Finite Group Databasebeta

← #49130874

Subgroups — #49130874

Order 36224.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D283C211yesmaximal, characteristic, complemented
2#31826972C211yesmaximal, characteristic, complemented
3C2×#22583581C211yesmaximal, characteristic, complemented
4#31826715C212yesmaximal, complemented
5#31826708C212yesmaximal, complemented
6#31826698C212yesmaximal, complemented
7#31826691C212yesmaximal, complemented
8#31826507C211yesmaximal, characteristic, complemented
9#31826470C211yesmaximal, characteristic, complemented
10#31826466C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C22×C22⋊D283—21no—
13C22×C4×D283C2211yescharacteristic
14C2×#17985376C2211yescharacteristic
15C2×#17985375C2212yescomplemented
16M16×D283—24no—
17C2×C8×D283C2212yescomplemented
18#22583591—21no—
19#22583586C2211yescharacteristic, complemented
20#22583586C2211yescharacteristic, complemented
21#22583581—21no—
22#22583581—21no—
23#22583581C2212yes—
24#22583581C2212yes—
25#22583581C2211yescharacteristic, complemented
26#22583581C2211yescharacteristic, complemented
27#22583578—21no—
28C4×C22⋊D283—21no—
29C4×C22⋊D283—21no—
30#22583569C2211yescharacteristic
31C2×C4⋊Dic283C2211yescharacteristic
32C2×C22⋊Dic283—21no—
33#22583544C2212yescomplemented
34#22583542C2212yescomplemented
35#22583534C2212yescomplemented
36#22583531C2212yescomplemented
37#22583531C2212yescomplemented
38#22583531C2212yescomplemented
39C2×#41×C283C2211yescharacteristic
40C22×D4×C283C2211yescharacteristic
41AGL(1,ℤ/8)×C283—24no—
42C2×SD16×C283C2212yescomplemented
43C2×D8×C283C2212yescomplemented
44C2×M16×C283C2211yescharacteristic
45C23×D283D411yescharacteristic
46C2×C22⋊D283—21no—
47C2×C22⋊D283—21no—
48C2×C22⋊D283—22no—
49C2×C22⋊D283—22no—
50C2×C22⋊D283—44no—
51C22×Dic283—21no—
52C22×Dic283D411yescharacteristic
53C2×C4×D283—22no—
54C2×C4×D283—22no—
55C2×C4×D283D411yescharacteristic
56C2×C4×D283D411yescharacteristic
57C2×C4×D283C2311yescharacteristic
58C2×C4×D283C2311yescharacteristic
59#17985376—22no—
60#17985375—24no—
61C2×C283⋊C8C2312yes—
62C8×D283—24no—
63C4⋊Dic283—21no—
64C4⋊Dic283C2312yes—
65C4⋊Dic283C2311yescharacteristic
66C4⋊Dic283C2311yescharacteristic
67Dic283⋊C4—21no—
68Dic283⋊C4—21no—
69C22⋊Dic283—22no—
70C22⋊Dic283—22no—
71C22⋊Dic283—21no—
72C22⋊Dic283—21no—
73C22⋊Dic283—21no—
74(C2×D283)⋊C4—21no—
75(C2×D283)⋊C4—21no—
76C4×Dic283—21no—
77C24×C283—21no—
78#41×C283—22no—
79#41×C283—22no—
80C2×Q8×C283C2311yescharacteristic
81C2×D4×C283—21no—
82C2×D4×C283—22no—
83C2×D4×C283—21no—
84C2×D4×C283—22no—
85C2×D4×C283C2312yes—
86C2×D4×C283C2311yescharacteristic
87C22×C4×C283—21no—
88C22×C4×C283C2311yescharacteristic
89SD16×C283—24no—
90D8×C283—24no—
91M16×C283D414yescomplemented
92C2×C8×C283C2312yes—
93C22×D283—22no—
94C22×D283—22no—
95C22×D283—41no—
96C22×D283C2×D411yescharacteristic
97C22×D283C2×D411yescharacteristic
98C22⋊D283—42no—
99C22⋊D283—44no—
100C22⋊D283—42no—
101C22⋊D283—44no—
102C2×Dic283—22no—
103C2×Dic283—21no—
104C2×Dic283—21no—
105C2×Dic283—22no—
106C2×Dic283—42no—
107C2×Dic283C2×D411yescharacteristic
108C2×Dic283C2×D411yescharacteristic
109C4×D283—22no—
110C4×D283—22no—
111C4×D283—21no—
112C4×D283—21no—
113C4×D283—21no—
114C4×D283—21no—
115C283⋊C8—22no—
116C23×C283—21no—
117C23×C283—22no—
118C23×C283—42no—
119C23×C283—42no—
120C23×C283—42no—
121C23×C283C2×D411yescharacteristic
122Q8×C283—21no—
123Q8×C283—21no—
124D4×C283—21no—
125D4×C283—22no—
126D4×C283—22no—
127D4×C283—21no—
128D4×C283—42no—
129D4×C283—42no—
130C2×C4×C283—21no—
131C2×C4×C283—42no—
132C2×C4×C283C2×D411yescharacteristic
133C2×C4×C283C2×D412yes—
134C2×C4×C283#4112yes—
135C2×C4×C283C2411yescharacteristic
136C8×C283C2×D414yescomplemented
137C22×C283—21no—
138C22×C283—21no—
139C22×C283—41no—
140C22×C283—42no—
141C22×C283—42no—
142C22×C283—41no—
143C22×C283—42no—
144C22×C283—41no—
145C22×C283—84no—
146C22×C283#12112yes—
147C22×C283C22×D411yescharacteristic
148C2×D283—21no—
149C2×D283—21no—
150C2×D283—21no—
151C2×D283—21no—
152C2×D283—21no—
153C2×D283—44no—
154C2×D283AGL(1,ℤ/8)12yescomplemented
155C4×C283—41no—
156C4×C283#12112yes—
157C4×C283C22×D411yescharacteristic
158C4×C283C2×#4111yescharacteristic
159Dic283—21no—
160Dic283—21no—
161Dic283—42no—
162Dic283—41no—
163Dic283—41no—
164C2×C283—22no—
165C2×C283—81no—
166C2×C283—82no—
167C2×C283C2×#12111yescharacteristic
168C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C283#57511yescharacteristic
170D283—22no—
171D283—41no—
172C283#305511yescharacteristic, complemented, Sylow
173#3055—2831nomaximal, Sylow
174C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
175C22×M16—2831no—
176#525—2831no—
177C2×#121—2831no—
178#469—2832no—
179#468—2832no—
180#466—2832no—
181#465—2832no—
182#428—2831no—
183#418—2831no—
184#417—2831no—
185C2×#41C2×D28311yescharacteristic
186C22×D4—5661no—
187C22×D4C2×D28311yescharacteristic
188C23×C4—2831no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D28312yescomplemented
191C2×D8C2×D28312yescomplemented
192C2×M16—2831no—
193C2×M16—2832no—
194C2×M16—5664no—
195C2×M16C2×D28311yescharacteristic
196C22×C8—2832no—
197#123—5661no—
198C22⋊Q8—2831no—
199C22⋊Q8—2831no—
200#121—2832no—
201#121—2832no—
202#121—2831no—
203#121—2831no—
204#121—5661no—
205#121—5661no—
206#120—5661no—
207C4×D4—5661no—
208C4×D4—5661no—
209#117—2831no—
210C2×C4⋊C4—2831no—
211C2×C22⋊C4—5661no—
212#107—2832no—
213#106—2832no—
214#103—2832no—
215#102—2832no—
216#102—2832no—
217#102—2832no—
218C24—21no—
219C24—2831no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D28311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—5661no—
228C2×D4—5661no—
229C2×D4—5662no—
230C2×D4—5662no—
231C2×D4—11324no—
232C2×D4C22×D28311yescharacteristic
233C2×D4C22×D28312yes—
234C22×C4—21no—
235C22×C4—2831no—
236C22×C4—2831no—
237C22×C4—2831no—
238C22×C4—2831no—
239C22×C4—2831no—
240C22×C4—5662no—
241C22×C4—5662no—
242C22×C4—5661no—
243C22×C4C22×D28311yescharacteristic
244SD16—24no—
245D8—24no—
246M16—5662no—
247M16—5664no—
248M16C22⋊D28314yescomplemented
249C2×C8—2832no—
250C2×C8—5664no—
251C2×C8C22×D28312yes—
252C4⋊C4—2831no—
253C4⋊C4—2831no—
254C4⋊C4—2832no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C22⋊C4—5661no—
259C22⋊C4—5661no—
260C22⋊C4—5662no—
261C22⋊C4—5662no—
262C22⋊C4—5661no—
263C22⋊C4—5661no—
264C22⋊C4—5661no—
265C42—5661no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—2831no—
272C23—2831no—
273C23—5662no—
274C23—5662no—
275C23—11321no—
276C23D4×D28311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—11324no—
286D4—11322no—
287D4—11324no—
288D4—11322no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—2831no—
292C2×C4—2831no—
293C2×C4—5662no—
294C2×C4—5662no—
295C2×C4—5662no—
296C2×C4—5661no—
297C2×C4—5661no—
298C2×C4—5661no—
299C2×C4—5661no—
300C2×C4—5661no—
301C2×C4—5661no—
302C2×C4—5662no—
303C2×C4—11322no—
304C2×C4D4×D28311yescharacteristic
305C2×C4C2×C22⋊D28312yes—
306C2×C4#1798539112yes—
307C2×C4C23×D28311yescharacteristic
308C8—5662no—
309C8C2×C22⋊D28314yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—2832no—
320C22—5661no—
321C22—5661no—
322C22—5661no—
323C22—5661no—
324C22—5661no—
325C22—11324no—
326C22#2258358112yes—
327C22C2×D4×D28311yescharacteristic
328C4—41no—
329C4—5661no—
330C4—5661no—
331C4—11322no—
332C4—11321no—
333C4—11321no—
334C4#2258358112yes—
335C4C22×C22⋊D28311yescharacteristic
336C4C2×#1798539111yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—5662no—
341C2—11321no—
342C2C2×#2258358111yescharacteristic
343C2AGL(1,ℤ/8)×D28311yescharacteristic
344C2#3182726111yescharacteristic
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#3055
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×M16×D283C2semidirect
2. #31826972C2semidirect
3. C2×#22583581C2semidirect
4. #31826715C2semidirect
5. #31826708C2semidirect
6. #31826698C2semidirect
7. #31826691C2semidirect
8. #31826507C2semidirect
9. #31826470C2semidirect
10. #31826466C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
15. C2×#17985375C22semidirect
17. C2×C8×D283C22semidirect
19. #22583586C22semidirect
20. #22583586C22semidirect
25. #22583581C22semidirect
26. #22583581C22semidirect
33. #22583544C22semidirect
34. #22583542C22semidirect
35. #22583534C22semidirect
36. #22583531C22semidirect
37. #22583531C22semidirect
38. #22583531C22semidirect
42. C2×SD16×C283C22semidirect
43. C2×D8×C283C22semidirect
91. M16×C283D4semidirect
136. C8×C283C2×D4semidirect
154. C2×D283AGL(1,ℤ/8)semidirect
172. C283#3055semidirect
174. C2×AGL(1,ℤ/8)D283semidirect
190. C2×SD16C2×D283semidirect
191. C2×D8C2×D283semidirect
248. M16C22⋊D283semidirect
309. C8C2×C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2832
#318269722
C2×#225835812
#318267152
#318267082
#318266982
#318266912
#318265072
#318264702
#318264662
C2×AGL(1,ℤ/8)×C2832
#3055283