Finite Group Databasebeta

← #27154879

Subgroups — #27154879

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#20261549C211yesmaximal, characteristic, complemented
2#20261537C211yesmaximal, characteristic, complemented
3#20261177C212yesmaximal, complemented
4#20261175C211yesmaximal, characteristic, complemented
5#20261170C211yesmaximal, characteristic, complemented
6#20261169C212yesmaximal, complemented
7#20261166C212yesmaximal, complemented
8C2×#15690755C211yesmaximal, characteristic, complemented
9SD32×D97C212yesmaximal, complemented
10#20261151C211yesmaximal, characteristic, complemented
11#508×C97C211yesmaximal, characteristic, complemented
12#15690857—21no—
13#15690856—21no—
14#15690840—21no—
15#15690839—22no—
16#15690837—22no—
17#15690835—21no—
18#15690833C2212yes—
19#15690829C2212yescomplemented
20C2×Dic388C2211yescharacteristic, complemented
21Q16×D97C2212yescomplemented
22C2×D776C2211yescharacteristic
23D8×D97C2212yes—
24#15690816C2211yescharacteristic
25#15690756C2212yescomplemented
26#15690755C2212yescomplemented
27#15690755C2212yescomplemented
28#15690754C2212yescomplemented
29#15690753C2212yescomplemented
30#15690751C2212yescomplemented
31#15690750C2211yescharacteristic, complemented
32#15690749C2212yescomplemented
33C16×D97C2212yescomplemented
34#135×C97C2211yescharacteristic
35#135×C97C2211yescharacteristic, complemented
36Q32×C97C2211yescharacteristic, complemented
37SD32×C97C2211yescharacteristic, complemented
38SD32×C97C2211yescharacteristic, complemented
39D16×C97C2211yescharacteristic, complemented
40C2×C16×C97C2211yescharacteristic, complemented
41#13095512—22no—
42#13095511—41no—
43#13095511—41no—
44#13095511D411yescharacteristic
45#13095510—22no—
46#13095510—42no—
47C2×Dic194—21no—
48C2×Dic194—41no—
49Q8×D97—22no—
50C2×D388—21no—
51C2×D388—41no—
52D4×D97—22no—
53D4×D97—42no—
54Dic388—21no—
55Dic388C2312yes—
56#13095501—22no—
57#13095499—22no—
58#13095498—22no—
59D776—21no—
60D776C2312yes—
61#13095496—22no—
62#13095495D411yescharacteristic
63#13095494D412yes—
64C8×D97C2312yes—
65#13095486C2312yes—
66#41×C97—21no—
67#41×C97—21no—
68Q16×C97C2311yescharacteristic
69Q16×C97C2311yescharacteristic
70SD16×C97—21no—
71SD16×C97—21no—
72D8×C97C2311yescharacteristic
73D8×C97C2311yescharacteristic
74C2×C8×C97C2311yescharacteristic
75C16×C97C2311yescharacteristic
76C16×C97C2311yescharacteristic
77C22×D97—41no—
78C22×D97—42no—
79Q8×C97—21no—
80Q8×C97—21no—
81D4×C97—21no—
82D4×C97—21no—
83D4×C97—41no—
84D4×C97—41no—
85C2×C4×C97—41no—
86C2×C4×C97—41no—
87C2×C4×C97C2×D411yescharacteristic
88C22⋊D97—42no—
89C22⋊D97—42no—
90C22⋊D97D812yes—
91C2×Dic97—41no—
92C2×Dic97—42no—
93D388—22no—
94D388—42no—
95D388—41no—
96D388D811yescharacteristic
97C4×D97—42no—
98C4×D97—42no—
99C4×D97C2×D412yes—
100Dic194—22no—
101Dic194—41no—
102Dic194—42no—
103Dic194D811yescharacteristic
104C8×C97C2×D411yescharacteristic
105C8×C97C2411yescharacteristic
106#11715895C2×D412yes—
107C22×C97—41no—
108C22×C97—41no—
109C22×C97C2×D811yescharacteristic
110C2×D97—42no—
111C2×D97—162no—
112C2×D97—161no—
113C2×D97C2×D812yes—
114C4×C97—41no—
115C4×C97—41no—
116C4×C97C2×D811yescharacteristic
117C4×C97C22×D411yescharacteristic
118Dic97—42no—
119Dic97C2×D812yes—
120C2×C97—21no—
121C2×C97—81no—
122C2×C97—81no—
123C2×C97C22×D811yescharacteristic
124D97—22no—
125D97—82no—
126#3416—971nomaximal, Sylow
127C97#341611yescharacteristic, complemented, Sylow
128#578—971no—
129#576—971no—
130#510—971no—
131#510—972no—
132#509—971no—
133#509—972no—
134#508—972no—
135#508D9711yescharacteristic, complemented
136C2×SD32—972no—
137C2×SD32—971no—
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#135C2×D9711yescharacteristic
148#135C2×D9711yescharacteristic, complemented
149C2×Q16—972no—
150C2×Q16—971no—
151C2×D8—972no—
152C2×D8—971no—
153#131—971no—
154Q32—972no—
155Q32C2×D9711yescharacteristic, complemented
156SD32—972no—
157SD32—972no—
158SD32—972no—
159SD32—972no—
160SD32C2×D9711yescharacteristic, complemented
161SD32C2×D9711yescharacteristic, complemented
162D16—972no—
163D16C2×D9711yescharacteristic, complemented
164M32—971no—
165M32—972no—
166C2×C16—972no—
167C2×C16C2×D9711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—971no—
171#41—3881no—
172#41—3881no—
173#41—3882no—
174#41—1942no—
175#41—1942no—
176C2×Q8—3881no—
177C2×Q8—1942no—
178C2×Q8—1941no—
179C2×D4—3881no—
180C2×D4—3882no—
181C2×D4—1942no—
182C2×D4—1941no—
183Q16—972no—
184Q16—1942no—
185Q16—1941no—
186Q16C22×D9711yescharacteristic
187Q16C22×D9711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—1942no—
191SD16—1942no—
192D8—972no—
193D8—1942no—
194D8—1941no—
195D8C22×D9711yescharacteristic
196D8C22×D9711yescharacteristic
197M16—971no—
198M16—972no—
199C2×C8—972no—
200C2×C8C22×D9711yescharacteristic
201C16—972no—
202C16C22×D9711yescharacteristic
203C16C22×D9711yescharacteristic
204C23—3882no—
205C23—3881no—
206Q8—21no—
207Q8—21no—
208Q8—971no—
209Q8—3882no—
210Q8—3881no—
211Q8—1942no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—972no—
217D4—971no—
218D4—3882no—
219D4—3882no—
220D4—3882no—
221D4—3881no—
222D4—1942no—
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—41no—
236C22—15521no—
237C22—15522no—
238C22—972no—
239C22—3882no—
240C22D8×D9711yescharacteristic
241C4—41no—
242C4—41no—
243C4—972no—
244C4—3882no—
245C4D8×D9711yescharacteristic
246C4C2×D4×D9711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—7762no—
251C2—1942no—
252C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC2C2×D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#508×C97C2
socleC2×C97C22×D8
hypercenter#508D97
hypocenterC97#3416
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #20261549C2semidirect
2. #20261537C2semidirect
3. #20261177C2semidirect
4. #20261175C2semidirect
5. #20261170C2semidirect
6. #20261169C2semidirect
7. #20261166C2semidirect
8. C2×#15690755C2semidirect
9. SD32×D97C2semidirect
10. #20261151C2semidirect
11. #508×C97C2semidirect
19. #15690829C22semidirect
20. C2×Dic388C22semidirect
21. Q16×D97C22semidirect
25. #15690756C22semidirect
26. #15690755C22semidirect
27. #15690755C22semidirect
28. #15690754C22semidirect
29. #15690753C22semidirect
30. #15690751C22semidirect
31. #15690750C22semidirect
32. #15690749C22semidirect
33. C16×D97C22semidirect
35. #135×C97C22semidirect
36. Q32×C97C22semidirect
37. SD32×C97C22semidirect
38. SD32×C97C22semidirect
39. D16×C97C22semidirect
40. C2×C16×C97C22semidirect
127. C97#3416semidirect
135. #508D97semidirect
148. #135C2×D97semidirect
155. Q32C2×D97semidirect
160. SD32C2×D97semidirect
161. SD32C2×D97semidirect
163. D16C2×D97semidirect
167. C2×C16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615492
#202615372
#202611772
#202611752
#202611702
#202611692
#202611662
C2×#156907552
SD32×D972
#202611512
#508×C972
#341697