Finite Group Databasebeta

← #12801

Subgroups — #12801

Order 256.

250 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×#124C211yesmaximal, characteristic
2C2×#355C212yesmaximal
3#1801C212yesmaximal, complemented
4#1394C212yesmaximal, complemented
5C2×#124C2211yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C411yescharacteristic
8#423—22no—
9C2×#101C2214yes—
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#388—22no—
13#382C411yescharacteristic
14C4×C22⋊C4—22no—
15C43C411yescharacteristic
16#355—24no—
17#343—22no—
18C22×Q8C2311yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C2×M16—24no—
22#124—22no—
23#124D412yes—
24#124Q812yes—
25C4×Q8—41no—
26C4×Q8—41no—
27C4×D4—41no—
28C4×D4—41no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4D412yes—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—22no—
35C2×C42—21no—
36C2×C42—21no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42D412yes—
41C4⋊C8—42no—
42C4⋊C8—42no—
43#101—24no—
44#101—24no—
45#95—22no—
46#95—22no—
47C24—21no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8C2×D411yescharacteristic
51C2×Q8C2×Q811yescharacteristic
52C2×Q8#4112yes—
53C2×D4—21no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4C4⋊C412yes—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—44no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4C22⋊C412yes—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D412yes—
68M16—44no—
69M16—44no—
70C2×C8—44no—
71C4⋊C4—41no—
72C4⋊C4—42no—
73C4⋊C4—41no—
74C22⋊C4—24no—
75C22⋊C4—24no—
76C22⋊C4—44no—
77C22⋊C4—42no—
78C42—22no—
79C42—22no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—41no—
84C42—42no—
85C42—42no—
86C42C4⋊C412yes—
87C23—22no—
88C23—42no—
89C23—42no—
90C23—41no—
91C23—41no—
92C23C2×C22⋊C411yescharacteristic
93Q8—41no—
94Q8—41no—
95Q8—42no—
96D4—42no—
97D4—42no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—22no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—82no—
116C2×C4—82no—
117C2×C4#9911yescharacteristic
118C2×C4#9911yescharacteristic
119C2×C4C2×C4⋊C411yescharacteristic
120C2×C4#11711yescharacteristic
121C2×C4#12112yes—
122C2×C4C22⋊Q812yes—
123C8—84no—
124C22—21no—
125C22—21no—
126C22—44no—
127C22—42no—
128C22—82no—
129C22—81no—
130C22—81no—
131C22#38011yescharacteristic
132C22C2×#9911yescharacteristic
133C22#41011yescharacteristic
134C4—22no—
135C4—21no—
136C4—21no—
137C4—22no—
138C4—21no—
139C4—42no—
140C4—42no—
141C4—41no—
142C4—42no—
143C4—81no—
144C4#35512yescomplemented
145C2—21no—
146C2—21no—
147C2—82no—
148C2#179911yescharacteristic
149C2C2×#35511yescharacteristic
150C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1801C2semidirect
4. #1394C2semidirect
144. C4#355semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
C2×#3552
#18012
#13942