Finite Group Databasebeta

← #11473

Subgroups — #11473

Order 256.

274 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 65 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#2321C211yesmaximal, characteristic
2#1460C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1326C212yesmaximal, complemented
5#1316C212yesmaximal, complemented
6C2×#120C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C22⋊C4C411yescharacteristic
9#431—21no—
10C2×C22⋊C8—21no—
11#390—21no—
12#388—21no—
13#388—21no—
14#386—21no—
15#386C411yescharacteristic
16#385C2211yescharacteristic
17#379C2211yescharacteristic
18C4×C22⋊C4C411yescharacteristic
19#324—21no—
20#323—21no—
21#323C2212yes—
22#323C2212yes—
23C25C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C22×C8—41no—
30#120—22no—
31#120C814yescomplemented
32C4×D4—41no—
33C4×D4—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—41no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C22⋊C4C2311yescharacteristic
48C2×C42—21no—
49C2×C42D411yescharacteristic
50C4⋊C8—41no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8—41no—
54C4×C8—41no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24—21no—
60C24—41no—
61C24—41no—
62C24—81no—
63C24—81no—
64C24C22⋊C411yescharacteristic
65C24C2×C812yescomplemented
66C24C22×C411yescharacteristic
67C2×D4—24no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—81no—
71C2×D4—81no—
72C2×D4C2×C812yes—
73C2×D4M1612yes—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—41no—
82C22×C4—81no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—82no—
93C4⋊C4—41no—
94C4⋊C4—81no—
95C22⋊C4—21no—
96C22⋊C4—24no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—42no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C22⋊C4C2×C812yescomplemented
107C42—21no—
108C42—21no—
109C42—81no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—22no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—42no—
124C23—42no—
125C23—41no—
126C23—41no—
127C23—82no—
128C23—82no—
129C23—82no—
130C23—82no—
131C23—84no—
132C23C22⋊C812yes—
133C23C2×C22⋊C411yescharacteristic
134C23C22×C811yescharacteristic
135C23C2×M1611yescharacteristic
136D4—42no—
137D4—42no—
138D4—84no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—82no—
162C2×C4C22⋊C812yes—
163C2×C4#9911yescharacteristic
164C2×C4#9911yescharacteristic
165C8—82no—
166C8—82no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—22no—
172C22—42no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—84no—
187C22—84no—
188C22—81no—
189C22—81no—
190C22C2×C22⋊C811yescharacteristic
191C22C2×#9911yescharacteristic
192C22#41311yescharacteristic
193C4—21no—
194C4—21no—
195C4—22no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—81no—
201C4—81no—
202C2—21no—
203C2—21no—
204C2—41no—
205C2—41no—
206C2—42no—
207C2—82no—
208C2#145911yescharacteristic
209C2#212111yescharacteristic
210C2#212211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
prime radicalC2×#120C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1460C2semidirect
3. #1459C2semidirect
4. #1326C2semidirect
5. #1316C2semidirect
31. #120C8semidirect
65. C24C2×C8semidirect
106. C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#23212
#14602
#14592
#13262
#13162