Finite Group Databasebeta

← #25760

Subgroups — #25760

Order 256.

757 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 167 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×D4×Q8C211yesmaximal, characteristic
2C2×#494C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#451C214yesmaximal, complemented
5#2542C211yesmaximal, characteristic
6#2001C214yesmaximal, complemented
7#1981C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic
9C4×#103C211yesmaximal, characteristic, complemented
10C23×Q8C2212yes—
11C22×Q16C2212yes—
12C22×Q16C2214yes—
13D4×Q8—24no—
14D4×Q8—24no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—22no—
19C2×C22⋊Q8C2212yes—
20C2×C22⋊Q8C2214yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#494—24no—
25#462—24no—
26#451—216no—
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2214yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×C22⋊C8C2214yes—
32C2×C4×C8C2211yescharacteristic
33#399—22no—
34#390—21no—
35#384—21no—
36C4×C4⋊C4C2211yescharacteristic
37#340C2212yes—
38C22×Q8—24no—
39C22×Q8—24no—
40C22×Q8—22no—
41C22×Q8—28no—
42C22×Q8—44no—
43C22×Q8C2312yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4D411yescharacteristic
47C23×C4—22no—
48C23×C4C2312yes—
49C2×Q16—28no—
50C2×Q16—24no—
51C2×Q16—28no—
52C2×Q16—28no—
53C2×Q16—44no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—28no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61C22⋊Q8—216no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C22⋊Q8—28no—
65C22⋊Q8—44no—
66C4×Q8—22no—
67C4×Q8D414yes—
68C4×D4—28no—
69C4×D4—22no—
70C4×D4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4D412yes—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—22no—
88#103—28no—
89#103—24no—
90#103D418yescomplemented
91C22⋊C8—28no—
92C4×C8—22no—
93#95—21no—
94C24C2×D412yes—
95C2×Q8—28no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—24no—
100C2×Q8—22no—
101C2×Q8—24no—
102C2×Q8—22no—
103C2×Q8—24no—
104C2×Q8—24no—
105C2×Q8—48no—
106C2×Q8—416no—
107C2×Q8—48no—
108C2×Q8—48no—
109C2×Q8C2×D414yes—
110C2×Q8C2×D414yes—
111C2×Q8C2×D412yes—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4Q1614yescomplemented
115C22×C4—21no—
116C22×C4—28no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—48no—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130Q16—416no—
131Q16—48no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—24no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C4⋊C4—28no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—48no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4C2×D414yes—
148C4⋊C4C2×D412yes—
149C22⋊C4—28no—
150C22⋊C4—24no—
151C22⋊C4—48no—
152C42—24no—
153C42—42no—
154C42C2×D412yes—
155C42C2×D412yes—
156C23—22no—
157C23—24no—
158C23—22no—
159C23—24no—
160C23C2×Q1614yes—
161C23C22×D411yescharacteristic
162Q8—22no—
163Q8—24no—
164Q8—28no—
165Q8—416no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—44no—
171Q8#12714yes—
172D4—28no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—416no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4#12012yes—
192C2×C4#12012yes—
193C2×C4C2×Q1612yes—
194C2×C4#13712yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D412yes—
197C2×C4C22×D411yescharacteristic
198C2×C4H(5,𝔽2)12yes—
199C8—24no—
200C8—42no—
201C22—24no—
202C22—28no—
203C22—42no—
204C22—42no—
205C22—44no—
206C22C2×#12011yescharacteristic
207C22D4212yes—
208C22C22×Q1611yescharacteristic
209C22C2×#13711yescharacteristic
210C22#57812yes—
211C4—24no—
212C4—22no—
213C4—42no—
214C4—44no—
215C4—42no—
216C4—44no—
217C4—82no—
218C4#45114yescomplemented
219C4C2×#12711yescharacteristic
220C4#53511yescharacteristic
221C4D4212yes—
222C2—44no—
223C2#243311yescharacteristic
224C2C2×#45111yescharacteristic
225C2#301111yescharacteristic
226C2#327612yes—
227C2D4×Q1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#494C2semidirect
3. C2×#462C2semidirect
4. C2×#451C2semidirect
6. #2001C2semidirect
7. #1981C2semidirect
9. C4×#103C2semidirect
90. #103D4semidirect
114. C2×D4Q16semidirect
218. C4#451semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4942
C2×#4622
C2×#4512
#25422
#20012
#19812
#19542
C4×#1032