Finite Group Databasebeta

← #25571

Subgroups — #25571

Order 256.

693 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 147 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×#547C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4#2542C211yesmaximal, characteristic, complemented
5#2000C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1872C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11C22×SD16C2212yes—
12#547—24no—
13#547—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#123—22no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2214yes—
19C2×#121C2212yes—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23#487—24no—
24#461—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2214yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2214yes—
30C2×C4×C8C2211yescharacteristic
31#399—22no—
32#390—21no—
33#384—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C2×#41—28no—
38C2×#41—44no—
39C2×#41—44no—
40C22×Q8—21no—
41C22×Q8C2312yes—
42C22×D4—22no—
43C22×D4D411yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4—21no—
47C23×C4C2312yes—
48C2×SD16—24no—
49C2×SD16—44no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8D414yes—
56#123—216no—
57C22⋊Q8—28no—
58C22⋊Q8—28no—
59C22⋊Q8—44no—
60#121—24no—
61#121—24no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4D414yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4D412yes—
81C2×C42—21no—
82C2×C42C2311yescharacteristic
83C4⋊C8—22no—
84#103—24no—
85#103—28no—
86#102—24no—
87C22⋊C8—44no—
88C4×C8—22no—
89#95—21no—
90C24—21no—
91C24C2×D412yes—
92#41—432no—
93C2×Q8—24no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—24no—
97C2×Q8—24no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×D4—28no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4SD1614yescomplemented
109C2×D4C2×D412yes—
110C22×C4—28no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128SD16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—24no—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—48no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4C2×D412yes—
144C4⋊C4C2×D414yes—
145C4⋊C4#4114yescomplemented
146C22⋊C4—28no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—48no—
150C42—42no—
151C42C2×D412yes—
152C42C2×D412yes—
153C23—22no—
154C23—24no—
155C23—22no—
156C23—24no—
157C23—22no—
158C23—42no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23C2×SD1614yes—
163C23C22×D411yescharacteristic
164Q8—48no—
165Q8—42no—
166Q8—42no—
167Q8—44no—
168D4—22no—
169D4—22no—
170D4—28no—
171D4—416no—
172D4—42no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—416no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—48no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4#12012yes—
193C2×C4#12012yes—
194C2×C4C2×SD1612yes—
195C2×C4AGL(1,ℤ/8)12yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C22×D411yescharacteristic
198C2×C4C2×#4112yes—
199C2×C4H(5,𝔽2)12yes—
200C8—42no—
201C8—42no—
202C22—24no—
203C22—28no—
204C22—44no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22C2×#12011yescharacteristic
213C22#54612yes—
214C22C22×SD1611yescharacteristic
215C22C2×AGL(1,ℤ/8)11yescharacteristic
216C22#57812yes—
217C4—22no—
218C4—44no—
219C4—42no—
220C4—42no—
221C4—44no—
222C4—82no—
223C4#45014yes—
224C4#53211yescharacteristic
225C4#53511yescharacteristic
226C4#55012yes—
227C2—44no—
228C2—82no—
229C2#243111yescharacteristic
230C2C2×#45011yescharacteristic
231C2#301111yescharacteristic
232C2#329812yes—
233C2#333012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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×#487C2semidirect
3. C2×#461C2semidirect
4. #2542C2semidirect
5. #2000C2semidirect
6. #1988C2semidirect
7. #1954C2semidirect
8. #1872C2semidirect
9. C4×#102C2semidirect
108. C2×D4SD16semidirect
145. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4872
C2×#4612
#25422
#20002
#19882
#19542
#18722
C4×#1022