Finite Group Databasebeta

← #9649

Subgroups — #9649

Order 256.

468 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#390C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#323C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7C22×C22⋊C4C412yes—
8C22×C22⋊C4C2212yes—
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12C2×C22⋊C8C2212yes—
13C2×C4×C8—21no—
14#390—21no—
15#390—21no—
16#390—21no—
17#390—21no—
18#336—21no—
19#336—21no—
20#323—28no—
21C25C2×C412yes—
22C22×D4—21no—
23C22×D4—21no—
24C22×D4—22no—
25C22×D4—42no—
26C22×D4—42no—
27C22×D4C814yescomplemented
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C412yes—
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33C22×C8—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—42no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—24no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—24no—
43C2×C22⋊C4—44no—
44C2×C42—42no—
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C4⋊C8—42no—
50C22⋊C8—22no—
51C22⋊C8—24no—
52C22⋊C8D414yescomplemented
53C4×C8—42no—
54C24—22no—
55C24—22no—
56C24—24no—
57C24—24no—
58C24—42no—
59C24—42no—
60C24—42no—
61C24—42no—
62C24C2×C814yes—
63C24C22×C411yescharacteristic
64C2×D4—24no—
65C2×D4—24no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—48no—
69C2×D4—48no—
70C2×D4—42no—
71C2×D4—42no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C2×C812yes—
88C22×C4M1612yes—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D412yes—
92C22×C4#4112yes—
93C2×C8—24no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—42no—
99C4⋊C4—42no—
100C4⋊C4—42no—
101C22⋊C4—44no—
102C22⋊C4—44no—
103C22⋊C4—44no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—22no—
107C42—22no—
108C42—41no—
109C42—41no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—24no—
115C23—28no—
116C23—44no—
117C23—44no—
118C23—44no—
119C23—44no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—44no—
127C23—84no—
128C23—84no—
129C23—84no—
130C23—84no—
131C23C2×C22⋊C411yescharacteristic
132C23C4×D412yes—
133C23C22×C811yescharacteristic
134C23C2×M1611yescharacteristic
135C23#13112yes—
136D4—48no—
137D4—48no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—44no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—84no—
158C2×C4C22⋊C812yes—
159C2×C4C22⋊C812yes—
160C2×C4#9912yes—
161C2×C4#10012yes—
162C2×C4#10012yes—
163C2×C4#12112yes—
164C2×C4#12411yescharacteristic
165C2×C4#12711yescharacteristic
166C8—44no—
167C8—82no—
168C22—22no—
169C22—22no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—22no—
176C22—44no—
177C22—44no—
178C22—44no—
179C22—88no—
180C22—88no—
181C22—82no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22#39011yescharacteristic
186C22C2×C22⋊C811yescharacteristic
187C22#40811yescharacteristic
188C22C2×#9911yescharacteristic
189C22#41011yescharacteristic
190C22C2×#10011yescharacteristic
191C22C2×#10011yescharacteristic
192C22C8×D412yes—
193C22#43612yes—
194C4—22no—
195C4—21no—
196C4—21no—
197C4—22no—
198C4—22no—
199C4—41no—
200C4—82no—
201C4#32314yescomplemented
202C2—22no—
203C2—22no—
204C2—84no—
205C2C2×#32311yescharacteristic
206C2#146011yescharacteristic
207C2#146911yescharacteristic
208C2#147111yescharacteristic
209C2#195411yescharacteristic
210C2#196411yescharacteristic
211C2#196511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC23×D4C4
LCSG ▷ C23 ▷ C2 ▷ 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
2. #1797C2semidirect
3. C4×C22⋊C8C2semidirect
4. C2×#323C2semidirect
27. C22×D4C8semidirect
52. C22⋊C8D4semidirect
201. C4#323semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#17972
C4×C22⋊C82
C2×#3232