Finite Group Databasebeta

← #29633

Subgroups — #29633

Order 256.

630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 127 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×#536C211yesmaximal, characteristic
2C2×#452C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic, complemented
4#2625C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2016C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1818C211yesmaximal, characteristic, complemented
10#1788C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×Q16C2212yes—
15C22×Q16C2211yescharacteristic
16C22×SD16C2211yescharacteristic
17#536—24no—
18#536—24no—
19C2×C4⋊Q8C2212yes—
20C2×#124C2212yes—
21C2×#123—22no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#452—24no—
27#451—24no—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#400—21no—
38#398—21no—
39#397—22no—
40#391C2212yes—
41#389C2212yes—
42#380C2211yescharacteristic
43#336C2212yes—
44#336C2212yes—
45C2×#41—22no—
46C2×#41—42no—
47C2×#41—42no—
48C2×#41D414yes—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8D414yes—
54C22×Q8C2311yescharacteristic
55C22×Q8C2312yes—
56C22×Q8C2311yescharacteristic
57C22×D4—21no—
58C22×D4C2311yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C2×Q16—24no—
63C2×Q16—22no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×SD16—22no—
68C2×SD16—42no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2312yes—
72C4⋊Q8—24no—
73C4⋊Q8—24no—
74#124—24no—
75#124—24no—
76#123—216no—
77C22⋊Q8—28no—
78C22⋊Q8—28no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81#117—22no—
82#117—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4D412yes—
96C2×C42C2312yes—
97#103—24no—
98#103—22no—
99#103—42no—
100#103—42no—
101#103—42no—
102#102—22no—
103#102—42no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106#95—21no—
107#95—22no—
108#95—22no—
109#95—22no—
110C24—21no—
111C24C2×D411yescharacteristic
112#41—44no—
113#41—44no—
114#41—48no—
115C2×Q8—24no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8C2×D414yes—
133C2×Q8C2×D412yes—
134C2×D4—24no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—44no—
139C2×D4C2×D412yes—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2×D412yes—
156C22×C4C2411yescharacteristic
157Q16—48no—
158Q16—44no—
159SD16—44no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C4⋊C4—28no—
168C4⋊C4—28no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—82no—
177C22⋊C4—28no—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—42no—
181C22⋊C4—81no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—42no—
190C23—41no—
191C23—42no—
192C23—41no—
193C23#12012yes—
194C23C22×D411yescharacteristic
195Q8—24no—
196Q8—22no—
197Q8—24no—
198Q8—22no—
199Q8—48no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203Q8—41no—
204Q8—41no—
205Q8—42no—
206D4—22no—
207D4—22no—
208D4—48no—
209D4—42no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4#12012yes—
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4H(5,𝔽2)12yes—
238C2×C4H(5,𝔽2)12yes—
239C8—41no—
240C8—41no—
241C8—42no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22—82no—
252C22C2×#12011yescharacteristic
253C22#53512yes—
254C22#57711yescharacteristic
255C22#57811yescharacteristic
256C22#57812yes—
257C4—22no—
258C4—42no—
259C4—44no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—81no—
265C4#53411yescharacteristic
266C4#53411yescharacteristic
267C4#53412yes—
268C2—42no—
269C2—82no—
270C2#261311yescharacteristic
271C2#301111yescharacteristic
272C2#301311yescharacteristic
273C2#322312yes—
274C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#451C2semidirect
4. #2625C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #2016C2semidirect
8. #2015C2semidirect
9. #1818C2semidirect
11. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4522
C2×#4512
#26252
#20432
#20422
#20162
#20152
#18182
#17882
#17872