Finite Group Databasebeta

← #24045

Subgroups — #24045

Order 256.

705 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 137 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×#550C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2041C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×M16C2211yescharacteristic
15#550—24no—
16#550—24no—
17C2×#127—21no—
18C2×#127C2211yescharacteristic
19C2×#123—21no—
20C2×#121—21no—
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#481—24no—
29C2×#106C2212yes—
30#417—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2212yes—
39#398—21no—
40#397—21no—
41#389C2212yes—
42#381—21no—
43#380C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41—42no—
54C2×#41D414yes—
55C22×Q8C2311yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4D411yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—42no—
72C22×C8C2312yes—
73C22×C8C2312yes—
74#127—28no—
75#127—22no—
76#127—22no—
77#123—44no—
78#121—28no—
79#121—28no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121D414yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×D4—28no—
87C4×D4—22no—
88C4×D4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2312yes—
96C2×C4⋊C4C2312yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106#106—24no—
107#103—24no—
108#103—24no—
109#102—24no—
110#102—24no—
111#102—42no—
112#102—42no—
113C22⋊C8—22no—
114C22⋊C8—42no—
115#95—22no—
116#95—21no—
117#95—22no—
118C24—21no—
119C24—21no—
120C24C2×D411yescharacteristic
121C24C2×D411yescharacteristic
122#41—28no—
123#41—44no—
124#41—48no—
125#41—48no—
126#41—44no—
127#41—44no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8C2×D412yes—
131C2×Q8#4114yes—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—41no—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—82no—
152C2×D4—82no—
153C2×D4—82no—
154C2×D4C2×D412yes—
155C2×D4C2×D412yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
180M16—44no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4C2×D412yes—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C42—24no—
204C42—21no—
205C42—21no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23#12012yes—
226C23C22×D411yescharacteristic
227Q8—22no—
228Q8—24no—
229Q8#12314yescomplemented
230D4—22no—
231D4—22no—
232D4—24no—
233D4—44no—
234D4—48no—
235D4—48no—
236D4—48no—
237D4—41no—
238D4—41no—
239D4—42no—
240D4—84no—
241D4—84no—
242D4—84no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—84no—
269C2×C4#12012yes—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4112yes—
274C2×C4H(5,𝔽2)12yes—
275C8—42no—
276C8—42no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22C2×#12011yescharacteristic
298C22#54612yes—
299C22C2×AGL(1,ℤ/8)11yescharacteristic
300C22#57511yescharacteristic
301C22#57712yes—
302C4—22no—
303C4—24no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—82no—
309C4—82no—
310C4C2×#12311yescharacteristic
311C4#53411yescharacteristic
312C4#54612yes—
313C2—42no—
314C2—81no—
315C2—82no—
316C2—82no—
317C2#240511yescharacteristic
318C2#300611yescharacteristic
319C2#300911yescharacteristic
320C2#331712yes—
321C2#333212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#550C2
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×#481C2semidirect
3. C2×#417C2semidirect
4. #2486C2semidirect
5. #2041C2semidirect
6. #2034C2semidirect
7. #1895C2semidirect
8. #1870C2semidirect
9. #1865C2semidirect
10. #1788C2semidirect
11. #1788C2semidirect
229. Q8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4812
C2×#4172
#24862
#20412
#20342
#18952
#18702
#18652
#17882