Finite Group Databasebeta

← #49130501

Subgroups — #49130501

Order 36224.

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

normal subgroups only

This group has 758 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
1C22×#17985392C211yesmaximal, characteristic, complemented
2C2×#22583640C211yesmaximal, characteristic, complemented
3C2×#22583635C211yesmaximal, characteristic, complemented
4#31826625C214yesmaximal, complemented
5#31826608C214yesmaximal, complemented
6#31826473C211yesmaximal, characteristic, complemented
7#31826469C211yesmaximal, characteristic, complemented
8#31826429C211yesmaximal, characteristic, complemented
9#525×C283C211yesmaximal, characteristic, complemented
10C2×#17985392—24no—
11C2×#17985392—24no—
12C2×#17985392—22no—
13C2×#17985392C2214yes—
14C22×Dic566C2211yescharacteristic
15C22×D1132C2211yescharacteristic
16C22×C22⋊D283—21no—
17C22×C4×D283—21no—
18#22583640—24no—
19#22583635—24no—
20C2×#17985382C2212yescomplemented
21C2×#17985380C2212yescomplemented
22C2×#17985379C2212yescomplemented
23C2×#17985377C2212yescomplemented
24C2×#17985376C2211yescharacteristic
25C2×#17985376C2211yescharacteristic
26#22583533C2214yescomplemented
27#22583530C2214yescomplemented
28C22⋊(C283⋊C8)C2214yescomplemented
29C2×#41×C283C2211yescharacteristic
30C23×C4×C283C2211yescharacteristic
31#123×C283—21no—
32C22⋊Q8×C283C2212yescomplemented
33#121×C283C2212yescomplemented
34#120×C283—21no—
35C4×D4×C283—22no—
36#117×C283C2211yescharacteristic
37C23×D283—21no—
38#17985392—216no—
39#17985392—28no—
40#17985392—28no—
41C2×Dic566—22no—
42C2×Dic566—21no—
43C2×Dic566D414yescomplemented
44C2×Dic566C2312yes—
45C2×D1132—22no—
46C2×D1132—21no—
47C2×D1132D414yescomplemented
48C2×D1132C2312yes—
49C2×C22⋊D283—28no—
50C2×C22⋊D283—24no—
51C22×Dic283—21no—
52C2×C4×D283—28no—
53C2×C4×D283—24no—
54#17985382—24no—
55#17985380—24no—
56#17985379—24no—
57#17985377—24no—
58#17985376—22no—
59#17985376—22no—
60C2×C283⋊C8C2312yes—
61C2×C283⋊C8C2312yes—
62C24×C283D411yescharacteristic
63#41×C283—22no—
64#41×C283—22no—
65C2×Q8×C283C2311yescharacteristic
66C2×D4×C283—22no—
67C2×D4×C283—21no—
68C2×D4×C283C2311yescharacteristic
69C22×C4×C283—22no—
70C22×C4×C283—22no—
71C22×C4×C283—21no—
72C22×C4×C283D411yescharacteristic
73C22×C4×C283D412yes—
74C22×C4×C283C2312yes—
75C22×C4×C283C2311yescharacteristic
76C4⋊C4×C283—22no—
77C4⋊C4×C283C2312yes—
78C22⋊C4×C283—21no—
79C22⋊C4×C283—22no—
80C22⋊C4×C283—22no—
81C42×C283—21no—
82C22×D283—24no—
83C22×D283—22no—
84C22×D283—44no—
85Dic566—24no—
86Dic566—22no—
87Dic566C2×D414yes—
88D1132—24no—
89D1132—22no—
90D1132C2×D414yes—
91C22⋊D283—216no—
92C2×Dic283—24no—
93C2×Dic283—22no—
94C4×D283—216no—
95C283⋊C8—22no—
96C283⋊C8—22no—
97C23×C283—22no—
98C23×C283—21no—
99C23×C283—42no—
100C23×C283C2×D412yes—
101C23×C283C2×D411yescharacteristic
102Q8×C283—21no—
103Q8×C283—21no—
104D4×C283—21no—
105D4×C283—21no—
106D4×C283—44no—
107D4×C283—42no—
108C2×C4×C283—22no—
109C2×C4×C283—24no—
110C2×C4×C283—22no—
111C2×C4×C283—22no—
112C2×C4×C283—22no—
113C2×C4×C283—21no—
114C2×C4×C283—42no—
115C2×C4×C283C2×D412yes—
116C2×C4×C283C2×D412yes—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283C2411yescharacteristic
120C22×C283—22no—
121C22×C283—22no—
122C22×C283—22no—
123C22×C283—22no—
124C22×C283—44no—
125C22×C283—41no—
126C22×C283—42no—
127C22×C283#12012yes—
128C22×C283C22×D411yescharacteristic
129C2×D283—24no—
130C2×D283—48no—
131C2×D283—44no—
132C4×C283—22no—
133C4×C283—41no—
134C4×C283—42no—
135C4×C283#12012yes—
136C4×C283C22×D411yescharacteristic
137C4×C283C22×D411yescharacteristic
138Dic283—24no—
139C2×C283—22no—
140C2×C283—42no—
141C2×C283—81no—
142C2×C283C2×#12011yescharacteristic
143C2×C283#57512yes—
144D283—44no—
145C283#300511yescharacteristic, complemented, Sylow
146#3005—2831nomaximal, Sylow
147C22×#41—2831no—
148C2×#137—2831no—
149C2×AGL(1,ℤ/8)—2831no—
150#525D28311yescharacteristic, complemented
151#452—2834no—
152#449—2834no—
153#419—2831no—
154#418—2831no—
155#407—2831no—
156C2×#41—2834no—
157C2×#41—5664no—
158C2×#41—5664no—
159C2×#41—5662no—
160C2×#41C2×D28311yescharacteristic
161C22×Q8—2831no—
162C22×D4—2831no—
163C22×D4—5661no—
164C23×C4—5661no—
165C23×C4C2×D28311yescharacteristic
166#137—5664no—
167AGL(1,ℤ/8)—5664no—
168C2×Q16—2832no—
169C2×SD16—2832no—
170C2×SD16—2832no—
171C2×D8—2832no—
172C2×M16—2831no—
173C2×M16—2831no—
174#123—21no—
175C22⋊Q8C2×D28312yescomplemented
176#121C2×D28312yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D28311yescharacteristic
180#103—2834no—
181#102—2834no—
182C22⋊C8—2834no—
183C24—5661no—
184C24C22⋊D28311yescharacteristic
185#41—22no—
186#41—22no—
187#41—5668no—
188#41—5668no—
189#41—56616no—
190C2×Q8—2834no—
191C2×Q8—2832no—
192C2×Q8—5662no—
193C2×Q8—5661no—
194C2×Q8C22×D28311yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—2834no—
198C2×D4—2832no—
199C2×D4—5668no—
200C2×D4—5664no—
201C2×D4—5662no—
202C2×D4—5661no—
203C2×D4C22×D28311yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—5668no—
208C22×C4—5661no—
209C22×C4—5664no—
210C22×C4C22⋊D28312yes—
211C22×C4C22⋊D28311yescharacteristic
212C22×C4C22×D28312yes—
213C22×C4C22×D28311yescharacteristic
214Q16—5664no—
215SD16—5664no—
216SD16—5664no—
217D8—5664no—
218M16—5662no—
219M16—5662no—
220C2×C8—2832no—
221C2×C8—2832no—
222C4⋊C4—22no—
223C4⋊C4C22×D28312yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—5664no—
232C23—5662no—
233C23—11324no—
234C23C2×C22⋊D28312yes—
235C23C2×C22⋊D28311yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—2834no—
239Q8—5664no—
240Q8—5662no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—2834no—
246D4—56616no—
247D4—5664no—
248D4—5662no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—56616no—
257C2×C4—5664no—
258C2×C4—5662no—
259C2×C4D4×D28312yes—
260C2×C4D4×D28312yes—
261C2×C4C2×C22⋊D28312yes—
262C2×C4C2×C22⋊D28311yescharacteristic
263C2×C4C23×D28311yescharacteristic
264C8—5662no—
265C8—5662no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—5664no—
274C22—11328no—
275C22—11324no—
276C22#2258357812yes—
277C22C22×C22⋊D28311yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—5664no—
282C4#2258357812yes—
283C4C2×D4×D28311yescharacteristic
284C4C2×D4×D28311yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—11324no—
289C2C2×#2258357811yescharacteristic
290C2#3182726312yes—
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#525×C283C2
socleC22×C283C22×D4
hypercenter#525D283
hypocenterC283#3005
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#17985392C2semidirect
2. C2×#22583640C2semidirect
3. C2×#22583635C2semidirect
4. #31826625C2semidirect
5. #31826608C2semidirect
6. #31826473C2semidirect
7. #31826469C2semidirect
8. #31826429C2semidirect
9. #525×C283C2semidirect
20. C2×#17985382C22semidirect
21. C2×#17985380C22semidirect
22. C2×#17985379C22semidirect
23. C2×#17985377C22semidirect
26. #22583533C22semidirect
27. #22583530C22semidirect
28. C22⋊(C283⋊C8)C22semidirect
32. C22⋊Q8×C283C22semidirect
33. #121×C283C22semidirect
43. C2×Dic566D4semidirect
47. C2×D1132D4semidirect
145. C283#3005semidirect
150. #525D283semidirect
175. C22⋊Q8C2×D283semidirect
176. #121C2×D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853922
C2×#225836402
C2×#225836352
#318266252
#318266082
#318264732
#318264692
#318264292
#525×C2832
#3005283