Finite Group Databasebeta

← #12964

Subgroups — #12964

Order 256.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#2366C211yesmaximal, characteristic
2C2×#353C211yesmaximal, characteristic
3#1887C211yesmaximal, characteristic
4#1867C212yesmaximal, complemented
5#1390C212yesmaximal, complemented
6C2×#127C2211yescharacteristic
7#518C2211yescharacteristic
8C2×C4×D4—22no—
9#418—21no—
10C2×#100C2211yescharacteristic
11C2×#99C2212yes—
12#390—22no—
13#386—21no—
14#386C2212yes—
15#385—22no—
16C4×C4⋊C4—21no—
17#353C414yescomplemented
18#342—22no—
19#339—21no—
20C22×D4—22no—
21C22×D4D411yescharacteristic
22C22×D4C2311yescharacteristic
23C23×C4—21no—
24C23×C4D412yes—
25C2×M16—21no—
26#127—22no—
27#127C2×C412yes—
28#127Q812yes—
29C4×D4—41no—
30C4×D4—41no—
31C4×D4—48no—
32#117—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—42no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4D412yes—
42C2×C42—22no—
43C2×C42D411yescharacteristic
44#102—41no—
45#102—41no—
46#100—22no—
47#99—22no—
48#99C2×C414yescomplemented
49#95—22no—
50#95—42no—
51C24—21no—
52C24C2×D412yes—
53C2×D4—24no—
54C2×D4—24no—
55C2×D4—21no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—44no—
59C2×D4—44no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4C4⋊C412yescomplemented
63C2×D4C22×C411yescharacteristic
64C2×D4C2×Q811yescharacteristic
65C2×D4#4112yes—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—44no—
72C22×C4—44no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—81no—
76C22×C4C2×D411yescharacteristic
77M16—41no—
78M16—41no—
79C2×C8—41no—
80C4⋊C4—24no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—42no—
84C4⋊C4—84no—
85C22⋊C4—24no—
86C22⋊C4—44no—
87C22⋊C4—44no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—41no—
91C22⋊C4—42no—
92C22⋊C4—81no—
93C42—21no—
94C42—44no—
95C42—41no—
96C42C4⋊C412yescomplemented
97C23—22no—
98C23—22no—
99C23—22no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—42no—
104C23—41no—
105C23—41no—
106C23C4×D412yes—
107C23#12011yescharacteristic
108C23C22⋊Q812yes—
109D4—41no—
110D4—41no—
111D4—42no—
112D4—42no—
113D4—48no—
114D4—48no—
115D4—81no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—24no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—42no—
126C2×C4—44no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—82no—
132C2×C4—84no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4C2×C4⋊C411yescharacteristic
138C2×C4#12311yescharacteristic
139C8—81no—
140C22—21no—
141C22—21no—
142C22—22no—
143C22—22no—
144C22—42no—
145C22—42no—
146C22—44no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22—82no—
151C22—82no—
152C22—81no—
153C22—81no—
154C22#38511yescharacteristic
155C22B(4,𝔽2)11yescharacteristic
156C22#45811yescharacteristic
157C4—24no—
158C4—21no—
159C4—21no—
160C4—44no—
161C4—82no—
162C4—84no—
163C4—81no—
164C4—81no—
165C2—21no—
166C2—21no—
167C2—42no—
168C2—42no—
169C2—82no—
170C2#186711yescharacteristic
171C2#219611yescharacteristic
172C2#219811yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×D4C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#127C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1867C2semidirect
5. #1390C2semidirect
17. #353C4semidirect
48. #99C2×C4semidirect
62. C2×D4C4⋊C4semidirect
96. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23662
C2×#3532
#18872
#18672
#13902