Finite Group Databasebeta

← #23977

Subgroups — #23977

Order 256.

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

normal subgroups only

This group has 643 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×#555C211yesmaximal, characteristic
2C2×#448C211yesmaximal, characteristic, complemented
3C2×#419C211yesmaximal, characteristic, complemented
4#2409C211yesmaximal, characteristic, complemented
5#2024C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1877C212yesmaximal, complemented
9#1873C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1825C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2212yes—
14C22×M16C2211yescharacteristic
15#555—24no—
16#555—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124—21no—
19C2×#124—21no—
20C2×C22⋊Q8C2212yes—
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×#117—21no—
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#448—24no—
30#419—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37#395—21no—
38#393—21no—
39#391C2212yes—
40#386—21no—
41#385—21no—
42#380C2211yescharacteristic
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8—24no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2312yes—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4D411yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C2×SD16—24no—
63C2×SD16—24no—
64C2×SD16—42no—
65C2×M16—22no—
66C2×M16—42no—
67C22×C8C2312yes—
68C22×C8C2312yes—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71#124—28no—
72#124—28no—
73C22⋊Q8—24no—
74C22⋊Q8—42no—
75#121—28no—
76#121—22no—
77#121—22no—
78#121—22no—
79#121D414yes—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84#117—28no—
85#117—22no—
86#117—22no—
87#117—22no—
88#117D414yes—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108#103—24no—
109#103—24no—
110#103—24no—
111#102—42no—
112C22⋊C8—22no—
113C22⋊C8—42no—
114#95—21no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C24C2×D411yescharacteristic
120C2×Q8—24no—
121C2×Q8—22no—
122C2×Q8—28no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8—42no—
131C2×Q8—82no—
132C2×Q8#4114yes—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—82no—
142C2×D4—82no—
143C2×D4C2×D412yes—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167SD16—48no—
168M16—44no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—44no—
183C4⋊C4C2×D412yes—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—22no—
193C22⋊C4—44no—
194C22⋊C4—42no—
195C42—24no—
196C42—24no—
197C42—21no—
198C42—21no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23#12012yes—
214C23C22×D411yescharacteristic
215Q8—24no—
216Q8—24no—
217Q8—48no—
218Q8—42no—
219Q8—42no—
220Q8—84no—
221D4—24no—
222D4—44no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—84no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4#12012yes—
256C2×C4#13712yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4112yes—
260C2×C4H(5,𝔽2)12yes—
261C8—42no—
262C8—42no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22C2×#12011yescharacteristic
279C22#54612yes—
280C22C2×#13711yescharacteristic
281C22#57511yescharacteristic
282C22#57712yes—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—44no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—42no—
291C4—82no—
292C4#52511yescharacteristic
293C4#53511yescharacteristic
294C4#54612yes—
295C2—42no—
296C2—81no—
297C2—82no—
298C2#240411yescharacteristic
299C2#300711yescharacteristic
300C2#300911yescharacteristic
301C2#332112yes—
302C2#332912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
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×#448C2semidirect
3. C2×#419C2semidirect
4. #2409C2semidirect
5. #2024C2semidirect
6. #2000C2semidirect
7. #1895C2semidirect
8. #1877C2semidirect
9. #1873C2semidirect
10. #1870C2semidirect
11. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4482
C2×#4192
#24092
#20242
#20002
#18952
#18772
#18732
#18702
#18252