Finite Group Databasebeta

← #25583

Subgroups — #25583

Order 256.

631 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 133 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×#552C211yesmaximal, characteristic, complemented
2C2×#490C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic
5#2000C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal, complemented
10#1874C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13#552—24no—
14#552—24no—
15C2×#127—21no—
16C2×#127C2211yescharacteristic
17C2×#126—21no—
18C2×#125—21no—
19C2×#125C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2212yes—
24C2×#121C2212yes—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30#490—24no—
31#460—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#392—21no—
41#392—21no—
42#390—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#340C2212yes—
47C22×Q8C2311yescharacteristic
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4D411yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62#127—22no—
63#127—22no—
64#127—44no—
65#126—28no—
66#125—28no—
67#125—22no—
68#125D414yescomplemented
69#123—28no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74#121—24no—
75#121—24no—
76#121—42no—
77#121—42no—
78C4×Q8—22no—
79C4×Q8D414yescomplemented
80C4×D4—22no—
81C4×D4—22no—
82#117—28no—
83#117—22no—
84#117—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2312yes—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C4⋊C8—22no—
110#103—24no—
111#102—24no—
112#102—24no—
113#102—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#97—22no—
117#95—21no—
118C24—22no—
119C24C2×D411yescharacteristic
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8C2×D412yes—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—41no—
134C2×D4—44no—
135C2×D4—44no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4—82no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—48no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C4⋊C4#4114yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C42—24no—
196C42—24no—
197C42—42no—
198C42C2×D412yes—
199C42C2×D412yes—
200C23—21no—
201C23—22no—
202C23—21no—
203C23—22no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23AGL(1,ℤ/8)12yes—
217C23C22×D411yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221D4—44no—
222D4—42no—
223D4—42no—
224D4—42no—
225D4—42no—
226D4—84no—
227D4—84no—
228D4—88no—
229D4—88no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4#12012yes—
258C2×C4#12012yes—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C2×#4112yes—
262C2×C4H(5,𝔽2)12yes—
263C8—42no—
264C8—42no—
265C22—22no—
266C22—24no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22C2×#12011yescharacteristic
283C22#54612yes—
284C22C2×AGL(1,ℤ/8)11yescharacteristic
285C22#57511yescharacteristic
286C22#57712yes—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—42no—
295C4—44no—
296C4#53211yescharacteristic
297C4#53511yescharacteristic
298C4#55012yes—
299C2—42no—
300C2—81no—
301C2—162no—
302C2#243111yescharacteristic
303C2#300611yescharacteristic
304C2#301011yescharacteristic
305C2#331012yes—
306C2#331712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
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×#552C2semidirect
2. C2×#490C2semidirect
3. C2×#460C2semidirect
5. #2000C2semidirect
6. #2000C2semidirect
7. #1989C2semidirect
8. #1955C2semidirect
9. #1875C2semidirect
10. #1874C2semidirect
11. #1763C2semidirect
68. #125D4semidirect
79. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4902
C2×#4602
#25412
#20002
#19892
#19552
#18752
#18742
#17632