57C4×D4D412yes—
58#117—24no—
59#117—22no—
60#117C2×C412yes—
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C4⋊C4D412yes—
63C2×C42—22no—
64C2×C42D412yes—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67C4⋊C8—22no—
68C22⋊C8—22no—
69C4×C8—22no—
70C4×C8—22no—
71#41—28no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—24no—
75C2×Q8C22⋊C412yes—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4C22⋊C412yes—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4C22⋊C412yes—
83C22×C4C22⋊C412yes—
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—41no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C4⋊C4—24no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊C412yes—
103C4⋊C4C22⋊C412yes—
104C22⋊C4—24no—
105C22⋊C4—24no—
106C22⋊C4—24no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C42—22no—
111C42—22no—
112C42—22no—
113C42—22no—
114C42—22no—
115C42C2×D412yes—
116C42C2×D412yes—
117C23—22no—
118C23C2×C22⋊C411yescharacteristic
119Q8—24no—
120Q8—44no—
121Q8—82no—
122D4—24no—
123D4—44no—
124D4—82no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—82no—
139C2×C4C2×C22⋊C412yes—
140C2×C4#12011yescharacteristic
141C2×C4#12012yes—
142C2×C4#12011yescharacteristic
143C2×C4#13114yes—
144C8—42no—
145C8—82no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22#37911yescharacteristic
153C22#40812yes—
154C4—21no—
155C4—22no—
156C4—42no—
157C4—42no—
158C4—42no—
159C4—42no—
160C4—42no—
161C4#40814yes—
162C4#45411yescharacteristic
163C4#45611yescharacteristic
164C2—22no—
165C2—82no—
166C2#178011yescharacteristic
167C2#179311yescharacteristic
168C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#537C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1984C2semidirect
3. #1477C2semidirect
4. #1476C2semidirect
5. #1461C2semidirect

Maximal subgroups

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

subgroupindex
#34612
#19842
#14772
#14762
#14612