Finite Group Databasebeta

← #11845

Subgroups — #11845

Order 256.

253 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 51 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#2561C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic
3C2×#323C211yesmaximal, characteristic
4#1351C212yesmaximal, complemented
5#1344C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic, complemented
7C22×C22⋊C4C2211yescharacteristic
8C2×C22⋊C8—21no—
9#397C411yescharacteristic, complemented
10#397C2211yescharacteristic
11#393C411yescharacteristic, complemented
12#393C2211yescharacteristic
13#388—21no—
14#388—21no—
15#381—21no—
16#379—21no—
17#379—21no—
18C4×C22⋊C4C411yescharacteristic, complemented
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C22×Q8C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C22×C8—41no—
28C22⋊Q8—41no—
29C22⋊Q8—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—41no—
40C2×C22⋊C4—41no—
41C2×C22⋊C4—41no—
42C2×C22⋊C4—41no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C22⋊C4C2311yescharacteristic
47C2×C42—21no—
48C22⋊C8—22no—
49C22⋊C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95C2×C411yescharacteristic
55#95C2×C411yescharacteristic
56C24—21no—
57C24—21no—
58C24—22no—
59C24—41no—
60C24—42no—
61C24—81no—
62C24C22⋊C411yescharacteristic
63C24C22⋊C411yescharacteristic
64C24C2×D411yescharacteristic
65C2×Q8—41no—
66C2×Q8—81no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—81no—
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C2×C8—42no—
83C2×C8—82no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—81no—
98C22⋊C4—81no—
99C22⋊C4—82no—
100C22⋊C4—82no—
101C42—81no—
102C23—22no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—42no—
112C23—42no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—42no—
118C23—42no—
119C23—81no—
120C23—81no—
121C23—84no—
122C23—84no—
123C23#9911yescharacteristic
124C23#9911yescharacteristic
125C23C2×C22⋊C411yescharacteristic
126C23#13512yes—
127Q8—82no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—42no—
133C2×C4—41no—
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—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—82no—
154C2×C4—82no—
155C8—82no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—22no—
163C22—21no—
164C22—42no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22—84no—
175C22—81no—
176C22—84no—
177C22#35412yes—
178C22C2×#9911yescharacteristic
179C22#41611yescharacteristic
180C22#42111yescharacteristic
181C4—41no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C4—162no—
190C2—21no—
191C2—21no—
192C2—22no—
193C2—42no—
194C2—41no—
195C2—81no—
196C2#151611yescharacteristic
197C2#212111yescharacteristic
198C2C2×#35411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1351C2semidirect
5. #1344C2semidirect
6. C2×C22⋊Q8C4semidirect
9. #397C4semidirect
11. #393C4semidirect
18. C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#25612
#18192
C2×#3232
#13512
#13442