Finite Group Databasebeta

← #49131709

Subgroups — #49131709

Order 36224.

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

normal subgroups only

This group has 656 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
1C2×#22583638C211yesmaximal, characteristic, complemented
2C2×#22583636C211yesmaximal, characteristic, complemented
3C2×#22583635C211yesmaximal, characteristic, complemented
4#31827011C211yesmaximal, characteristic, complemented
5C2×#22583606C211yesmaximal, characteristic, complemented
6#31826853C212yesmaximal, complemented
7#31826851C212yesmaximal, complemented
8#31826840C212yesmaximal, complemented
9#31826836C212yesmaximal, complemented
10M16×Dic283C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×D1132C2211yescharacteristic
14C22×C22⋊D283—22no—
15#22583638—24no—
16#22583636—24no—
17#22583635—24no—
18C2×#17985381C2212yescomplemented
19C2×#17985380C2212yescomplemented
20C2×#17985379C2212yescomplemented
21C2×D2264C2212yescomplemented
22C2×#17985377C2212yescomplemented
23C2×#17985377C2212yescomplemented
24C2×#17985376C2211yescharacteristic
25#22583608C2211yescharacteristic, complemented
26#22583606—22no—
27#22583606C2212yes—
28#22583606C2212yes—
29#22583606C2211yescharacteristic, complemented
30#22583599C2211yescharacteristic, complemented
31#22583597C2211yescharacteristic, complemented
32#22583582—22no—
33C4×C22⋊D283—22no—
34C2×C4×Dic283C2211yescharacteristic
35#22583520C2212yescomplemented
36C8×Dic283C2212yescomplemented
37C2×#41×C283C2211yescharacteristic
38C22×D4×C283C2211yescharacteristic
39AGL(1,ℤ/8)×C283—24no—
40C2×SD16×C283C2212yescomplemented
41C2×D8×C283C2212yescomplemented
42C2×M16×C283C2211yescharacteristic
43C23×D283—21no—
44#17985392—22no—
45#17985392—22no—
46C2×Dic566C2311yescharacteristic
47C2×D1132—22no—
48C2×D1132—22no—
49C2×D1132—21no—
50C2×D1132C2311yescharacteristic
51C2×D1132C2312yes—
52C2×C22⋊D283—22no—
53C2×C22⋊D283—24no—
54C2×C22⋊D283—24no—
55C2×C22⋊D283—21no—
56C2×C22⋊D283—48no—
57C22×Dic283D412yes—
58C2×C4×D283—21no—
59#17985381—24no—
60#17985380—24no—
61#17985379—24no—
62D2264—24no—
63#17985377—24no—
64#17985377—24no—
65#17985376D414yescomplemented
66C2×C283⋊C8C2312yes—
67Dic283⋊C4—22no—
68C22⋊Dic283—22no—
69(C2×D283)⋊C4—22no—
70C4×Dic283D412yes—
71C4×Dic283C2311yescharacteristic
72C4×Dic283C2311yescharacteristic
73C24×C283—21no—
74#41×C283—22no—
75#41×C283—22no—
76C2×Q8×C283C2311yescharacteristic
77C2×D4×C283—21no—
78C2×D4×C283—22no—
79C2×D4×C283—22no—
80C2×D4×C283—21no—
81C2×D4×C283C2311yescharacteristic
82C2×D4×C283C2312yes—
83C22×C4×C283—21no—
84C22×C4×C283C2311yescharacteristic
85SD16×C283—24no—
86D8×C283—24no—
87M16×C283D414yescomplemented
88C2×C8×C283C2312yes—
89C22×D283—21no—
90C22×D283—22no—
91C22×D283—44no—
92C22×D283—42no—
93Dic566—21no—
94Dic566—21no—
95D1132—21no—
96D1132—22no—
97D1132—21no—
98D1132—22no—
99D1132—42no—
100C22⋊D283—48no—
101C22⋊D283—42no—
102C22⋊D283—44no—
103C22⋊D283—48no—
104C2×Dic283—24no—
105C2×Dic283—21no—
106C2×Dic283C2×D412yes—
107C2×Dic283C2×D412yes—
108C4×D283—42no—
109C283⋊C8C2×D414yescomplemented
110C23×C283—21no—
111C23×C283—22no—
112C23×C283—44no—
113C23×C283—42no—
114C23×C283C2×D411yescharacteristic
115Q8×C283—21no—
116Q8×C283—21no—
117D4×C283—21no—
118D4×C283—22no—
119D4×C283—22no—
120D4×C283—21no—
121D4×C283—42no—
122D4×C283—42no—
123C2×C4×C283—21no—
124C2×C4×C283—42no—
125C2×C4×C283C2×D411yescharacteristic
126C2×C4×C283C2×D412yes—
127C2×C4×C283C2×D412yes—
128C2×C4×C283C2411yescharacteristic
129C8×C283C2×D414yescomplemented
130C22×C283—22no—
131C22×C283—41no—
132C22×C283—42no—
133C22×C283—44no—
134C22×C283—42no—
135C22×C283—84no—
136C22×C283#12712yes—
137C22×C283C22×D411yescharacteristic
138C2×D283—41no—
139C2×D283—42no—
140C2×D283—44no—
141C2×D283—42no—
142C2×D283—84no—
143C4×C283—41no—
144C4×C283#12712yes—
145C4×C283C22×D411yescharacteristic
146C4×C283C22×D411yescharacteristic
147Dic283—22no—
148Dic283—41no—
149Dic283AGL(1,ℤ/8)14yescomplemented
150C2×C283—22no—
151C2×C283—81no—
152C2×C283—82no—
153C2×C283C2×#12711yescharacteristic
154C2×C283C2×AGL(1,ℤ/8)12yes—
155D283—81no—
156D283—82no—
157C283#315111yescharacteristic, complemented, Sylow
158#3151—2831nomaximal, Sylow
159C2×AGL(1,ℤ/8)—2831no—
160C2×AGL(1,ℤ/8)—2831no—
161C2×AGL(1,ℤ/8)—2831no—
162C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
163#532—2831no—
164C2×#127—2831no—
165#496—2832no—
166#496—2832no—
167#493—2832no—
168#492—2832no—
169C4×M16—2831no—
170C2×#41—2831no—
171C2×#41C2×D28311yescharacteristic
172C22×D4—2831no—
173C22×D4—5662no—
174C22×D4C2×D28311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5664no—
177AGL(1,ℤ/8)—5664no—
178AGL(1,ℤ/8)—5664no—
179C2×SD16—2832no—
180C2×SD16—2832no—
181C2×SD16—2832no—
182C2×SD16C2×D28312yescomplemented
183C2×D8—2832no—
184C2×D8—2832no—
185C2×D8—2832no—
186C2×D8C2×D28312yescomplemented
187C2×M16—2831no—
188C2×M16C2×D28311yescharacteristic
189C4⋊Q8—2831no—
190#127—2832no—
191#127—2832no—
192#127—2831no—
193#127—5662no—
194#124—2831no—
195#124—2831no—
196#121—5662no—
197C4×D4—5662no—
198C2×C42—2831no—
199#97—2832no—
200C4×C8—2832no—
201C24—21no—
202C24—5661no—
203#41—22no—
204#41—22no—
205#41—5662no—
206#41—5662no—
207C2×Q8—2831no—
208C2×Q8C22×D28311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2831no—
214C2×D4—2832no—
215C2×D4—5662no—
216C2×D4—5664no—
217C2×D4—5664no—
218C2×D4—5661no—
219C2×D4—5662no—
220C2×D4—5661no—
221C2×D4—5662no—
222C2×D4—11328no—
223C2×D4C22×D28311yescharacteristic
224C2×D4C22×D28312yes—
225C22×C4—21no—
226C22×C4—2832no—
227C22×C4—5661no—
228C22×C4C22×D28311yescharacteristic
229SD16—24no—
230SD16—5664no—
231SD16—5664no—
232SD16—5664no—
233D8—24no—
234D8—5664no—
235D8—5664no—
236D8—5664no—
237M16—2834no—
238M16C22⋊D28314yescomplemented
239C2×C8—2832no—
240C2×C8C22×D28312yes—
241C4⋊C4—5662no—
242C22⋊C4—5662no—
243C22⋊C4—5662no—
244C42—2832no—
245C42—2831no—
246C42—2831no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—5661no—
252C23—5662no—
253C23—11324no—
254C23—11322no—
255C23D4×D28311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—5661no—
259Q8—5661no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—5661no—
267D4—5662no—
268D4—5661no—
269D4—5662no—
270D4—11328no—
271D4—11324no—
272D4—11328no—
273D4—11322no—
274D4—11322no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2832no—
278C2×C4—2832no—
279C2×C4—5664no—
280C2×C4—5661no—
281C2×C4—11322no—
282C2×C4D4×D28311yescharacteristic
283C2×C4D4×D28312yes—
284C2×C4C2×C22⋊D28312yes—
285C2×C4C23×D28311yescharacteristic
286C8—2834no—
287C8C2×C22⋊D28314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—11324no—
295C22—11322no—
296C22—11321no—
297C22—11322no—
298C22—22644no—
299C22#2258360612yes—
300C22C2×D4×D28311yescharacteristic
301C4—41no—
302C4—2834no—
303C4—5662no—
304C4—11321no—
305C4#2258360612yes—
306C4C2×D4×D28311yescharacteristic
307C4C22×C22⋊D28311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—22642no—
312C2—22641no—
313C2C2×#2258360611yescharacteristic
314C2AGL(1,ℤ/8)×D28312yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3151
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583638C2semidirect
2. C2×#22583636C2semidirect
3. C2×#22583635C2semidirect
4. #31827011C2semidirect
5. C2×#22583606C2semidirect
6. #31826853C2semidirect
7. #31826851C2semidirect
8. #31826840C2semidirect
9. #31826836C2semidirect
10. M16×Dic283C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
18. C2×#17985381C22semidirect
19. C2×#17985380C22semidirect
20. C2×#17985379C22semidirect
21. C2×D2264C22semidirect
22. C2×#17985377C22semidirect
23. C2×#17985377C22semidirect
25. #22583608C22semidirect
29. #22583606C22semidirect
30. #22583599C22semidirect
31. #22583597C22semidirect
35. #22583520C22semidirect
36. C8×Dic283C22semidirect
40. C2×SD16×C283C22semidirect
41. C2×D8×C283C22semidirect
65. #17985376D4semidirect
87. M16×C283D4semidirect
109. C283⋊C8C2×D4semidirect
129. C8×C283C2×D4semidirect
149. Dic283AGL(1,ℤ/8)semidirect
157. C283#3151semidirect
162. C2×AGL(1,ℤ/8)D283semidirect
182. C2×SD16C2×D283semidirect
186. C2×D8C2×D283semidirect
238. M16C22⋊D283semidirect
287. C8C2×C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836382
C2×#225836362
C2×#225836352
#318270112
C2×#225836062
#318268532
#318268512
#318268402
#318268362
M16×Dic2832
C2×AGL(1,ℤ/8)×C2832
#3151283