Finite Group Databasebeta

← #508×D101 #27184246

Subgroups — #508×D101 #27184246

Order 12928.

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×D101C212yesmaximal, complemented
2#20265610C211yesmaximal, characteristic, complemented
3#20265609C211yesmaximal, characteristic, complemented
4#20265608C212yesmaximal, complemented
5#20265607C212yesmaximal, complemented
6#20265606C211yesmaximal, characteristic, complemented
7Q32×D101C211yesmaximal, characteristic, complemented
8SD32×D101C212yesmaximal, complemented
9D16×D101C211yesmaximal, characteristic, complemented
10C2×C16×D101C211yesmaximal, characteristic, complemented
11#508×C101C211yesmaximal, characteristic, complemented
12#41×D101—22no—
13#15692278C2212yescomplemented
14#15692277C2212yescomplemented
15#15692276—22no—
16#15692275—22no—
17#15692274C2212yescomplemented
18Q16×D101C2212yes—
19SD16×D101—22no—
20D8×D101C2212yes—
21C2×C8×D101C2211yescharacteristic
22Dic808C2211yescharacteristic, complemented
23#15692201C2212yescomplemented
24#15692200C2212yescomplemented
25#15692199C2212yescomplemented
26#15692198C2212yescomplemented
27D1616C2211yescharacteristic, complemented
28#15692196C2212yescomplemented
29C2×#13108049C2211yescharacteristic, complemented
30C16×D101C2211yescharacteristic
31C16×D101C2211yescharacteristic
32C16×D101C2211yescharacteristic, complemented
33C16×D101C2211yescharacteristic, complemented
34#135×C101C2212yescomplemented
35Q32×C101C2211yescharacteristic, complemented
36SD32×C101C2212yescomplemented
37D16×C101C2211yescharacteristic, complemented
38C2×C16×C101C2211yescharacteristic, complemented
39#13108075—22no—
40#13108074—22no—
41#13108074—42no—
42#13108073—22no—
43#13108073—42no—
44Q8×D101—22no—
45D4×D101—22no—
46D4×D101—42no—
47C2×C4×D101—42no—
48C2×C4×D101D411yescharacteristic
49Dic404C2312yes—
50#13108064—22no—
51#13108063—22no—
52#13108062—22no—
53#13108061—22no—
54D808C2312yes—
55#13108059—22no—
56C2×#11717579D411yescharacteristic
57C8×D101D411yescharacteristic
58C8×D101D411yescharacteristic
59C8×D101C2311yescharacteristic
60C8×D101C2311yescharacteristic
61#13108049C2311yescharacteristic
62#13108049C2311yescharacteristic
63#41×C101—22no—
64Q16×C101C2312yes—
65SD16×C101—22no—
66D8×C101C2312yes—
67C2×C8×C101C2311yescharacteristic
68C16×C101C2311yescharacteristic
69C16×C101C2311yescharacteristic
70C22×D101—42no—
71C22×D101D811yescharacteristic
72Q8×C101—22no—
73D4×C101—22no—
74D4×C101—42no—
75C2×C4×C101—42no—
76C2×C4×C101C2×D411yescharacteristic
77C22⋊D101—42no—
78C22⋊D101—42no—
79C22⋊D101—42no—
80C2×Dic101—42no—
81C2×Dic101D811yescharacteristic
82D404—22no—
83D404—42no—
84C4×D101—42no—
85C4×D101—42no—
86C4×D101—42no—
87C4×D101D811yescharacteristic
88C4×D101D811yescharacteristic
89C4×D101C2×D411yescharacteristic
90C4×D101C2×D411yescharacteristic
91Dic202—22no—
92Dic202—42no—
93C8×C101C2×D411yescharacteristic
94C8×C101C2411yescharacteristic
95#11717579C2×D411yescharacteristic
96#11717579C2×D411yescharacteristic
97C22×C101—42no—
98C22×C101C2×D811yescharacteristic
99C2×D101—22no—
100C2×D101—42no—
101C2×D101—84no—
102C2×D101C2×D811yescharacteristic
103C2×D101C2×D811yescharacteristic
104C4×C101—42no—
105C4×C101C2×D811yescharacteristic
106C4×C101C22×D411yescharacteristic
107Dic101—42no—
108Dic101C2×D811yescharacteristic
109Dic101C2×D811yescharacteristic
110C2×C101—21no—
111C2×C101—82no—
112C2×C101C22×D811yescharacteristic
113D101—21no—
114D101—82no—
115D101#50812yesdirect product factor
116C2×#508—1011nomaximal, Sylow
117C101C2×#50811yescharacteristic, complemented, Sylow
118C2×#135—1012no—
119#508—1012no—
120#508—1011no—
121#508—1012no—
122#508—1011no—
123#508—1011no—
124#508D10111yescharacteristic, direct product factor
125C2×Q32—1011no—
126C2×SD32—1012no—
127C2×D16—1011no—
128C22×C16—1011no—
129C2×#41—2022no—
130#135—1012no—
131#135—1012no—
132#135—1012no—
133#135—2022no—
134#135—2022no—
135#135C2×D10112yescomplemented
136C2×Q16—1012no—
137C2×SD16—2022no—
138C2×D8—1012no—
139C22×C8—1011no—
140Q32—1012no—
141Q32—1011no—
142Q32C2×D10111yescharacteristic, complemented
143SD32—1012no—
144SD32—1012no—
145SD32—1012no—
146SD32C2×D10112yescomplemented
147D16—1012no—
148D16—1011no—
149D16C2×D10111yescharacteristic, complemented
150C2×C16—1011no—
151C2×C16—1011no—
152C2×C16—1011no—
153C2×C16—1011no—
154C2×C16—1011no—
155C2×C16C2×D10111yescharacteristic, complemented
156#41—22no—
157#41—4042no—
158#41—4042no—
159#41—2022no—
160#41—2022no—
161#41—2022no—
162C2×Q8—2022no—
163C2×D4—4042no—
164C2×D4—2022no—
165C22×C4—1011no—
166C22×C4—4042no—
167Q16—1012no—
168Q16—2022no—
169Q16C22×D10112yes—
170SD16—22no—
171SD16—2022no—
172SD16—2022no—
173SD16—2022no—
174D8—1012no—
175D8—2022no—
176D8C22×D10112yes—
177C2×C8—1011no—
178C2×C8—1011no—
179C2×C8—1011no—
180C2×C8—1011no—
181C2×C8—1011no—
182C2×C8C22×D10111yescharacteristic
183C16—1011no—
184C16—1011no—
185C16C22×D10111yescharacteristic
186C16C22×D10111yescharacteristic
187C23—1011no—
188C23—4042no—
189Q8—22no—
190Q8—4042no—
191Q8—2022no—
192D4—22no—
193D4—42no—
194D4—4042no—
195D4—4042no—
196D4—4042no—
197D4—4042no—
198D4—2022no—
199C2×C4—42no—
200C2×C4—1011no—
201C2×C4—1011no—
202C2×C4—1011no—
203C2×C4—1011no—
204C2×C4—1011no—
205C2×C4—4042no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—4042no—
209C2×C4D4×D10111yescharacteristic
210C8—1011no—
211C8—1011no—
212C8D4×D10111yescharacteristic
213C8C23×D10111yescharacteristic
214C22—42no—
215C22—8084no—
216C22—1011no—
217C22—1011no—
218C22—4042no—
219C22—2022no—
220C22D8×D10111yescharacteristic
221C4—42no—
222C4—1011no—
223C4—1011no—
224C4—4042no—
225C4D8×D10111yescharacteristic
226C4C2×D4×D10111yescharacteristic
227C2—21no—
228C2—82no—
229C2—8082no—
230C2—1012no—
231C2—2021no—
232C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC4D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#508×C101C2
socleC2×C101C22×D8
hypercenter#508D101
hypocenterC101C2×#508
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D101C2semidirect
2. #20265610C2semidirect
3. #20265609C2semidirect
4. #20265608C2semidirect
5. #20265607C2semidirect
6. #20265606C2semidirect
7. Q32×D101C2semidirect
8. SD32×D101C2semidirect
9. D16×D101C2semidirect
10. C2×C16×D101C2semidirect
11. #508×C101C2semidirect
13. #15692278C22semidirect
14. #15692277C22semidirect
17. #15692274C22semidirect
22. Dic808C22semidirect
23. #15692201C22semidirect
24. #15692200C22semidirect
25. #15692199C22semidirect
26. #15692198C22semidirect
27. D1616C22semidirect
28. #15692196C22semidirect
29. C2×#13108049C22semidirect
32. C16×D101C22semidirect
33. C16×D101C22semidirect
34. #135×C101C22semidirect
35. Q32×C101C22semidirect
36. SD32×C101C22semidirect
37. D16×C101C22semidirect
38. C2×C16×C101C22semidirect
115. D101#508direct (N × H)
117. C101C2×#508semidirect
124. #508D101direct (N × H)
135. #135C2×D101semidirect
142. Q32C2×D101semidirect
146. SD32C2×D101semidirect
149. D16C2×D101semidirect
155. C2×C16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#135×D1012
#202656102
#202656092
#202656082
#202656072
#202656062
Q32×D1012
SD32×D1012
D16×D1012
C2×C16×D1012
#508×C1012
C2×#508101