Finite Group Databasebeta

← #23936

Subgroups — #23936

Order 256.

760 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 141 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×#529C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C2×#390C211yesmaximal, characteristic, complemented
5#2025C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#1880C212yesmaximal, complemented
8#1879C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic, complemented
19C2×#124C2211yescharacteristic, complemented
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic, complemented
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic, complemented
26C2×#117—21no—
27C22×C4⋊C4C2211yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C42C2211yescharacteristic
31#450—24no—
32#447—24no—
33C2×#103C2212yescomplemented
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yescomplemented
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#390—21no—
40#390—21no—
41#390C2212yes—
42#390C2212yes—
43#340C2212yescomplemented
44#340C2212yescomplemented
45#336C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4—24no—
51C22×D4—21no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4—42no—
55C22×D4C2311yescharacteristic
56C22×D4C2312yes—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C2×SD16—24no—
63C2×SD16—24no—
64C2×SD16—42no—
65C2×D8—24no—
66C2×D8—24no—
67C2×D8—42no—
68C22×C8C2312yes—
69C22×C8C2312yes—
70#126—28no—
71#125—22no—
72#125—22no—
73#124—22no—
74#124—22no—
75#123—28no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78#121—22no—
79#121D414yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—28no—
83C4×D4—22no—
84C4×D4—22no—
85#117—28no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—44no—
106C2×C42—21no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#103—42no—
114#102—24no—
115#102—24no—
116#102—42no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119C24—24no—
120C24—21no—
121C24—22no—
122C24—42no—
123C24—42no—
124C24—42no—
125C24—42no—
126C24—42no—
127C24—42no—
128C24C2×D411yescharacteristic
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8C2×D412yes—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4C2×D412yes—
150C2×D4#4114yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177SD16—48no—
178D8—48no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4C2×D412yes—
195C4⋊C4C2×D412yes—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C42—24no—
206C42—22no—
207C42—22no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—48no—
219C23—48no—
220C23—48no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—84no—
234C23—84no—
235C23—84no—
236C23—84no—
237C23#12012yes—
238C23#13512yes—
239C23AGL(1,ℤ/8)12yes—
240C23C22×D411yescharacteristic
241Q8—24no—
242Q8—42no—
243D4—24no—
244D4—24no—
245D4—24no—
246D4—48no—
247D4—48no—
248D4—42no—
249D4—42no—
250D4—42no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—84no—
282C2×C4#12012yes—
283C2×C4AGL(1,ℤ/8)12yes—
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C2×C4C2×#4112yes—
287C2×C4H(5,𝔽2)12yes—
288C8—42no—
289C8—42no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—816no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—82no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22#44914yes—
312C22C2×#12011yescharacteristic
313C22#52511yescharacteristic
314C22#53511yescharacteristic
315C22C2×#13511yescharacteristic
316C22C2×AGL(1,ℤ/8)11yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22#57511yescharacteristic
319C4—22no—
320C4—22no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—42no—
327C4—82no—
328C4#54612yes—
329C4#54612yes—
330C2—24no—
331C2—82no—
332C2—84no—
333C2#240411yescharacteristic
334C2C2×#44911yescharacteristic
335C2#300611yescharacteristic
336C2#310511yescharacteristic
337C2#311411yescharacteristic
338C2#321111yescharacteristic
339C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#529C2semidirect
2. C2×#450C2semidirect
4. C2×#390C2semidirect
5. #2025C2semidirect
6. #2012C2semidirect
7. #1880C2semidirect
8. #1879C2semidirect
9. #1874C2semidirect
10. #1872C2semidirect
11. #1817C2semidirect
18. C2×#125C22semidirect
19. C2×#124C22semidirect
23. C2×C4×Q8C22semidirect
25. C2×C4×D4C22semidirect
33. C2×#103C22semidirect
36. C2×#102C22semidirect
43. #340C22semidirect
44. #340C22semidirect
45. #336C22semidirect
46. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4502
C2×#4472
C2×#3902
#20252
#20122
#18802
#18792
#18742
#18722
#18172