Finite Group Databasebeta

← #27184274

Subgroups — #27184274

Order 12928.

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#20265990C211yesmaximal, characteristic, complemented
2#20265978C211yesmaximal, characteristic, complemented
3#20265618C211yesmaximal, characteristic, complemented
4#20265616C212yesmaximal, complemented
5#20265614C212yesmaximal, complemented
6#20265613C211yesmaximal, characteristic, complemented
7#20265609C211yesmaximal, characteristic, complemented
8#20265608C212yesmaximal, complemented
9SD32×D101C212yesmaximal, complemented
10#20265593C211yesmaximal, characteristic, complemented
11C2×SD32×C101C211yesmaximal, characteristic, complemented
12#15692303—21no—
13#15692300—21no—
14#15692287—22no—
15#15692286—21no—
16#15692281—21no—
17#15692279—22no—
18#15692278C2212yescomplemented
19#15692277C2211yescharacteristic
20#15692277C2211yescharacteristic, complemented
21#15692274C2212yes—
22Q16×D101C2212yescomplemented
23D8×D101C2212yes—
24#15692261C2211yescharacteristic
25Dic808C2211yescharacteristic, complemented
26#15692201C2212yescomplemented
27#15692200C2211yescharacteristic, complemented
28#15692200C2211yescharacteristic, complemented
29#15692199C2212yescomplemented
30#15692198C2212yescomplemented
31D1616C2211yescharacteristic, complemented
32#15692196C2212yescomplemented
33#15692195C2211yescharacteristic, complemented
34#15692194C2212yescomplemented
35C16×D101C2212yescomplemented
36C2×Q16×C101C2211yescharacteristic, complemented
37C2×D8×C101C2211yescharacteristic
38SD32×C101C2212yescomplemented
39SD32×C101C2212yescomplemented
40C2×C16×C101C2211yescharacteristic, complemented
41#13108075—22no—
42#13108075—41no—
43#13108074—21no—
44#13108074—21no—
45#13108074—42no—
46#13108074D411yescharacteristic
47#13108073—22no—
48#13108073—41no—
49Q8×D101—22no—
50Q8×D101—41no—
51C2×C22⋊D101—42no—
52D4×D101—22no—
53D4×D101—41no—
54Dic404C2311yescharacteristic
55Dic404C2311yescharacteristic
56#13108064—22no—
57#13108063—21no—
58#13108063—21no—
59#13108062—22no—
60#13108061—22no—
61D808C2311yescharacteristic
62D808C2311yescharacteristic
63#13108059—22no—
64#13108058D411yescharacteristic
65#13108057D412yes—
66C8×D101C2312yes—
67#13108049C2312yes—
68C2×Q8×C101—21no—
69C2×D4×C101—21no—
70Q16×C101—21no—
71Q16×C101C2312yes—
72D8×C101—21no—
73D8×C101C2312yes—
74C2×C8×C101C2311yescharacteristic
75C16×C101C2311yescharacteristic
76C16×C101C2311yescharacteristic
77C23×C101—41no—
78C22×D101—42no—
79Q8×C101—22no—
80Q8×C101—41no—
81D4×C101—22no—
82D4×C101—41no—
83C2×C4×C101—41no—
84C2×C4×C101C2×D411yescharacteristic
85C22⋊D101—41no—
86C22⋊D101—41no—
87C22⋊D101—42no—
88C22⋊D101—42no—
89C22⋊D101D812yes—
90C2×Dic101—42no—
91D404—21no—
92D404—21no—
93D404—42no—
94D404D811yescharacteristic
95C4×D101—42no—
96C4×D101—41no—
97C4×D101—41no—
98C4×D101—42no—
99C4×D101C2×D412yes—
100Dic202—21no—
101Dic202—21no—
102Dic202—42no—
103Dic202D811yescharacteristic
104C8×C101C2×D411yescharacteristic
105C8×C101C2411yescharacteristic
106#11717579C2×D412yes—
107C22×C101—42no—
108C22×C101—161no—
109C22×C101C2×D811yescharacteristic
110C2×D101—41no—
111C2×D101—41no—
112C2×D101—162no—
113C2×D101C2×D812yes—
114C4×C101—42no—
115C4×C101C2×D811yescharacteristic
116C4×C101C22×D411yescharacteristic
117Dic101—41no—
118Dic101—41no—
119Dic101C2×D812yes—
120C2×C101—21no—
121C2×C101—82no—
122C2×C101C22×D811yescharacteristic
123D101—22no—
124D101—81no—
125D101—81no—
126#3416—1011nomaximal, Sylow
127C101#341611yescharacteristic, complemented, Sylow
128#578—1011no—
129#576—1011no—
130#510—1011no—
131#510—1012no—
132#509—1011no—
133#509—1012no—
134#508—1012no—
135#508—1011no—
136C2×SD32—1012no—
137C2×SD32D10111yescharacteristic, complemented
138#504—1011no—
1392−1+4—2021no—
140H(5,𝔽2)—2021no—
141#137—2022no—
142#137—2021no—
143AGL(1,ℤ/8)—2022no—
144AGL(1,ℤ/8)—2021no—
145#135—1012no—
146#135—1012no—
147#135—1011no—
148#135—1011no—
149C2×Q16—1012no—
150C2×Q16C2×D10111yescharacteristic, complemented
151C2×D8—1012no—
152C2×D8C2×D10111yescharacteristic
153#131—1011no—
154Q32—1012no—
155Q32—1011no—
156SD32—1012no—
157SD32—1012no—
158SD32—1011no—
159SD32—1011no—
160SD32C2×D10112yescomplemented
161SD32C2×D10112yescomplemented
162D16—1012no—
163D16—1011no—
164M32—1011no—
165M32—1012no—
166C2×C16—1012no—
167C2×C16C2×D10111yescharacteristic, complemented
168#41—1011no—
169#41—4042no—
170#41—4041no—
171#41—4041no—
172#41—2022no—
173#41—2022no—
174#41—2021no—
175#41—2021no—
176C2×Q8—21no—
177C2×Q8—4041no—
178C2×Q8—2022no—
179C2×D4—21no—
180C2×D4—4041no—
181C2×D4—4042no—
182C2×D4—2022no—
183Q16—21no—
184Q16—1011no—
185Q16—1011no—
186Q16—2022no—
187Q16C22×D10112yes—
188SD16—2022no—
189SD16—2022no—
190SD16—2021no—
191SD16—2021no—
192D8—21no—
193D8—1011no—
194D8—1011no—
195D8—2022no—
196D8C22×D10112yes—
197M16—1011no—
198M16—1012no—
199C2×C8—1012no—
200C2×C8C22×D10111yescharacteristic
201C16—1012no—
202C16C22×D10111yescharacteristic
203C16C22×D10111yescharacteristic
204C23—41no—
205C23—4042no—
206Q8—22no—
207Q8—41no—
208Q8—1011no—
209Q8—4042no—
210Q8—2021no—
211Q8—2021no—
212D4—22no—
213D4—41no—
214D4—1012no—
215D4—1011no—
216D4—4042no—
217D4—4042no—
218D4—4042no—
219D4—4041no—
220D4—4041no—
221D4—2021no—
222D4—2021no—
223C2×C4—41no—
224C2×C4—1012no—
225C2×C4—4042no—
226C2×C4—4042no—
227C2×C4—4042no—
228C2×C4—4041no—
229C2×C4—4041no—
230C2×C4D4×D10111yescharacteristic
231C8—1012no—
232C8D4×D10111yescharacteristic
233C8C23×D10111yescharacteristic
234C22—42no—
235C22—161no—
236C22—16162no—
237C22—1012no—
238C22—4041no—
239C22—4041no—
240C22D8×D10111yescharacteristic
241C4—42no—
242C4—1012no—
243C4—4041no—
244C4—4041no—
245C4D8×D10111yescharacteristic
246C4C2×D4×D10111yescharacteristic
247C2—21no—
248C2—82no—
249C2—8081no—
250C2—8081no—
251C2—2022no—
252C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
FittingC2×SD32×C101C2
socleC2×C101C22×D8
hypercenterC2×SD32D101
hypocenterC101#3416
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #20265990C2semidirect
2. #20265978C2semidirect
3. #20265618C2semidirect
4. #20265616C2semidirect
5. #20265614C2semidirect
6. #20265613C2semidirect
7. #20265609C2semidirect
8. #20265608C2semidirect
9. SD32×D101C2semidirect
10. #20265593C2semidirect
11. C2×SD32×C101C2semidirect
18. #15692278C22semidirect
20. #15692277C22semidirect
22. Q16×D101C22semidirect
25. Dic808C22semidirect
26. #15692201C22semidirect
27. #15692200C22semidirect
28. #15692200C22semidirect
29. #15692199C22semidirect
30. #15692198C22semidirect
31. D1616C22semidirect
32. #15692196C22semidirect
33. #15692195C22semidirect
34. #15692194C22semidirect
35. C16×D101C22semidirect
36. C2×Q16×C101C22semidirect
38. SD32×C101C22semidirect
39. SD32×C101C22semidirect
40. C2×C16×C101C22semidirect
127. C101#3416semidirect
137. C2×SD32D101semidirect
150. C2×Q16C2×D101semidirect
160. SD32C2×D101semidirect
161. SD32C2×D101semidirect
167. C2×C16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659902
#202659782
#202656182
#202656162
#202656142
#202656132
#202656092
#202656082
SD32×D1012
#202655932
C2×SD32×C1012
#3416101