Finite Group Databasebeta

← #12730

Subgroups — #12730

Order 256.

333 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 69 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#3509C211yesmaximal, characteristic
2#2133C212yesmaximal
3C2×#356C212yesmaximal, complemented
4#1830C212yesmaximal, complemented
5#557—22no—
6#556—22no—
7#552—24no—
8#541—22no—
9#536C411yescharacteristic
10#536C2211yescharacteristic
11#533—22no—
12#533C411yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14#519C411yescharacteristic
15#519C2211yescharacteristic
16C4×#41C411yescharacteristic
17C2×#101C2214yes—
18#356—22no—
19#356—22no—
20#356—22no—
21#355—22no—
22#344—24no—
23C2×#41C2×C411yescharacteristic
24C22×Q8C2×C411yescharacteristic
25C22×Q8C2311yescharacteristic
26C2×M16—24no—
27C4⋊Q8—21no—
28C4⋊Q8—22no—
29C4⋊Q8—21no—
30C4⋊Q8D412yes—
31C4⋊Q8D412yes—
32C4⋊Q8D412yes—
33#126—24no—
34#126—24no—
35#125—24no—
36#125—24no—
37#125—24no—
38#125—21no—
39#124—21no—
40#124D412yes—
41#123—22no—
42#123—22no—
43C22⋊Q8—24no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46C22⋊Q8—22no—
47C4×Q8—24no—
48C4×Q8—21no—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×D4—21no—
55C4×D4—22no—
56#117—24no—
57#117—22no—
58#117—21no—
59#117C2×C411yescharacteristic
60#117C2×C411yescharacteristic
61#117C2×C411yescharacteristic
62#117D414yes—
63C2×C4⋊C4—22no—
64C2×C42—21no—
65C2×C42C2×C411yescharacteristic
66#101—24no—
67#101—24no—
68#41—42no—
69C2×Q8—22no—
70C2×Q8—21no—
71C2×Q8—42no—
72C2×Q8—42no—
73C2×Q8C22⋊C412yes—
74C2×Q8C22⋊C411yescharacteristic
75C2×Q8C2×D412yes—
76C2×Q8C2×D412yes—
77C2×D4—21no—
78C2×D4C22⋊C411yescharacteristic
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4C22×C411yescharacteristic
84C22×C4C2×D412yes—
85M16—44no—
86M16—44no—
87C2×C8—44no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C22⋊C4—22no—
102C22⋊C4—24no—
103C22⋊C4—21no—
104C22⋊C4—24no—
105C22⋊C4C22⋊C412yes—
106C22⋊C4C22⋊C412yes—
107C42—22no—
108C42—24no—
109C42—21no—
110C42—21no—
111C42—21no—
112C42—24no—
113C42C22⋊C411yescharacteristic
114C42C22⋊C411yescharacteristic
115C42C22⋊C412yes—
116C23—21no—
117C23C2×C22⋊C411yescharacteristic
118Q8—42no—
119Q8—42no—
120Q8—42no—
121Q8—41no—
122Q8—44no—
123Q8—42no—
124D4—41no—
125D4—42no—
126C2×C4—21no—
127C2×C4—24no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4C2×C22⋊C411yescharacteristic
140C2×C4C2×C22⋊C411yescharacteristic
141C2×C4#12014yes—
142C8—84no—
143C22—41no—
144C22—42no—
145C22—41no—
146C22#37911yescharacteristic
147C22#41012yes—
148C4—22no—
149C4—24no—
150C4—21no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—42no—
155C4—42no—
156C4—41no—
157C4—42no—
158C2—21no—
159C2—22no—
160C2—81no—
161C2#178211yescharacteristic
subgroupG/H
center, socleC2#1782
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC2×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ #519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#356C2semidirect
4. #1830C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#21332
C2×#3562
#18302