Finite Group Databasebeta

← #29597

Subgroups — #29597

Order 256.

721 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 143 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×C22⋊Q8C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4#2848C211yesmaximal, characteristic
5#2036C214yesmaximal, complemented
6#2001C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic, complemented
8#1787C211yesmaximal, characteristic, complemented
9#1779C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×Q16C2212yescomplemented
13C2×C22⋊Q8—24no—
14C2×C22⋊Q8—22no—
15C2×C22⋊Q8—24no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2214yes—
18C2×C22⋊Q8C2212yes—
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4—21no—
22#484—24no—
23#451—24no—
24C2×#107C2212yescomplemented
25C2×#103C2212yescomplemented
26C2×#103C2212yescomplemented
27C2×#103C2214yescomplemented
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2214yes—
31#385—21no—
32#385—22no—
33#380C2211yescharacteristic
34#380C2212yes—
35#379—21no—
36#340C2214yescomplemented
37C25D411yescharacteristic
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8—44no—
41C22×Q8D414yes—
42C22×Q8C2311yescharacteristic
43C22×Q8C2312yes—
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—44no—
49C23×C4D411yescharacteristic
50C23×C4D412yes—
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C2×Q16—24no—
54C2×Q16—44no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C22⋊Q8—216no—
58C22⋊Q8—28no—
59C22⋊Q8—28no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—44no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D414yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79#107—24no—
80#103—24no—
81#103—24no—
82#103—44no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85C22⋊C8—44no—
86#95—22no—
87#95—21no—
88C24—22no—
89C24—22no—
90C24—42no—
91C24—42no—
92C24Q1614yescomplemented
93C24C2×D412yes—
94C24C2×D411yescharacteristic
95C2×Q8—24no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—24no—
102C2×Q8—48no—
103C2×Q8—44no—
104C2×Q8—48no—
105C2×Q8C2×D414yes—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—48no—
124C22×C4—48no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2411yescharacteristic
132Q16—48no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—24no—
136C2×C8—24no—
137C4⋊C4—216no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4C2×D414yes—
146C22⋊C4—216no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—28no—
160C23—48no—
161C23—44no—
162C23—48no—
163C23—44no—
164C23#12012yes—
165C23C2×Q1614yes—
166C23C2×Q1612yes—
167C23#13712yes—
168C23C22×D411yescharacteristic
169Q8—24no—
170Q8—24no—
171Q8—48no—
172Q8—44no—
173Q8—42no—
174Q8—42no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—48no—
184C2×C4—48no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4#12012yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4H(5,𝔽2)14yes—
202C8—42no—
203C8—42no—
204C22—22no—
205C22—24no—
206C22—24no—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—24no—
211C22—416no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22#45114yes—
217C22C2×#12011yescharacteristic
218C22#53412yes—
219C22C22×Q1611yescharacteristic
220C22C2×#13711yescharacteristic
221C22#57512yes—
222C4—22no—
223C4—42no—
224C4—44no—
225C4—44no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4#53412yes—
230C4#53512yes—
231C2—24no—
232C2—44no—
233C2#261311yescharacteristic
234C2C2×#45111yescharacteristic
235C2#300511yescharacteristic
236C2#318912yes—
237C2#319612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#451C2semidirect
5. #2036C2semidirect
6. #2001C2semidirect
7. #1824C2semidirect
8. #1787C2semidirect
12. C22×Q16C22semidirect
24. C2×#107C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
36. #340C22semidirect
92. C24Q16semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4842
C2×#4512
#28482
#20362
#20012
#18242
#17872
#17792