Finite Group Databasebeta

← #23908

Subgroups — #23908

Order 256.

907 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 907 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#3485C211yesmaximal, characteristic, complemented
2C2×#532C211yesmaximal, characteristic, complemented
3C2×#455C211yesmaximal, characteristic, complemented
4C2×#453C211yesmaximal, characteristic, complemented
5#3005C212yesmaximal, complemented
6#2005C212yesmaximal, complemented
7#2003C212yesmaximal, complemented
8#1896C212yesmaximal, complemented
9#1887C211yesmaximal, characteristic, complemented
10#1885C211yesmaximal, characteristic, complemented
11#1785C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×#41C2211yescharacteristic, complemented
15C2×#137C2212yes—
16C2×AGL(1,ℤ/8)C2212yes—
17#547—24no—
18#546—24no—
19#536C2212yes—
20#534—22no—
21#532—24no—
22#532—24no—
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#127C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×#124C2211yescharacteristic, complemented
27#525C2212yes—
28C2×#123—22no—
29C2×#121—22no—
30#518C2211yescharacteristic
31C2×C4×D4—22no—
32C22×C42C2211yescharacteristic, complemented
33#455—24no—
34#453—24no—
35#452—24no—
36#449—24no—
37C2×#104C2212yes—
38C2×#104C2212yes—
39#419C2212yescomplemented
40#418C2212yescomplemented
41C2×#101C2212yescomplemented
42C2×#100C2211yescharacteristic
43C2×#100C2211yescharacteristic
44#407C2212yescomplemented
45#339C2212yescomplemented
46#339C2212yescomplemented
47H(5,𝔽2)—22no—
48H(5,𝔽2)—22no—
49H(5,𝔽2)—42no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—24no—
56C2×#41—24no—
57C2×#41—42no—
58C2×#41—42no—
59C2×#41C2312yes—
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—22no—
63C22×D4—21no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4D412yes—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4D412yes—
72C23×C4C2311yescharacteristic
73#137—24no—
74#137—42no—
75AGL(1,ℤ/8)—24no—
76AGL(1,ℤ/8)—42no—
77C2×Q16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×D8—22no—
81C2×M16C2312yes—
82C2×M16C2312yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86#127—22no—
87#127D414yescomplemented
88#124—22no—
89#124—22no—
90#124—22no—
91#124—22no—
92#124D414yescomplemented
93#123—28no—
94#123—24no—
95#123—22no—
96#123—24no—
97C22⋊Q8—24no—
98C22⋊Q8D414yescomplemented
99#121—216no—
100#121—28no—
101#121D414yescomplemented
102#120—24no—
103#120—22no—
104C4×D4—216no—
105C4×D4—24no—
106C4×D4—24no—
107#117C2312yes—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C42—21no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#104—24no—
120#104—24no—
121#103—24no—
122#102—24no—
123#101—42no—
124#100—22no—
125#100—22no—
126C22⋊C8—24no—
127C24—22no—
128C24—21no—
129C24C2×D412yes—
130C24C2×D411yescharacteristic
131#41—22no—
132#41—22no—
133#41—22no—
134#41—22no—
135#41—48no—
136#41—48no—
137#41—48no—
138#41—48no—
139#41—44no—
140#41—44no—
141C2×Q8—22no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8C2×D412yes—
149C2×Q8C2×D412yes—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—28no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—48no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4C2×D412yes—
171C2×D4C2×D412yes—
172C2×D4C2×D412yes—
173C2×D4C2×D412yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4C2×D412yes—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D414yes—
194C22×C4C2411yescharacteristic
195Q16—44no—
196SD16—44no—
197SD16—44no—
198D8—44no—
199M16—22no—
200M16—22no—
201M16—22no—
202M16—22no—
203C2×C8—22no—
204C2×C8—22no—
205C4⋊C4—28no—
206C4⋊C4—28no—
207C4⋊C4—24no—
208C4⋊C4C2×D414yes—
209C22⋊C4—24no—
210C22⋊C4—28no—
211C22⋊C4—28no—
212C22⋊C4—28no—
213C22⋊C4—24no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C42—24no—
217C42—21no—
218C42—21no—
219C42—22no—
220C42C2×D412yes—
221C42C2×D412yes—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—84no—
235C23#12012yes—
236C23#12014yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—24no—
240Q8—22no—
241Q8—42no—
242Q8—42no—
243Q8—42no—
244D4—24no—
245D4—24no—
246D4—22no—
247D4—24no—
248D4—22no—
249D4—416no—
250D4—48no—
251D4—48no—
252D4—48no—
253D4—42no—
254D4—42no—
255D4—42no—
256D4—84no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—84no—
280C2×C4#12012yes—
281C2×C4C22×D411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D412yes—
284C2×C4H(5,𝔽2)12yes—
285C8—42no—
286C8—42no—
287C22—22no—
288C22—22no—
289C22—24no—
290C22—24no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22—42no—
297C22—42no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—164no—
303C22C2×#12011yescharacteristic
304C22C2×#12011yescharacteristic
305C22#53411yescharacteristic
306C4—24no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—44no—
313C4D4212yes—
314C4D4212yes—
315C2—21no—
316C2—21no—
317C2—42no—
318C2—44no—
319C2—82no—
320C2—82no—
321C2#240311yescharacteristic
322C2#301812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3485C2semidirect
2. C2×#532C2semidirect
3. C2×#455C2semidirect
4. C2×#453C2semidirect
5. #3005C2semidirect
6. #2005C2semidirect
7. #2003C2semidirect
8. #1896C2semidirect
9. #1887C2semidirect
10. #1885C2semidirect
11. #1785C2semidirect
14. C22×#41C22semidirect
23. C2×C4⋊Q8C22semidirect
26. C2×#124C22semidirect
32. C22×C42C22semidirect
39. #419C22semidirect
40. #418C22semidirect
41. C2×#101C22semidirect
44. #407C22semidirect
45. #339C22semidirect
46. #339C22semidirect
87. #127D4semidirect
92. #124D4semidirect
98. C22⋊Q8D4semidirect
101. #121D4semidirect

Maximal subgroups

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

subgroupindex
#34852
C2×#5322
C2×#4552
C2×#4532
#30052
#20052
#20032
#18962
#18872
#18852
#17852