Finite Group Databasebeta

← #25580

Subgroups — #25580

Order 256.

581 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 131 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×#556C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#464C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal, complemented
10#1873C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14#556—24no—
15#556—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#124—21no—
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2212yes—
22C2×#121C2212yes—
23C2×C4×Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×#117C2212yes—
27#489—24no—
28#464—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#399—21no—
39#392—21no—
40#390—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—22no—
49C22×D4D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4C2312yes—
52C2×Q16—22no—
53C2×Q16—42no—
54C2×SD16—22no—
55C2×SD16—42no—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22×C8C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8D414yescomplemented
61#126—216no—
62#124—28no—
63#124—22no—
64#124—22no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—42no—
69#121—24no—
70#121—42no—
71C4×Q8—28no—
72C4×Q8—22no—
73C4×Q8D414yescomplemented
74C4×D4—22no—
75C4×D4—22no—
76#117—24no—
77#117—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4D412yes—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—22no—
99#103—24no—
100#103—24no—
101#102—22no—
102#102—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—22no—
106#95—21no—
107C24—21no—
108C24C2×D412yes—
109C2×Q8—24no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—44no—
118C2×Q8C2×D412yes—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—42no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—82no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143Q16—44no—
144SD16—44no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D414yes—
162C4⋊C4#4114yes—
163C22⋊C4—28no—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C22⋊C4—44no—
167C22⋊C4—44no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—28no—
171C42—24no—
172C42—81no—
173C42C2×D412yes—
174C42C2×D412yes—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—41no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23C22×D411yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—48no—
191Q8—42no—
192Q8—41no—
193Q8—41no—
194Q8—42no—
195D4—44no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—88no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4#12012yes—
223C2×C4#12012yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C2×#4112yes—
227C2×C4H(5,𝔽2)12yes—
228C8—41no—
229C8—41no—
230C8—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—82no—
242C22—82no—
243C22C2×#12011yescharacteristic
244C22#54612yes—
245C22#57512yes—
246C22#57711yescharacteristic
247C22#57811yescharacteristic
248C4—22no—
249C4—22no—
250C4—42no—
251C4—42no—
252C4—44no—
253C4—42no—
254C4—44no—
255C4—81no—
256C4#53211yescharacteristic
257C4#53511yescharacteristic
258C4#55012yes—
259C2—82no—
260C2—161no—
261C2#243111yescharacteristic
262C2#300511yescharacteristic
263C2#301311yescharacteristic
264C2#331112yes—
265C2#331312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
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
1. C2×#556C2semidirect
2. C2×#489C2semidirect
3. C2×#464C2semidirect
5. #2001C2semidirect
6. #2000C2semidirect
7. #1987C2semidirect
8. #1955C2semidirect
9. #1875C2semidirect
10. #1873C2semidirect
11. #1763C2semidirect
60. C4⋊Q8D4semidirect
73. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4892
C2×#4642
#25402
#20012
#20002
#19872
#19552
#18752
#18732
#17632