Finite Group Databasebeta

← #49131746

Subgroups — #49131746

Order 36224.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22583630C211yesmaximal, characteristic, complemented
3C22×#17985377C211yesmaximal, characteristic, complemented
4#31827032C211yesmaximal, characteristic, complemented
5#31827029C211yesmaximal, characteristic, complemented
6#31826853C212yesmaximal, complemented
7#31826851C212yesmaximal, complemented
8#31826846C212yesmaximal, complemented
9#31826840C212yesmaximal, complemented
10#31826423C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×D1132C2211yescharacteristic
14#22583638—24no—
15#22583630—24no—
16C2×#17985382C2211yescharacteristic, complemented
17C2×#17985381C2212yescomplemented
18C2×#17985380C2211yescharacteristic, complemented
19C2×#17985379C2211yescharacteristic, complemented
20C2×D2264C2212yescomplemented
21C2×#17985377—24no—
22C2×#17985377C2212yes—
23C2×#17985377C2212yes—
24C2×#17985377C2211yescharacteristic, complemented
25C22×C283⋊C8C2211yescharacteristic
26#22583606C2212yescomplemented
27#22583606C2212yescomplemented
28#22583599C2212yescomplemented
29#22583597C2212yescomplemented
30#22583584—22no—
31#22583581—22no—
32#22583579—22no—
33#22583578—22no—
34#22583569C2211yescharacteristic
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×D1132C2312yes—
51C2×D1132C2311yescharacteristic
52C2×C22⋊D283—24no—
53C2×C22⋊D283—24no—
54C2×C22⋊D283—21no—
55C2×C4×D283—21no—
56#17985382—22no—
57#17985381—24no—
58#17985380—22no—
59#17985379—22no—
60D2264—24no—
61#17985377—24no—
62#17985377—24no—
63#17985377—22no—
64C2×C283⋊C8D414yescomplemented
65C2×C283⋊C8C2311yescharacteristic
66C2×C283⋊C8C2311yescharacteristic
67C4⋊Dic283D412yes—
68C22⋊Dic283—24no—
69C22⋊Dic283D412yes—
70(C2×D283)⋊C4—24no—
71C4×Dic283C2312yes—
72C24×C283—21no—
73#41×C283—22no—
74#41×C283—22no—
75C2×Q8×C283C2311yescharacteristic
76C2×D4×C283—21no—
77C2×D4×C283—22no—
78C2×D4×C283—22no—
79C2×D4×C283—21no—
80C2×D4×C283C2312yes—
81C2×D4×C283C2311yescharacteristic
82C22×C4×C283—21no—
83C22×C4×C283C2311yescharacteristic
84SD16×C283—24no—
85D8×C283—24no—
86M16×C283D414yescomplemented
87C2×C8×C283C2312yes—
88C22×D283—22no—
89C22×D283—21no—
90C22×D283—42no—
91C22×D283—82no—
92Dic566—21no—
93Dic566—21no—
94D1132—22no—
95D1132—21no—
96D1132—21no—
97D1132—22no—
98D1132—42no—
99C22⋊D283—42no—
100C22⋊D283—84no—
101C22⋊D283—84no—
102C2×Dic283—21no—
103C2×Dic283C2×D414yes—
104C4×D283—42no—
105C283⋊C8C2×D412yes—
106C283⋊C8C2×D412yes—
107C23×C283—21no—
108C23×C283—22no—
109C23×C283—42no—
110C23×C283—82no—
111C23×C283C2×D411yescharacteristic
112Q8×C283—21no—
113Q8×C283—21no—
114D4×C283—21no—
115D4×C283—22no—
116D4×C283—22no—
117D4×C283—21no—
118D4×C283—42no—
119D4×C283—42no—
120C2×C4×C283—21no—
121C2×C4×C283—42no—
122C2×C4×C283C2×D411yescharacteristic
123C2×C4×C283C2×D412yes—
124C2×C4×C283C2×D412yes—
125C2×C4×C283C2411yescharacteristic
126C8×C283C2×D414yes—
127C22×C283—22no—
128C22×C283—41no—
129C22×C283—42no—
130C22×C283—44no—
131C22×C283—42no—
132C22×C283—84no—
133C22×C283#12712yes—
134C22×C283C22×D411yescharacteristic
135C2×D283—41no—
136C2×D283—42no—
137C2×D283—42no—
138C2×D283—44no—
139C2×D283—84no—
140C4×C283—41no—
141C4×C283#12712yes—
142C4×C283C22×D411yescharacteristic
143C4×C283C22×D411yescharacteristic
144Dic283—24no—
145Dic283—41no—
146C2×C283—22no—
147C2×C283—81no—
148C2×C283—82no—
149C2×C283C2×#12711yescharacteristic
150C2×C283#57612yes—
151D283—81no—
152D283—82no—
153C283#315511yescharacteristic, complemented, Sylow
154#3155—2831nomaximal, Sylow
155C2×AGL(1,ℤ/8)—2831no—
156C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
157C2×#135—2831no—
158C22×D8—2831no—
159#535—2831no—
160#535—2831no—
161#496—2832no—
162#496—2832no—
163#495—2832no—
164#493—2832no—
165#405—2831no—
166C2×#41—2831no—
167C2×#41C2×D28311yescharacteristic
168C22×D4—2831no—
169C22×D4C2×D28311yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—5664no—
172#135—5664no—
173C2×Q16—2831no—
174C2×SD16—2831no—
175C2×SD16—2831no—
176C2×SD16—2832no—
177C2×SD16C2×D28312yescomplemented
178C2×D8—2832no—
179C2×D8—2832no—
180C2×D8—2831no—
181C2×D8—2832no—
182C2×D8—5664no—
183C2×D8C2×D28312yescomplemented
184C2×M16C2×D28311yescharacteristic
185C22×C8—2831no—
186#127—2832no—
187#127—2832no—
188#124—2832no—
189#124—2832no—
190#121—5662no—
191#121—5662no—
192#120—5662no—
193#120—5662no—
194#117—2831no—
195#97—2832no—
196C4×C8—2832no—
197C24—21no—
198C24—5661no—
199#41—22no—
200#41—22no—
201#41—5662no—
202#41—5662no—
203C2×Q8—2831no—
204C2×Q8C22×D28311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2832no—
210C2×D4—2831no—
211C2×D4—5664no—
212C2×D4—5664no—
213C2×D4—5661no—
214C2×D4—5662no—
215C2×D4—5661no—
216C2×D4—5662no—
217C2×D4C22×D28312yes—
218C2×D4C22×D28311yescharacteristic
219C22×C4—21no—
220C22×C4—5661no—
221C22×C4C22×D28311yescharacteristic
222Q16—5662no—
223SD16—24no—
224SD16—5662no—
225SD16—5662no—
226SD16—5664no—
227D8—24no—
228D8—5664no—
229D8—5664no—
230D8—5662no—
231D8—5664no—
232M16C22⋊D28314yescomplemented
233C2×C8—2834no—
234C2×C8—2831no—
235C2×C8—2831no—
236C2×C8C22×D28312yes—
237C4⋊C4—2832no—
238C22⋊C4—2832no—
239C22⋊C4—5664no—
240C22⋊C4—5664no—
241C42—2832no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—5662no—
247C23—5661no—
248C23—11322no—
249C23—22642no—
250C23D4×D28311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—5661no—
254Q8—5661no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—5662no—
262D4—5661no—
263D4—5661no—
264D4—5662no—
265D4—11322no—
266D4—11322no—
267D4—22644no—
268D4—22644no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2834no—
272C2×C4—5661no—
273C2×C4—11322no—
274C2×C4D4×D28311yescharacteristic
275C2×C4D4×D28312yes—
276C2×C4C2×C22⋊D28312yes—
277C2×C4C23×D28311yescharacteristic
278C8—2832no—
279C8—2832no—
280C8C2×C22⋊D28314yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—11321no—
288C22—11322no—
289C22—11322no—
290C22—11324no—
291C22—22644no—
292C22#2258360612yes—
293C22C2×D4×D28311yescharacteristic
294C4—41no—
295C4—5664no—
296C4—11321no—
297C4#2258360612yes—
298C4C2×D4×D28311yescharacteristic
299C4C22×C22⋊D28311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—22642no—
304C2—22641no—
305C2C2×#2258360611yescharacteristic
306C2#3182726912yes—
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#3155
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×#22583630C2semidirect
3. C22×#17985377C2semidirect
4. #31827032C2semidirect
5. #31827029C2semidirect
6. #31826853C2semidirect
7. #31826851C2semidirect
8. #31826846C2semidirect
9. #31826840C2semidirect
10. #31826423C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
16. C2×#17985382C22semidirect
17. C2×#17985381C22semidirect
18. C2×#17985380C22semidirect
19. C2×#17985379C22semidirect
20. C2×D2264C22semidirect
24. C2×#17985377C22semidirect
26. #22583606C22semidirect
27. #22583606C22semidirect
28. #22583599C22semidirect
29. #22583597C22semidirect
35. #22583520C22semidirect
36. C8×Dic283C22semidirect
40. C2×SD16×C283C22semidirect
41. C2×D8×C283C22semidirect
64. C2×C283⋊C8D4semidirect
86. M16×C283D4semidirect
153. C283#3155semidirect
156. C2×AGL(1,ℤ/8)D283semidirect
177. C2×SD16C2×D283semidirect
183. C2×D8C2×D283semidirect
232. M16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836382
C2×#225836302
C22×#179853772
#318270322
#318270292
#318268532
#318268512
#318268462
#318268402
#318264232
C2×AGL(1,ℤ/8)×C2832
#3155283