Finite Group Databasebeta

← #25713

Subgroups — #25713

Order 256.

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

normal subgroups only

This group has 551 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×#553C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2563C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12#553—24no—
13#553—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—21no—
16C2×#125C2211yescharacteristic
17C2×#123—21no—
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#501—24no—
27#474—24no—
28C2×#107C2212yes—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37#393—21no—
38#393—21no—
39#390—21no—
40#382—21no—
41#380C2212yes—
42#380C2212yes—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#340C2212yes—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60#126—28no—
61#125—22no—
62#125D414yes—
63#123—28no—
64#121—28no—
65#121—22no—
66#121—22no—
67#121—22no—
68#121—22no—
69C4×D4—28no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4D414yes—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
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—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99#107—24no—
100#107—24no—
101#106—24no—
102C4⋊C8—22no—
103#102—24no—
104#102—24no—
105C22⋊C8—24no—
106C22⋊C8—24no—
107#97—22no—
108#95—21no—
109#95—22no—
110#95—22no—
111C24—21no—
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C2×Q8—42no—
115C2×Q8—82no—
116C2×D4—24no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—41no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—82no—
127C2×D4C2×D412yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8#4114yes—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—24no—
180C42—42no—
181C42C2×D412yes—
182C42C2×D412yes—
183C23—21no—
184C23—22no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—22no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23—42no—
197C23#13712yes—
198C23C22×D411yescharacteristic
199Q8—84no—
200D4—22no—
201D4—22no—
202D4—44no—
203D4—48no—
204D4—42no—
205D4—84no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4#12012yes—
235C2×C4#12012yes—
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4112yes—
239C2×C42−1+412yes—
240C8—24no—
241C8—42no—
242C22—24no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—42no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22—82no—
256C22—82no—
257C22C2×#12011yescharacteristic
258C22#54712yes—
259C22C2×#13711yescharacteristic
260C22#57511yescharacteristic
261C22#57612yes—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—82no—
271C4#53211yescharacteristic
272C4#53611yescharacteristic
273C4#54812yes—
274C2—42no—
275C2—81no—
276C2—82no—
277C2#243211yescharacteristic
278C2#300711yescharacteristic
279C2#300811yescharacteristic
280C2#334612yes—
281C2#334712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#553C2
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
2. C2×#501C2semidirect
3. C2×#474C2semidirect
4. #2563C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1989C2semidirect
8. #1955C2semidirect
9. #1854C2semidirect
10. #1852C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#5012
C2×#4742
#25632
#20342
#20332
#19892
#19552
#18542
#18522
#17622