Finite Group Databasebeta

← #49130527

Subgroups — #49130527

Order 36224.

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

normal subgroups only

This group has 784 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×#22583656C211yesmaximal, characteristic, complemented
2C2×#22583636C211yesmaximal, characteristic, complemented
3C22×#17985377C211yesmaximal, characteristic, complemented
4#31826608C214yesmaximal, complemented
5#31826598C214yesmaximal, complemented
6#31826469C211yesmaximal, characteristic, complemented
7#31826459C211yesmaximal, characteristic, complemented
8#31826432C211yesmaximal, characteristic, complemented
9#535×C283C211yesmaximal, characteristic, complemented
10#22583656—24no—
11#22583656—24no—
12C2×#17985393—21no—
13C2×#17985392C2211yescharacteristic
14C2×#17985392C2212yescomplemented
15C22×D1132C2211yescharacteristic
16C22×D1132C2212yescomplemented
17C2×D4×D283—22no—
18C2×D4×D283—21no—
19#22583636—24no—
20C2×#17985380C2212yescomplemented
21C2×#17985377—24no—
22C2×#17985377C2214yescomplemented
23C2×#17985377C2212yescomplemented
24C2×#17985376C2211yescharacteristic, complemented
25C22×C283⋊C8C2211yescharacteristic, complemented
26#22583533C2212yescomplemented
27#22583530C2212yescomplemented
28#22583530C2214yescomplemented
29C22⋊(C283⋊C8)C2214yescomplemented
30C2×#41×C283C2211yescharacteristic
31C22×D4×C283C2211yescharacteristic, complemented
32#127×C283—21no—
33#124×C283—21no—
34#121×C283C2214yescomplemented
35#120×C283—22no—
36#117×C283C2211yescharacteristic, complemented
37C23×D283—21no—
38C23×D283—22no—
39#17985393—28no—
40#17985392—24no—
41#17985392—24no—
42#17985392—22no—
43#17985392D414yescomplemented
44C2×Dic566C2311yescharacteristic
45C2×D1132—24no—
46C2×D1132—24no—
47C2×D1132—21no—
48C2×D1132—22no—
49C2×D1132—22no—
50C2×D1132D414yescomplemented
51C2×D1132C2312yes—
52C2×D1132C2311yescharacteristic
53C2×C22⋊D283—21no—
54C2×C22⋊D283—22no—
55D4×D283—216no—
56D4×D283—28no—
57C2×C4×D283—21no—
58C2×C4×D283—22no—
59#17985380—24no—
60#17985377—28no—
61#17985377—24no—
62#17985376—22no—
63C2×C283⋊C8—22no—
64C2×C283⋊C8C2312yes—
65C2×C283⋊C8C2312yes—
66C24×C283—21no—
67#41×C283—22no—
68#41×C283—22no—
69C2×Q8×C283D411yescharacteristic
70C2×D4×C283—24no—
71C2×D4×C283—22no—
72C2×D4×C283—22no—
73C2×D4×C283—21no—
74C2×D4×C283D412yes—
75C2×D4×C283D411yescharacteristic
76C2×D4×C283C2312yes—
77C22×C4×C283C2312yes—
78C22×C4×C283C2311yescharacteristic
79C4⋊C4×C283C2312yes—
80C22⋊C4×C283—21no—
81C22⋊C4×C283—24no—
82C42×C283—21no—
83C22×D283—21no—
84C22×D283—22no—
85C22×D283—28no—
86C22×D283—24no—
87C22×D283—82no—
88C22×D283—84no—
89Dic566—21no—
90Dic566C2×D412yes—
91D1132—28no—
92D1132—24no—
93D1132—22no—
94D1132—21no—
95D1132C2×D412yes—
96D1132C2×D414yes—
97C22⋊D283—28no—
98C22⋊D283—24no—
99C2×Dic283—21no—
100C4×D283—28no—
101C4×D283—24no—
102C283⋊C8—22no—
103C283⋊C8—22no—
104C23×C283—22no—
105C23×C283—42no—
106C23×C283—82no—
107C23×C283C2×D412yes—
108C23×C283C2×D411yescharacteristic
109Q8×C283—22no—
110D4×C283—24no—
111D4×C283—22no—
112D4×C283—22no—
113D4×C283—22no—
114D4×C283—42no—
115D4×C283—84no—
116C2×C4×C283—22no—
117C2×C4×C283—22no—
118C2×C4×C283—22no—
119C2×C4×C283C2×D412yes—
120C2×C4×C283C2×D412yes—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D411yescharacteristic
123C2×C4×C283C2411yescharacteristic
124C22×C283—22no—
125C22×C283—22no—
126C22×C283—24no—
127C22×C283—42no—
128C22×C283—44no—
129C22×C283—84no—
130C22×C283#12012yes—
131C22×C283C22×D411yescharacteristic
132C2×D283—24no—
133C2×D283—22no—
134C2×D283—44no—
135C2×D283—42no—
136C2×D283—84no—
137C2×D283—88no—
138C4×C283—22no—
139C4×C283—42no—
140C4×C283#12012yes—
141C4×C283C22×D411yescharacteristic
142C4×C283C22×D411yescharacteristic
143Dic283—22no—
144C2×C283—22no—
145C2×C283—42no—
146C2×C283—82no—
147C2×C283C2×#12011yescharacteristic
148C2×C283#57612yes—
149D283—44no—
150D283—42no—
151C283#300811yescharacteristic, complemented, Sylow
152#3008—2831nomaximal, Sylow
153C2×H(5,𝔽2)—2831no—
154C2×AGL(1,ℤ/8)—2831no—
155C22×D8—2831no—
156#535D28311yescharacteristic, complemented
157#449—2834no—
158#447—2834no—
159#418—2831no—
160#416—2831no—
161#408—2831no—
162H(5,𝔽2)—5664no—
163H(5,𝔽2)—5664no—
164C2×#41—2831no—
165C2×#41—2832no—
166C2×#41—5661no—
167C2×#41C2×D28311yescharacteristic
168C22×D4—2831no—
169C22×D4—2832no—
170C22×D4—5662no—
171C22×D4—5661no—
172C22×D4C2×D28311yescharacteristic, complemented
173AGL(1,ℤ/8)—5664no—
174C2×SD16—2832no—
175C2×D8—2834no—
176C2×D8—2832no—
177C2×D8—5664no—
178C2×M16—2831no—
179C22×C8—2831no—
180#127—21no—
181#124—21no—
182#121C2×D28314yescomplemented
183#120—22no—
184#117C2×D28311yescharacteristic, complemented
185#103—2832no—
186#102—2832no—
187#102—2834no—
188C22⋊C8—2834no—
189C24—21no—
190C24—5662no—
191C24—5661no—
192#41—22no—
193#41—22no—
194#41—2834no—
195#41—5664no—
196#41—5664no—
197#41—5668no—
198#41—5662no—
199C2×Q8—2831no—
200C2×Q8C22⋊D28311yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—2834no—
206C2×D4—2832no—
207C2×D4—2831no—
208C2×D4—5664no—
209C2×D4—5664no—
210C2×D4—56616no—
211C2×D4—5668no—
212C2×D4—5662no—
213C2×D4—5661no—
214C2×D4—5662no—
215C2×D4—5662no—
216C2×D4—5661no—
217C2×D4C22⋊D28312yes—
218C2×D4C22⋊D28311yescharacteristic
219C2×D4C22×D28312yes—
220C22×C4—5662no—
221C22×C4—5661no—
222C22×C4C22×D28312yes—
223C22×C4C22×D28311yescharacteristic
224SD16—5664no—
225D8—5668no—
226D8—5664no—
227M16—5662no—
228C2×C8—2832no—
229C2×C8—2832no—
230C2×C8—5662no—
231C4⋊C4C22×D28312yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—5668no—
239C23—5662no—
240C23—5661no—
241C23—5664no—
242C23—22642no—
243C23—22644no—
244C23C2×C22⋊D28312yes—
245C23C2×C22⋊D28311yescharacteristic
246Q8—22no—
247Q8—2832no—
248Q8—5661no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—2834no—
256D4—2832no—
257D4—5668no—
258D4—5668no—
259D4—5664no—
260D4—5664no—
261D4—5661no—
262D4—5662no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—5668no—
267C2×C4—5661no—
268C2×C4—5664no—
269C2×C4D4×D28312yes—
270C2×C4D4×D28312yes—
271C2×C4C2×C22⋊D28312yes—
272C2×C4C2×C22⋊D28311yescharacteristic
273C2×C4C23×D28311yescharacteristic
274C8—5662no—
275C8—5662no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—5664no—
283C22—5662no—
284C22—11324no—
285C22—11322no—
286C22—22644no—
287C22—22648no—
288C22#2258357812yes—
289C22C22×C22⋊D28311yescharacteristic
290C4—22no—
291C4—42no—
292C4—5662no—
293C4#2258357812yes—
294C4C2×D4×D28311yescharacteristic
295C4C2×D4×D28311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—11324no—
300C2—11322no—
301C2C2×#2258357811yescharacteristic
302C2#3182726812yes—
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#535×C283C2
socleC22×C283C22×D4
hypercenter#535D283
hypocenterC283#3008
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22583656C2semidirect
2. C2×#22583636C2semidirect
3. C22×#17985377C2semidirect
4. #31826608C2semidirect
5. #31826598C2semidirect
6. #31826469C2semidirect
7. #31826459C2semidirect
8. #31826432C2semidirect
9. #535×C283C2semidirect
14. C2×#17985392C22semidirect
16. C22×D1132C22semidirect
20. C2×#17985380C22semidirect
22. C2×#17985377C22semidirect
23. C2×#17985377C22semidirect
24. C2×#17985376C22semidirect
25. C22×C283⋊C8C22semidirect
26. #22583533C22semidirect
27. #22583530C22semidirect
28. #22583530C22semidirect
29. C22⋊(C283⋊C8)C22semidirect
31. C22×D4×C283C22semidirect
34. #121×C283C22semidirect
36. #117×C283C22semidirect
43. #17985392D4semidirect
50. C2×D1132D4semidirect
151. C283#3008semidirect
156. #535D283semidirect
172. C22×D4C2×D283semidirect
182. #121C2×D283semidirect
184. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C2×#225836362
C22×#179853772
#318266082
#318265982
#318264692
#318264592
#318264322
#535×C2832
#3008283