Finite Group Databasebeta

← #49130614

Subgroups — #49130614

Order 36224.

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

normal subgroups only

This group has 724 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
2#31827265C212yesmaximal, complemented
3#31827011C211yesmaximal, characteristic, complemented
4#31826650C212yesmaximal, complemented
5#31826645C212yesmaximal, complemented
6#31826640C212yesmaximal, complemented
7#31826635C212yesmaximal, complemented
8C2×#22583536C212yesmaximal, complemented
9#413×C283C211yesmaximal, characteristic, complemented
10#22583658C2212yes—
11#22583657—22no—
12#22583656—22no—
13#22583655—22no—
14#22583655C2212yes—
15C2×#17985392—22no—
16C2×#17985392C2212yes—
17C2×#17985392C2211yescharacteristic, complemented
18#41×D283—24no—
19#22583643—22no—
20#22583643C2214yescomplemented
21#22583638—22no—
22#22583638C2214yescomplemented
23#22583632—24no—
24#22583608C2211yescharacteristic, complemented
25#22583606C2211yescharacteristic, complemented
26#22583599C2211yescharacteristic, complemented
27#22583597C2211yescharacteristic, complemented
28#22583582—22no—
29C4×C22⋊D283—22no—
30C2×C4×Dic283C2211yescharacteristic, complemented
31#22583536C2214yescomplemented
32#22583536C2214yescomplemented
33C2×#41×C283C2211yescharacteristic
34C2×M16×C283C2212yescomplemented
35#101×C283C2212yescomplemented
36#100×C283C2212yescomplemented
37#17985393—24no—
38#17985392—24no—
39#17985392—24no—
40#17985392—22no—
41#17985392—24no—
42#17985392—22no—
43#17985392—22no—
44#17985392—22no—
45#17985392D414yescomplemented
46#17985392C2314yes—
47#17985391—28no—
48#17985391—24no—
49C2×Dic566—22no—
50C2×Dic566D412yescomplemented
51C2×Dic566C2311yescharacteristic
52Q8×D283—24no—
53C2×D1132—22no—
54C2×D1132D412yescomplemented
55C2×D1132C2311yescharacteristic
56C2×C22⋊D283—24no—
57C2×C22⋊D283—22no—
58C2×C22⋊D283—22no—
59C2×C22⋊D283—21no—
60D4×D283—28no—
61D4×D283—24no—
62C22×Dic283—21no—
63C2×C4×D283—24no—
64C2×C4×D283—22no—
65C2×C4×D283—21no—
66Dic1132—24no—
67#17985381—24no—
68#17985381—24no—
69D2264—24no—
70Dic283⋊C4—22no—
71C22⋊Dic283—22no—
72(C2×D283)⋊C4—22no—
73C4×Dic283—21no—
74C4×Dic283C2311yescharacteristic
75C4×Dic283C2311yescharacteristic
76#41×C283—22no—
77#41×C283—22no—
78C2×Q8×C283C2311yescharacteristic
79C2×D4×C283D412yes—
80C2×D4×C283C2311yescharacteristic
81C22×C4×C283D412yes—
82C22×C4×C283C2311yescharacteristic
83M16×C283—22no—
84M16×C283C2314yes—
85C2×C8×C283—22no—
86C22×D283—24no—
87C22×D283—22no—
88C22×D283—21no—
89Dic566—24no—
90Dic566—22no—
91Dic566—22no—
92Dic566C2×D414yes—
93D1132—24no—
94D1132—22no—
95D1132—22no—
96D1132C2×D414yes—
97C22⋊D283—28no—
98C22⋊D283—24no—
99C22⋊D283—24no—
100C22⋊D283—44no—
101C22⋊D283—42no—
102C2×Dic283—24no—
103C2×Dic283—22no—
104C2×Dic283—21no—
105C2×Dic283—21no—
106C2×Dic283—21no—
107C2×Dic283—42no—
108C4×D283—28no—
109C4×D283—24no—
110C4×D283—24no—
111C4×D283—41no—
112C4×D283—41no—
113C23×C283C2×D412yes—
114C23×C283C2×D411yescharacteristic
115Q8×C283—22no—
116D4×C283—24no—
117D4×C283—22no—
118C2×C4×C283—22no—
119C2×C4×C283—22no—
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2×D411yescharacteristic
124C2×C4×C283C2411yescharacteristic
125C8×C283—24no—
126C22×C283—22no—
127C22×C283—41no—
128C22×C283—42no—
129C22×C283#12012yes—
130C22×C283C22×D411yescharacteristic
131C2×D283—24no—
132C2×D283—42no—
133C2×D283—41no—
134C2×D283—82no—
135C2×D283—84no—
136C4×C283—22no—
137C4×C283#12011yescharacteristic
138C4×C283#12011yescharacteristic
139C4×C283C22×D412yes—
140Dic283—22no—
141Dic283—24no—
142Dic283—41no—
143Dic283—41no—
144C2×C283—21no—
145C2×C283—22no—
146C2×C283—42no—
147C2×C283C2×#12011yescharacteristic
148D283—44no—
149D283—81no—
150C283#301811yescharacteristic, complemented, Sylow
151#3018—2831nomaximal, Sylow
152#585—2831no—
153#575—2832no—
154#532—2831no—
155#456—2832no—
156#455—2832no—
157#454—2832no—
158#453—2832no—
159C2×#104—2832no—
160#413D28311yescharacteristic, complemented
1612−1+4—2832no—
1622−1+4—5662no—
163H(5,𝔽2)—2832no—
164H(5,𝔽2)—5662no—
165H(5,𝔽2)—5662no—
166C2×#41—2832no—
167C2×#41—2831no—
168C2×#41—5664no—
169C2×#41—5662no—
170C2×#41C2×D28311yescharacteristic
171#137—2834no—
172#137—5662no—
173AGL(1,ℤ/8)—2834no—
174AGL(1,ℤ/8)—5662no—
175#135—5664no—
176C2×M16C2×D28312yescomplemented
177C4⋊Q8—2831no—
178#127—2831no—
179#124—2831no—
180#124—2831no—
181#121—5662no—
182C4×D4—5662no—
183C2×C42—2831no—
184#104—2834no—
185#104—2834no—
186#101C2×D28312yescomplemented
187#100C2×D28312yescomplemented
188#41—22no—
189#41—22no—
190#41—2834no—
191#41—2834no—
192#41—5664no—
193#41—5664no—
194#41—5664no—
195#41—5668no—
196#41—5664no—
197#41—5662no—
198#41—5662no—
199#41—5664no—
200#41—5662no—
201#41—5662no—
202C2×Q8—2832no—
203C2×Q8—2831no—
204C2×Q8—5662no—
205C2×Q8—5664no—
206C2×Q8C22×D28311yescharacteristic
207C2×D4—2832no—
208C2×D4—2831no—
209C2×D4—5662no—
210C2×D4—5664no—
211C2×D4—5668no—
212C2×D4—5662no—
213C2×D4—5664no—
214C2×D4—5662no—
215C2×D4—5661no—
216C2×D4D113212yes—
217C2×D4C22×D28311yescharacteristic
218C22×C4—5664no—
219C22×C4—5662no—
220C22×C4—5661no—
221C22×C4—5661no—
222C22×C4D113212yes—
223C22×C4C22×D28311yescharacteristic
224Q16—5664no—
225SD16—5664no—
226SD16—5664no—
227D8—5664no—
228M16—22no—
229M16C22×D28314yes—
230C2×C8—22no—
231C4⋊C4—5662no—
232C22⋊C4—5662no—
233C22⋊C4—5662no—
234C42—2831no—
235C42—2831no—
236C42—5661no—
237C23—5664no—
238C23—5662no—
239C23—5661no—
240C23C2×D113212yes—
241C23C2×D113211yescharacteristic
242Q8—22no—
243Q8—2834no—
244Q8—5664no—
245Q8—5662no—
246Q8—5662no—
247D4—24no—
248D4—22no—
249D4—2834no—
250D4—5664no—
251D4—5668no—
252D4—5664no—
253D4—5664no—
254D4—5662no—
255D4—5662no—
256D4—11324no—
257D4—11322no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—5668no—
261C2×C4—5664no—
262C2×C4—5662no—
263C2×C4—5661no—
264C2×C4—5664no—
265C2×C4—5664no—
266C2×C4—5661no—
267C2×C4—5661no—
268C2×C4—11322no—
269C2×C4—11321no—
270C2×C4—11321no—
271C2×C4D4×D28312yes—
272C2×C4D4×D28312yes—
273C2×C4C2×D113212yes—
274C2×C4C2×D113211yescharacteristic
275C2×C4C23×D28311yescharacteristic
276C8—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—5664no—
281C22—11322no—
282C22—11321no—
283C22—22642no—
284C22—22644no—
285C22#2258358012yes—
286C22C22×D113211yescharacteristic
287C4—22no—
288C4—5662no—
289C4—5664no—
290C4—11321no—
291C4—11321no—
292C4#2258358011yescharacteristic
293C4#2258358011yescharacteristic
294C4C2×D4×D28312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—11324no—
299C2—22641no—
300C2C2×#2258358011yescharacteristic
subgroupG/H
centerC4#22583580
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#413×C283C2
socleC2×C283C2×#120
hypercenter#413D283
hypocenterC283#3018
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #31827301C2semidirect
2. #31827265C2semidirect
3. #31827011C2semidirect
4. #31826650C2semidirect
5. #31826645C2semidirect
6. #31826640C2semidirect
7. #31826635C2semidirect
8. C2×#22583536C2semidirect
9. #413×C283C2semidirect
17. C2×#17985392C22semidirect
20. #22583643C22semidirect
22. #22583638C22semidirect
24. #22583608C22semidirect
25. #22583606C22semidirect
26. #22583599C22semidirect
27. #22583597C22semidirect
30. C2×C4×Dic283C22semidirect
31. #22583536C22semidirect
32. #22583536C22semidirect
34. C2×M16×C283C22semidirect
35. #101×C283C22semidirect
36. #100×C283C22semidirect
45. #17985392D4semidirect
50. C2×Dic566D4semidirect
54. C2×D1132D4semidirect
150. C283#3018semidirect
160. #413D283semidirect
176. C2×M16C2×D283semidirect
186. #101C2×D283semidirect
187. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318272652
#318270112
#318266502
#318266452
#318266402
#318266352
C2×#225835362
#413×C2832
#3018283