Finite Group Databasebeta

← #25735

Subgroups — #25735

Order 256.

733 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 151 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×#497C211yesmaximal, characteristic, complemented
3C2×#494C211yesmaximal, characteristic, complemented
4C2×#452C212yesmaximal
5C2×#451C212yesmaximal, complemented
6#2621C211yesmaximal, characteristic, complemented
7#2016C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1960C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×Q16C2212yes—
15C22×Q16C2212yes—
16C22×Q16C2212yes—
17C22×SD16C2212yes—
18#536—24no—
19#536—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8C2212yes—
28C2×#117C2211yescharacteristic
29#497—24no—
30#494—24no—
31#452—28no—
32#451—28no—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#398—22no—
41#397—22no—
42#391C2212yes—
43#382C2211yescharacteristic
44#336C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—42no—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8—21no—
52C22×Q8—24no—
53C22×Q8—42no—
54C22×Q8C2312yes—
55C22×Q8C2311yescharacteristic
56C22×Q8C2311yescharacteristic
57C22×Q8C2311yescharacteristic
58C22×D4—21no—
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×Q16—24no—
64C2×Q16—24no—
65C2×Q16—24no—
66C2×Q16—24no—
67C2×Q16—24no—
68C2×Q16—24no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×Q16—42no—
72C2×SD16—24no—
73C2×SD16—24no—
74C2×SD16—42no—
75C22×C8C2312yes—
76C22×C8C2312yes—
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8D414yes—
83#124—22no—
84#124—22no—
85#124—22no—
86#124D414yes—
87#123—216no—
88C22⋊Q8—216no—
89C22⋊Q8—24no—
90C22⋊Q8—24no—
91#117—22no—
92#117—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4D412yes—
98C2×C4⋊C4C2312yes—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D412yes—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#103—24no—
106#103—24no—
107#103—24no—
108#102—24no—
109C22⋊C8D418yes—
110#97—22no—
111C4×C8—22no—
112#95—22no—
113#95—24no—
114C24—21no—
115C24C2×D411yescharacteristic
116#41—48no—
117#41—48no—
118C2×Q8—24no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—48no—
135C2×Q8—48no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8C2×D412yes—
139C2×Q8C2×D412yes—
140C2×Q8C2×D412yes—
141C2×D4—24no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4C2×D412yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D414yes—
166C22×C4C2411yescharacteristic
167Q16—48no—
168Q16—48no—
169Q16—48no—
170SD16—48no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8C2×D414yes—
176C2×C8C2×D414yes—
177C4⋊C4—28no—
178C4⋊C4—28no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C22⋊C4—28no—
185C22⋊C4—28no—
186C22⋊C4—24no—
187C22⋊C4—44no—
188C42—21no—
189C42—21no—
190C42C2×D412yes—
191C42C2×D412yes—
192C23—21no—
193C23—21no—
194C23—24no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—44no—
199C23#12712yes—
200C23C22×D411yescharacteristic
201Q8—24no—
202Q8—24no—
203Q8—24no—
204Q8—48no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—42no—
212D4—24no—
213D4—48no—
214D4—42no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4#12014yes—
239C2×C4#12712yes—
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D412yes—
243C2×C4H(5,𝔽2)12yes—
244C8—24no—
245C8—24no—
246C22—24no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—44no—
252C22—82no—
253C22—82no—
254C22C2×#12011yescharacteristic
255C22C2×#12711yescharacteristic
256C22#53511yescharacteristic
257C22#57814yes—
258C4—22no—
259C4—24no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—82no—
265C4—82no—
266C4D4212yes—
267C4D4212yes—
268C2—42no—
269C2—82no—
270C2#243311yescharacteristic
271C2#301112yes—
272C2#315612yes—
273C2#322312yes—
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
2. C2×#497C2semidirect
3. C2×#494C2semidirect
5. C2×#451C2semidirect
6. #2621C2semidirect
7. #2016C2semidirect
8. #2015C2semidirect
9. #1960C2semidirect
10. #1958C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4972
C2×#4942
C2×#4522
C2×#4512
#26212
#20162
#20152
#19602
#19582
#17502