Finite Group Databasebeta

← #49130935

Subgroups — #49130935

Order 36224.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22583618C211yesmaximal, characteristic, complemented
2#31827043C211yesmaximal, characteristic, complemented
3#31827032C211yesmaximal, characteristic, complemented
4#31826715C212yesmaximal, complemented
5#31826708C212yesmaximal, complemented
6#31826698C212yesmaximal, complemented
7#31826691C212yesmaximal, complemented
8#31826507C211yesmaximal, characteristic, complemented
9#31826461C211yesmaximal, characteristic, complemented
10C2×#22583531C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13#22583618—24no—
14C2×#17985375C2212yescomplemented
15C22×C283⋊C8C2211yescharacteristic
16C2×C8×D283C2212yescomplemented
17#22583606—21no—
18#22583597—21no—
19#22583589—21no—
20#22583586C2212yescomplemented
21#22583584—21no—
22#22583582—22no—
23#22583581C2212yes—
24#22583581C2212yes—
25#22583581C2212yescomplemented
26#22583579—22no—
27#22583569C2211yescharacteristic
28C2×C4⋊Dic283C2211yescharacteristic
29#22583544C2212yescomplemented
30#22583542C2212yescomplemented
31#22583534C2211yescharacteristic, complemented
32#22583534C2211yescharacteristic, complemented
33#22583531C2212yes—
34#22583531C2212yes—
35#22583531C2211yescharacteristic, complemented
36#22583531C2211yescharacteristic, complemented
37C2×#41×C283C2211yescharacteristic
38C22×D4×C283C2211yescharacteristic
39AGL(1,ℤ/8)×C283—24no—
40C2×SD16×C283C2212yes—
41C2×D8×C283C2212yes—
42C2×M16×C283C2211yescharacteristic
43#17985392—22no—
44#17985392—22no—
45C2×Dic566D411yescharacteristic
46C2×D1132D411yescharacteristic
47C2×C22⋊D283—22no—
48C2×C22⋊D283—22no—
49C2×C22⋊D283—22no—
50C2×C22⋊D283D412yes—
51C22×Dic283—21no—
52C2×C4×D283C2312yes—
53#17985375—24no—
54C2×C283⋊C8—22no—
55C2×C283⋊C8C2311yescharacteristic
56C2×C283⋊C8C2311yescharacteristic
57C8×D283—24no—
58C4⋊Dic283—21no—
59C4⋊Dic283C2312yes—
60C4⋊Dic283C2311yescharacteristic
61C4⋊Dic283C2311yescharacteristic
62Dic283⋊C4—22no—
63C22⋊Dic283—22no—
64C22⋊Dic283—22no—
65C22⋊Dic283—22no—
66C22⋊Dic283—21no—
67(C2×D283)⋊C4—22no—
68C4×Dic283—21no—
69C24×C283—21no—
70#41×C283—22no—
71#41×C283—22no—
72C2×Q8×C283C2311yescharacteristic
73C2×D4×C283—21no—
74C2×D4×C283—22no—
75C2×D4×C283—21no—
76C2×D4×C283—22no—
77C2×D4×C283C2312yes—
78C2×D4×C283C2311yescharacteristic
79C22×C4×C283—21no—
80C22×C4×C283C2311yescharacteristic
81SD16×C283—24no—
82D8×C283—24no—
83M16×C283D414yescomplemented
84C2×C8×C283C2312yes—
85C22×D283C2×D412yes—
86Dic566—22no—
87D1132—22no—
88C22⋊D283—24no—
89C22⋊D283—82no—
90C22⋊D283—82no—
91C22⋊D283—82no—
92C2×Dic283—21no—
93C2×Dic283—21no—
94C2×Dic283—22no—
95C2×Dic283—81no—
96C2×Dic283C2×D412yes—
97C4×D283—22no—
98C4×D283—22no—
99C283⋊C8—21no—
100C283⋊C8—21no—
101C23×C283—21no—
102C23×C283—22no—
103C23×C283—42no—
104C23×C283—81no—
105C23×C283—81no—
106C23×C283C2×D411yescharacteristic
107Q8×C283—21no—
108Q8×C283—21no—
109D4×C283—21no—
110D4×C283—22no—
111D4×C283—22no—
112D4×C283—21no—
113D4×C283—42no—
114D4×C283—42no—
115C2×C4×C283—21no—
116C2×C4×C283—42no—
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283C2×D412yes—
119C2×C4×C283#4112yes—
120C2×C4×C283C2411yescharacteristic
121C8×C283C2×D414yes—
122C22×C283—21no—
123C22×C283—21no—
124C22×C283—41no—
125C22×C283—42no—
126C22×C283—42no—
127C22×C283—41no—
128C22×C283—41no—
129C22×C283—42no—
130C22×C283—84no—
131C22×C283#12112yes—
132C22×C283C22×D411yescharacteristic
133C2×D283—22no—
134C2×D283—22no—
135C2×D283—22no—
136C4×C283—41no—
137C4×C283#12112yes—
138C4×C283C22×D411yescharacteristic
139C4×C283C2×#4111yescharacteristic
140Dic283—22no—
141Dic283—42no—
142Dic283—41no—
143Dic283—41no—
144C2×C283—22no—
145C2×C283—81no—
146C2×C283—82no—
147C2×C283C2×#12111yescharacteristic
148C2×C283#57611yescharacteristic
149C2×C283#57711yescharacteristic
150D283—42no—
151C283#306211yescharacteristic, complemented, Sylow
152#3062—2831nomaximal, Sylow
153C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
154C2×#131—2831no—
155#537—2831no—
156#535—2831no—
157#469—2832no—
158#468—2832no—
159#466—2832no—
160#465—2832no—
161#428—2831no—
162#416—2831no—
163C2×#102—2831no—
164C2×#41—2831no—
165C2×#41C2×D28311yescharacteristic
166C22×D4C2×D28311yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D28312yes—
169C2×D8C2×D28312yes—
170#131—5664no—
171C2×M16—2832no—
172C2×M16C2×D28311yescharacteristic
173C22×C8—2831no—
174C22×C8—2832no—
175#127—5661no—
176#124—5661no—
177C22⋊Q8—2832no—
178C22⋊Q8—5661no—
179#121—2832no—
180#121—2832no—
181#121—2832no—
182#121—5661no—
183#121—5662no—
184#120—5662no—
185#117—2831no—
186C2×C4⋊C4—2831no—
187#107—2832no—
188#106—2832no—
189#103—2831no—
190#103—2831no—
191#102—2832no—
192#102—2832no—
193#102—2831no—
194#102—2831no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—5662no—
199#41—5662no—
200C2×Q8—2831no—
201C2×Q8C22×D28311yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—2832no—
207C2×D4—2831no—
208C2×D4—5662no—
209C2×D4—5662no—
210C2×D4—5662no—
211C2×D4C22×D28311yescharacteristic
212C2×D4C22×D28312yes—
213C22×C4—21no—
214C22×C4—2832no—
215C22×C4—5661no—
216C22×C4C22×D28311yescharacteristic
217SD16—24no—
218D8—24no—
219M16—5664no—
220M16C22⋊D28314yescomplemented
221C2×C8—2831no—
222C2×C8—2831no—
223C2×C8—5662no—
224C2×C8—5664no—
225C2×C8C22×D28312yes—
226C4⋊C4—2832no—
227C4⋊C4—2831no—
228C4⋊C4—2831no—
229C4⋊C4—5662no—
230C4⋊C4—5661no—
231C22⋊C4—5662no—
232C22⋊C4—5662no—
233C22⋊C4—5662no—
234C22⋊C4—5662no—
235C22⋊C4—5661no—
236C42—5661no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—2832no—
243C23D4×D28311yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—5662no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—5664no—
254D4—5662no—
255D4—22642no—
256D4—22642no—
257D4—22642no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2832no—
261C2×C4—5662no—
262C2×C4—5662no—
263C2×C4—5661no—
264C2×C4—5661no—
265C2×C4—5662no—
266C2×C4—22641no—
267C2×C4D4×D28311yescharacteristic
268C2×C4C2×C22⋊D28312yes—
269C2×C4#1798539112yes—
270C2×C4C23×D28311yescharacteristic
271C8—5661no—
272C8—5661no—
273C8C2×C22⋊D28314yes—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—84no—
283C22—5662no—
284C22—5662no—
285C22—5662no—
286C22#2258358112yes—
287C22C2×D4×D28311yescharacteristic
288C4—41no—
289C4—5662no—
290C4—11322no—
291C4—11321no—
292C4—11321no—
293C4#2258358112yes—
294C4C22×C22⋊D28311yescharacteristic
295C4C2×#1798539111yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—11322no—
300C2C2×#2258358111yescharacteristic
301C2#3182726911yescharacteristic
302C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3062
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583618C2semidirect
2. #31827043C2semidirect
3. #31827032C2semidirect
4. #31826715C2semidirect
5. #31826708C2semidirect
6. #31826698C2semidirect
7. #31826691C2semidirect
8. #31826507C2semidirect
9. #31826461C2semidirect
10. C2×#22583531C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
14. C2×#17985375C22semidirect
16. C2×C8×D283C22semidirect
20. #22583586C22semidirect
25. #22583581C22semidirect
29. #22583544C22semidirect
30. #22583542C22semidirect
31. #22583534C22semidirect
32. #22583534C22semidirect
35. #22583531C22semidirect
36. #22583531C22semidirect
83. M16×C283D4semidirect
151. C283#3062semidirect
153. C2×AGL(1,ℤ/8)D283semidirect
220. M16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836182
#318270432
#318270322
#318267152
#318267082
#318266982
#318266912
#318265072
#318264612
C2×#225835312
C2×AGL(1,ℤ/8)×C2832
#3062283