Finite Group Databasebeta

← #25679

Subgroups — #25679

Order 256.

557 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 135 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×#529C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#489C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic, complemented
5#2035C212yesmaximal, complemented
6#2033C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13#529—24no—
14#529—24no—
15C2×#126—21no—
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26C22×C42C2211yescharacteristic
27#501—24no—
28#489—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#391—21no—
39#389—21no—
40#389C2212yes—
41#384C2211yescharacteristic
42#380—21no—
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#126—28no—
60#125—22no—
61#125D414yes—
62#124—22no—
63#124D414yes—
64#123—28no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—44no—
68#121—22no—
69#121—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—28no—
73C4×D4—22no—
74C4×D4—22no—
75#117—28no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—24no—
102#106—24no—
103#103—24no—
104#103—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—24no—
108#97—22no—
109#97—22no—
110#95—22no—
111#95—22no—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8C2×D412yes—
121C2×D4—24no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C2×D412yes—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2411yescharacteristic
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8#4114yes—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—44no—
180C42—24no—
181C42—22no—
182C42—22no—
183C42—21no—
184C42—21no—
185C42C2×D412yes—
186C42C2×D412yes—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23AGL(1,ℤ/8)12yes—
201C23C22×D411yescharacteristic
202Q8—24no—
203Q8—42no—
204Q8—84no—
205D4—24no—
206D4—48no—
207D4—42no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4#12012yes—
241C2×C4#12012yes—
242C2×C4#13712yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C2×#4112yes—
246C2×C42−1+412yes—
247C8—24no—
248C8—42no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22C2×#12011yescharacteristic
264C22#53211yescharacteristic
265C22#53611yescharacteristic
266C22C2×AGL(1,ℤ/8)11yescharacteristic
267C22C2×#13711yescharacteristic
268C22#57511yescharacteristic
269C22#57511yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—42no—
275C4—41no—
276C4—42no—
277C4—42no—
278C4—82no—
279C4—82no—
280C4#54712yes—
281C4#54812yes—
282C2—42no—
283C2—82no—
284C2#243211yescharacteristic
285C2#300611yescharacteristic
286C2#300711yescharacteristic
287C2#317211yescharacteristic
288C2#318011yescharacteristic
289C2#323811yescharacteristic
290C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#501C2semidirect
3. C2×#489C2semidirect
4. #2857C2semidirect
5. #2035C2semidirect
7. #1992C2semidirect
8. #1959C2semidirect
9. #1864C2semidirect
10. #1863C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#5012
C2×#4892
#28572
#20352
#20332
#19922
#19592
#18642
#18632
#17492