Finite Group Databasebeta

← #27154881

Subgroups — #27154881

Order 12416.

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#20261548C211yesmaximal, characteristic, complemented
2#20261537C211yesmaximal, characteristic, complemented
3#20261177C212yesmaximal, complemented
4#20261173C211yesmaximal, characteristic, complemented
5#20261172C212yesmaximal, complemented
6#20261168C212yesmaximal, complemented
7#20261165C211yesmaximal, characteristic, complemented
8C2×#15690753C211yesmaximal, characteristic, complemented
9SD32×D97C212yesmaximal, complemented
10#20261153C211yesmaximal, characteristic, complemented
11#510×C97C211yesmaximal, characteristic, complemented
12#15690858—21no—
13#15690856—21no—
14#15690842—22no—
15#15690841—21no—
16#15690837—22no—
17#15690835—21no—
18#15690833C2212yes—
19#15690833C2212yescomplemented
20#15690832C2211yescharacteristic, complemented
21Q16×D97C2212yescomplemented
22C2×D776C2211yescharacteristic
23D8×D97C2212yes—
24#15690816C2211yescharacteristic
25#15690756C2211yescharacteristic, complemented
26#15690755C2212yescomplemented
27#15690754C2211yescharacteristic, complemented
28#15690753C2212yescomplemented
29#15690753C2212yescomplemented
30#15690753C2211yescharacteristic, complemented
31D1552C2212yescomplemented
32#15690751C2211yescharacteristic, complemented
33#15690749C2212yescomplemented
34C2×#13095486C2211yescharacteristic, complemented
35C16×D97C2212yescomplemented
36#135×C97C2211yescharacteristic
37C2×Q16×C97C2211yescharacteristic, complemented
38Q32×C97C2212yescomplemented
39SD32×C97C2212yescomplemented
40M32×C97C2211yescharacteristic, complemented
41#13095512—22no—
42#13095512—22no—
43#13095512—41no—
44#13095511—21no—
45#13095511—42no—
46#13095511—41no—
47#13095511D411yescharacteristic
48Q8×D97—22no—
49Q8×D97—41no—
50C2×D388—21no—
51C2×D388—41no—
52D4×D97—22no—
53D4×D97—42no—
54Dic388C2311yescharacteristic
55#13095501—22no—
56#13095500—21no—
57#13095498—22no—
58#13095498—22no—
59D776—21no—
60D776C2311yescharacteristic
61D776C2312yes—
62#13095496—22no—
63#13095495D411yescharacteristic
64#13095494D412yes—
65C8×D97C2312yes—
66#13095486C2311yescharacteristic
67#13095486C2311yescharacteristic
68#41×C97—21no—
69C2×Q8×C97—21no—
70Q16×C97—21no—
71Q16×C97C2312yes—
72Q16×C97C2311yescharacteristic
73SD16×C97—21no—
74D8×C97C2311yescharacteristic
75C2×C8×C97C2311yescharacteristic
76C16×C97C2312yes—
77C22×D97—41no—
78C22×D97—42no—
79Q8×C97—21no—
80Q8×C97—22no—
81Q8×C97—41no—
82D4×C97—21no—
83D4×C97—41no—
84C2×C4×C97—41no—
85C2×C4×C97—41no—
86C2×C4×C97C2×D411yescharacteristic
87C22⋊D97—41no—
88C22⋊D97—42no—
89C22⋊D97D812yes—
90D388—22no—
91D388—21no—
92D388—42no—
93D388—42no—
94D388—41no—
95D388D811yescharacteristic
96C4×D97—42no—
97C4×D97—41no—
98C4×D97—42no—
99C4×D97—42no—
100C4×D97C2×D412yes—
101Dic194—21no—
102Dic194—42no—
103Dic194D811yescharacteristic
104C8×C97C2×D411yescharacteristic
105C8×C97C2411yescharacteristic
106#11715895C2×D412yes—
107C22×C97—41no—
108C22×C97C2×D811yescharacteristic
109C2×D97—41no—
110C2×D97—42no—
111C2×D97—162no—
112C2×D97—161no—
113C2×D97C2×D812yes—
114C4×C97—42no—
115C4×C97—41no—
116C4×C97C2×D811yescharacteristic
117C4×C97C22×D411yescharacteristic
118Dic97—41no—
119Dic97C2×D812yes—
120C2×C97—21no—
121C2×C97—81no—
122C2×C97C22×D811yescharacteristic
123D97—22no—
124D97—81no—
125D97—82no—
126#3416—971nomaximal, Sylow
127C97#341611yescharacteristic, complemented, Sylow
128#578—971no—
129#576—971no—
130#510—972no—
131#510D9711yescharacteristic, complemented
132#509—972no—
133#509—971no—
134#508—971no—
135#508—972no—
136C2×SD32—971no—
137C2×SD32—972no—
138#504—971no—
1392−1+4—1941no—
140H(5,𝔽2)—1941no—
141#137—1942no—
142#137—1941no—
143AGL(1,ℤ/8)—1942no—
144AGL(1,ℤ/8)—1941no—
145#135—972no—
146#135—972no—
147#135—971no—
148#135C2×D9711yescharacteristic
149C2×Q16—972no—
150C2×Q16C2×D9711yescharacteristic, complemented
151C2×D8—972no—
152C2×D8—971no—
153#131—971no—
154Q32—971no—
155Q32C2×D9712yescomplemented
156SD32—972no—
157SD32—972no—
158SD32—971no—
159SD32—971no—
160SD32—972no—
161SD32C2×D9712yescomplemented
162D16—971no—
163D16—972no—
164M32—972no—
165M32C2×D9711yescharacteristic, complemented
166C2×C16—971no—
167C2×C16—972no—
168#41—21no—
169#41—971no—
170#41—3882no—
171#41—3881no—
172#41—3881no—
173#41—1942no—
174#41—1942no—
175#41—1941no—
176C2×Q8—21no—
177C2×Q8—3881no—
178C2×Q8—1942no—
179C2×D4—3881no—
180C2×D4—3882no—
181C2×D4—1942no—
182C2×D4—1941no—
183Q16—21no—
184Q16—971no—
185Q16—1942no—
186Q16C22×D9712yes—
187Q16C22×D9711yescharacteristic
188SD16—21no—
189SD16—1942no—
190SD16—1942no—
191SD16—1941no—
192D8—971no—
193D8—972no—
194D8—1942no—
195D8—1941no—
196D8C22×D9711yescharacteristic
197M16—971no—
198M16—972no—
199C2×C8—972no—
200C2×C8C22×D9711yescharacteristic
201C16—971no—
202C16—971no—
203C16C22×D9712yes—
204C23—3882no—
205C23—3881no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—971no—
210Q8—3882no—
211Q8—1941no—
212D4—21no—
213D4—41no—
214D4—972no—
215D4—971no—
216D4—3882no—
217D4—3882no—
218D4—3882no—
219D4—3881no—
220D4—3881no—
221D4—1942no—
222D4—1941no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—972no—
226C2×C4—3882no—
227C2×C4—3882no—
228C2×C4—3882no—
229C2×C4—3881no—
230C2×C4D4×D9711yescharacteristic
231C8—972no—
232C8D4×D9711yescharacteristic
233C8C23×D9711yescharacteristic
234C22—41no—
235C22—15521no—
236C22—15522no—
237C22—972no—
238C22—3881no—
239C22—3882no—
240C22D8×D9711yescharacteristic
241C4—41no—
242C4—42no—
243C4—972no—
244C4—3881no—
245C4D8×D9711yescharacteristic
246C4C2×D4×D9711yescharacteristic
247C2—21no—
248C2—81no—
249C2—7762no—
250C2—7761no—
251C2—1942no—
252C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC2C2×D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#510×C97C2
socleC2×C97C22×D8
hypercenter#510D97
hypocenterC97#3416
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #20261548C2semidirect
2. #20261537C2semidirect
3. #20261177C2semidirect
4. #20261173C2semidirect
5. #20261172C2semidirect
6. #20261168C2semidirect
7. #20261165C2semidirect
8. C2×#15690753C2semidirect
9. SD32×D97C2semidirect
10. #20261153C2semidirect
11. #510×C97C2semidirect
19. #15690833C22semidirect
20. #15690832C22semidirect
21. Q16×D97C22semidirect
25. #15690756C22semidirect
26. #15690755C22semidirect
27. #15690754C22semidirect
28. #15690753C22semidirect
29. #15690753C22semidirect
30. #15690753C22semidirect
31. D1552C22semidirect
32. #15690751C22semidirect
33. #15690749C22semidirect
34. C2×#13095486C22semidirect
35. C16×D97C22semidirect
37. C2×Q16×C97C22semidirect
38. Q32×C97C22semidirect
39. SD32×C97C22semidirect
40. M32×C97C22semidirect
127. C97#3416semidirect
131. #510D97semidirect
150. C2×Q16C2×D97semidirect
155. Q32C2×D97semidirect
161. SD32C2×D97semidirect
165. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615482
#202615372
#202611772
#202611732
#202611722
#202611682
#202611652
C2×#156907532
SD32×D972
#202611532
#510×C972
#341697