Finite Group Databasebeta

← #23864

Subgroups — #23864

Order 256.

975 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 975 conjugacy classes of subgroups — too many to draw. Only its 153 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×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3C2×#449C212yesmaximal, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5#2408C211yesmaximal, characteristic, complemented
6#2013C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1894C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic, complemented
14C23×D4C2211yescharacteristic
15C2×AGL(1,ℤ/8)—24no—
16C2×AGL(1,ℤ/8)—28no—
17C22×SD16C2212yes—
18C22×SD16C2212yes—
19C22×D8C2212yes—
20C22×M16C2211yescharacteristic
21#547—24no—
22#547—24no—
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#123—21no—
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic, complemented
28C2×#121—21no—
29C2×#121C2212yes—
30C2×#120—21no—
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic, complemented
33C22×C4⋊C4—21no—
34#450—24no—
35#449—28no—
36C2×#103C2212yescomplemented
37C2×#103C2212yescomplemented
38C2×#102C2212yes—
39C2×#102C2212yescomplemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2212yescomplemented
42#394—21no—
43#392—21no—
44#390C2212yes—
45#380C2211yescharacteristic
46#379—21no—
47#340C2212yescomplemented
48#336C2212yescomplemented
49C25—21no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—24no—
54C2×#41—24no—
55C2×#41—42no—
56C2×#41—42no—
57C2×#41D414yescomplemented
58C22×Q8C2311yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—24no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—42no—
68C22×D4D411yescharacteristic
69C22×D4C2312yes—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76AGL(1,ℤ/8)—232no—
77C2×SD16—24no—
78C2×SD16—24no—
79C2×SD16—24no—
80C2×SD16—24no—
81C2×SD16—24no—
82C2×SD16—42no—
83C2×D8—24no—
84C2×D8—24no—
85C2×D8D418yescomplemented
86C2×M16—22no—
87C2×M16—24no—
88C22×C8C2312yes—
89C22×C8C2312yes—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92#123—28no—
93#123—28no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8D414yescomplemented
98#121—28no—
99#121—24no—
100#121—42no—
101#120—44no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2312yes—
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42C2311yescharacteristic
124#103—24no—
125#103—42no—
126#102—24no—
127#102—24no—
128C22⋊C8—22no—
129C22⋊C8—24no—
130#95—21no—
131C24—21no—
132C24—22no—
133C24—44no—
134C24—42no—
135C24—42no—
136C24—42no—
137C24—82no—
138C24C2×D411yescharacteristic
139C24C2×D411yescharacteristic
140#41—28no—
141#41—44no—
142#41—48no—
143#41—48no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8C2×D412yes—
153C2×Q8C2×D412yes—
154C2×D4—28no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—48no—
167C2×D4—41no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—82no—
176C2×D4C2×D414yes—
177C2×D4C2×D414yes—
178C2×D4C2×D412yes—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D411yescharacteristic
202C22×C4C2411yescharacteristic
203SD16—216no—
204SD16—48no—
205D8—216no—
206M16—28no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8C2×D414yes—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4C2×D412yes—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C23—22no—
234C23—22no—
235C23—24no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—21no—
240C23—24no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—48no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—84no—
256C23—84no—
257C23—88no—
258C23—88no—
259C23#12012yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—24no—
264Q8—41no—
265Q8—41no—
266Q8—42no—
267D4—24no—
268D4—22no—
269D4—24no—
270D4—24no—
271D4—44no—
272D4—48no—
273D4—48no—
274D4—48no—
275D4—42no—
276D4—42no—
277D4—88no—
278D4#12014yescomplemented
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4#12012yes—
306C2×C4AGL(1,ℤ/8)12yescomplemented
307C2×C4C22×D411yescharacteristic
308C2×C4C22×D412yes—
309C2×C4C22×D411yescharacteristic
310C2×C4H(5,𝔽2)12yes—
311C8—24no—
312C8—42no—
313C22—22no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—44no—
323C22—41no—
324C22—44no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—44no—
329C22—42no—
330C22—44no—
331C22—816no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22C2×#12011yescharacteristic
337C22D4212yes—
338C22C2×AGL(1,ℤ/8)11yescharacteristic
339C22#57511yescharacteristic
340C22#57712yes—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—41no—
348C4—82no—
349C4C2×#12011yescharacteristic
350C4#53411yescharacteristic
351C4D4212yes—
352C2—42no—
353C2—44no—
354C2—81no—
355C2—84no—
356C2#240311yescharacteristic
357C2#300611yescharacteristic
358C2#301011yescharacteristic
359C2#326712yes—
360C2#327212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C22×AGL(1,ℤ/8)C2semidirect
2. C2×#547C2semidirect
3. C2×#449C2semidirect
4. C2×#450C2semidirect
5. #2408C2semidirect
6. #2013C2semidirect
7. #2000C2semidirect
8. #1894C2semidirect
9. #1879C2semidirect
10. #1870C2semidirect
11. #1788C2semidirect
13. C22×#41C22semidirect
23. C2×C4⋊Q8C22semidirect
27. C2×C22⋊Q8C22semidirect
32. C2×C4×D4C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
39. C2×#102C22semidirect
41. C2×C22⋊C8C22semidirect
47. #340C22semidirect
48. #336C22semidirect
57. C2×#41D4semidirect
85. C2×D8D4semidirect
97. C22⋊Q8D4semidirect
278. D4#120semidirect
306. C2×C4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#5472
C2×#4492
C2×#4502
#24082
#20132
#20002
#18942
#18792
#18702
#17882