Finite Group Databasebeta

← #49134407

Subgroups — #49134407

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#31827279C211yesmaximal, characteristic, complemented
2#31827266C211yesmaximal, characteristic, complemented
3#31826908C211yesmaximal, characteristic, complemented
4#31826904C212yesmaximal, complemented
5#31826899C212yesmaximal, complemented
6#31826895C211yesmaximal, characteristic, complemented
7#31826894C212yesmaximal, complemented
8C2×#22583554C211yesmaximal, characteristic, complemented
9SD32×D283C212yesmaximal, complemented
10#31826883C211yesmaximal, characteristic, complemented
11#509×C283C211yesmaximal, characteristic, complemented
12#22583657—21no—
13#22583655—21no—
14#22583640—21no—
15#22583639—22no—
16#22583636—21no—
17#22583634—22no—
18#22583632C2211yescharacteristic
19#22583629C2212yes—
20#22583629C2212yescomplemented
21C2×Dic1132C2211yescharacteristic, complemented
22Q16×D283C2212yescomplemented
23D8×D283C2212yes—
24#22583616C2211yescharacteristic
25Dic2264C2212yescomplemented
26#22583556C2211yescharacteristic, complemented
27#22583555C2212yescomplemented
28#22583554C2212yescomplemented
29#22583554C2212yescomplemented
30#22583554C2211yescharacteristic, complemented
31#22583553C2211yescharacteristic, complemented
32#22583551C2211yescharacteristic, complemented
33#22583549C2212yescomplemented
34C2×C283⋊C16C2211yescharacteristic, complemented
35C16×D283C2212yescomplemented
36#135×C283C2211yescharacteristic, complemented
37C2×D8×C283C2211yescharacteristic
38SD32×C283C2212yescomplemented
39D16×C283C2212yescomplemented
40M32×C283C2211yescharacteristic, complemented
41#17985392—21no—
42#17985392—41no—
43#17985392D411yescharacteristic
44#17985391—22no—
45#17985391—22no—
46#17985391—41no—
47#17985391—42no—
48C2×Dic566—21no—
49C2×Dic566—41no—
50Q8×D283—22no—
51C2×C22⋊D283—42no—
52D4×D283—22no—
53D4×D283—41no—
54Dic1132—21no—
55Dic1132C2312yes—
56Dic1132C2311yescharacteristic
57#17985382—22no—
58#17985381—21no—
59#17985380—22no—
60#17985380—22no—
61D2264C2311yescharacteristic
62#17985377—22no—
63#17985376D411yescharacteristic
64#17985375D412yes—
65C8×D283C2312yes—
66C283⋊C16C2311yescharacteristic
67C283⋊C16C2311yescharacteristic
68#41×C283—21no—
69C2×D4×C283—21no—
70Q16×C283C2311yescharacteristic
71SD16×C283—21no—
72D8×C283—21no—
73D8×C283C2312yes—
74D8×C283C2311yescharacteristic
75C2×C8×C283C2311yescharacteristic
76C16×C283C2312yes—
77C22×D283—42no—
78Dic566—21no—
79Dic566—22no—
80Dic566—41no—
81Dic566—42no—
82Dic566D811yescharacteristic
83D1132—21no—
84D1132D811yescharacteristic
85C22⋊D283—41no—
86C22⋊D283—42no—
87C22⋊D283—42no—
88C22⋊D283—42no—
89C22⋊D283D812yes—
90C2×Dic283—41no—
91C2×Dic283—42no—
92C2×Dic283—42no—
93C4×D283—42no—
94C4×D283—41no—
95C4×D283C2×D412yes—
96C283⋊C8C2×D412yes—
97C23×C283—41no—
98Q8×C283—21no—
99D4×C283—22no—
100D4×C283—21no—
101D4×C283—41no—
102D4×C283—41no—
103C2×C4×C283—41no—
104C2×C4×C283C2×D411yescharacteristic
105C8×C283C2×D411yescharacteristic
106C8×C283C2411yescharacteristic
107C22×C283—41no—
108C22×C283—42no—
109C22×C283—161no—
110C22×C283C2×D811yescharacteristic
111C2×D283—41no—
112C2×D283—162no—
113C2×D283C2×D812yes—
114C4×C283—41no—
115C4×C283C2×D811yescharacteristic
116C4×C283C22×D411yescharacteristic
117Dic283—41no—
118Dic283—42no—
119Dic283C2×D812yes—
120C2×C283—21no—
121C2×C283—82no—
122C2×C283—81no—
123C2×C283C22×D811yescharacteristic
124D283—22no—
125D283—81no—
126C283#341611yescharacteristic, complemented, Sylow
127#3416—2831nomaximal, Sylow
128#578—2831no—
129#576—2831no—
130#510—2832no—
131#510—2831no—
132#509—2832no—
133#509D28311yescharacteristic, complemented
134#508—2831no—
135#508—2832no—
136C2×SD32—2831no—
137C2×SD32—2832no—
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#135C2×D28311yescharacteristic, complemented
149C2×Q16—2832no—
150C2×Q16—2831no—
151C2×D8—2832no—
152C2×D8C2×D28311yescharacteristic
153#131—2831no—
154Q32—2831no—
155Q32—2832no—
156SD32—2831no—
157SD32—2832no—
158SD32—2832no—
159SD32—2831no—
160SD32—2832no—
161SD32C2×D28312yescomplemented
162D16—2831no—
163D16C2×D28312yescomplemented
164M32—2832no—
165M32C2×D28311yescharacteristic, complemented
166C2×C16—2831no—
167C2×C16—2832no—
168#41—21no—
169#41—2831no—
170#41—5662no—
171#41—5662no—
172#41—5661no—
173#41—11321no—
174#41—11322no—
175#41—11321no—
176C2×Q8—5662no—
177C2×Q8—5661no—
178C2×Q8—11321no—
179C2×D4—21no—
180C2×D4—5662no—
181C2×D4—11321no—
182C2×D4—11322no—
183Q16—2832no—
184Q16—2831no—
185Q16—5662no—
186Q16—5661no—
187Q16C22×D28311yescharacteristic
188SD16—21no—
189SD16—5662no—
190SD16—5662no—
191SD16—5661no—
192D8—21no—
193D8—2831no—
194D8—5662no—
195D8C22×D28311yescharacteristic
196D8C22×D28312yes—
197M16—2831no—
198M16—2832no—
199C2×C8—2832no—
200C2×C8C22×D28311yescharacteristic
201C16—2831no—
202C16—2831no—
203C16C22×D28312yes—
204C23—41no—
205C23—11322no—
206Q8—21no—
207Q8—2831no—
208Q8—5661no—
209Q8—5662no—
210Q8—11322no—
211Q8—11321no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2832no—
217D4—2831no—
218D4—5661no—
219D4—11322no—
220D4—11322no—
221D4—11322no—
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—41no—
235C22—42no—
236C22—161no—
237C22—2832no—
238C22—11321no—
239C22—45282no—
240C22D8×D28311yescharacteristic
241C4—41no—
242C4—2832no—
243C4—11321no—
244C4—11322no—
245C4D8×D28311yescharacteristic
246C4C2×D4×D28311yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—5662no—
251C2—22641no—
252C2C2×D8×D28311yescharacteristic
subgroupG/H
centerC2C2×D8×D283
commutator, cosocleC8×C283C24
FrattiniC8C23×D283
Fitting#509×C283C2
socleC2×C283C22×D8
hypercenter#509D283
hypocenterC283#3416
LCSG ▷ C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31827279C2semidirect
2. #31827266C2semidirect
3. #31826908C2semidirect
4. #31826904C2semidirect
5. #31826899C2semidirect
6. #31826895C2semidirect
7. #31826894C2semidirect
8. C2×#22583554C2semidirect
9. SD32×D283C2semidirect
10. #31826883C2semidirect
11. #509×C283C2semidirect
20. #22583629C22semidirect
21. C2×Dic1132C22semidirect
22. Q16×D283C22semidirect
25. Dic2264C22semidirect
26. #22583556C22semidirect
27. #22583555C22semidirect
28. #22583554C22semidirect
29. #22583554C22semidirect
30. #22583554C22semidirect
31. #22583553C22semidirect
32. #22583551C22semidirect
33. #22583549C22semidirect
34. C2×C283⋊C16C22semidirect
35. C16×D283C22semidirect
36. #135×C283C22semidirect
38. SD32×C283C22semidirect
39. D16×C283C22semidirect
40. M32×C283C22semidirect
126. C283#3416semidirect
133. #509D283semidirect
148. #135C2×D283semidirect
161. SD32C2×D283semidirect
163. D16C2×D283semidirect
165. M32C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272792
#318272662
#318269082
#318269042
#318268992
#318268952
#318268942
C2×#225835542
SD32×D2832
#318268832
#509×C2832
#3416283