Finite Group Databasebeta

← #24381

Subgroups — #24381

Order 256.

829 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#465C212yesmaximal, complemented
3C2×#448C211yesmaximal, characteristic, complemented
4C22×C22⋊C8C211yesmaximal, characteristic, complemented
5#2847C211yesmaximal, characteristic
6#2035C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1892C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1853C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C22×SD16C2212yescomplemented
15C23×C8C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—28no—
19C2×C22⋊Q8C2212yes—
20C2×C22⋊Q8C2212yes—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#120—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2211yescharacteristic
27C22×C22⋊C4—21no—
28#465—28no—
29#448—24no—
30C2×#106C2212yescomplemented
31C2×#103C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#103C2212yescomplemented
34C2×#102C2212yescomplemented
35C2×C22⋊C8—24no—
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2211yescharacteristic
39#386—21no—
40#386—21no—
41#386—21no—
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#340C2212yescomplemented
46#340C2212yescomplemented
47C25D411yescharacteristic
48C22×Q8—24no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2312yes—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16—42no—
73C22×C8—24no—
74C22×C8—24no—
75C22×C8—42no—
76C22×C8C2312yes—
77C22×C8C2312yes—
78C22⋊Q8—28no—
79C22⋊Q8—28no—
80C22⋊Q8—24no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83C22⋊Q8D418yescomplemented
84#121—22no—
85#121—24no—
86#121—22no—
87#121—42no—
88#120—44no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2312yes—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#106—24no—
108#103—24no—
109#103—24no—
110#103—42no—
111#102—24no—
112C22⋊C8—22no—
113C22⋊C8—24no—
114C22⋊C8D414yescomplemented
115#95—21no—
116#95—21no—
117#95—21no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24SD1614yescomplemented
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C24C2×D411yescharacteristic
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—48no—
136C2×Q8—48no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8C2×D414yes—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D414yes—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4#4112yes—
190C22×C4C2411yescharacteristic
191SD16—48no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—48no—
198C2×C8C2×D414yes—
199C4⋊C4—24no—
200C4⋊C4—28no—
201C4⋊C4—28no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4C2×D414yes—
208C22⋊C4—28no—
209C22⋊C4—28no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—28no—
231C23—28no—
232C23—44no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#12012yes—
245C23#12112yes—
246C23C2×SD1612yes—
247C23C2×SD1612yes—
248C23C2×SD1612yes—
249C23#13512yes—
250C23#13712yes—
251C23C22×D411yescharacteristic
252Q8—24no—
253Q8—24no—
254Q8—48no—
255Q8—42no—
256Q8—42no—
257D4—42no—
258D4—42no—
259D4—84no—
260D4—84no—
261D4—84no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4#12012yes—
296C2×C4#12112yes—
297C2×C4C22×D411yescharacteristic
298C2×C4C22×D412yes—
299C2×C4C2×#4111yescharacteristic
300C2×C4H(5,𝔽2)12yes—
301C8—24no—
302C8—42no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—416no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#44814yescomplemented
331C22#46514yescomplemented
332C22C2×#12011yescharacteristic
333C22C2×#12111yescharacteristic
334C22#53411yescharacteristic
335C22C22×SD1611yescharacteristic
336C22C2×#13511yescharacteristic
337C22C2×#13711yescharacteristic
338C22#57511yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—44no—
343C4—44no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4D4212yes—
348C4#54612yes—
349C2—24no—
350C2—24no—
351C2—42no—
352C2—161no—
353C2#241011yescharacteristic
354C2C2×#44811yescharacteristic
355C2#300211yescharacteristic
356C2C2×#46511yescharacteristic
357C2#305611yescharacteristic
358C2#318711yescharacteristic
359C2#319011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊Q8C2semidirect
2. C2×#465C2semidirect
3. C2×#448C2semidirect
4. C22×C22⋊C8C2semidirect
6. #2035C2semidirect
7. #2000C2semidirect
8. #1892C2semidirect
9. #1874C2semidirect
10. #1853C2semidirect
11. #1787C2semidirect
14. C22×SD16C22semidirect
30. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
34. C2×#102C22semidirect
45. #340C22semidirect
46. #340C22semidirect
83. C22⋊Q8D4semidirect
114. C22⋊C8D4semidirect
125. C24SD16semidirect
330. C22#448semidirect
331. C22#465semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4652
C2×#4482
C22×C22⋊C82
#28472
#20352
#20002
#18922
#18742
#18532
#17872