Finite Group Databasebeta

← #24248

Subgroups — #24248

Order 256.

970 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 147 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×D42C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic, complemented
3C2×#447C212yesmaximal, complemented
4C2×#442C211yesmaximal, characteristic, complemented
5#2535C211yesmaximal, characteristic, complemented
6#2034C212yesmaximal, complemented
7#1999C212yesmaximal, complemented
8#1961C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1786C212yesmaximal, complemented
12C23×D4—21no—
13C23×D4C2212yes—
14C22×D8C2211yescharacteristic
15C22×D8C2212yescomplemented
16D42—24no—
17D42—24no—
18C2×#127C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2212yes—
21C2×#121C2212yescomplemented
22C2×#120—22no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic, complemented
26#459—24no—
27#447—28no—
28#442—24no—
29C2×#106C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic, complemented
34C2×#102C2212yescomplemented
35C2×#102C2212yescomplemented
36C2×#102C2212yescomplemented
37C2×C22⋊C8C2212yescomplemented
38C2×C22⋊C8C2212yescomplemented
39C2×#97C2211yescharacteristic, complemented
40#397—21no—
41#392—21no—
42#386—22no—
43#382C2211yescharacteristic, complemented
44#380C2212yescomplemented
45#340C2212yescomplemented
46C25—22no—
47C22×D4—28no—
48C22×D4—22no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—42no—
58C22×D4—42no—
59C22×D4—48no—
60C22×D4D411yescharacteristic
61C22×D4C2312yes—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2312yes—
67C2×D8—22no—
68C2×D8—22no—
69C2×D8—24no—
70C2×D8—24no—
71C2×D8—42no—
72C2×D8D414yescomplemented
73C22×C8C2312yes—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#127—22no—
77#127D414yescomplemented
78#121—28no—
79#121—24no—
80#121—24no—
81#121—24no—
82#121—42no—
83#120—216no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4D414yescomplemented
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4D412yes—
100C2×C42C2311yescharacteristic
101#106—22no—
102C4⋊C8—22no—
103#102—22no—
104#102—24no—
105#102—24no—
106#102—24no—
107#102—22no—
108#102—42no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111#97—22no—
112#95—22no—
113#95—21no—
114#95—21no—
115C24—21no—
116C24—28no—
117C24—21no—
118C24—22no—
119C24—44no—
120C24—44no—
121C24—44no—
122C24—44no—
123C24—82no—
124C24—82no—
125C24C2×D412yes—
126C2×D4—24no—
127C2×D4—28no—
128C2×D4—28no—
129C2×D4—22no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—48no—
149C2×D4—48no—
150C2×D4—432no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—48no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—82no—
158C2×D4—82no—
159C2×D4C2×D414yes—
160C2×D4C2×D412yes—
161C2×D4C2×D412yes—
162C2×D4C2×D412yescomplemented
163C2×D4#4112yes—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—82no—
177C22×C4C2×D412yes—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181D8—28no—
182D8—48no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—21no—
191C2×C8C2×D412yescomplemented
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—22no—
198C4⋊C4C2×D412yes—
199C22⋊C4—28no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42C2×D412yes—
206C42C2×D412yes—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—24no—
217C23—22no—
218C23—416no—
219C23—416no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—88no—
233C23—88no—
234C23—88no—
235C23—88no—
236C23—88no—
237C23AGL(1,ℤ/8)14yes—
238C23C22×D411yescharacteristic
239D4—28no—
240D4—22no—
241D4—22no—
242D4—22no—
243D4—22no—
244D4—24no—
245D4—22no—
246D4—24no—
247D4—48no—
248D4—416no—
249D4—416no—
250D4—42no—
251D4—41no—
252D4—42no—
253D4—41no—
254D4—84no—
255D4—88no—
256D4#12114yes—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—84no—
275C2×C4#12012yes—
276C2×C4#12012yes—
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C22×D411yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C8—22no—
284C8—41no—
285C8—42no—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—44no—
296C22—42no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—832no—
304C22—81no—
305C22—81no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22C2×#12011yescharacteristic
312C22D4211yescharacteristic
313C22#54611yescharacteristic
314C22C2×AGL(1,ℤ/8)12yes—
315C22#57612yes—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—42no—
320C4—81no—
321C4—82no—
322C4C2×#12111yescharacteristic
323C4#53511yescharacteristic
324C4D4211yescharacteristic
325C4#54611yescharacteristic
326C2—44no—
327C2—44no—
328C2—82no—
329C2—84no—
330C2#240811yescharacteristic
331C2#300311yescharacteristic
332C2#300811yescharacteristic
333C2#326412yes—
334C2#331812yes—
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. C2×D42C2semidirect
2. C2×#459C2semidirect
3. C2×#447C2semidirect
4. C2×#442C2semidirect
5. #2535C2semidirect
6. #2034C2semidirect
7. #1999C2semidirect
8. #1961C2semidirect
9. #1955C2semidirect
10. #1804C2semidirect
11. #1786C2semidirect
15. C22×D8C22semidirect
21. C2×#121C22semidirect
25. C2×C4×D4C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
33. C2×#102C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
38. C2×C22⋊C8C22semidirect
39. C2×#97C22semidirect
43. #382C22semidirect
44. #380C22semidirect
45. #340C22semidirect
72. C2×D8D4semidirect
77. #127D4semidirect
89. C4×D4D4semidirect
162. C2×D4C2×D4semidirect
191. C2×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C2×#4592
C2×#4472
C2×#4422
#25352
#20342
#19992
#19612
#19552
#18042
#17862