Finite Group Databasebeta

← #25641

Subgroups — #25641

Order 256.

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

normal subgroups only

This group has 586 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×#559C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4#2568C211yesmaximal, characteristic, complemented
5#2015C212yesmaximal, complemented
6#2014C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C22×SD16C2212yes—
15#559—24no—
16#559—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#497—24no—
30#461—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#97C2211yescharacteristic
37#400—21no—
38#400—21no—
39#391—21no—
40#391C2212yes—
41#391C2212yes—
42#382—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8D411yescharacteristic
48C22×Q8C2312yes—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C2×Q16—24no—
57C2×Q16—24no—
58C2×Q16—42no—
59C2×SD16—24no—
60C2×SD16—24no—
61C2×SD16—24no—
62C2×SD16—42no—
63C2×SD16—42no—
64C2×SD16—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69#127—22no—
70#127—22no—
71#124—28no—
72#124—22no—
73#124—22no—
74#124—22no—
75#124D414yes—
76#123—28no—
77#123—28no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—28no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4D414yes—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C4⋊C8—22no—
107#103—24no—
108#103—42no—
109#102—24no—
110#102—42no—
111#97—22no—
112#95—21no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24—21no—
119C2×Q8—22no—
120C2×Q8—22no—
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—41no—
131C2×Q8C2×D412yes—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4C2×D412yes—
144C2×D4C2×D412yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160Q16—48no—
161SD16—48no—
162SD16—48no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8#4114yes—
169C4⋊C4—24no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4C2×D412yes—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C42—24no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—42no—
191C42C2×D412yes—
192C42C2×D412yes—
193C23—22no—
194C23—22no—
195C23—42no—
196C23—42no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23C22×D411yescharacteristic
204Q8—24no—
205Q8—44no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—42no—
212D4—22no—
213D4—22no—
214D4—24no—
215D4—48no—
216D4—41no—
217D4—41no—
218D4—42no—
219D4—84no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—84no—
243C2×C4—84no—
244C2×C4#12012yes—
245C2×C4#12012yes—
246C2×C4#13712yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C2×#4112yes—
250C2×C4H(5,𝔽2)12yes—
251C8—24no—
252C8—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22C2×#12011yescharacteristic
264C22#55012yes—
265C22C2×#13711yescharacteristic
266C22#57511yescharacteristic
267C22#57712yes—
268C4—22no—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—42no—
276C4—82no—
277C4—82no—
278C4#53211yescharacteristic
279C4#53511yescharacteristic
280C4#54612yes—
281C2—82no—
282C2—82no—
283C2#243111yescharacteristic
284C2#300711yescharacteristic
285C2#300911yescharacteristic
286C2#340012yes—
287C2#340112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ 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×#461C2semidirect
4. #2568C2semidirect
5. #2015C2semidirect
7. #1985C2semidirect
8. #1983C2semidirect
9. #1878C2semidirect
10. #1877C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4972
C2×#4612
#25682
#20152
#20142
#19852
#19832
#18782
#18772
#17622