Finite Group Databasebeta

← #25680

Subgroups — #25680

Order 256.

516 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#489C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic, complemented
5#2035C212yesmaximal, complemented
6#2034C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12#542—24no—
13#542—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—21no—
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#498—24no—
27#489—24no—
28C2×#106C2212yes—
29C2×#106C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#400—21no—
38#399—21no—
39#395—21no—
40#393—21no—
41#389C2212yes—
42#382C2211yescharacteristic
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8D414yes—
60#126—28no—
61#124—28no—
62#124—22no—
63#124D414yes—
64#123—28no—
65C22⋊Q8—28no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—22no—
69#121—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—22no—
73C4×D4—22no—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#106—24no—
99#106—24no—
100#103—24no—
101#103—24no—
102#102—24no—
103#102—24no—
104C22⋊C8—24no—
105#97—22no—
106C4×C8—22no—
107#95—22no—
108#95—21no—
109#95—21no—
110#95—22no—
111C24—21no—
112C24C2×D411yescharacteristic
113C2×Q8—24no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—42no—
118C2×Q8—82no—
119C2×Q8—82no—
120C2×Q8C2×D412yes—
121C2×D4—24no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—82no—
126C2×D4C2×D412yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—82no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8#4114yes—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—82no—
173C42—24no—
174C42—21no—
175C42—21no—
176C42C2×D412yes—
177C42C2×D412yes—
178C23—21no—
179C23—22no—
180C23—21no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23C22×D411yescharacteristic
189Q8—24no—
190Q8—42no—
191Q8—84no—
192Q8—84no—
193D4—24no—
194D4—42no—
195D4—84no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—84no—
221C2×C4#12012yes—
222C2×C4#12012yes—
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C42−1+412yes—
227C8—24no—
228C8—42no—
229C22—42no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22C2×#12011yescharacteristic
240C22#53211yescharacteristic
241C22#53611yescharacteristic
242C22#57712yes—
243C22#57712yes—
244C4—22no—
245C4—22no—
246C4—41no—
247C4—42no—
248C4—41no—
249C4—42no—
250C4—42no—
251C4—41no—
252C4—82no—
253C4—82no—
254C4#54712yes—
255C4#54812yes—
256C2—81no—
257C2—82no—
258C2#243211yescharacteristic
259C2#300911yescharacteristic
260C2#301011yescharacteristic
261C2#317312yes—
262C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#498C2semidirect
3. C2×#489C2semidirect
4. #2628C2semidirect
5. #2035C2semidirect
7. #1993C2semidirect
8. #1958C2semidirect
9. #1865C2semidirect
10. #1864C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4982
C2×#4892
#26282
#20352
#20342
#19932
#19582
#18652
#18642
#17502