Finite Group Databasebeta

← #49134403

Subgroups — #49134403

Order 36224.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#31827278C211yesmaximal, characteristic, complemented
2#31827266C211yesmaximal, characteristic, complemented
3#31826906C211yesmaximal, characteristic, complemented
4#31826904C212yesmaximal, complemented
5#31826902C212yesmaximal, complemented
6#31826901C211yesmaximal, characteristic, complemented
7#31826897C211yesmaximal, characteristic, complemented
8#31826896C212yesmaximal, complemented
9SD32×D283C212yesmaximal, complemented
10#31826881C211yesmaximal, characteristic, complemented
11C2×SD32×C283C211yesmaximal, characteristic, complemented
12#22583658—21no—
13#22583655—21no—
14#22583642—22no—
15#22583641—21no—
16#22583636—21no—
17#22583634—22no—
18#22583633C2212yescomplemented
19#22583632C2211yescharacteristic
20#22583632C2211yescharacteristic, complemented
21#22583629C2212yes—
22Q16×D283C2212yescomplemented
23D8×D283C2212yes—
24#22583616C2211yescharacteristic
25Dic2264C2211yescharacteristic, complemented
26#22583556C2212yescomplemented
27#22583555C2211yescharacteristic, complemented
28#22583555C2211yescharacteristic, complemented
29#22583554C2212yescomplemented
30#22583553C2212yescomplemented
31D4528C2211yescharacteristic, complemented
32#22583551C2212yescomplemented
33#22583550C2211yescharacteristic, complemented
34#22583549C2212yescomplemented
35C16×D283C2212yescomplemented
36C2×Q16×C283C2211yescharacteristic, complemented
37C2×D8×C283C2211yescharacteristic
38SD32×C283C2212yescomplemented
39SD32×C283C2212yescomplemented
40C2×C16×C283C2211yescharacteristic, complemented
41#17985393—22no—
42#17985393—41no—
43#17985392—21no—
44#17985392—21no—
45#17985392—42no—
46#17985392D411yescharacteristic
47#17985391—22no—
48#17985391—41no—
49Q8×D283—22no—
50Q8×D283—41no—
51C2×C22⋊D283—42no—
52D4×D283—22no—
53D4×D283—41no—
54Dic1132C2311yescharacteristic
55Dic1132C2311yescharacteristic
56#17985382—22no—
57#17985381—21no—
58#17985381—21no—
59#17985380—22no—
60#17985379—22no—
61D2264C2311yescharacteristic
62D2264C2311yescharacteristic
63#17985377—22no—
64#17985376D411yescharacteristic
65#17985375D412yes—
66C8×D283C2312yes—
67C283⋊C16C2312yes—
68C2×Q8×C283—21no—
69C2×D4×C283—21no—
70Q16×C283—21no—
71Q16×C283C2312yes—
72D8×C283—21no—
73D8×C283C2312yes—
74C2×C8×C283C2311yescharacteristic
75C16×C283C2311yescharacteristic
76C16×C283C2311yescharacteristic
77C22×D283—42no—
78Dic566—21no—
79Dic566—21no—
80Dic566—42no—
81Dic566D811yescharacteristic
82D1132—21no—
83D1132—21no—
84D1132—42no—
85D1132D811yescharacteristic
86C22⋊D283—41no—
87C22⋊D283—41no—
88C22⋊D283—42no—
89C22⋊D283—42no—
90C22⋊D283D812yes—
91C2×Dic283—42no—
92C4×D283—42no—
93C4×D283—41no—
94C4×D283—41no—
95C4×D283—42no—
96C4×D283C2×D412yes—
97C283⋊C8C2×D412yes—
98C23×C283—41no—
99Q8×C283—22no—
100Q8×C283—41no—
101D4×C283—22no—
102D4×C283—41no—
103C2×C4×C283—41no—
104C2×C4×C283C2×D411yescharacteristic
105C8×C283C2×D411yescharacteristic
106C8×C283C2411yescharacteristic
107C22×C283—42no—
108C22×C283—161no—
109C22×C283C2×D811yescharacteristic
110C2×D283—41no—
111C2×D283—41no—
112C2×D283—162no—
113C2×D283C2×D812yes—
114C4×C283—42no—
115C4×C283C2×D811yescharacteristic
116C4×C283C22×D411yescharacteristic
117Dic283—41no—
118Dic283—41no—
119Dic283C2×D812yes—
120C2×C283—21no—
121C2×C283—82no—
122C2×C283C22×D811yescharacteristic
123D283—22no—
124D283—81no—
125D283—81no—
126C283#341611yescharacteristic, complemented, Sylow
127#3416—2831nomaximal, Sylow
128#578—2831no—
129#576—2831no—
130#510—2831no—
131#510—2832no—
132#509—2831no—
133#509—2832no—
134#508—2832no—
135#508—2831no—
136C2×SD32—2832no—
137C2×SD32D28311yescharacteristic, complemented
138#504—2831no—
1392−1+4—5661no—
140H(5,𝔽2)—5661no—
141#137—5662no—
142#137—5661no—
143AGL(1,ℤ/8)—5662no—
144AGL(1,ℤ/8)—5661no—
145#135—2832no—
146#135—2832no—
147#135—2831no—
148#135—2831no—
149C2×Q16—2832no—
150C2×Q16C2×D28311yescharacteristic, complemented
151C2×D8—2832no—
152C2×D8C2×D28311yescharacteristic
153#131—2831no—
154Q32—2832no—
155Q32—2831no—
156SD32—2832no—
157SD32—2832no—
158SD32—2831no—
159SD32—2831no—
160SD32C2×D28312yescomplemented
161SD32C2×D28312yescomplemented
162D16—2832no—
163D16—2831no—
164M32—2831no—
165M32—2832no—
166C2×C16—2832no—
167C2×C16C2×D28311yescharacteristic, complemented
168#41—2831no—
169#41—5662no—
170#41—5662no—
171#41—5661no—
172#41—5661no—
173#41—11322no—
174#41—11321no—
175#41—11321no—
176C2×Q8—21no—
177C2×Q8—5662no—
178C2×Q8—11321no—
179C2×D4—21no—
180C2×D4—5662no—
181C2×D4—11321no—
182C2×D4—11322no—
183Q16—21no—
184Q16—2831no—
185Q16—2831no—
186Q16—5662no—
187Q16C22×D28312yes—
188SD16—5662no—
189SD16—5662no—
190SD16—5661no—
191SD16—5661no—
192D8—21no—
193D8—2831no—
194D8—2831no—
195D8—5662no—
196D8C22×D28312yes—
197M16—2831no—
198M16—2832no—
199C2×C8—2832no—
200C2×C8C22×D28311yescharacteristic
201C16—2832no—
202C16C22×D28311yescharacteristic
203C16C22×D28311yescharacteristic
204C23—41no—
205C23—11322no—
206Q8—22no—
207Q8—41no—
208Q8—2831no—
209Q8—5661no—
210Q8—5661no—
211Q8—11322no—
212D4—22no—
213D4—41no—
214D4—2832no—
215D4—2831no—
216D4—5661no—
217D4—5661no—
218D4—11322no—
219D4—11322no—
220D4—11322no—
221D4—11321no—
222D4—11321no—
223C2×C4—41no—
224C2×C4—2832no—
225C2×C4—11322no—
226C2×C4—11322no—
227C2×C4—11322no—
228C2×C4—11321no—
229C2×C4—11321no—
230C2×C4D4×D28311yescharacteristic
231C8—2832no—
232C8D4×D28311yescharacteristic
233C8C23×D28311yescharacteristic
234C22—42no—
235C22—161no—
236C22—2832no—
237C22—11321no—
238C22—11321no—
239C22—45282no—
240C22D8×D28311yescharacteristic
241C4—42no—
242C4—2832no—
243C4—11321no—
244C4—11321no—
245C4D8×D28311yescharacteristic
246C4C2×D4×D28311yescharacteristic
247C2—21no—
248C2—82no—
249C2—5662no—
250C2—22641no—
251C2—22641no—
252C2C2×D8×D28311yescharacteristic
subgroupG/H
centerC2C2×D8×D283
commutator, cosocleC8×C283C24
FrattiniC8C23×D283
FittingC2×SD32×C283C2
socleC2×C283C22×D8
hypercenterC2×SD32D283
hypocenterC283#3416
LCSG ▷ C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #31827278C2semidirect
2. #31827266C2semidirect
3. #31826906C2semidirect
4. #31826904C2semidirect
5. #31826902C2semidirect
6. #31826901C2semidirect
7. #31826897C2semidirect
8. #31826896C2semidirect
9. SD32×D283C2semidirect
10. #31826881C2semidirect
11. C2×SD32×C283C2semidirect
18. #22583633C22semidirect
20. #22583632C22semidirect
22. Q16×D283C22semidirect
25. Dic2264C22semidirect
26. #22583556C22semidirect
27. #22583555C22semidirect
28. #22583555C22semidirect
29. #22583554C22semidirect
30. #22583553C22semidirect
31. D4528C22semidirect
32. #22583551C22semidirect
33. #22583550C22semidirect
34. #22583549C22semidirect
35. C16×D283C22semidirect
36. C2×Q16×C283C22semidirect
38. SD32×C283C22semidirect
39. SD32×C283C22semidirect
40. C2×C16×C283C22semidirect
126. C283#3416semidirect
137. C2×SD32D283semidirect
150. C2×Q16C2×D283semidirect
160. SD32C2×D283semidirect
161. SD32C2×D283semidirect
167. C2×C16C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272782
#318272662
#318269062
#318269042
#318269022
#318269012
#318268972
#318268962
SD32×D2832
#318268812
C2×SD32×C2832
#3416283