Finite Group Databasebeta

← #7715

Subgroups — #7715

Order 256.

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

normal subgroups only

This group has 232 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#2855C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C4⋊C8—21no—
9C2×C4⋊C8—21no—
10C2×C22⋊C8—21no—
11C2×C22⋊C8—21no—
12#391C411yescharacteristic
13#391C2211yescharacteristic
14#389C411yescharacteristic
15#389C2211yescharacteristic
16#387—21no—
17#387—21no—
18#385—21no—
19#381—21no—
20#380C411yescharacteristic
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C22×C8—22no—
28C22×C8—22no—
29C22⋊Q8—42no—
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×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C42—42no—
39C2×C42C2×C411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42C2311yescharacteristic
43C4⋊C8—42no—
44C4⋊C8—42no—
45C22⋊C8—42no—
46C22⋊C8—42no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54C24C22⋊C411yescharacteristic
55C2×Q8—22no—
56C2×Q8—41no—
57C2×Q8—41no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4SD1612yes—
75C22×C4Q1612yes—
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C2×C8—42no—
80C2×C8—42no—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—42no—
84C2×C8—42no—
85C4⋊C4—22no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C42—22no—
95C42—22no—
96C42—41no—
97C42—41no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23#10312yes—
105C23C2×C22⋊C411yescharacteristic
106Q8—82no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
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—42no—
121C2×C4—42no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4#9912yes—
138C2×C4#10312yes—
139C2×C4#13112yes—
140C2×C4C2×SD1611yescharacteristic
141C2×C4C2×Q1611yescharacteristic
142C2×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C4#13711yescharacteristic
144C8—82no—
145C8—82no—
146C22—22no—
147C22—22no—
148C22—22no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22#40811yescharacteristic
154C22C2×#9911yescharacteristic
155C22#41311yescharacteristic
156C22C2×#10311yescharacteristic
157C22#41711yescharacteristic
158C22#42111yescharacteristic
159C22#42111yescharacteristic
160C4—22no—
161C4—22no—
162C4—21no—
163C4—21no—
164C4—21no—
165C4—22no—
166C4—21no—
167C4—41no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C2—42no—
173C2#146811yescharacteristic
174C2#148111yescharacteristic
175C2#149511yescharacteristic
176C2#151311yescharacteristic
177C2#152011yescharacteristic
178C2#153511yescharacteristic
179C2#155011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1272C2semidirect
5. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28552
#17972
#12722
#12712