Finite Group Databasebeta

← #21877

Subgroups — #21877

Order 256.

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

normal subgroups only

This group has 549 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
1C2×#539C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic, complemented
3#2313C211yesmaximal, characteristic, complemented
4#1996C212yesmaximal, complemented
5#1964C212yesmaximal
6#1899C214yesmaximal, complemented
7#1799C214yesmaximal, complemented
8#539—24no—
9#539—24no—
10C2×#126C412yescomplemented
11C2×#125C412yescomplemented
12C2×#123C412yescomplemented
13C2×#123C2212yes—
14C2×C22⋊Q8C412yescomplemented
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×#117C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C2×#99C2218yes—
21#389—24no—
22#387—24no—
23#381C2212yes—
24#380C2212yes—
25#380C2212yes—
26#380C2212yes—
27C4×C22⋊C4C2212yes—
28#342C2218yes—
29C22×Q8C2×C411yescharacteristic
30C22×D4C2×C411yescharacteristic
31C22×D4C2312yes—
32C23×C4C2×C412yes—
33C23×C4C2311yescharacteristic
34C23×C4C2311yescharacteristic
35C23×C4C2311yescharacteristic
36#126—28no—
37#125—28no—
38#123—24no—
39#123—24no—
40#123—28no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43C22⋊Q8—28no—
44#121—22no—
45#121—22no—
46C4×D4—24no—
47C4×D4—24no—
48#117—44no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4C2×C412yes—
54C2×C4⋊C4C2×C411yescharacteristic
55C2×C4⋊C4C2×C412yes—
56C2×C4⋊C4C2×C412yes—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C412yes—
59C2×C4⋊C4C2×C412yes—
60C2×C4⋊C4C2×C412yes—
61C2×C4⋊C4D414yes—
62C2×C4⋊C4D414yes—
63C2×C22⋊C4C2×C412yes—
64C2×C22⋊C4C2×C412yes—
65C2×C22⋊C4C2×C412yes—
66C2×C22⋊C4C2×C412yes—
67C2×C22⋊C4C2314yes—
68C2×C22⋊C4C2314yes—
69C2×C22⋊C4C2312yes—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C2×C412yes—
73#99—48no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—24no—
79C24C22×C412yes—
80C24C2411yescharacteristic
81C2×Q8—22no—
82C2×Q8—81no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—24no—
86C2×D4—24no—
87C2×D4—22no—
88C2×D4—81no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—24no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—81no—
95C22×C4—81no—
96C22×C4—82no—
97C22×C4C22×C412yes—
98C22×C4C22×C412yes—
99C22×C4C22×C412yes—
100C22×C4C22×C412yes—
101C22×C4C22×C412yes—
102C22×C4C22×C411yescharacteristic
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—24no—
116C4⋊C4—44no—
117C4⋊C4—82no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C4⋊C4C22⋊C418yes—
121C4⋊C4C22⋊C418yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—28no—
130C22⋊C4—24no—
131C42—24no—
132C42—44no—
133C42—82no—
134C23—22no—
135C23—24no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23C23×C411yescharacteristic
141C23C22×D411yescharacteristic
142C23C22×D411yescharacteristic
143C23H(5,𝔽2)12yes—
144C232−1+412yes—
145Q8—82no—
146D4—42no—
147D4—42no—
148D4—44no—
149D4—82no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—84no—
174C2×C4C2×C22⋊C414yes—
175C2×C4C2×C22⋊C414yes—
176C2×C4C2×C22⋊C412yes—
177C2×C4C2×C22⋊C412yes—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—84no—
190C22C22×C22⋊C411yescharacteristic
191C22#52012yes—
192C22#53511yescharacteristic
193C22#53611yescharacteristic
194C22#53712yes—
195C4—24no—
196C4—24no—
197C4—22no—
198C4—22no—
199C4—42no—
200C4—42no—
201C4—42no—
202C4—84no—
203C4—84no—
204C2—21no—
205C2—22no—
206C2—22no—
207C2—21no—
208C2—82no—
209C2#231111yescharacteristic
210C2#288311yescharacteristic
211C2#288411yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
prime radicalC2×#121C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2316C2semidirect
3. #2313C2semidirect
4. #1996C2semidirect
6. #1899C2semidirect
7. #1799C2semidirect
10. C2×#126C4semidirect
11. C2×#125C4semidirect
12. C2×#123C4semidirect
14. C2×C22⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#23162
#23132
#19962
#19642
#18992
#17992