Finite Group Databasebeta

← #18580

Subgroups — #18580

Order 256.

481 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 117 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#3523C211yesmaximal, characteristic, complemented
2#3217C211yesmaximal, characteristic
3#3138C211yesmaximal, characteristic, complemented
4C2×#460C211yesmaximal, characteristic, complemented
5#1628C212yesmaximal, complemented
6#1627C212yesmaximal, complemented
7#1626C212yesmaximal, complemented
8#1626C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic, complemented
10#1491C211yesmaximal, characteristic, complemented
11#1483C211yesmaximal, characteristic, complemented
12C22×SD16—21no—
13#557—21no—
14#557C2211yescharacteristic, complemented
15#557C2211yescharacteristic, complemented
16#556—21no—
17#556C2211yescharacteristic, complemented
18#552—22no—
19#552—21no—
20D4×Q8—21no—
21D4×Q8C2211yescharacteristic, complemented
22#541—22no—
23#541—21no—
24#533—21no—
25#533C2211yescharacteristic
26C2×#127C2211yescharacteristic
27#529—21no—
28#529C2211yescharacteristic
29#519—21no—
30C2×C4×Q8C2211yescharacteristic
31#496—21no—
32#490—22no—
33#490—21no—
34#486C2212yes—
35#486C2212yes—
36#460—21no—
37#460—21no—
38#460C2212yes—
39#460C2212yes—
40#460C2212yes—
41#459C2212yes—
42#450—21no—
43#447—21no—
44#432C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#418—22no—
47C2×#102—21no—
48C4×M16C2211yescharacteristic
49#332C2212yescomplemented
50#332C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52Q8⋊C8C2212yescomplemented
53Q8⋊C8C2212yescomplemented
54D4⋊C8C2212yescomplemented
55C22×Q8—21no—
56C22×Q8D411yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C2×SD16—22no—
61C2×SD16—22no—
62C2×SD16—22no—
63C2×SD16—44no—
64C2×D8—22no—
65C2×M16—21no—
66C22×C8—21no—
67C4⋊Q8—22no—
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8C2311yescharacteristic
72C4⋊Q8C2311yescharacteristic
73#127—21no—
74#127—21no—
75#127—21no—
76#127C2312yes—
77#126—22no—
78#126—22no—
79#126—22no—
80#126—22no—
81#126—21no—
82#125—22no—
83#125—22no—
84#125—22no—
85#125—21no—
86#125—22no—
87#125—22no—
88#125—21no—
89#125D411yescharacteristic
90#125D412yes—
91#124—21no—
92#124D411yescharacteristic
93#123—22no—
94#123—21no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—21no—
99C22⋊Q8—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C22⋊Q8D411yescharacteristic
103#121D411yescharacteristic
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8D411yescharacteristic
116C4×Q8D411yescharacteristic
117C4×Q8C2311yescharacteristic
118C4×Q8C2311yescharacteristic
119C4×Q8C2311yescharacteristic
120C4×D4—22no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4C2311yescharacteristic
124#117—22no—
125#117—22no—
126#117—22no—
127#117—21no—
128#117—21no—
129#117D412yes—
130C2×C4⋊C4—21no—
131C2×C4⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134C4⋊C8—21no—
135C4⋊C8C2312yes—
136C4⋊C8C2312yes—
137#102—22no—
138#102—22no—
139#102—24no—
140#102—24no—
141#102—22no—
142#102—22no—
143C22⋊C8—21no—
144#97—21no—
145#97—21no—
146C4×C8C2312yes—
147C24—41no—
148C2×Q8—22no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8C2×D411yescharacteristic
157C2×Q8C2×D411yescharacteristic
158C2×Q8C2×D411yescharacteristic
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—42no—
164C2×D4—44no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—82no—
170C2×D4C2×D411yescharacteristic
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178SD16—44no—
179SD16—44no—
180SD16—82no—
181D8—82no—
182M16—42no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—42no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D412yes—
211C4⋊C4C2×D412yes—
212C4⋊C4C2×D411yescharacteristic
213C4⋊C4C2×D411yescharacteristic
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—21no—
218C22⋊C4—22no—
219C22⋊C4—21no—
220C22⋊C4—21no—
221C22⋊C4—21no—
222C22⋊C4—21no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42C2×D411yescharacteristic
233C42C2×D411yescharacteristic
234C42C2×D411yescharacteristic
235C42C2411yescharacteristic
236C23—21no—
237C23—42no—
238C23—82no—
239C23—82no—
240C23—82no—
241C23#12011yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—21no—
245Q8—22no—
246Q8—42no—
247Q8—44no—
248Q8—41no—
249Q8—41no—
250Q8—42no—
251Q8AGL(1,ℤ/8)12yes—
252D4—22no—
253D4—44no—
254D4—42no—
255D4—42no—
256D4—42no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—82no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4#12011yescharacteristic
281C2×C4#12011yescharacteristic
282C2×C4#12011yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C8—42no—
287C8—42no—
288C8—42no—
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—164no—
298C22#45412yes—
299C22C2×#12011yescharacteristic
300C4—21no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4C2×AGL(1,ℤ/8)11yescharacteristic
314C4#57511yescharacteristic
315C4#57712yes—
316C2—22no—
317C2—81no—
318C2—162no—
319C2#300611yescharacteristic
320C2#301011yescharacteristic
321C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3217C2
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. #3523C2semidirect
3. #3138C2semidirect
4. C2×#460C2semidirect
5. #1628C2semidirect
6. #1627C2semidirect
7. #1626C2semidirect
8. #1626C2semidirect
9. #1552C2semidirect
10. #1491C2semidirect
11. #1483C2semidirect
14. #557C22semidirect
15. #557C22semidirect
17. #556C22semidirect
21. D4×Q8C22semidirect
49. #332C22semidirect
50. #332C22semidirect
51. Q8⋊C8C22semidirect
52. Q8⋊C8C22semidirect
53. Q8⋊C8C22semidirect
54. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35232
#32172
#31382
C2×#4602
#16282
#16272
#16262
#15522
#14912
#14832