Finite Group Databasebeta

← #29661

Subgroups — #29661

Order 256.

636 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 145 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×#533C211yesmaximal, characteristic, complemented
2C2×#460C212yesmaximal, complemented
3#2856C211yesmaximal, characteristic
4#2023C212yesmaximal, complemented
5#2013C212yesmaximal, complemented
6#2000C214yesmaximal, complemented
7#1803C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic, complemented
9C22×SD16C2214yes—
10#533—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#125—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17C2×#121—21no—
18C2×#121—21no—
19C2×#121C2212yes—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×#117—21no—
23C22×C42C2211yescharacteristic
24#460—28no—
25C2×C4⋊C8C2212yes—
26C2×#103C2214yes—
27C2×#102C2214yes—
28C2×C22⋊C8C2212yes—
29#390—21no—
30#390—21no—
31#390C2212yes—
32#384C2211yescharacteristic
33#340C2214yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×Q8C2312yes—
37C22×D4—22no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4C2312yes—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C2×SD16—28no—
45C2×SD16—44no—
46C2×SD16—44no—
47C22×C8C2314yes—
48C4⋊Q8—24no—
49C4⋊Q8—24no—
50#127—24no—
51#125—28no—
52C22⋊Q8—28no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55#121—44no—
56#121—44no—
57#121—44no—
58C4×Q8—28no—
59C4×Q8—24no—
60C4×Q8D418yescomplemented
61#117—28no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C42—22no—
76C2×C42—24no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C4⋊C8—24no—
82#103—44no—
83#102—28no—
84C22⋊C8—42no—
85C24—22no—
86C24C2×D411yescharacteristic
87C2×Q8—24no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8C2×D414yes—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—41no—
121C22×C4SD1614yescomplemented
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128SD16—416no—
129C2×C8—24no—
130C2×C8—24no—
131C2×C8—24no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—28no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C42—22no—
147C42—22no—
148C42—28no—
149C42—22no—
150C42C2×D414yes—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23C2×SD1612yes—
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—24no—
168Q8—48no—
169Q8—42no—
170Q8—42no—
171D4—44no—
172D4—44no—
173D4—84no—
174D4—88no—
175D4—84no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4#12012yes—
199C2×C4#12012yes—
200C2×C4C2×SD1612yes—
201C2×C4C2×SD1612yes—
202C2×C4AGL(1,ℤ/8)12yes—
203C2×C4#13712yes—
204C2×C4C22×D412yes—
205C2×C4H(5,𝔽2)11yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C2×C42−1+411yescharacteristic
208C8—44no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22C2×#12011yescharacteristic
224C22#53511yescharacteristic
225C22#53711yescharacteristic
226C22C22×SD1611yescharacteristic
227C22C2×AGL(1,ℤ/8)11yescharacteristic
228C22C2×#13711yescharacteristic
229C22#57511yescharacteristic
230C4—22no—
231C4—22no—
232C4—22no—
233C4—41no—
234C4—44no—
235C4—44no—
236C4—82no—
237C4#44814yes—
238C4#53512yes—
239C4#53712yes—
240C2—42no—
241C2—162no—
242C2#261511yescharacteristic
243C2C2×#44811yescharacteristic
244C2#300611yescharacteristic
245C2#320811yescharacteristic
246C2#321611yescharacteristic
247C2#324611yescharacteristic
248C2#325411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#533C2semidirect
2. C2×#460C2semidirect
4. #2023C2semidirect
5. #2013C2semidirect
6. #2000C2semidirect
7. #1803C2semidirect
8. #1797C2semidirect
60. C4×Q8D4semidirect
121. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4602
#28562
#20232
#20132
#20002
#18032
#17972