Finite Group Databasebeta

← #49133047

Subgroups — #49133047

Order 36224.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#31827273C212yesmaximal, complemented
2#31827267C211yesmaximal, characteristic, complemented
3#31826850C211yesmaximal, characteristic, complemented
4#31826737C211yesmaximal, characteristic, complemented
5#31826735C211yesmaximal, characteristic, complemented
6#31826730C212yesmaximal, complemented
7#31826641C212yesmaximal, complemented
8#31826638C212yesmaximal, complemented
9#31826582C211yesmaximal, characteristic, complemented
10#31826580C211yesmaximal, characteristic, complemented
11#578×C283C211yesmaximal, characteristic, complemented
12#22583656C2212yes—
13#22583641—22no—
14#22583641C2212yes—
15#22583637—22no—
16#22583635—22no—
17#22583635C2212yes—
18#22583635C2212yescomplemented
19#22583633—21no—
20#22583630—22no—
21#22583630C2211yescharacteristic
22C2×#17985379—22no—
23C2×#17985379C2212yescomplemented
24C2×D2264C2211yescharacteristic
25D8×D283—21no—
26#22583617C2212yescomplemented
27#22583616C2211yescharacteristic
28#22583599C2212yescomplemented
29#22583546C2211yescharacteristic, complemented
30#22583545C2211yescharacteristic, complemented
31#22583537C2211yescharacteristic, complemented
32#22583537C2212yescomplemented
33#22583536C2211yescharacteristic
34#22583529C2212yescomplemented
35#22583526C2212yescomplemented
36C8×Dic283C2211yescharacteristic, complemented
372−1+4×C283C2212yescomplemented
38#137×C283—22no—
39#137×C283C2212yescomplemented
40#135×C283—21no—
41#135×C283C2211yescharacteristic
42C2×Q16×C283—21no—
43C2×Q16×C283C2211yescharacteristic, complemented
44#131×C283C2211yescharacteristic, complemented
45#17985393—22no—
46#17985392—22no—
47#17985392C2311yescharacteristic
48C2×D1132—22no—
49C2×D1132C2312yes—
50D4×D283—22no—
51D4×D283—42no—
52#17985382—22no—
53#17985382D411yescharacteristic
54#17985380D411yescharacteristic
55#17985379—22no—
56#17985379—22no—
57#17985379—22no—
58#17985379—22no—
59#17985379—22no—
60#17985379D411yescharacteristic
61D2264—21no—
62D2264—21no—
63#17985377—22no—
64#17985377—22no—
65#17985377D411yescharacteristic
66#17985376—22no—
67#17985376C2311yescharacteristic
68#17985376C2312yes—
69#17985375—21no—
70C2×C283⋊C8—22no—
71C2×C283⋊C8C2311yescharacteristic
72C8×D283—21no—
73(C2×D283)⋊C4—42no—
74C4×Dic283C2311yescharacteristic
75#41×C283—22no—
76#41×C283—22no—
77#41×C283—42no—
78#41×C283C2312yes—
79#41×C283C2311yescharacteristic
80C2×Q8×C283—22no—
81C2×Q8×C283—22no—
82C2×Q8×C283C2312yes—
83Q16×C283—22no—
84Q16×C283—21no—
85Q16×C283—21no—
86Q16×C283D411yescharacteristic
87SD16×C283—22no—
88SD16×C283D412yes—
89D8×C283D411yescharacteristic
90M16×C283—21no—
91M16×C283C2311yescharacteristic
92C2×C8×C283—21no—
93C2×C8×C283C2311yescharacteristic
94C22×D283—22no—
95C22×D283—42no—
96Dic566C2×D411yescharacteristic
97D1132—22no—
98D1132—22no—
99D1132—42no—
100D1132C2×D411yescharacteristic
101C22⋊D283—21no—
102C22⋊D283—42no—
103C2×Dic283—21no—
104C4×D283—21no—
105C4×D283—42no—
106C283⋊C8—21no—
107C283⋊C8—22no—
108C283⋊C8C2×D411yescharacteristic
109C283⋊C8C2×D411yescharacteristic
110Q8×C283—22no—
111Q8×C283—22no—
112Q8×C283—42no—
113Q8×C283C2×D411yescharacteristic
114Q8×C283C2×D412yes—
115D4×C283—21no—
116D4×C283—22no—
117D4×C283—42no—
118D4×C283C2×D411yescharacteristic
119D4×C283C2×D412yes—
120C2×C4×C283—21no—
121C2×C4×C283—22no—
122C2×C4×C283—22no—
123C2×C4×C283—42no—
124C2×C4×C283—42no—
125C2×C4×C283C2411yescharacteristic
126C8×C283—21no—
127C8×C283C2×D411yescharacteristic
128C8×C283C2×D411yescharacteristic
129C22×C283—21no—
130C22×C283—22no—
131C22×C283H(5,𝔽2)11yescharacteristic
132C2×D283—21no—
133C2×D283—42no—
134C2×D283—82no—
135C2×D283—162no—
136C4×C283—21no—
137C4×C283—22no—
138C4×C283—42no—
139C4×C283C22×D411yescharacteristic
140C4×C283C22×D411yescharacteristic
141Dic283—21no—
142Dic283—41no—
143C2×C283—21no—
144C2×C283—42no—
145C2×C283—41no—
146C2×C283D4211yescharacteristic
147D283—41no—
148D283—82no—
149C283#329011yescharacteristic, complemented, Sylow
150#3290—2831nomaximal, Sylow
151#578D28311yescharacteristic, complemented
152#577—2832no—
153#576—2831no—
154#495—2831no—
155#473—2831no—
156#472—2831no—
157#471—2832no—
158#455—2832no—
159#454—2832no—
160#443—2831no—
161#443—2831no—
1622−1+4C2×D28312yescomplemented
163H(5,𝔽2)—2832no—
164#137—22no—
165#137—2832no—
166#137—5662no—
167#137C2×D28312yescomplemented
168AGL(1,ℤ/8)—2832no—
169AGL(1,ℤ/8)—2832no—
170AGL(1,ℤ/8)—5662no—
171AGL(1,ℤ/8)—5662no—
172#135—21no—
173#135—2831no—
174#135—5662no—
175#135—5661no—
176#135C2×D28311yescharacteristic
177C2×Q16—21no—
178C2×Q16C2×D28311yescharacteristic, complemented
179C2×SD16—2832no—
180C2×SD16—5662no—
181C2×D8—2831no—
182C2×D8—5661no—
183#131—2832no—
184#131—2831no—
185#131C2×D28311yescharacteristic, complemented
186#124—2832no—
187#108—2831no—
188#108—2831no—
189#104—2831no—
190#104—2832no—
191#104—2831no—
192#101—2832no—
193#100—2832no—
194C4×C8—2831no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—2831no—
199#41—5662no—
200#41—5662no—
201#41C22×D28311yescharacteristic
202#41C22×D28312yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D28312yes—
206C2×D4—2832no—
207C2×D4—5662no—
208C2×D4—5662no—
209C2×D4—11322no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—2831no—
214Q16—5662no—
215Q16C22⋊D28311yescharacteristic
216SD16—22no—
217SD16—2831no—
218SD16—2831no—
219SD16—5662no—
220SD16—5662no—
221SD16—5662no—
222SD16—5662no—
223SD16—5662no—
224SD16C22⋊D28312yes—
225D8—2831no—
226D8—5662no—
227D8—5662no—
228D8—5661no—
229D8—5661no—
230D8C22⋊D28311yescharacteristic
231M16—21no—
232M16—2832no—
233M16—2831no—
234M16—5662no—
235M16—5661no—
236M16C22×D28311yescharacteristic
237C2×C8—21no—
238C2×C8—2831no—
239C2×C8—5662no—
240C2×C8—5661no—
241C2×C8C22×D28311yescharacteristic
242C22⋊C4—11322no—
243C42—2831no—
244C23—5662no—
245C23—11322no—
246Q8—22no—
247Q8—22no—
248Q8—42no—
249Q8—2831no—
250Q8D4×D28311yescharacteristic
251Q8C2×C22⋊D28312yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—2831no—
256D4—5661no—
257D4—5662no—
258D4—5662no—
259D4—11322no—
260D4—11322no—
261D4D4×D28311yescharacteristic
262D4C2×C22⋊D28312yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—5661no—
269C2×C4—5661no—
270C2×C4—11322no—
271C2×C4C23×D28311yescharacteristic
272C8—21no—
273C8—2831no—
274C8—2831no—
275C8—5662no—
276C8—5661no—
277C8C2×C22⋊D28311yescharacteristic
278C8C2×C22⋊D28311yescharacteristic
279C22—21no—
280C22—22no—
281C22—5661no—
282C22—11322no—
283C22—45282no—
284C22—22642no—
285C22#2258365511yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—5661no—
290C4—11321no—
291C4C2×D4×D28311yescharacteristic
292C4C22×C22⋊D28311yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—11321no—
297C2—22642no—
298C2D4×C22⋊D28311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#578×C283C2
socleC2×C283D42
hypercenter#578D283
hypocenterC283#3290
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #31827273C2semidirect
2. #31827267C2semidirect
3. #31826850C2semidirect
4. #31826737C2semidirect
5. #31826735C2semidirect
6. #31826730C2semidirect
7. #31826641C2semidirect
8. #31826638C2semidirect
9. #31826582C2semidirect
10. #31826580C2semidirect
11. #578×C283C2semidirect
18. #22583635C22semidirect
23. C2×#17985379C22semidirect
26. #22583617C22semidirect
28. #22583599C22semidirect
29. #22583546C22semidirect
30. #22583545C22semidirect
31. #22583537C22semidirect
32. #22583537C22semidirect
34. #22583529C22semidirect
35. #22583526C22semidirect
36. C8×Dic283C22semidirect
37. 2−1+4×C283C22semidirect
39. #137×C283C22semidirect
43. C2×Q16×C283C22semidirect
44. #131×C283C22semidirect
149. C283#3290semidirect
151. #578D283semidirect
162. 2−1+4C2×D283semidirect
167. #137C2×D283semidirect
178. C2×Q16C2×D283semidirect
185. #131C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272732
#318272672
#318268502
#318267372
#318267352
#318267302
#318266412
#318266382
#318265822
#318265802
#578×C2832
#3290283