Finite Group Databasebeta

← #18662

Subgroups — #18662

Order 256.

630 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 129 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#3496C211yesmaximal, characteristic, complemented
2#3245C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1647C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1625C212yesmaximal, complemented
8#1619C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1492C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13C2×AGL(1,ℤ/8)—21no—
14#547—21no—
15#547C2212yes—
16#546—22no—
17#546—22no—
18D42—21no—
19D42C2212yes—
20#537—21no—
21#534—22no—
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24#532C2211yescharacteristic, complemented
25C2×#121—22no—
26C2×C4×D4C2211yescharacteristic
27#500C2211yescharacteristic, complemented
28#493C2211yescharacteristic, complemented
29#486C2211yescharacteristic, complemented
30#486C2211yescharacteristic, complemented
31#474—21no—
32#474C2212yescomplemented
33#466—22no—
34#466—22no—
35#463—22no—
36#461C2212yescomplemented
37#459—21no—
38#459C2212yescomplemented
39#459C2212yescomplemented
40C4×D8—22no—
41#423C2211yescharacteristic, complemented
42C2×C4⋊C8C2211yescharacteristic, complemented
43#417—21no—
44C2×C4×C8C2211yescharacteristic, complemented
45#332C2212yescomplemented
46#331C2212yescomplemented
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51H(5,𝔽2)—44no—
52H(5,𝔽2)—44no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—21no—
56C2×#41—21no—
57C2×#41—21no—
58C22×D4—24no—
59C22×D4—22no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4D411yescharacteristic
63C23×C4—21no—
64AGL(1,ℤ/8)—44no—
65C2×SD16—22no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8—22no—
69C2×M16—21no—
70C22×C8—21no—
71C22×C8—21no—
72C4⋊Q8D411yescharacteristic
73C4⋊Q8C2311yescharacteristic
74C4⋊Q8C2311yescharacteristic
75#127D411yescharacteristic
76#127C2311yescharacteristic
77#127C2311yescharacteristic
78#124—21no—
79#124—21no—
80#124—21no—
81#123—24no—
82#123—24no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85#121—24no—
86#121—24no—
87#121—22no—
88#121—22no—
89#121—21no—
90#121—21no—
91#121—22no—
92#121—22no—
93#121—21no—
94#121—21no—
95#121D412yes—
96#121D412yes—
97#120—24no—
98#120—24no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4D412yes—
106C4×D4D412yes—
107C4×D4C2312yes—
108C4×D4C2312yes—
109C2×C4⋊C4—22no—
110C2×C4⋊C4D411yescharacteristic
111C2×C22⋊C4—24no—
112C2×C22⋊C4—21no—
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—22no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2312yes—
119C4⋊C8C2312yes—
120#103—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126C4×C8—21no—
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C24—21no—
130C24—24no—
131#41—44no—
132#41—48no—
133#41—44no—
134#41—42no—
135#41—42no—
136#41—42no—
137#41—42no—
138#41—42no—
139#41—42no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—24no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—44no—
158C2×D4—48no—
159C2×D4—416no—
160C2×D4—44no—
161C2×D4—41no—
162C2×D4—82no—
163C2×D4—82no—
164C2×D4D814yescomplemented
165C2×D4C2×D412yes—
166C2×D4C2×D412yes—
167C2×D4C2×D412yes—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180SD16—44no—
181D8—44no—
182D8—44no—
183D8—82no—
184M16—42no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—42no—
190C2×C8—81no—
191C4⋊C4—24no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×D412yes—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42C2×D411yescharacteristic
213C42C2411yescharacteristic
214C23—22no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—48no—
219C23—41no—
220C23—81no—
221C23—81no—
222C23—84no—
223C23—84no—
224C23#12011yescharacteristic
225Q8—41no—
226Q8—41no—
227Q8—42no—
228Q8—41no—
229Q8—41no—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—24no—
234D4—42no—
235D4—42no—
236D4—48no—
237D4—48no—
238D4—44no—
239D4—44no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—42no—
249D4—81no—
250D4—84no—
251D4—82no—
252D4C2×D814yes—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4#12011yescharacteristic
270C2×C4#12011yescharacteristic
271C2×C4#12011yescharacteristic
272C2×C4C2×D812yes—
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C8—22no—
278C8—41no—
279C8—42no—
280C8—42no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—84no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—168no—
297C22#44712yes—
298C22C2×#12011yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—81no—
306C4#44712yes—
307C4C22×D811yescharacteristic
308C4C2×AGL(1,ℤ/8)11yescharacteristic
309C4#57611yescharacteristic
310C4#57711yescharacteristic
311C2—22no—
312C2—44no—
313C2—82no—
314C2—82no—
315C2—161no—
316C2C2×#44711yescharacteristic
317C2#301211yescharacteristic
318C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. #3245C2semidirect
3. #3169C2semidirect
4. #3036C2semidirect
5. #1647C2semidirect
6. #1639C2semidirect
7. #1625C2semidirect
8. #1619C2semidirect
9. #1547C2semidirect
10. #1492C2semidirect
11. C2×D4⋊C8C2semidirect
24. #532C22semidirect
27. #500C22semidirect
28. #493C22semidirect
29. #486C22semidirect
30. #486C22semidirect
32. #474C22semidirect
36. #461C22semidirect
38. #459C22semidirect
39. #459C22semidirect
41. #423C22semidirect
42. C2×C4⋊C8C22semidirect
44. C2×C4×C8C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect
164. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
#34962
#32452
#31692
#30362
#16472
#16392
#16252
#16192
#15472
#14922
C2×D4⋊C82