Finite Group Databasebeta

← #49130635

Subgroups — #49130635

Order 36224.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#31827301C211yesmaximal, characteristic, complemented
2C2×#22583638C212yesmaximal, complemented
3#31827029C211yesmaximal, characteristic, complemented
4#31826640C214yesmaximal, complemented
5#31826635C214yesmaximal, complemented
6#31826481C212yesmaximal, complemented
7#413×C283C211yesmaximal, characteristic, complemented
8#22583658C2212yescomplemented
9#22583657—22no—
10#22583656—22no—
11#22583655—22no—
12#22583655C2212yescomplemented
13C2×#17985392—22no—
14C2×#17985392C2212yes—
15#41×D283—24no—
16C22×D1132C2211yescharacteristic, complemented
17#22583638—24no—
18#22583638C2218yescomplemented
19C2×#17985381—22no—
20C2×D2264—22no—
21#22583606C2212yescomplemented
22#22583599C2212yescomplemented
23#22583584—22no—
24#22583578—22no—
25#22583569C2211yescharacteristic, complemented
26#22583536C2218yescomplemented
27C2×#41×C283C2211yescharacteristic
28C2×M16×C283C2212yescomplemented
29#101×C283C2212yescomplemented
30#100×C283C2212yescomplemented
31C23×D283—21no—
32#17985393—24no—
33#17985392—24no—
34#17985392—24no—
35#17985392—22no—
36#17985392—24no—
37#17985392—22no—
38#17985392D414yescomplemented
39#17985392C2314yes—
40#17985391—28no—
41#17985391—24no—
42C2×Dic566—22no—
43C2×Dic566D412yescomplemented
44Q8×D283—24no—
45C2×D1132—22no—
46C2×D1132—22no—
47C2×D1132—22no—
48C2×D1132—21no—
49C2×D1132D412yescomplemented
50C2×D1132C2312yes—
51C2×C22⋊D283—24no—
52C2×C22⋊D283—22no—
53C2×C22⋊D283—24no—
54D4×D283—28no—
55D4×D283—24no—
56C2×C4×D283—24no—
57C2×C4×D283—22no—
58#17985381—28no—
59D2264—28no—
60C4⋊Dic283—21no—
61C22⋊Dic283—21no—
62(C2×D283)⋊C4—24no—
63C4×Dic283C2312yes—
64#41×C283—22no—
65#41×C283—22no—
66C2×Q8×C283C2311yescharacteristic
67C2×D4×C283D412yes—
68C2×D4×C283C2311yescharacteristic
69C22×C4×C283D412yes—
70C22×C4×C283C2311yescharacteristic
71M16×C283—22no—
72M16×C283C2314yes—
73C2×C8×C283—22no—
74C22×D283—24no—
75C22×D283—22no—
76C22×D283—22no—
77C22×D283—42no—
78C22×D283—82no—
79Dic566—24no—
80Dic566—22no—
81Dic566C2×D414yes—
82D1132—24no—
83D1132—22no—
84D1132—24no—
85D1132—41no—
86D1132—41no—
87D1132C2×D414yes—
88C22⋊D283—28no—
89C22⋊D283—24no—
90C22⋊D283—24no—
91C22⋊D283—84no—
92C2×Dic283—24no—
93C2×Dic283—22no—
94C2×Dic283—22no—
95C4×D283—28no—
96C4×D283—24no—
97C4×D283—24no—
98C23×C283C2×D412yes—
99C23×C283C2×D411yescharacteristic
100Q8×C283—22no—
101D4×C283—24no—
102D4×C283—22no—
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283C2×D412yes—
106C2×C4×C283C2×D412yes—
107C2×C4×C283C2×D412yes—
108C2×C4×C283C2×D411yescharacteristic
109C2×C4×C283C2411yescharacteristic
110C8×C283—24no—
111C22×C283—22no—
112C22×C283—41no—
113C22×C283—42no—
114C22×C283#12012yes—
115C22×C283C22×D411yescharacteristic
116C2×D283—24no—
117C2×D283—44no—
118C2×D283—42no—
119C2×D283—82no—
120C2×D283—84no—
121C2×D283—84no—
122C4×C283—22no—
123C4×C283#12011yescharacteristic
124C4×C283#12011yescharacteristic
125C4×C283C22×D412yes—
126Dic283—24no—
127Dic283—42no—
128C2×C283—21no—
129C2×C283—22no—
130C2×C283—42no—
131C2×C283C2×#12011yescharacteristic
132D283—44no—
133D283—82no—
134C283#302111yescharacteristic, complemented, Sylow
135#3021—2831nomaximal, Sylow
136#585—2831no—
137C2×AGL(1,ℤ/8)—2832no—
138#535—2831no—
139#454—2834no—
140#453—2834no—
141#421—2832no—
142#413D28311yescharacteristic, complemented
1432−1+4—2832no—
1442−1+4—5662no—
145H(5,𝔽2)—2832no—
146H(5,𝔽2)—5662no—
147H(5,𝔽2)—5662no—
148C2×#41—2832no—
149C2×#41—5664no—
150C2×#41—5662no—
151C2×#41C2×D28311yescharacteristic
152C22×D4—2831no—
153AGL(1,ℤ/8)—2838no—
154AGL(1,ℤ/8)—5664no—
155C2×SD16—5662no—
156C2×D8—5662no—
157C2×M16C2×D28312yescomplemented
158#127—2832no—
159#124—2832no—
160#121—5662no—
161#120—5662no—
162#117—2831no—
163#104—2838no—
164#101C2×D28312yescomplemented
165#100C2×D28312yescomplemented
166C24—5661no—
167#41—22no—
168#41—22no—
169#41—2834no—
170#41—2834no—
171#41—5664no—
172#41—5664no—
173#41—5664no—
174#41—5668no—
175#41—5664no—
176#41—5662no—
177#41—5662no—
178#41—5664no—
179C2×Q8—2832no—
180C2×Q8—5662no—
181C2×Q8—5664no—
182C2×Q8C22×D28311yescharacteristic
183C2×D4—2832no—
184C2×D4—2832no—
185C2×D4—5662no—
186C2×D4—5664no—
187C2×D4—5668no—
188C2×D4—5662no—
189C2×D4—5664no—
190C2×D4—5664no—
191C2×D4—5662no—
192C2×D4—5662no—
193C2×D4—5661no—
194C2×D4D113212yes—
195C2×D4C22×D28311yescharacteristic
196C22×C4—5664no—
197C22×C4—5662no—
198C22×C4D113212yes—
199C22×C4C22×D28311yescharacteristic
200SD16—5668no—
201D8—5668no—
202M16—22no—
203M16C22×D28314yes—
204C2×C8—22no—
205C4⋊C4—5661no—
206C22⋊C4—5661no—
207C22⋊C4—5664no—
208C42—2832no—
209C23—5664no—
210C23—5662no—
211C23—5662no—
212C23—11322no—
213C23—22642no—
214C23C2×D113212yes—
215C23C2×D113211yescharacteristic
216Q8—22no—
217Q8—2834no—
218Q8—5664no—
219Q8—5662no—
220D4—24no—
221D4—22no—
222D4—2834no—
223D4—5664no—
224D4—5668no—
225D4—5664no—
226D4—5664no—
227D4—5662no—
228D4—5664no—
229D4—11321no—
230D4—11321no—
231D4—22644no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—5668no—
235C2×C4—5664no—
236C2×C4—5662no—
237C2×C4—5664no—
238C2×C4—5664no—
239C2×C4—5662no—
240C2×C4D4×D28312yes—
241C2×C4D4×D28312yes—
242C2×C4C2×D113212yes—
243C2×C4C2×D113211yescharacteristic
244C2×C4C23×D28311yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—5664no—
250C22—11324no—
251C22—11322no—
252C22—22642no—
253C22—22644no—
254C22—22644no—
255C22#2258358012yes—
256C22C22×D113211yescharacteristic
257C4—22no—
258C4—5664no—
259C4—11322no—
260C4#2258358011yescharacteristic
261C4#2258358011yescharacteristic
262C4C2×D4×D28312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—11324no—
267C2—22642no—
268C2C2×#2258358011yescharacteristic
subgroupG/H
centerC2C2×#22583580
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#413×C283C2
socleC2×C283C2×#120
hypercenter#413D283
hypocenterC283#3021
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #31827301C2semidirect
2. C2×#22583638C2semidirect
3. #31827029C2semidirect
4. #31826640C2semidirect
5. #31826635C2semidirect
6. #31826481C2semidirect
7. #413×C283C2semidirect
8. #22583658C22semidirect
12. #22583655C22semidirect
16. C22×D1132C22semidirect
18. #22583638C22semidirect
21. #22583606C22semidirect
22. #22583599C22semidirect
25. #22583569C22semidirect
26. #22583536C22semidirect
28. C2×M16×C283C22semidirect
29. #101×C283C22semidirect
30. #100×C283C22semidirect
38. #17985392D4semidirect
43. C2×Dic566D4semidirect
49. C2×D1132D4semidirect
134. C283#3021semidirect
142. #413D283semidirect
157. C2×M16C2×D283semidirect
164. #101C2×D283semidirect
165. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
C2×#225836382
#318270292
#318266402
#318266352
#318264812
#413×C2832
#3021283