Finite Group Databasebeta

← #27181613

Subgroups — #27181613

Order 12928.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692281C212yesmaximal, complemented
2C2×#15692277C211yesmaximal, characteristic, complemented
3#20265744C212yesmaximal, complemented
4#20265565C214yesmaximal, complemented
5#20265560C212yesmaximal, complemented
6#20265552C212yesmaximal, complemented
7#20265132C211yesmaximal, characteristic, complemented
8C22×D8×C101C211yesmaximal, characteristic, complemented
9C2×#13108074C2212yes—
10#15692281—28no—
11#15692277—24no—
12C2×Dic404C2211yescharacteristic, complemented
13C2×#13108063C2212yescomplemented
14C2×#13108062C2214yescomplemented
15C2×D808C2211yescharacteristic, complemented
16C2×#13108059C2214yescomplemented
17C2×#13108058C2211yescharacteristic
18#15692251C2214yescomplemented
19#15692242C2214yescomplemented
20#15692226—24no—
21#15692224—24no—
22#15692214C2211yescharacteristic
23#15692165C2212yescomplemented
24C8×Dic101C2212yescomplemented
25C22×D4×C101C2212yes—
26C2×D8×C101—24no—
27C2×D8×C101C2212yes—
28C2×D8×C101C2212yes—
29C22×C8×C101C2211yescharacteristic
30#13108074—24no—
31#13108074—24no—
32C2×Dic202C2312yes—
33C2×D404C2312yes—
34C2×C22⋊D101—22no—
35C2×C22⋊D101—28no—
36C2×C4×D101—22no—
37Dic404—22no—
38#13108063—24no—
39#13108062—28no—
40D808—22no—
41#13108059—28no—
42#13108058D414yescomplemented
43C2×#11717579C2312yes—
44C4⋊Dic101D412yes—
45C22⋊Dic101—28no—
46C22⋊Dic101D412yes—
47C4×Dic101C2312yes—
48C24×C101—22no—
49C2×D4×C101—24no—
50C2×D4×C101—24no—
51C2×D4×C101—22no—
52C2×D4×C101C2314yes—
53C22×C4×C101C2311yescharacteristic
54D8×C101—24no—
55D8×C101—24no—
56C2×C8×C101D414yescomplemented
57C2×C8×C101C2311yescharacteristic
58C2×C8×C101C2311yescharacteristic
59C23×C101—24no—
60C23×C101—44no—
61C23×C101—84no—
62C23×C101C2×D411yescharacteristic
63C22×D101—22no—
64D4×C101—24no—
65D4×C101—24no—
66D4×C101—44no—
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D412yes—
69C2×C4×C101C2×D412yes—
70C2×C4×C101C2411yescharacteristic
71C22⋊D101—44no—
72C22⋊D101—88no—
73C2×Dic101—22no—
74C2×Dic101C2×D414yes—
75D404—22no—
76D404—22no—
77C4×D101—44no—
78Dic202—22no—
79Dic202—22no—
80C8×C101C2×D412yes—
81C8×C101C2×D412yes—
82#11717579C2×D414yes—
83C22×C101—22no—
84C22×C101—44no—
85C22×C101—48no—
86C22×C101—88no—
87C22×C101#12712yes—
88C22×C101C22×D411yescharacteristic
89C2×D101—42no—
90C2×D101—44no—
91C4×C101#12712yes—
92C4×C101C22×D411yescharacteristic
93C4×C101C22×D411yescharacteristic
94Dic101—24no—
95Dic101—42no—
96C2×C101—22no—
97C2×C101—84no—
98C2×C101C2×#12711yescharacteristic
99C2×C101#57612yes—
100D101—82no—
101#3155—1011nomaximal, Sylow
102C101#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1012no—
104C2×#135—1011no—
105C22×D8D10111yescharacteristic, complemented
106#535—1012no—
107#496—1014no—
108#495—1012no—
109#493—1012no—
110#405—1011no—
111C2×#41—1012no—
112C22×D4C2×D10112yes—
113AGL(1,ℤ/8)—2028no—
114#135—2024no—
115C2×Q16—1011no—
116C2×SD16—1014no—
117C2×SD16—1012no—
118C2×D8—24no—
119C2×D8—1014no—
120C2×D8—1011no—
121C2×D8C2×D10112yes—
122C2×D8C2×D10112yes—
123C2×M16—1011no—
124C22×C8C2×D10111yescharacteristic
125#127—1014no—
126#124—1014no—
127#121—2024no—
128#120—2024no—
129#117—1011no—
130#97—1012no—
131C4×C8—1012no—
132C24—22no—
133#41—2024no—
134#41—2024no—
135C2×Q8—1012no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—1012no—
140C2×D4—2028no—
141C2×D4—2022no—
142C2×D4C22×D10114yes—
143C22×C4—2022no—
144C22×C4C22×D10111yescharacteristic
145Q16—2022no—
146SD16—2028no—
147SD16—2024no—
148D8—24no—
149D8—24no—
150D8—2028no—
151D8—2022no—
152M16—1014no—
153C2×C8—1012no—
154C2×C8C22⋊D10114yescomplemented
155C2×C8C22×D10111yescharacteristic
156C2×C8C22×D10111yescharacteristic
157C4⋊C4—1012no—
158C22⋊C4—1012no—
159C22⋊C4—2028no—
160C42—1012no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—2022no—
165C23D4×D10111yescharacteristic
166Q8—2022no—
167Q8—2022no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—8088no—
172D4—4044no—
173D4—2022no—
174D4—2022no—
175C2×C4—1014no—
176C2×C4—4044no—
177C2×C4—2022no—
178C2×C4D4×D10112yes—
179C2×C4D4×D10111yescharacteristic
180C2×C4C2×C22⋊D10112yes—
181C2×C4C23×D10111yescharacteristic
182C8—1014no—
183C8C2×C22⋊D10112yes—
184C8C2×C22⋊D10112yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—4042no—
190C22—4044no—
191C22#1569225112yes—
192C22C2×D4×D10111yescharacteristic
193C4—4042no—
194C4—2024no—
195C4#1569225112yes—
196C4C2×D4×D10111yescharacteristic
197C4C22×C22⋊D10111yescharacteristic
198C2—22no—
199C2—84no—
200C2—8082no—
201C2C2×#1569225111yescharacteristic
202C2#2026597812yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×D8×C101C2
socleC22×C101C22×D4
hypercenterC22×D8D101
hypocenterC101#3155
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×#15692277C2semidirect
3. #20265744C2semidirect
4. #20265565C2semidirect
5. #20265560C2semidirect
6. #20265552C2semidirect
7. #20265132C2semidirect
8. C22×D8×C101C2semidirect
12. C2×Dic404C22semidirect
13. C2×#13108063C22semidirect
14. C2×#13108062C22semidirect
15. C2×D808C22semidirect
16. C2×#13108059C22semidirect
18. #15692251C22semidirect
19. #15692242C22semidirect
23. #15692165C22semidirect
24. C8×Dic101C22semidirect
42. #13108058D4semidirect
56. C2×C8×C101D4semidirect
102. C101#3155semidirect
105. C22×D8D101semidirect
154. C2×C8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C2×#156922772
#202657442
#202655652
#202655602
#202655522
#202651322
C22×D8×C1012
#3155101