Finite Group Databasebeta

← #12708

Subgroups — #12708

Order 256.

501 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 73 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#3487C211yesmaximal, characteristic
2#2127C214yesmaximal, complemented
3#1782C212yesmaximal, complemented
4C2×2−1+4C411yescharacteristic, complemented
5C22×#41C2211yescharacteristic
6#550—24no—
7#548—24no—
8#535—22no—
9#535C2212yes—
10#525—22no—
11#525C412yescomplemented
12#519C411yescharacteristic, complemented
13C2×#99—22no—
14C2×#99C2214yes—
15#381—22no—
16#354—24no—
17#353—24no—
182−1+4—44no—
19C2×#41—22no—
20C2×#41—24no—
21C2×#41—22no—
22C2×#41—44no—
23C2×#41C2×C412yes—
24C22×Q8—22no—
25C22×Q8C2×C411yescharacteristic
26C22×D4D412yes—
27C22×D4C2311yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D412yes—
30#127—24no—
31#127D414yes—
32#124—24no—
33#124D414yes—
34#123—22no—
35C22⋊Q8—24no—
36C22⋊Q8—22no—
37#121—24no—
38#121—28no—
39#121—22no—
40#121—24no—
41#120—22no—
42#120—24no—
43C4×Q8—24no—
44C4×D4—28no—
45C4×D4—24no—
46#117—22no—
47#117C2×C412yes—
48C2×C22⋊C4—24no—
49#99—28no—
50#99—44no—
51#95—44no—
52C24C22⋊C411yescharacteristic
53C24C2×D412yes—
54#41—44no—
55#41—44no—
56#41—416no—
57#41—48no—
58#41—44no—
59C2×Q8—22no—
60C2×Q8—24no—
61C2×Q8—22no—
62C2×Q8—48no—
63C2×Q8C22⋊C412yes—
64C2×D4—22no—
65C2×D4—22no—
66C2×D4—24no—
67C2×D4—24no—
68C2×D4—28no—
69C2×D4—24no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4C22⋊C412yes—
73C2×D4C2×D414yes—
74C22×C4—24no—
75C22×C4—24no—
76C22×C4—24no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—44no—
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C412yes—
84C22×C4C22×C411yescharacteristic
85C4⋊C4—24no—
86C4⋊C4—24no—
87C4⋊C4—22no—
88C22⋊C4—24no—
89C22⋊C4—24no—
90C22⋊C4—22no—
91C22⋊C4—28no—
92C22⋊C4—44no—
93C22⋊C4—44no—
94C42—24no—
95C42C22⋊C414yes—
96C23—22no—
97C23—22no—
98C23—21no—
99C23—22no—
100C23—22no—
101C23—82no—
102C23—81no—
103C23C2×C22⋊C411yescharacteristic
104C23#12014yes—
105Q8—28no—
106Q8—42no—
107Q8—44no—
108D4—44no—
109D4—48no—
110D4—42no—
111D4—44no—
112D4—44no—
113D4—88no—
114D4—84no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—48no—
123C2×C4—44no—
124C2×C4—48no—
125C2×C4—44no—
126C2×C4—84no—
127C2×C4—88no—
128C2×C4#9914yes—
129C2×C4C2×C22⋊C412yes—
130C22—24no—
131C22—41no—
132C22—44no—
133C22—44no—
134C22—44no—
135C22—42no—
136C22—42no—
137C22—41no—
138C22—82no—
139C22—82no—
140C22#37911yescharacteristic
141C22C2×#9912yes—
142C4—24no—
143C4—22no—
144C4—44no—
145C4—41no—
146C4—42no—
147C4—164no—
148C2—21no—
149C2—22no—
150C2—44no—
151C2—82no—
152C2—81no—
153C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radical#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2127C2semidirect
3. #1782C2semidirect
4. C2×2−1+4C4semidirect
11. #525C4semidirect
12. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34872
#21272
#17822