Finite Group Databasebeta

← #49131735

Subgroups — #49131735

Order 36224.

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

normal subgroups only

This group has 564 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×#22583643C211yesmaximal, characteristic, complemented
2C2×#22583630C211yesmaximal, characteristic, complemented
3C22×#17985380C211yesmaximal, characteristic, complemented
4#31827036C211yesmaximal, characteristic, complemented
5#31827032C211yesmaximal, characteristic, complemented
6#31826856C212yesmaximal, complemented
7#31826853C212yesmaximal, complemented
8#31826848C212yesmaximal, complemented
9#31826836C212yesmaximal, complemented
10#31826423C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×Dic566C2211yescharacteristic
14#22583643—24no—
15#22583630—24no—
16C2×Dic1132C2212yescomplemented
17C2×#17985382C2211yescharacteristic, complemented
18C2×#17985381C2212yescomplemented
19C2×#17985380—24no—
20C2×#17985380C2212yes—
21C2×#17985380C2212yes—
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic, complemented
24C2×#17985377C2211yescharacteristic, complemented
25C22×C283⋊C8C2211yescharacteristic
26#22583608C2212yescomplemented
27#22583606C2212yes—
28#22583597C2212yes—
29#22583597C2212yescomplemented
30#22583591—22no—
31#22583589—22no—
32#22583581—22no—
33#22583579—22no—
34#22583569C2211yescharacteristic
35#22583520C2212yescomplemented
36C8×Dic283C2212yescomplemented
37C2×#41×C283C2211yescharacteristic
38C22×D4×C283C2211yescharacteristic
39AGL(1,ℤ/8)×C283—24no—
40C2×SD16×C283C2212yes—
41C2×D8×C283C2212yes—
42C2×M16×C283C2211yescharacteristic
43#17985392—22no—
44#17985392—22no—
45C2×Dic566—21no—
46C2×Dic566—22no—
47C2×Dic566—22no—
48C2×Dic566C2311yescharacteristic
49C2×Dic566C2312yes—
50C2×D1132C2311yescharacteristic
51C2×C22⋊D283—24no—
52C2×C22⋊D283—21no—
53C22×Dic283—21no—
54C2×C4×D283—21no—
55Dic1132—24no—
56#17985382—22no—
57#17985381—24no—
58#17985380—24no—
59#17985380—24no—
60#17985380—22no—
61#17985379—22no—
62#17985377—22no—
63C2×C283⋊C8D414yescomplemented
64C2×C283⋊C8C2311yescharacteristic
65C2×C283⋊C8C2311yescharacteristic
66C4⋊Dic283D412yes—
67Dic283⋊C4—24no—
68C22⋊Dic283—24no—
69C22⋊Dic283—24no—
70C22⋊Dic283D412yes—
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—21no—
89Dic566—22no—
90Dic566—21no—
91Dic566—22no—
92Dic566—21no—
93Dic566—42no—
94D1132—21no—
95D1132—21no—
96C22⋊D283—42no—
97C22⋊D283—84no—
98C2×Dic283—21no—
99C2×Dic283—22no—
100C2×Dic283—42no—
101C2×Dic283C2×D414yes—
102C4×D283—42no—
103C283⋊C8C2×D412yes—
104C283⋊C8C2×D412yes—
105C23×C283—21no—
106C23×C283—22no—
107C23×C283—42no—
108C23×C283—82no—
109C23×C283C2×D411yescharacteristic
110Q8×C283—21no—
111Q8×C283—21no—
112D4×C283—21no—
113D4×C283—22no—
114D4×C283—22no—
115D4×C283—21no—
116D4×C283—42no—
117D4×C283—42no—
118C2×C4×C283—21no—
119C2×C4×C283—42no—
120C2×C4×C283C2×D411yescharacteristic
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2411yescharacteristic
124C8×C283C2×D414yes—
125C22×C283—22no—
126C22×C283—41no—
127C22×C283—42no—
128C22×C283—42no—
129C22×C283—44no—
130C22×C283—84no—
131C22×C283#12712yes—
132C22×C283C22×D411yescharacteristic
133C2×D283—41no—
134C2×D283—42no—
135C4×C283—41no—
136C4×C283#12712yes—
137C4×C283C22×D411yescharacteristic
138C4×C283C22×D411yescharacteristic
139Dic283—24no—
140Dic283—41no—
141Dic283—42no—
142C2×C283—22no—
143C2×C283—81no—
144C2×C283—82no—
145C2×C283C2×#12711yescharacteristic
146C2×C283#57712yes—
147D283—81no—
148C283#315411yescharacteristic, complemented, Sylow
149#3154—2831nomaximal, Sylow
150C2×#137—2831no—
151C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
152C2×#135—2831no—
153C22×SD16—2831no—
154#536—2831no—
155#535—2831no—
156#497—2832no—
157#496—2832no—
158#495—2832no—
159#492—2832no—
160#405—2831no—
161C2×#41—2831no—
162C2×#41C2×D28311yescharacteristic
163C22×Q8—2831no—
164C22×D4C2×D28311yescharacteristic
165#137—5664no—
166AGL(1,ℤ/8)—24no—
167#135—5664no—
168C2×Q16—2831no—
169C2×Q16—2832no—
170C2×SD16—2831no—
171C2×SD16—2832no—
172C2×SD16—2832no—
173C2×SD16—2831no—
174C2×SD16—2832no—
175C2×SD16—5664no—
176C2×SD16C2×D28312yes—
177C2×D8—2831no—
178C2×D8C2×D28312yes—
179C2×M16C2×D28311yescharacteristic
180C22×C8—2831no—
181C4⋊Q8—2832no—
182#127—2832no—
183#124—2832no—
184#124—2832no—
185#123—5662no—
186C22⋊Q8—5662no—
187#121—5662no—
188#120—5662no—
189#117—2831no—
190#97—2832no—
191C4×C8—2832no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—5662no—
196#41—5662no—
197C2×Q8—2831no—
198C2×Q8—2832no—
199C2×Q8—5661no—
200C2×Q8—5662no—
201C2×Q8—5662no—
202C2×Q8C22×D28311yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—2831no—
208C2×D4—5664no—
209C2×D4—5661no—
210C2×D4C22×D28311yescharacteristic
211C2×D4C22×D28312yes—
212C22×C4—21no—
213C22×C4—5661no—
214C22×C4—5661no—
215C22×C4C22×D28311yescharacteristic
216Q16—5662no—
217Q16—5664no—
218SD16—24no—
219SD16—5664no—
220SD16—5664no—
221SD16—5662no—
222SD16—5662no—
223SD16—5664no—
224D8—24no—
225D8—5662no—
226M16C22⋊D28314yescomplemented
227C2×C8—2834no—
228C2×C8—2831no—
229C2×C8—2831no—
230C2×C8C22×D28312yes—
231C4⋊C4—2832no—
232C4⋊C4—5664no—
233C22⋊C4—2832no—
234C22⋊C4—5664no—
235C22⋊C4—5664no—
236C42—2832no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—5661no—
242C23D4×D28311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—5662no—
246Q8—5661no—
247Q8—5662no—
248Q8—5661no—
249Q8—11322no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—5661no—
257D4—5661no—
258D4—11322no—
259D4—22644no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—2834no—
263C2×C4—5661no—
264C2×C4—5662no—
265C2×C4—11322no—
266C2×C4—11322no—
267C2×C4D4×D28311yescharacteristic
268C2×C4D4×D28312yes—
269C2×C4C2×C22⋊D28312yes—
270C2×C4C23×D28311yescharacteristic
271C8—2832no—
272C8—2832no—
273C8C2×C22⋊D28314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—11321no—
281C22—11322no—
282C22#2258360612yes—
283C22C2×D4×D28311yescharacteristic
284C4—41no—
285C4—5664no—
286C4—11321no—
287C4—11322no—
288C4#2258360612yes—
289C4C2×D4×D28311yescharacteristic
290C4C22×C22⋊D28311yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—22641no—
295C2C2×#2258360611yescharacteristic
296C2#3182727512yes—
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#3154
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×#22583643C2semidirect
2. C2×#22583630C2semidirect
3. C22×#17985380C2semidirect
4. #31827036C2semidirect
5. #31827032C2semidirect
6. #31826856C2semidirect
7. #31826853C2semidirect
8. #31826848C2semidirect
9. #31826836C2semidirect
10. #31826423C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
16. C2×Dic1132C22semidirect
17. C2×#17985382C22semidirect
18. C2×#17985381C22semidirect
22. C2×#17985380C22semidirect
23. C2×#17985379C22semidirect
24. C2×#17985377C22semidirect
26. #22583608C22semidirect
29. #22583597C22semidirect
35. #22583520C22semidirect
36. C8×Dic283C22semidirect
63. C2×C283⋊C8D4semidirect
86. M16×C283D4semidirect
148. C283#3154semidirect
151. C2×AGL(1,ℤ/8)D283semidirect
226. M16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836432
C2×#225836302
C22×#179853802
#318270362
#318270322
#318268562
#318268532
#318268482
#318268362
#318264232
C2×AGL(1,ℤ/8)×C2832
#3154283