Finite Group Databasebeta

← #508×D97 #27154850

Subgroups — #508×D97 #27154850

Order 12416.

368 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D97C212yesmaximal, complemented
2#20261168C211yesmaximal, characteristic, complemented
3#20261167C211yesmaximal, characteristic, complemented
4#20261166C212yesmaximal, complemented
5#20261165C212yesmaximal, complemented
6#20261164C211yesmaximal, characteristic, complemented
7Q32×D97C211yesmaximal, characteristic, complemented
8SD32×D97C212yesmaximal, complemented
9D16×D97C211yesmaximal, characteristic, complemented
10C2×C16×D97C211yesmaximal, characteristic, complemented
11#508×C97C211yesmaximal, characteristic, complemented
12#41×D97—22no—
13#15690833C2212yescomplemented
14#15690832C2212yescomplemented
15#15690831—22no—
16#15690830—22no—
17#15690829C2212yescomplemented
18Q16×D97C2212yes—
19SD16×D97—22no—
20D8×D97C2212yes—
21C2×C8×D97C2211yescharacteristic
22Dic776C2211yescharacteristic, complemented
23#15690756C2212yescomplemented
24#15690755C2212yescomplemented
25#15690754C2212yescomplemented
26#15690753C2212yescomplemented
27D1552C2211yescharacteristic, complemented
28#15690751C2212yescomplemented
29C2×#13095486C2211yescharacteristic, complemented
30C16×D97C2211yescharacteristic
31C16×D97C2211yescharacteristic
32C16×D97C2211yescharacteristic, complemented
33C16×D97C2211yescharacteristic, complemented
34#135×C97C2212yescomplemented
35Q32×C97C2211yescharacteristic, complemented
36SD32×C97C2212yescomplemented
37D16×C97C2211yescharacteristic, complemented
38C2×C16×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095511—22no—
41#13095511—42no—
42#13095510—22no—
43#13095510—42no—
44Q8×D97—22no—
45D4×D97—22no—
46D4×D97—42no—
47C2×C4×D97—42no—
48C2×C4×D97D411yescharacteristic
49Dic388C2312yes—
50#13095501—22no—
51#13095500—22no—
52#13095499—22no—
53#13095498—22no—
54D776C2312yes—
55#13095496—22no—
56C2×#11715895D411yescharacteristic
57C8×D97D411yescharacteristic
58C8×D97D411yescharacteristic
59C8×D97C2311yescharacteristic
60C8×D97C2311yescharacteristic
61#13095486C2311yescharacteristic
62#13095486C2311yescharacteristic
63#41×C97—22no—
64Q16×C97C2312yes—
65SD16×C97—22no—
66D8×C97C2312yes—
67C2×C8×C97C2311yescharacteristic
68C16×C97C2311yescharacteristic
69C16×C97C2311yescharacteristic
70C22×D97—42no—
71C22×D97D811yescharacteristic
72Q8×C97—22no—
73D4×C97—22no—
74D4×C97—42no—
75C2×C4×C97—42no—
76C2×C4×C97C2×D411yescharacteristic
77C22⋊D97—42no—
78C22⋊D97—42no—
79C22⋊D97—42no—
80C2×Dic97—42no—
81C2×Dic97D811yescharacteristic
82D388—22no—
83D388—42no—
84C4×D97—42no—
85C4×D97—42no—
86C4×D97—42no—
87C4×D97D811yescharacteristic
88C4×D97D811yescharacteristic
89C4×D97C2×D411yescharacteristic
90C4×D97C2×D411yescharacteristic
91Dic194—22no—
92Dic194—42no—
93C8×C97C2×D411yescharacteristic
94C8×C97C2411yescharacteristic
95#11715895C2×D411yescharacteristic
96#11715895C2×D411yescharacteristic
97C22×C97—42no—
98C22×C97C2×D811yescharacteristic
99C2×D97—22no—
100C2×D97—42no—
101C2×D97—84no—
102C2×D97C2×D811yescharacteristic
103C2×D97C2×D811yescharacteristic
104C4×C97—42no—
105C4×C97C2×D811yescharacteristic
106C4×C97C22×D411yescharacteristic
107Dic97—42no—
108Dic97C2×D811yescharacteristic
109Dic97C2×D811yescharacteristic
110C2×C97—21no—
111C2×C97—82no—
112C2×C97C22×D811yescharacteristic
113D97—21no—
114D97—82no—
115D97#50812yesdirect product factor
116C2×#508—971nomaximal, Sylow
117C97C2×#50811yescharacteristic, complemented, Sylow
118C2×#135—972no—
119#508—972no—
120#508—971no—
121#508—972no—
122#508—971no—
123#508—971no—
124#508D9711yescharacteristic, direct product factor
125C2×Q32—971no—
126C2×SD32—972no—
127C2×D16—971no—
128C22×C16—971no—
129C2×#41—1942no—
130#135—972no—
131#135—972no—
132#135—972no—
133#135—1942no—
134#135—1942no—
135#135C2×D9712yescomplemented
136C2×Q16—972no—
137C2×SD16—1942no—
138C2×D8—972no—
139C22×C8—971no—
140Q32—972no—
141Q32—971no—
142Q32C2×D9711yescharacteristic, complemented
143SD32—972no—
144SD32—972no—
145SD32—972no—
146SD32C2×D9712yescomplemented
147D16—972no—
148D16—971no—
149D16C2×D9711yescharacteristic, complemented
150C2×C16—971no—
151C2×C16—971no—
152C2×C16—971no—
153C2×C16—971no—
154C2×C16—971no—
155C2×C16C2×D9711yescharacteristic, complemented
156#41—22no—
157#41—3882no—
158#41—3882no—
159#41—1942no—
160#41—1942no—
161#41—1942no—
162C2×Q8—1942no—
163C2×D4—3882no—
164C2×D4—1942no—
165C22×C4—971no—
166C22×C4—3882no—
167Q16—972no—
168Q16—1942no—
169Q16C22×D9712yes—
170SD16—22no—
171SD16—1942no—
172SD16—1942no—
173SD16—1942no—
174D8—972no—
175D8—1942no—
176D8C22×D9712yes—
177C2×C8—971no—
178C2×C8—971no—
179C2×C8—971no—
180C2×C8—971no—
181C2×C8—971no—
182C2×C8C22×D9711yescharacteristic
183C16—971no—
184C16—971no—
185C16C22×D9711yescharacteristic
186C16C22×D9711yescharacteristic
187C23—971no—
188C23—3882no—
189Q8—22no—
190Q8—3882no—
191Q8—1942no—
192D4—22no—
193D4—42no—
194D4—3882no—
195D4—3882no—
196D4—3882no—
197D4—3882no—
198D4—1942no—
199C2×C4—42no—
200C2×C4—971no—
201C2×C4—971no—
202C2×C4—971no—
203C2×C4—971no—
204C2×C4—971no—
205C2×C4—3882no—
206C2×C4—3882no—
207C2×C4—3882no—
208C2×C4—3882no—
209C2×C4D4×D9711yescharacteristic
210C8—971no—
211C8—971no—
212C8D4×D9711yescharacteristic
213C8C23×D9711yescharacteristic
214C22—42no—
215C22—7764no—
216C22—971no—
217C22—971no—
218C22—3882no—
219C22—1942no—
220C22D8×D9711yescharacteristic
221C4—42no—
222C4—971no—
223C4—971no—
224C4—3882no—
225C4D8×D9711yescharacteristic
226C4C2×D4×D9711yescharacteristic
227C2—21no—
228C2—82no—
229C2—7762no—
230C2—972no—
231C2—1941no—
232C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC4D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#508×C97C2
socleC2×C97C22×D8
hypercenter#508D97
hypocenterC97C2×#508
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D97C2semidirect
2. #20261168C2semidirect
3. #20261167C2semidirect
4. #20261166C2semidirect
5. #20261165C2semidirect
6. #20261164C2semidirect
7. Q32×D97C2semidirect
8. SD32×D97C2semidirect
9. D16×D97C2semidirect
10. C2×C16×D97C2semidirect
11. #508×C97C2semidirect
13. #15690833C22semidirect
14. #15690832C22semidirect
17. #15690829C22semidirect
22. Dic776C22semidirect
23. #15690756C22semidirect
24. #15690755C22semidirect
25. #15690754C22semidirect
26. #15690753C22semidirect
27. D1552C22semidirect
28. #15690751C22semidirect
29. C2×#13095486C22semidirect
32. C16×D97C22semidirect
33. C16×D97C22semidirect
34. #135×C97C22semidirect
35. Q32×C97C22semidirect
36. SD32×C97C22semidirect
37. D16×C97C22semidirect
38. C2×C16×C97C22semidirect
115. D97#508direct (N × H)
117. C97C2×#508semidirect
124. #508D97direct (N × H)
135. #135C2×D97semidirect
142. Q32C2×D97semidirect
146. SD32C2×D97semidirect
149. D16C2×D97semidirect
155. C2×C16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#135×D972
#202611682
#202611672
#202611662
#202611652
#202611642
Q32×D972
SD32×D972
D16×D972
C2×C16×D972
#508×C972
C2×#50897