Finite Group Databasebeta

← #49131723

Subgroups — #49131723

Order 36224.

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

normal subgroups only

This group has 572 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×#22583640C211yesmaximal, characteristic, complemented
3C2×#22583636C211yesmaximal, characteristic, complemented
4#31827011C211yesmaximal, characteristic, complemented
5C2×#22583597C211yesmaximal, characteristic, complemented
6#31826856C212yesmaximal, complemented
7#31826853C212yesmaximal, complemented
8#31826848C212yesmaximal, complemented
9#31826846C212yesmaximal, complemented
10M16×Dic283C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×Dic566C2211yescharacteristic
14#22583643—24no—
15#22583640—24no—
16#22583636—24no—
17C2×Dic1132C2212yescomplemented
18C2×#17985382C2212yescomplemented
19C2×#17985381C2212yescomplemented
20C2×#17985380C2212yes—
21C2×#17985380C2212yescomplemented
22C2×#17985377C2212yes—
23C2×#17985376C2211yescharacteristic
24#22583608C2211yescharacteristic, complemented
25#22583606C2211yescharacteristic, complemented
26#22583599C2211yescharacteristic, complemented
27#22583597—22no—
28#22583597C2212yes—
29#22583597C2212yes—
30#22583597C2211yescharacteristic, complemented
31#22583582—22no—
32C4×C22⋊D283—22no—
33C2×C22⋊Dic283—22no—
34C2×C4×Dic283C2211yescharacteristic
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×Dic566C2312yes—
49C2×Dic566C2311yescharacteristic
50C2×D1132C2311yescharacteristic
51C2×C22⋊D283—22no—
52C2×C22⋊D283—21no—
53C22×Dic283—21no—
54C22×Dic283D412yes—
55C2×C4×D283—21no—
56Dic1132—24no—
57#17985382—24no—
58#17985381—24no—
59#17985380—24no—
60#17985380—24no—
61#17985377—24no—
62#17985376D414yescomplemented
63C2×C283⋊C8C2312yes—
64Dic283⋊C4—22no—
65C22⋊Dic283—24no—
66C22⋊Dic283—24no—
67C22⋊Dic283—22no—
68(C2×D283)⋊C4—22no—
69C4×Dic283D412yes—
70C4×Dic283C2311yescharacteristic
71C4×Dic283C2311yescharacteristic
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—21no—
90Dic566—22no—
91Dic566—22no—
92Dic566—21no—
93Dic566—42no—
94D1132—21no—
95D1132—21no—
96C22⋊D283—44no—
97C22⋊D283—42no—
98C2×Dic283—22no—
99C2×Dic283—21no—
100C2×Dic283—42no—
101C2×Dic283—42no—
102C2×Dic283C2×D412yes—
103C2×Dic283C2×D412yes—
104C4×D283—42no—
105C283⋊C8C2×D414yes—
106C23×C283—21no—
107C23×C283—22no—
108C23×C283—44no—
109C23×C283—42no—
110C23×C283C2×D411yescharacteristic
111Q8×C283—21no—
112Q8×C283—21no—
113D4×C283—21no—
114D4×C283—22no—
115D4×C283—22no—
116D4×C283—21no—
117D4×C283—42no—
118D4×C283—42no—
119C2×C4×C283—21no—
120C2×C4×C283—42no—
121C2×C4×C283C2×D411yescharacteristic
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2×D412yes—
124C2×C4×C283C2411yescharacteristic
125C8×C283C2×D414yes—
126C22×C283—22no—
127C22×C283—41no—
128C22×C283—42no—
129C22×C283—44no—
130C22×C283—42no—
131C22×C283—84no—
132C22×C283#12712yes—
133C22×C283C22×D411yescharacteristic
134C2×D283—41no—
135C2×D283—42no—
136C4×C283—41no—
137C4×C283#12712yes—
138C4×C283C22×D411yescharacteristic
139C4×C283C22×D411yescharacteristic
140Dic283—22no—
141Dic283—22no—
142Dic283—41no—
143Dic283—42no—
144C2×C283—22no—
145C2×C283—81no—
146C2×C283—82no—
147C2×C283C2×#12711yescharacteristic
148C2×C283#57512yes—
149D283—81no—
150C283#315311yescharacteristic, complemented, Sylow
151#3153—2831nomaximal, Sylow
152C2×#137—2831no—
153C2×#137—2831no—
154C2×AGL(1,ℤ/8)—2831no—
155C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
156#532—2831no—
157C2×#124—2831no—
158#497—2832no—
159#496—2832no—
160#495—2832no—
161#495—2832no—
162C4×M16—2831no—
163C2×#41—2831no—
164C2×#41C2×D28311yescharacteristic
165C22×Q8—2831no—
166C22×D4C2×D28311yescharacteristic
167#137—5664no—
168#137—5664no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—5664no—
171C2×Q16—2832no—
172C2×Q16—2832no—
173C2×SD16—2832no—
174C2×SD16—2832no—
175C2×SD16—2832no—
176C2×SD16C2×D28312yes—
177C2×D8—2832no—
178C2×D8C2×D28312yes—
179C2×M16—2831no—
180C2×M16C2×D28311yescharacteristic
181C4⋊Q8—2831no—
182#127—2831no—
183#124—2831no—
184#124—2832no—
185#124—2832no—
186#124—2831no—
187#124—5662no—
188#121—5662no—
189C4×D4—5662no—
190C2×C22⋊C4—5662no—
191C2×C42—2831no—
192#97—2832no—
193C4×C8—2832no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—5662no—
198#41—5662no—
199C2×Q8—2832no—
200C2×Q8—2831no—
201C2×Q8—5661no—
202C2×Q8—5662no—
203C2×Q8—5662no—
204C2×Q8C22×D28311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2831no—
210C2×D4—5662no—
211C2×D4—5661no—
212C2×D4C22×D28312yes—
213C2×D4C22×D28311yescharacteristic
214C22×C4—21no—
215C22×C4—2832no—
216C22×C4—5661no—
217C22×C4—5661no—
218C22×C4C22×D28311yescharacteristic
219Q16—5664no—
220Q16—5664no—
221SD16—24no—
222SD16—5664no—
223SD16—5664no—
224SD16—5664no—
225D8—24no—
226D8—5664no—
227M16—2834no—
228M16C22⋊D28314yescomplemented
229C2×C8—2832no—
230C2×C8C22×D28312yes—
231C4⋊C4—5662no—
232C22⋊C4—5662no—
233C22⋊C4—5664no—
234C22⋊C4—5664no—
235C22⋊C4—5662no—
236C42—2832no—
237C42—2831no—
238C42—2831no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—5661no—
244C23D4×D28311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—5661no—
248Q8—5662no—
249Q8—5662no—
250Q8—5661no—
251Q8—11322no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—5661no—
259D4—5661no—
260D4—11324no—
261D4—11322no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2832no—
265C2×C4—2832no—
266C2×C4—5662no—
267C2×C4—5661no—
268C2×C4—11322no—
269C2×C4—11322no—
270C2×C4—11322no—
271C2×C4D4×D28311yescharacteristic
272C2×C4D4×D28312yes—
273C2×C4C2×C22⋊D28312yes—
274C2×C4C23×D28311yescharacteristic
275C8—2834no—
276C8C2×C22⋊D28314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—11321no—
284C22—11322no—
285C22#2258360612yes—
286C22C2×D4×D28311yescharacteristic
287C4—41no—
288C4—5662no—
289C4—5662no—
290C4—11321no—
291C4—11322no—
292C4#2258360612yes—
293C4C2×D4×D28311yescharacteristic
294C4C22×C22⋊D28311yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—22641no—
299C2C2×#2258360611yescharacteristic
300C2#3182726112yes—
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#3153
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×#22583640C2semidirect
3. C2×#22583636C2semidirect
4. #31827011C2semidirect
5. C2×#22583597C2semidirect
6. #31826856C2semidirect
7. #31826853C2semidirect
8. #31826848C2semidirect
9. #31826846C2semidirect
10. M16×Dic283C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
17. C2×Dic1132C22semidirect
18. C2×#17985382C22semidirect
19. C2×#17985381C22semidirect
21. C2×#17985380C22semidirect
24. #22583608C22semidirect
25. #22583606C22semidirect
26. #22583599C22semidirect
30. #22583597C22semidirect
35. #22583520C22semidirect
36. C8×Dic283C22semidirect
62. #17985376D4semidirect
86. M16×C283D4semidirect
150. C283#3153semidirect
155. C2×AGL(1,ℤ/8)D283semidirect
228. 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×#225836402
C2×#225836362
#318270112
C2×#225835972
#318268562
#318268532
#318268482
#318268462
M16×Dic2832
C2×AGL(1,ℤ/8)×C2832
#3153283