Finite Group Databasebeta

← #27184267

Subgroups — #27184267

Order 12928.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 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
1C2×#15692278C211yesmaximal, characteristic, complemented
2#135×D101C211yesmaximal, characteristic, complemented
3#20265619C212yesmaximal, complemented
4#20265618C211yesmaximal, characteristic, complemented
5#20265615C211yesmaximal, characteristic, complemented
6#20265614C212yesmaximal, complemented
7#20265612C211yesmaximal, characteristic, complemented
8#20265610C212yesmaximal, complemented
9#20265608C212yesmaximal, complemented
10M32×D101C211yesmaximal, characteristic, complemented
11#510×C101C211yesmaximal, characteristic, complemented
12C2×#13108075—21no—
13#41×D101—21no—
14#15692278—21no—
15#15692278—21no—
16#15692278C2212yes—
17#15692278C2212yescomplemented
18#15692278C2211yescharacteristic, complemented
19#15692277C2211yescharacteristic
20#15692276—21no—
21#15692275—21no—
22#15692274C2211yescharacteristic, complemented
23Q16×D101C2211yescharacteristic, complemented
24C2×#13108061—21no—
25SD16×D101—21no—
26C2×D808C2211yescharacteristic, complemented
27D8×D101C2211yescharacteristic, complemented
28C2×C8×D101C2211yescharacteristic
29#15692201C2212yescomplemented
30#15692200C2212yescomplemented
31#15692198C2212yescomplemented
32#15692198C2212yescomplemented
33D1616C2212yescomplemented
34#15692196C2212yescomplemented
35#15692195C2211yescharacteristic, complemented
36#15692194C2212yescomplemented
37C16×D101C2212yescomplemented
38#135×C101C2211yescharacteristic
39C2×Q16×C101C2211yescharacteristic, complemented
40Q32×C101C2212yescomplemented
41SD32×C101C2212yescomplemented
42M32×C101C2211yescharacteristic, complemented
43#13108075—21no—
44#13108075—22no—
45#13108075—22no—
46#13108075—41no—
47#13108075—41no—
48#13108074—21no—
49#13108074—41no—
50#13108073—21no—
51#13108073—41no—
52Q8×D101—21no—
53C2×D404—21no—
54C2×D404—41no—
55D4×D101—21no—
56D4×D101—41no—
57C2×C4×D101—41no—
58C2×C4×D101—41no—
59C2×C4×D101D411yescharacteristic
60Dic404C2311yescharacteristic
61#13108064—21no—
62#13108063—21no—
63#13108062—21no—
64#13108061—21no—
65#13108061—22no—
66#13108061—22no—
67D808—21no—
68D808C2312yes—
69D808C2311yescharacteristic
70#13108059—21no—
71C2×#11717579D411yescharacteristic
72C8×D101D411yescharacteristic
73C8×D101D411yescharacteristic
74C8×D101C2311yescharacteristic
75C8×D101C2311yescharacteristic
76#13108049C2312yes—
77#41×C101—21no—
78C2×Q8×C101—21no—
79Q16×C101—21no—
80Q16×C101C2312yes—
81Q16×C101C2311yescharacteristic
82SD16×C101—21no—
83D8×C101C2311yescharacteristic
84C2×C8×C101C2311yescharacteristic
85C16×C101C2312yes—
86C22×D101—41no—
87C22×D101—41no—
88C22×D101D811yescharacteristic
89Q8×C101—21no—
90Q8×C101—22no—
91Q8×C101—41no—
92D4×C101—21no—
93D4×C101—41no—
94C2×C4×C101—41no—
95C2×C4×C101—41no—
96C2×C4×C101C2×D411yescharacteristic
97C22⋊D101—41no—
98C22⋊D101—41no—
99C22⋊D101—41no—
100C2×Dic101—41no—
101C2×Dic101D811yescharacteristic
102D404—21no—
103D404—22no—
104D404—41no—
105D404—42no—
106D404—42no—
107D404—41no—
108C4×D101—41no—
109C4×D101—41no—
110C4×D101—41no—
111C4×D101—42no—
112C4×D101—42no—
113C4×D101D811yescharacteristic
114C4×D101D811yescharacteristic
115C4×D101C2×D411yescharacteristic
116C4×D101C2×D411yescharacteristic
117Dic202—21no—
118Dic202—41no—
119C8×C101C2×D411yescharacteristic
120C8×C101C2411yescharacteristic
121#11717579C2×D411yescharacteristic
122#11717579C2×D411yescharacteristic
123C22×C101—41no—
124C22×C101C2×D811yescharacteristic
125C2×D101—41no—
126C2×D101—42no—
127C2×D101—41no—
128C2×D101—82no—
129C2×D101—82no—
130C2×D101C2×D811yescharacteristic
131C2×D101C2×D811yescharacteristic
132C4×C101—41no—
133C4×C101—42no—
134C4×C101C2×D811yescharacteristic
135C4×C101C22×D411yescharacteristic
136Dic101—41no—
137Dic101C2×D811yescharacteristic
138Dic101C2×D811yescharacteristic
139C2×C101—21no—
140C2×C101—81no—
141C2×C101C22×D811yescharacteristic
142D101—21no—
143D101—21no—
144D101—82no—
145D101—81no—
146#3414—1011nomaximal, Sylow
147C101#341411yescharacteristic, complemented, Sylow
148C2×#135—1011no—
149C2×#135—1011no—
150#510—1011no—
151#510—1012no—
152#510D10111yescharacteristic, complemented
153#509—1011no—
154#509—1012no—
155#509—1011no—
156#508—1012no—
157#508—1012no—
158C2×M32—1011no—
159C2×#41—2021no—
160C2×#41—2021no—
161#135—1012no—
162#135—1011no—
163#135—1012no—
164#135—1011no—
165#135—1011no—
166#135—2021no—
167#135—2021no—
168#135—2021no—
169#135—2021no—
170#135C2×D10111yescharacteristic
171C2×Q16—1011no—
172C2×Q16C2×D10111yescharacteristic, complemented
173C2×SD16—2021no—
174C2×SD16—2021no—
175C2×D8—1011no—
176C2×D8—1011no—
177C22×C8—1011no—
178Q32—1012no—
179Q32C2×D10112yescomplemented
180SD32—1012no—
181SD32—1012no—
182SD32—1012no—
183SD32C2×D10112yescomplemented
184D16—1012no—
185D16—1012no—
186M32—1011no—
187M32—1012no—
188M32C2×D10111yescharacteristic, complemented
189C2×C16—1012no—
190#41—21no—
191#41—4041no—
192#41—4041no—
193#41—4041no—
194#41—4041no—
195#41—2021no—
196#41—2022no—
197#41—2021no—
198#41—2022no—
199#41—2021no—
200C2×Q8—21no—
201C2×Q8—2021no—
202C2×D4—4041no—
203C2×D4—4041no—
204C2×D4—2021no—
205C2×D4—2021no—
206C22×C4—1011no—
207C22×C4—4041no—
208C22×C4—4041no—
209Q16—21no—
210Q16—1011no—
211Q16—2021no—
212Q16C22×D10111yescharacteristic
213Q16C22×D10112yes—
214SD16—21no—
215SD16—2021no—
216SD16—2022no—
217SD16—2021no—
218SD16—2022no—
219SD16—2021no—
220D8—1012no—
221D8—1011no—
222D8—2021no—
223D8—2021no—
224D8C22×D10111yescharacteristic
225C2×C8—1011no—
226C2×C8—1011no—
227C2×C8—1011no—
228C2×C8—1011no—
229C2×C8—1011no—
230C2×C8C22×D10111yescharacteristic
231C16—1012no—
232C16C22×D10112yes—
233C23—1011no—
234C23—4041no—
235C23—4041no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—4041no—
240Q8—2021no—
241D4—21no—
242D4—41no—
243D4—4041no—
244D4—4042no—
245D4—4042no—
246D4—4041no—
247D4—4041no—
248D4—4041no—
249D4—4041no—
250D4—2021no—
251D4—2022no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1011no—
255C2×C4—1011no—
256C2×C4—1011no—
257C2×C4—1011no—
258C2×C4—1011no—
259C2×C4—4041no—
260C2×C4—4042no—
261C2×C4—4042no—
262C2×C4—4041no—
263C2×C4—4041no—
264C2×C4—4041no—
265C2×C4D4×D10111yescharacteristic
266C8—1011no—
267C8—1011no—
268C8D4×D10111yescharacteristic
269C8C23×D10111yescharacteristic
270C22—41no—
271C22—8082no—
272C22—8082no—
273C22—1011no—
274C22—1011no—
275C22—4041no—
276C22—4042no—
277C22—4041no—
278C22D8×D10111yescharacteristic
279C4—41no—
280C4—42no—
281C4—1011no—
282C4—1011no—
283C4—4041no—
284C4D8×D10111yescharacteristic
285C4C2×D4×D10111yescharacteristic
286C2—21no—
287C2—81no—
288C2—8082no—
289C2—8081no—
290C2—2021no—
291C2—2021no—
292C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#510×C101C2
socleC2×C101C22×D8
hypercenter#510D101
hypocenterC101#3414
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#15692278C2semidirect
2. #135×D101C2semidirect
3. #20265619C2semidirect
4. #20265618C2semidirect
5. #20265615C2semidirect
6. #20265614C2semidirect
7. #20265612C2semidirect
8. #20265610C2semidirect
9. #20265608C2semidirect
10. M32×D101C2semidirect
11. #510×C101C2semidirect
17. #15692278C22semidirect
18. #15692278C22semidirect
22. #15692274C22semidirect
23. Q16×D101C22semidirect
26. C2×D808C22semidirect
27. D8×D101C22semidirect
29. #15692201C22semidirect
30. #15692200C22semidirect
31. #15692198C22semidirect
32. #15692198C22semidirect
33. D1616C22semidirect
34. #15692196C22semidirect
35. #15692195C22semidirect
36. #15692194C22semidirect
37. C16×D101C22semidirect
39. C2×Q16×C101C22semidirect
40. Q32×C101C22semidirect
41. SD32×C101C22semidirect
42. M32×C101C22semidirect
147. C101#3414semidirect
152. #510D101semidirect
172. C2×Q16C2×D101semidirect
179. Q32C2×D101semidirect
183. SD32C2×D101semidirect
188. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922782
#135×D1012
#202656192
#202656182
#202656152
#202656142
#202656122
#202656102
#202656082
M32×D1012
#510×C1012
#3414101