Finite Group Databasebeta

← #27181375

Subgroups — #27181375

Order 12928.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265782C211yesmaximal, characteristic, complemented
2#20265501C212yesmaximal, complemented
3#20265477C212yesmaximal, complemented
4#20265372C212yesmaximal, complemented
5#20265311C212yesmaximal, complemented
6#20265287C212yesmaximal, complemented
7C4×#13108059C212yesmaximal, complemented
8#20265238C211yesmaximal, characteristic, complemented
9#518×C101C211yesmaximal, characteristic, complemented
10C2×#13108059C2212yes—
11C2×#13108059C2212yes—
12#15692252C2211yescharacteristic, complemented
13#15692247C2211yescharacteristic, complemented
14#15692240—22no—
15#15692229C2212yes—
16#15692228—22no—
17C4×D404C2212yescomplemented
18#15692188C2212yescomplemented
19#15692186C2212yescomplemented
20C4⋊#11717579C2212yescomplemented
21#15692176C2212yes—
22#15692176C2212yes—
23#15692175C2212yescomplemented
24#15692175C2212yescomplemented
25C22⋊#11717579C2212yes—
26#15692166C2211yescharacteristic, complemented
27C4×#11717579C2211yescharacteristic, complemented
28C22×D4×C101C2211yescharacteristic
29C4×D4×C101—22no—
30C4×D4×C101C2212yes—
31C4×D4×C101C2212yes—
32#117×C101C2211yescharacteristic
33#117×C101C2211yescharacteristic
34C2×C22⋊C4×C101—22no—
35C2×D404—22no—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C4×D101—22no—
39#13108059—24no—
40#13108059—24no—
41C2×#11717579C2312yes—
42C2×#11717579C2312yes—
43C4⋊Dic101C2312yes—
44#13108053—22no—
45#13108051—22no—
46#13108051—22no—
47C24×C101—21no—
48C2×D4×C101—24no—
49C2×D4×C101—21no—
50C2×D4×C101C2312yes—
51C22×C4×C101—22no—
52C22×C4×C101—22no—
53C22×C4×C101C2311yescharacteristic
54C4⋊C4×C101D412yes—
55C4⋊C4×C101C2312yes—
56C22⋊C4×C101—21no—
57C22⋊C4×C101—22no—
58C22⋊C4×C101—22no—
59C22⋊C4×C101D412yes—
60C42×C101—21no—
61C42×C101C2311yescharacteristic
62C42×C101C2311yescharacteristic
63C23×C101—24no—
64C23×C101—22no—
65C23×C101—82no—
66C23×C101C2×D411yescharacteristic
67C22×D101—22no—
68D4×C101—24no—
69D4×C101—22no—
70D4×C101#4114yes—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—82no—
74C2×C4×C101—82no—
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2×D412yes—
78C2×C4×C101C2411yescharacteristic
79C22⋊D101—82no—
80C2×Dic101—22no—
81D404—22no—
82D404—22no—
83D404—82no—
84C4×D101—82no—
85#11717579—22no—
86#11717579—22no—
87C22×C101—21no—
88C22×C101—24no—
89C22×C101—22no—
90C22×C101—42no—
91C22×C101—42no—
92C22×C101—84no—
93C22×C101C22×D411yescharacteristic
94C2×D101—42no—
95C2×D101—42no—
96C2×D101—42no—
97C4×C101—21no—
98C4×C101—22no—
99C4×C101—22no—
100C4×C101—42no—
101C4×C101C2×#4112yes—
102Dic101—42no—
103C2×C101—41no—
104C2×C101—44no—
105C2×C101#52511yescharacteristic
106C2×C101#57612yes—
107D101—82no—
108#3122—1011nomaximal, Sylow
109C101#312211yescharacteristic, complemented, Sylow
110#540—1011no—
111#518D10111yescharacteristic, complemented
112#482—1012no—
113#478—1012no—
114#459—1012no—
115#447—1012no—
116#442—1012no—
117C4×D8—1012no—
118#433—1011no—
119C22×D4C2×D10111yescharacteristic
120C2×D8—1012no—
121C2×D8—1012no—
122#127—1011no—
123#125—1011no—
124#123—2022no—
125#121—1012no—
126#121—2022no—
127C4×D4—22no—
128C4×D4—1012no—
129C4×D4C2×D10112yes—
130C4×D4C2×D10112yes—
131#117C2×D10111yescharacteristic
132#117C2×D10111yescharacteristic
133C2×C22⋊C4—22no—
134#107—1012no—
135#106—1012no—
136C4⋊C8—1012no—
137#102—1012no—
138#102—1012no—
139#102—1012no—
140#102—1012no—
141C22⋊C8—1012no—
142#97—1011no—
143C4×C8—1011no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—1012no—
148C2×D4—2022no—
149C2×D4—2022no—
150C2×D4C22×D10112yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—2022no—
154C22×C4C22×D10111yescharacteristic
155D8—2024no—
156D8—2024no—
157C2×C8—1012no—
158C2×C8—1012no—
159C4⋊C4—1012no—
160C4⋊C4—2022no—
161C4⋊C4C22⋊D10112yes—
162C4⋊C4C22×D10112yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—2022no—
167C22⋊C4—2022no—
168C22⋊C4C22⋊D10112yes—
169C42—21no—
170C42C22×D10111yescharacteristic
171C42C22×D10111yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—2022no—
176C23C2×C22⋊D10111yescharacteristic
177D4—24no—
178D4—22no—
179D4—8082no—
180D4—8082no—
181D4—2022no—
182D4—2022no—
183D4#1310807414yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—8082no—
189C2×C4—2022no—
190C2×C4C2×C22⋊D10111yescharacteristic
191C2×C4C2×C22⋊D10112yes—
192C2×C4C2×C22⋊D10112yes—
193C2×C4C23×D10111yescharacteristic
194C8—2022no—
195C8—2022no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—4042no—
203C22—4042no—
204C22—4042no—
205C22C22×C22⋊D10111yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—4042no—
211C4C2×#1310807412yes—
212C2—41no—
213C2—44no—
214C2—8082no—
215C2#2026568811yescharacteristic
216C2#2026598012yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#518×C101C2
socleC22×C101C22×D4
hypercenter#518D101
hypocenterC101#3122
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20265782C2semidirect
2. #20265501C2semidirect
3. #20265477C2semidirect
4. #20265372C2semidirect
5. #20265311C2semidirect
6. #20265287C2semidirect
7. C4×#13108059C2semidirect
8. #20265238C2semidirect
9. #518×C101C2semidirect
12. #15692252C22semidirect
13. #15692247C22semidirect
17. C4×D404C22semidirect
18. #15692188C22semidirect
19. #15692186C22semidirect
20. C4⋊#11717579C22semidirect
23. #15692175C22semidirect
24. #15692175C22semidirect
26. #15692166C22semidirect
27. C4×#11717579C22semidirect
109. C101#3122semidirect
111. #518D101semidirect

Maximal subgroups

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

subgroupindex
#202657822
#202655012
#202654772
#202653722
#202653112
#202652872
C4×#131080592
#202652382
#518×C1012
#3122101