Finite Group Databasebeta

← #27181595

Subgroups — #27181595

Order 12928.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15692287C212yesmaximal, complemented
2C2×#15692282C212yesmaximal, complemented
3#20265728C211yesmaximal, characteristic, complemented
4#124×D101C211yesmaximal, characteristic, complemented
5#20265569C211yesmaximal, characteristic, complemented
6#20265567C211yesmaximal, characteristic, complemented
7#20265566C211yesmaximal, characteristic, complemented
8#20265562C212yesmaximal, complemented
9#20265558C212yesmaximal, complemented
10#20265129C211yesmaximal, characteristic, complemented
11#497×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yes—
13C2×Q8×D101C2211yescharacteristic
14C2×D4×D101C2211yescharacteristic
15#15692287—28no—
16#15692282—28no—
17C2×#13108064C2212yescomplemented
18C2×#13108063C2212yescomplemented
19C2×#13108061C2212yes—
20C2×#13108061C2212yescomplemented
21C2×D808C2212yes—
22C2×#13108059C2212yes—
23C2×#13108057C2212yes—
24#15692252C2211yescharacteristic, complemented
25#15692244C2211yescharacteristic
26#15692244C2212yescomplemented
27#15692243C2211yescharacteristic
28#15692242C2211yescharacteristic
29#15692241—22no—
30#15692230—22no—
31#15692217—22no—
32C22⋊C4×D101—22no—
33C42×D101C2211yescharacteristic
34#15692166C2211yescharacteristic, complemented
35C8×Dic101C2212yescomplemented
36C2×Q16×C101C2212yescomplemented
37C2×SD16×C101C2212yes—
38C4⋊Q8×C101C2211yescharacteristic, complemented
39#124×C101C2211yescharacteristic
40#97×C101C2211yescharacteristic, complemented
41C23×D101—21no—
42#13108075—24no—
43#13108075—24no—
44C2×Dic202—21no—
45C2×Dic202C2311yescharacteristic
46Q8×D101—24no—
47C2×D404—22no—
48C2×D404—22no—
49C2×D404C2311yescharacteristic
50C2×D404C2312yes—
51C2×C22⋊D101—21no—
52D4×D101—24no—
53C2×C4×D101—21no—
54C2×C4×D101—22no—
55C2×C4×D101D412yes—
56C2×C4×D101C2311yescharacteristic
57#13108064—24no—
58#13108063—24no—
59#13108061—24no—
60#13108061—24no—
61D808—24no—
62#13108059—24no—
63#13108057D418yescomplemented
64C2×#11717579C2312yes—
65C22⋊Dic101—22no—
66#13108051—22no—
67#13108051—22no—
68#13108051—24no—
69C4×Dic101D412yes—
70C4×Dic101C2311yescharacteristic
71C2×Q8×C101C2311yescharacteristic
72C2×Q8×C101C2312yes—
73C2×D4×C101C2311yescharacteristic
74Q16×C101—24no—
75SD16×C101—24no—
76C2×C8×C101C2312yes—
77C4⋊C4×C101—22no—
78C22⋊C4×C101—22no—
79C42×C101C2311yescharacteristic
80C23×C101—21no—
81C22×D101—21no—
82C22×D101—22no—
83C22×D101—44no—
84C22×D101—42no—
85C22×D101C2×D411yescharacteristic
86Q8×C101—22no—
87Q8×C101—22no—
88Q8×C101—22no—
89D4×C101—22no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101C2×D412yes—
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—42no—
95C2×Dic101—21no—
96C2×Dic101C2×D412yes—
97C2×Dic101C2×D411yescharacteristic
98D404—22no—
99D404—22no—
100D404—22no—
101D404—44no—
102D404—44no—
103C4×D101—42no—
104C4×D101—42no—
105C4×D101—44no—
106C4×D101C2×D414yes—
107Dic202—22no—
108Dic202—42no—
109C8×C101C2×D414yes—
110#11717579C2×D414yes—
111C22×C101—41no—
112C22×C101—42no—
113C22×C101C22×D411yescharacteristic
114C2×D101—22no—
115C2×D101—42no—
116C2×D101—41no—
117C2×D101—44no—
118C2×D101—42no—
119C2×D101—84no—
120C2×D101#12712yes—
121C4×C101—22no—
122C4×C101—41no—
123C4×C101—42no—
124C4×C101C22×D412yes—
125Dic101—22no—
126Dic101—41no—
127Dic101#12712yes—
128C2×C101—81no—
129C2×C101C2×#12711yescharacteristic
130C2×C101#57512yes—
131D101—22no—
132D101—82no—
133D101—81no—
134#3153—1011nomaximal, Sylow
135C101#315311yescharacteristic, complemented, Sylow
136C2×#137—1012no—
137C2×AGL(1,ℤ/8)—1012no—
138#532—1011no—
139C2×#124—1011no—
140#497—1011no—
141#497D10111yescharacteristic, complemented
142#496—1011no—
143#496—1011no—
144#495—1012no—
145#495—1012no—
146C4×M16—1011no—
147C2×#41—1012no—
148C22×Q8—1011no—
149C22×D4—1011no—
150#137—2028no—
151AGL(1,ℤ/8)—2028no—
152C2×Q16—1012no—
153C2×Q16C2×D10112yescomplemented
154C2×SD16—1012no—
155C2×SD16—1012no—
156C2×SD16—1012no—
157C2×SD16C2×D10112yes—
158C2×D8—1012no—
159C2×D8—1012no—
160C2×M16—1012no—
161C4⋊Q8C2×D10111yescharacteristic, complemented
162#127—1011no—
163#124—1011no—
164#124—1012no—
165#124—1011no—
166#124—1011no—
167#124—2022no—
168#124C2×D10111yescharacteristic
169#121—2022no—
170C4×D4—2022no—
171C2×C22⋊C4—2022no—
172C2×C42—1011no—
173#97—1011no—
174#97C2×D10111yescharacteristic, complemented
175C4×C8—1012no—
176C24—2021no—
177#41—2024no—
178#41—2024no—
179C2×Q8—1011no—
180C2×Q8—2021no—
181C2×Q8—2024no—
182C2×Q8C22×D10111yescharacteristic
183C2×Q8C22×D10112yes—
184C2×D4—1011no—
185C2×D4—1012no—
186C2×D4—2022no—
187C2×D4—2024no—
188C2×D4—2021no—
189C2×D4—2022no—
190C2×D4C22×D10111yescharacteristic
191C22×C4—1011no—
192C22×C4—1012no—
193C22×C4—2022no—
194C22×C4—2021no—
195Q16—24no—
196Q16—2024no—
197SD16—24no—
198SD16—2024no—
199SD16—2024no—
200SD16—2024no—
201D8—2024no—
202D8—2024no—
203M16—1018no—
204C2×C8—1012no—
205C2×C8C22×D10112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—2022no—
209C22⋊C4—2024no—
210C22⋊C4—2022no—
211C22⋊C4—2022no—
212C42—1012no—
213C42—1011no—
214C42C22×D10111yescharacteristic
215C23—21no—
216C23—1011no—
217C23—4042no—
218C23—4044no—
219C23—2021no—
220C23—2022no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—4042no—
225Q8—2022no—
226D4—22no—
227D4—4044no—
228D4—4044no—
229D4—4042no—
230D4—2022no—
231D4—2022no—
232D4—2022no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—1011no—
236C2×C4—1014no—
237C2×C4—1012no—
238C2×C4—4042no—
239C2×C4—4044no—
240C2×C4—4042no—
241C2×C4—2021no—
242C2×C4D4×D10112yes—
243C2×C4C23×D10111yescharacteristic
244C8—1014no—
245C8D4×D10114yes—
246C22—41no—
247C22—42no—
248C22—8084no—
249C22—1012no—
250C22—4042no—
251C22—4041no—
252C22—4044no—
253C22—4042no—
254C22—2022no—
255C22C2×D4×D10111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—1012no—
260C4—4041no—
261C4—2022no—
262C4C2×D4×D10112yes—
263C2—81no—
264C2—8081no—
265C2—8082no—
266C2—2022no—
267C2#127×D10111yescharacteristic
268C2#2026597612yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#497×C101C2
socleC22×C101C22×D4
hypercenter#497D101
hypocenterC101#3153
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#15692287C2semidirect
2. C2×#15692282C2semidirect
3. #20265728C2semidirect
4. #124×D101C2semidirect
5. #20265569C2semidirect
6. #20265567C2semidirect
7. #20265566C2semidirect
8. #20265562C2semidirect
9. #20265558C2semidirect
10. #20265129C2semidirect
11. #497×C101C2semidirect
17. C2×#13108064C22semidirect
18. C2×#13108063C22semidirect
20. C2×#13108061C22semidirect
24. #15692252C22semidirect
26. #15692244C22semidirect
34. #15692166C22semidirect
35. C8×Dic101C22semidirect
36. C2×Q16×C101C22semidirect
38. C4⋊Q8×C101C22semidirect
40. #97×C101C22semidirect
63. #13108057D4semidirect
135. C101#3153semidirect
141. #497D101semidirect
153. C2×Q16C2×D101semidirect
161. C4⋊Q8C2×D101semidirect
174. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922872
C2×#156922822
#202657282
#124×D1012
#202655692
#202655672
#202655662
#202655622
#202655582
#202651292
#497×C1012
#3153101