Finite Group Databasebeta

← #22609

Subgroups — #22609

Order 256.

425 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C22×#106C211yesmaximal, characteristic
2#2345C211yesmaximal, characteristic
3#2344C211yesmaximal, characteristic
4#1941C212yesmaximal, complemented
5#1920C212yesmaximal, complemented
6#1919C212yesmaximal, complemented
7#1824C211yesmaximal, characteristic
8#1823C211yesmaximal, characteristic
9#1807C211yesmaximal, characteristic
10C4×#106C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic
12C23×C8C2211yescharacteristic
13C2×#117—21no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#106C418yescomplemented
17C2×#106C2212yes—
18C2×#106C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23C2×C4×C8C2212yes—
24#389C2212yes—
25#389C2212yes—
26#387—22no—
27#387—22no—
28#381—21no—
29#380—21no—
30#380C2211yescharacteristic
31#380C2211yescharacteristic
32C4×C4⋊C4C2212yes—
33C4×C4⋊C4C2212yes—
34C4×C22⋊C4—21no—
35C4×C22⋊C4C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42C23×C4—21no—
43C23×C4—21no—
44C23×C4C2311yescharacteristic
45C22×C8—24no—
46C22×C8C2×C414yes—
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50#117—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2×C414yes—
58C2×C4⋊C4C2×C414yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—22no—
70C2×C42—22no—
71C2×C42C2312yes—
72#106—24no—
73#106C2×C414yes—
74#106C2×C414yes—
75#106C2×C414yes—
76#106C2×C414yes—
77C22⋊C8—22no—
78C4×C8—24no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95D412yes—
86C24C2×D411yescharacteristic
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—24no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—82no—
100C22×C4C22×C412yes—
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4#4112yes—
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—28no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8C22×C412yes—
112C2×C8C22×C412yes—
113C2×C8#4114yes—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—44no—
125C4⋊C4—82no—
126C4⋊C4C22×C412yes—
127C4⋊C4C22×C412yes—
128C4⋊C4C22×C412yes—
129C4⋊C4C22×C412yes—
130C22⋊C4—22no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4SD1614yes—
134C42—22no—
135C42—22no—
136C42—44no—
137C42—82no—
138C23—22no—
139C23—22no—
140C23C4×D412yes—
141C23C2×SD1612yes—
142C23#13512yes—
143C23C22×D411yescharacteristic
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—48no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—84no—
171C2×C4C4×D412yes—
172C2×C4C2×SD1614yes—
173C2×C4C23×C411yescharacteristic
174C2×C4C2×#4112yes—
175C2×C4C2×#4111yescharacteristic
176C2×C42−1+412yes—
177C8—22no—
178C8—42no—
179C8#11714yescomplemented
180C22—24no—
181C22—24no—
182C22—22no—
183C22—22no—
184C22C4×SD1614yes—
185C22C2×C4×D411yescharacteristic
186C22#54712yes—
187C22C22×SD1611yescharacteristic
188C22C2×#13511yescharacteristic
189C22#57712yes—
190C4—24no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—82no—
198C4—82no—
199C4C2×#11711yescharacteristic
200C4#51911yescharacteristic
201C4#55212yes—
202C2—24no—
203C2#234411yescharacteristic
204C2C2×C4×SD1611yescharacteristic
205C2#294911yescharacteristic
206C2#333512yes—
207C2#333712yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1941C2semidirect
5. #1920C2semidirect
6. #1919C2semidirect
10. C4×#106C2semidirect
16. C2×#106C4semidirect
179. C8#117semidirect

Maximal subgroups

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

subgroupindex
C22×#1062
#23452
#23442
#19412
#19202
#19192
#18242
#18232
#18072
C4×#1062
C8×C22⋊C42