Finite Group Databasebeta

← #9913

Subgroups — #9913

Order 256.

274 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 77 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#2851C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1434C211yesmaximal, characteristic
4#1294C212yesmaximal
5#1294C212yesmaximal
6C2×#125C411yescharacteristic, complemented
7C22×C42C2211yescharacteristic
8C2×C4⋊C8—21no—
9C2×C22⋊C8C2212yes—
10#389C411yescharacteristic, complemented
11#389C411yescharacteristic, complemented
12#387—21no—
13#387—21no—
14#385—21no—
15#385—21no—
16#382C411yescharacteristic, complemented
17#380C2211yescharacteristic
18#380C2211yescharacteristic
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23C23×C4—22no—
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28#125—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42—42no—
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C4⋊C8—42no—
45C22⋊C8—24no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—22no—
53#95—42no—
54#95—42no—
55#95Q814yes—
56C24C2×D411yescharacteristic
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—24no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×Q812yes—
84C22×C4#4112yes—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—42no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C42—22no—
100C42—22no—
101C42—41no—
102C42—41no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23C2×C22⋊C411yescharacteristic
108C23C22⋊Q812yes—
109C23#13512yes—
110C23AGL(1,ℤ/8)11yescharacteristic
111C23#13711yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#9912yes—
152C2×C4C4×Q812yes—
153C2×C4#12411yescharacteristic
154C2×C4C4⋊Q811yescharacteristic
155C8—82no—
156C22—22no—
157C22—22no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—22no—
163C22—22no—
164C22#39111yescharacteristic
165C22C2×#9911yescharacteristic
166C22#41011yescharacteristic
167C22#41611yescharacteristic
168C22#41711yescharacteristic
169C22#42111yescharacteristic
170C22#42111yescharacteristic
171C22#47812yes—
172C22#47912yes—
173C4—21no—
174C4—22no—
175C4—21no—
176C4—22no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—84no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4—81no—
185C2—22no—
186C2—22no—
187C2#151211yescharacteristic
188C2#151311yescharacteristic
189C2#151611yescharacteristic
190C2#152111yescharacteristic
191C2#198911yescharacteristic
192C2#199511yescharacteristic
193C2#199611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#125C4semidirect
10. #389C4semidirect
11. #389C4semidirect
16. #382C4semidirect

Maximal subgroups

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

subgroupindex
#28512
#17972
#14342
#12942