Finite Group Databasebeta

← #49131724

Subgroups — #49131724

Order 36224.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22583642C212yesmaximal, complemented
2C2×#22583637C212yesmaximal, complemented
3#31827016C211yesmaximal, characteristic, complemented
4#124×D283C211yesmaximal, characteristic, complemented
5#31826857C211yesmaximal, characteristic, complemented
6#31826855C211yesmaximal, characteristic, complemented
7#31826854C211yesmaximal, characteristic, complemented
8#31826850C212yesmaximal, complemented
9#31826846C212yesmaximal, complemented
10#31826417C211yesmaximal, characteristic, complemented
11#497×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2212yes—
13C2×Q8×D283C2211yescharacteristic
14C2×D4×D283C2211yescharacteristic
15#22583642—28no—
16#22583637—28no—
17C2×#17985382C2212yescomplemented
18C2×#17985381C2212yescomplemented
19C2×#17985379C2212yes—
20C2×#17985379C2212yescomplemented
21C2×D2264C2212yes—
22C2×#17985377C2212yes—
23C2×#17985375C2212yes—
24#22583607C2211yescharacteristic, complemented
25#22583599C2211yescharacteristic
26#22583599C2212yescomplemented
27#22583598C2211yescharacteristic
28#22583597C2211yescharacteristic
29#22583596—22no—
30#22583585—22no—
31#22583572—22no—
32C22⋊C4×D283—22no—
33C42×D283C2211yescharacteristic
34#22583521C2211yescharacteristic, complemented
35C8×Dic283C2212yescomplemented
36C2×Q16×C283C2212yescomplemented
37C2×SD16×C283C2212yes—
38C4⋊Q8×C283C2211yescharacteristic, complemented
39#124×C283C2211yescharacteristic
40#97×C283C2211yescharacteristic, complemented
41C23×D283—21no—
42#17985393—24no—
43#17985393—24no—
44C2×Dic566—21no—
45C2×Dic566C2311yescharacteristic
46Q8×D283—24no—
47C2×D1132—22no—
48C2×D1132—22no—
49C2×D1132C2311yescharacteristic
50C2×D1132C2312yes—
51C2×C22⋊D283—21no—
52D4×D283—24no—
53C2×C4×D283—21no—
54C2×C4×D283—22no—
55C2×C4×D283D412yes—
56C2×C4×D283C2311yescharacteristic
57#17985382—24no—
58#17985381—24no—
59#17985379—24no—
60#17985379—24no—
61D2264—24no—
62#17985377—24no—
63#17985375D418yescomplemented
64C2×C283⋊C8C2312yes—
65C22⋊Dic283—22no—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—24no—
68(C2×D283)⋊C4—22no—
69C4×Dic283D412yes—
70C4×Dic283C2311yescharacteristic
71C2×Q8×C283C2311yescharacteristic
72C2×Q8×C283C2312yes—
73C2×D4×C283C2311yescharacteristic
74Q16×C283—24no—
75SD16×C283—24no—
76C2×C8×C283C2312yes—
77C4⋊C4×C283—22no—
78C22⋊C4×C283—22no—
79C42×C283C2311yescharacteristic
80C22×D283—21no—
81C22×D283—22no—
82C22×D283—44no—
83C22×D283—42no—
84C22×D283C2×D411yescharacteristic
85Dic566—22no—
86Dic566—42no—
87D1132—22no—
88D1132—22no—
89D1132—22no—
90D1132—44no—
91D1132—44no—
92C22⋊D283—42no—
93C2×Dic283—21no—
94C2×Dic283C2×D412yes—
95C2×Dic283C2×D411yescharacteristic
96C4×D283—42no—
97C4×D283—42no—
98C4×D283—44no—
99C4×D283C2×D414yes—
100C283⋊C8C2×D414yes—
101C23×C283—21no—
102Q8×C283—22no—
103Q8×C283—22no—
104Q8×C283—22no—
105D4×C283—22no—
106C2×C4×C283—21no—
107C2×C4×C283—22no—
108C2×C4×C283C2×D412yes—
109C2×C4×C283C2411yescharacteristic
110C8×C283C2×D414yes—
111C22×C283—41no—
112C22×C283—42no—
113C22×C283C22×D411yescharacteristic
114C2×D283—22no—
115C2×D283—42no—
116C2×D283—41no—
117C2×D283—44no—
118C2×D283—42no—
119C2×D283—84no—
120C2×D283#12712yes—
121C4×C283—22no—
122C4×C283—42no—
123C4×C283—41no—
124C4×C283C22×D412yes—
125Dic283—22no—
126Dic283—41no—
127Dic283#12712yes—
128C2×C283—81no—
129C2×C283C2×#12711yescharacteristic
130C2×C283#57512yes—
131D283—22no—
132D283—82no—
133D283—81no—
134C283#315311yescharacteristic, complemented, Sylow
135#3153—2831nomaximal, Sylow
136C2×#137—2832no—
137C2×AGL(1,ℤ/8)—2832no—
138#532—2831no—
139C2×#124—2831no—
140#497—2831no—
141#497D28311yescharacteristic, complemented
142#496—2831no—
143#496—2831no—
144#495—2832no—
145#495—2832no—
146C4×M16—2831no—
147C2×#41—2832no—
148C22×Q8—2831no—
149C22×D4—2831no—
150#137—5668no—
151AGL(1,ℤ/8)—5668no—
152C2×Q16—2832no—
153C2×Q16C2×D28312yescomplemented
154C2×SD16—2832no—
155C2×SD16—2832no—
156C2×SD16—2832no—
157C2×SD16C2×D28312yes—
158C2×D8—2832no—
159C2×D8—2832no—
160C2×M16—2832no—
161C4⋊Q8C2×D28311yescharacteristic, complemented
162#127—2831no—
163#124—2831no—
164#124—2832no—
165#124—2831no—
166#124—2831no—
167#124—5662no—
168#124C2×D28311yescharacteristic
169#121—5662no—
170C4×D4—5662no—
171C2×C22⋊C4—5662no—
172C2×C42—2831no—
173#97—2831no—
174#97C2×D28311yescharacteristic, complemented
175C4×C8—2832no—
176C24—5661no—
177#41—5664no—
178#41—5664no—
179C2×Q8—2831no—
180C2×Q8—5661no—
181C2×Q8—5664no—
182C2×Q8C22×D28311yescharacteristic
183C2×Q8C22×D28312yes—
184C2×D4—2831no—
185C2×D4—2832no—
186C2×D4—5662no—
187C2×D4—5664no—
188C2×D4—5661no—
189C2×D4—5662no—
190C2×D4C22×D28311yescharacteristic
191C22×C4—2831no—
192C22×C4—2832no—
193C22×C4—5662no—
194C22×C4—5661no—
195Q16—24no—
196Q16—5664no—
197SD16—24no—
198SD16—5664no—
199SD16—5664no—
200SD16—5664no—
201D8—5664no—
202D8—5664no—
203M16—2838no—
204C2×C8—2832no—
205C2×C8C22×D28312yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—5662no—
209C22⋊C4—5664no—
210C22⋊C4—5662no—
211C22⋊C4—5662no—
212C42—2832no—
213C42—2831no—
214C42C22×D28311yescharacteristic
215C23—21no—
216C23—2831no—
217C23—5661no—
218C23—5662no—
219C23—11322no—
220C23—11324no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—5662no—
225Q8—11322no—
226D4—22no—
227D4—5662no—
228D4—5662no—
229D4—5662no—
230D4—11324no—
231D4—11324no—
232D4—11322no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—2831no—
236C2×C4—2834no—
237C2×C4—2832no—
238C2×C4—5661no—
239C2×C4—11322no—
240C2×C4—11324no—
241C2×C4—11322no—
242C2×C4D4×D28312yes—
243C2×C4C23×D28311yescharacteristic
244C8—2834no—
245C8D4×D28314yes—
246C22—41no—
247C22—42no—
248C22—2832no—
249C22—5662no—
250C22—11322no—
251C22—11321no—
252C22—11324no—
253C22—11322no—
254C22—22644no—
255C22C2×D4×D28311yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—2832no—
260C4—5662no—
261C4—11321no—
262C4C2×D4×D28312yes—
263C2—81no—
264C2—5662no—
265C2—22641no—
266C2—22642no—
267C2#127×D28311yescharacteristic
268C2#3182726412yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#497×C283C2
socleC22×C283C22×D4
hypercenter#497D283
hypocenterC283#3153
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22583642C2semidirect
2. C2×#22583637C2semidirect
3. #31827016C2semidirect
4. #124×D283C2semidirect
5. #31826857C2semidirect
6. #31826855C2semidirect
7. #31826854C2semidirect
8. #31826850C2semidirect
9. #31826846C2semidirect
10. #31826417C2semidirect
11. #497×C283C2semidirect
17. C2×#17985382C22semidirect
18. C2×#17985381C22semidirect
20. C2×#17985379C22semidirect
24. #22583607C22semidirect
26. #22583599C22semidirect
34. #22583521C22semidirect
35. C8×Dic283C22semidirect
36. C2×Q16×C283C22semidirect
38. C4⋊Q8×C283C22semidirect
40. #97×C283C22semidirect
63. #17985375D4semidirect
134. C283#3153semidirect
141. #497D283semidirect
153. C2×Q16C2×D283semidirect
161. C4⋊Q8C2×D283semidirect
174. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836422
C2×#225836372
#318270162
#124×D2832
#318268572
#318268552
#318268542
#318268502
#318268462
#318264172
#497×C2832
#3153283